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Engineering Applications of Computational FluidMechanics

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A simple and efficient outflow boundary conditionfor the incompressible NavierndashStokes equations

Yibao Li Jung-II Choi Yongho Choic amp Junseok Kim

To cite this article Yibao Li Jung-II Choi Yongho Choic amp Junseok Kim (2017) A simpleand efficient outflow boundary condition for the incompressible NavierndashStokesequations Engineering Applications of Computational Fluid Mechanics 111 69-85 DOI1010801994206020161247296

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ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 2017VOL 11 NO 1 69ndash85httpdxdoiorg1010801994206020161247296

A simple and efficient outflow boundary condition for the incompressibleNavierndashStokes equations

Yibao Lia Jung-II Choib Yongho Choicc and Junseok Kimc

aSchool of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan PR China bDepartment of Computational Science and EngineeringYonsei University Seoul Republic of Korea cDepartment of Mathematics Korea University Seoul Republic of Korea

ABSTRACTMany researchers have proposed special treatments for outlet boundary conditions owing to lackof information at the outlet Among them the simplest method requires a large enough compu-tational domain to prevent or reduce numerical errors at the boundaries However an efficientmethod generally requires special treatment to overcome the problems raised by the outlet bound-ary condition used For example mass flux is not conserved and the fluid field is not divergence-freeat the outlet boundary Overcoming these problems requires additional computational cost Inthis paper we present a simple and efficient outflow boundary condition for the incompressibleNavierndashStokes equations aiming to reduce the computational domain for simulating flow inside along channel in the streamwise direction The proposed outflow boundary condition is based onthe transparent equation where a weak formulation is used The pressure boundary condition isderived by using theNavierndashStokes equations and the outlet flowboundary condition In the numer-ical algorithm a staggered marker-and-cell grid is used and temporal discretization is based on aprojectionmethod The intermediate velocity boundary condition is consistently adopted to handlethe velocityndashpressure coupling Characteristic numerical experiments are presented to demonstratethe robustness and accuracy of the proposed numerical scheme Furthermore the agreement ofcomputational results from small and large domains suggests that our proposed outflow boundarycondition can significantly reduce computational domain sizes

ARTICLE HISTORYReceived 28 February 2016Accepted 8 October 2016

KEYWORDSInflowndashoutflow boundarycondition marker-and-cellmesh incompressibleNavierndashStokes equationsprojection method

1 Introduction

Microfluidics which is mostly in a low Reynolds numberflow regime according to its velocity and length scalesoffers fundamentally new capabilities for the control ofconcentrations of molecules in space and time (White-sides 2006) There is also much research into sophis-ticated three-dimensional hydrodynamics and pollutantmodelling (Chau amp Jiang 2001 2004 Epely-ChauvinDe Cesare amp Schwindt 2014 Gholami Akbar Mina-tour Bonakdari amp Javadi 2014 Liu amp Yang 2014 Wu ampChau 2006) Thus microfluidics has become importantfor many lab-on-a-chip applications such as enzymatickinetics (Pabit amp Hagen 2002) protein folding (Gam-bin Simonnet VanDelinder Deniz amp Groisman 2010)single molecule detection (De Mello amp Edel 2007) andflow cytometry (Mao Lin Dong amp Huang 2009 Wanget al 2005 Yun et al 2010) The aspect ratio of atypical micro-channel is very large Therefore intro-ducing proper inflow and outflow boundary conditionsin numerical computations allows us to use a reduced

CONTACT Junseok Kim cfdkimkoreaackr Homepage httpmathkoreaackrsim cfdkim

computational domain for simulating flow inside sucha long channel in the streamwise direction For theinlet boundary conditions Han Lu and Bao (1994)performed a comparative study with different typesof boundary conditions at the inlet and found that aparabolic profile is simple but suitable for describing theinflow conditions However many researchers (DongKarniadakis amp Chryssostomidis 2014 Guermond ampShen 2003 Hagstrom 1991 Halpern 1986 Halpern ampSchatzman 1989 Hasan Anwer amp Sanghi 2005 Hed-strom 1979 Jin amp Braza 1993 Johansson 1993 Jor-dan 2014 Kim amp Moin 1985 Kirkpatrick amp Arm-field 2008 Lei Fedosov amp Karniadakis 2011 Liu 2009Mateescu Munoz amp Scholz 2010 Olrsquoshanskii ampStaroverov 2000 Poux Glockner Ahusborde amp Aza-iumlez 2012 Poux Glockner amp Azaiumlez 2011 RegulagaddaNaterer amp Dincer 2011 Rudy amp Strikwerda 1980 Saniamp Gresho 1994 Sani Shen Pironneau amp Gresho 2006Sohankar Norberg ampDavidson 1998 Thompson 1987)have proposed special treatments for outlet boundary

copy 2016 The Author(s) Published by Informa UK Limited trading as Taylor amp Francis GroupThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (httpcreativecommonsorglicensesby40) which permits unrestricted usedistribution and reproduction in any medium provided the original work is properly cited

70 Y LI ET AL

conditions owing to the lack of information at the outletFour different types of boundary conditions ndash Dirich-let (Han et al 1994) Neumann (Kim amp Moin 1985Kirkpatrick amp Armfield 2008) convective (Olrsquoshanskii ampStaroverov 2000) and nonreflecting (JinampBraza 1993) ndashare widely applied to describe flow conditions at theoutlet

Dirichlet and Neumann boundary conditions havebeen widely used for outflow conditions A Dirich-let boundary condition with a velocity profile (eg thePoiseuille profile in a channel) requires pre-knowledgeof the flow conditions at the outlet for specifying thevelocity profile explicitly while this naturally providesa global mass flux balance A Neumann boundary con-dition does not need pre-knowledge of the flow con-ditions however it requires a large enough computa-tional domain to prevent or reduce numerical errorsat the boundaries Because of this requirement addi-tional computational cost is expected for applying aNeumann boundary condition To overcome the limi-tations of Dirichlet and Neumann boundary conditionsat the outlet a convective boundary condition is oftenused for the outflow boundary condition Olrsquoshanskii andStaroverov (2000) proposed a convective boundary con-dition that is based on a first-order advection equation(Halpern amp Schatzman 1989) and a scaling method(Hagstrom 1991) Typically the advection velocity isassumed to be a constant or have a Poiseuille profile butis somewhat arbitrary Therefore the advection veloc-ity should be supplemented by a correction velocity ateach time step to conserve mass flux Nevertheless oncemass flux has been kept as a constant the convectiveboundary condition becomes a fixed Dirichlet bound-ary condition Another effective method derived froma wave equation is the nonreflecting boundary method(Hedstrom 1979 Jin amp Braza 1993 Johansson 1993Rudy amp Strikwerda 1980 Sani amp Gresho 1994 Thomp-son 1987) which performswell tominimize the spuriousartifacts at the outlet This kind of model is suitable forwake and jet flowwithmoderate andhighReynolds num-bers Recently Dong et al (2014) proposed an efficientoutflow boundary condition that can allow for the influxof kinetic energy into the domain but prevent uncon-trolled growth in the energy of the domain To decouplethe pressure and velocity computation they developeda rotational velocity-correction type strategy The pro-posed method is efficient and can maximize the domaintruncation without adversely affecting the flow physicsand enable simulations even with much higher Reynoldsnumber

In this paper we propose an efficient and simple trans-parent boundary condition to simulate channel-typeflows efficiently without trivial boundary treatment

especially for low Reynolds number A pressure bound-ary condition is also proposed to match with theNavierndashStokes equation In the numerical algorithm astaggered marker-and-cell grid is used and temporal dis-cretization is based on a projectionmethod The interme-diate velocity boundary condition is consistently adoptedfor handling the velocityndashpressure coupling Character-istic numerical experiments are presented to demon-strate the robustness and accuracy of the numericalscheme Furthermore the agreement of computationalresults from small and large domains suggests that ourproposed outflow boundary condition can significantlyreduce computational domain sizes

The paper is organized as follows in Section 2 velocityand pressure boundary conditions for an incompressibleNavierndashStokes equation are presented Section 3 outlinesthe numerical method Section 4 provides a variety ofresults Finally our conclusions are given in Section 5

2 Governing equations and boundaryconditions

The non-dimensional NavierndashStokes equation and thecontinuity equation for time-dependent incompressibleviscous flows can be written as

partupartt

+ u middot nablau = minusnablap + 1Re

u (1)

nabla middot u = 0 (2)

where Re = ρULμ is the Reynolds number ρ andμ arefluid density and viscosity respectivelyU and L are char-acteristic velocity and length scales respectively p(x t)is the pressure u(x t) = (u(x t) v(x t)) is the veloc-ity where u and v are its components in the x- andy-directions respectively x = (x y) are Cartesian coor-dinates on the domain sub R2 and t is the temporalvariable The domain boundary consists of part = partwall⋃

partin⋃

partout where partwall partin and partout are thewall inlet and outlet boundaries respectively Here thevelocities on the boundary are defined as

u|part =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

u|partwall is zerou|partin is a given functionu|partout is computed by an outflow

boundary condition

Figure 1 shows a schematic representation of theinflowndashoutflow conditions For simplicity of expositionwe shall describe the inlet and outlet in the x-directionThe other directions can be similarly defined

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 71

Figure 1 Schematic illustration of inflowndashoutflow boundaryconditions

21 Velocity boundary conditions

At the inlet the boundary condition can be chosen basedon the physical problem In our paper we consider inflowparallel to the wall Thus for partin we use the followingboundary condition

u = u(y t) and v = 0 (3)

where u(y t) is a given x-directional inflow velocity func-tion Jin and Braza (1993) developed a nonreflectingoutlet boundary condition for incompressible unsteadyNavierndashStokes calculations

ut + uux minus 1Re

uyy = 0 and vt + uvx minus 1Re

vyy = 0

(4)

Halpern and Schatzman (1989) proposed the convectiveboundary conditions

ut + U middot nablau = 0 (5)

where U(x t) = (U(x t)V(x t)) is a known flowOlrsquoshanskii and Staroverov (2000) proposed a simple con-vective boundary condition

ut + Uux = 0 and v = 0 (6)

The computed velocityU is supplemented by a correc-tion velocity to balance the mass flux ie

intpartin

u dy =int

partout

u dy

The function U is formally chosen as constant or as aPoiseuille profile If the outlet flow is unknown the veloc-ityU can be calculated by taking the integral of both sides

of Equation (6)

U =intpartout

ut dyintpartout

ux dy=

(intpartout

u dy)tint

partoutux dy

=

(intpartin

u dy)tint

partoutux dy

(7)

However when the inflow is a steady flow the con-stant flow rate at the inlet can lead to a zero velocityU at the outlet Thus the outflow boundary conditionEquation (6) will reduce to a Dirichlet boundary condi-tion ie ut = 0 It should be noted that for unsteady flowthe outflow is clearly not constant Sohankar et al (1998)also proposed to setU as a convective velocity but to add acorrection velocity to guarantee the balance between theinlet and outlet at each time step The correction veloc-ity is a constant over the entire outlet and is zero whenthe outflow reaches a steady state However the choice ofthe convective velocity is somewhat arbitrary Note thatin this approach to compare with other boundary condi-tions we will perform several tests by using a convectiveboundary condition The chosen convective velocity Uis a constant Poiseuille profile when the outflow is inthe steady state otherwise it will be computed by usingEquation (7)

The fluid flow at low Reynolds number is gener-ally characterized by smooth and constant fluid motionbecause viscous forces are dominant Therefore we canassume that the flow is onlymoved by the convection andthat itsmain direction is normal to the outflow boundaryThe first assumption implies that the transparent bound-ary condition ut + u middot nablau = 0 is satisfied and the secondassumption implies v=0 because the outflow boundaryis perpendicular to the x-direction in our paper There-fore to simulate the fluid flow at low Reynolds numberwithout trivial boundary treatment we propose to use asimple and efficient transparent boundary condition

ut + uux = 0 and v = 0 (8)

These conditions are admissible provided that the out-let boundary is not located quite nearby the downstreamNote that in solving the incompressible NavierndashStokesequations at the inflow and outflow boundaries boththe velocity-divergence and the velocityndashpressure formsshould be considered to balance themass flux This is alsoa new aspect of this paper The form (8) may suffer fromsome numerical difficulties which will be described inSection 31 Thus by using the divergence-free conditionnabla middot u = 0 we can rewrite Equation (8) as

ut + uux = ut + uux + u(ux + vy)

= ut + (u2)x = 0 and v = 0 (9)

72 Y LI ET AL

22 Pressure boundary conditions

Taking the inner product of both sides of Equation (1)with n which is the unit normal vector to the domainboundary we obtain

n middot nablap = n middot(

minusut minus u middot nablau + 1Re

u) (10)

On partwall we get

n middot nablap = n middot uRe

(11)

Here we have used the no-slip boundary condition ieu = 0 On partin by using Equation (10) we get

px = minusut minus uux minus vuy + 1Re

(uxx + uyy)

= minusut + 1Re

uyy (12)

Here we have used the inflow boundary condition (3)For the outlet boundary by substituting Equation (8) intoEquation (10) we get

px = 1Re

(uxx + uyy) (13)

Thus the pressure boundary condition is directlyderived by using the NavierndashStokes equation and out-let flow boundary conditions The pressure conditionis mainly balanced with the viscous stress terms at theboundary All the pressure boundary conditions are ofNeumann type Thus the method used with the projec-tion method would be convenient and simple It is note-worthy that the present form of the pressure boundarycondition ismore plausible because the flows inmicroflu-idics are mostly viscous dominated For high Reynoldsnumber flows the pressure boundary condition becomesa homogeneous Neumann boundary condition This isbecause the viscous term (uxx + uyy)Re is negligible

3 Numerical methods

We use a staggered grid where the pressure field isstored at cell centres and velocities at cell interfaces (seeFigure 2) Let h be a uniform mesh spacing The centreof each cell ij is located at (xi yj) = ((i minus 05)h (j minus05)h) on the computational domain = (0 Lx) times(0 Ly) for i = 1 Nx and j = 1 Ny Nx and Nyare the numbers of cells in the x- and y-directions respec-tively The cell edges are located at (xi+ 1

2 yj) = (ih (j minus

05)h) and (xi yj+ 12) = ((i minus 05)h jh)

At the beginning of each time step given un wewant to find un+1 and pn+1 by solving the following

Figure 2 Velocities defined at cell boundaries and the pressurefield defined at the cell centres

discretized equations in time with a projection method(Chorin 1968)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun

(14)

nablad middot un+1 = 0 (15)

Here nablad and d denote the centred differenceapproximations for the gradient and Laplacian operatorsrespectively

nabladp =(pi+1j minus pij

hpij+1 minus pij

h

)

du = (dui+ 12 jdvij+ 1

2)

=(ui+ 3

2 j+ uiminus 1

2 j+ ui+ 1

2 j+1 + ui+ 12 jminus1 minus 4ui+ 1

2 j

h2

vij+ 32

+ vijminus 12

+ vi+1j+ 12

+ viminus1j+ 12

minus 4vij+ 12

h2

)

The outline of the main procedure in one time step isas follows

Step 1 Initialize u0 to be the divergence-free velocityfield

Step 2 Update un+1 and n middot nabladpn+1 on the entireboundaries The computational details will begiven in Section 31

Step 3 Solve an intermediate velocity field uwithout thepressure gradient term

u minus un

t= minusun middot nabladun + 1

Redun (16)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 73

Step 4 Solve the Poisson equation for the pressure

dpn+1 = 1t

nablad middot u (17)

The resulting linear system of Equation (17)is solved using a multigrid method Note thatEquation (17) is derived by applying the diver-gence operator to Equation (18)

un+1 minus ut

= minusnabladpn+1 (18)

It should be pointed out that to find the solu-tion for pressure in Equation (17) we shouldknow the boundary condition of the intermedi-ate velocity Our proposed intermediate velocityboundary condition is described in Section 32

Step 5 Update the divergence-free velocity

un+1 = u minus tnabladpn+1 (19)

These steps complete one time step For more detailsee Li Jung Lee Lee and Kim (2012) In this study wefocus on introducing efficient outflow boundary condi-tions and pressure boundary conditions

31 Boundary conditions for velocity and pressuregradient

In this section we will give detailed descriptions ofboundary conditions for the velocity and pressure gra-dient on the entire boundary First we will considerthe velocity boundary condition The velocities un+1

on partin and partwall are given as a known functionand zero respectively The outflow boundary conditionEquation (8) can be discretizedwith anupwind scheme as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus tunNx+ 1

2 junxNx+ 1

2 j (20)

Here unxi+ 12 j

is computed using the upwind procedure

unxi+ 12 j

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

uni+ 1

2 jminus un

iminus 12 j

hif un

i+ 12 j

gt 0

uni+ 3

2 jminus un

i+ 12 j

hotherwise

Note that Equation (19) yields un+1Nx+ 1

2 j= 0 when

unNx+ 1

2 j= unyNx+ 1

2 j= 0 for every j = 1 Ny Thismeans

that once u is zero at the boundary uwill always be zeroTo avoid this trivial case we use the conservation form

of the convective terms in Equation (9) The discretizedform can be written as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus t

h

[(unNx+ 1

2 j)2 minus (unNxminus 1

2 j)2]

(21)

where we used v=0 on the outflow boundary condi-tion For the pressure boundary condition we discretizeEquation (12) on partin as

pn+1x 12 j

= minusun+1

12 j

minus un12 j

t+

un12 j+1

+ un12 jminus1

minus 2un12 j

h2Re (22)

At the wall without loss of generality we only describethe pressure in the y-directionDiscretizingEquation (11)we can get

pn+1yi 12

=vniminus 1

2+ vn

i 32minus 2vn

i 12h2Re

+vni+1 12

+ vniminus1 12

minus 2vni 12

h2Re

(23)

Since the no-slip boundary condition is used at the wallwe can get vi12 = 0 Meanwhile on the staggered gridvniminus12 can be simply computed as minusvni32 to matchthe no-slip boundary condition Thus in our approachEquation (11) can be reduced to

n middot nablapn+1 = 0

For partout the pressure boundary condition can bediscretized as

pn+1xNx+ 1

2 j=

unNx+ 1

2 j+1+ un

Nx+ 12 jminus1

minus 2unNx+ 1

2 j

h2Re

+2un

Nx+ 12 j

minus 5unNxminus 1

2 j+ 4un

Nxminus 32 j

minus unNxminus 5

2 j

h2Re

(24)

Note that since there is no information for uNx+ 32 j we

use a second-order one-sided approximation to uxx

32 Boundary condition for intermediate velocity

Since we have used the projection method to discretizethe time-dependent incompressible NavierndashStokesequation in time it is necessary to define the bound-ary condition for the intermediate velocity field In ourapproach we can directly compute ulowast Since we have cal-culatedun+1 andn middot nabladp in Step 2 of Section 3 we can use

74 Y LI ET AL

them straightforwardly From Equation (19) we have

u = un+1 + tnablapn+1 (25)

On partwall we get

n middot u = n middot (un+1 + tnabladpn+1) = 0 (26)

For the inflow boundary by recalling Equation (19)we have

u 12 j

= un+112 j

+ tpn+1x 12 j

= un12 j

+ th2Re

(un1

2 j+1 + un12 jminus1 minus 2un1

2 j

) (27)

Here we have used the Dirichlet boundary conditionfor velocity and the homogeneous Neumann boundarycondition for pressure On the outflow boundary weobtain

uNx+ 12 j

= un+1Nx+ 1

2 j+ tpn+1

xNx+ 12 j

= un+1Nx+ 1

2 j+ t

(unNx+ 1

2 j+1 + unNx+ 12 jminus1

minus2unNx+ 12 j

)(h2Re) + t

(2unNx+ 1

2 j

minus5unNxminus 12 j

+ 4unNxminus 32 j

minus unNxminus 52 j

)(h2Re)

(28)

Thus the intermediate velocity boundary condition isconsistently adopted for the velocityndashpressure couplingWith different projection methods the gradient pressureterm (nabladp) is added (Kim 2005 Kim amp Moin 1985Poux Glockner ampAzaiumlez 2011 TsengampHuang 2014) orsubtracted (Chorin 1968 Li Jung Lee Lee ampKim 2012Li Yun Lee Shin Jeong amp Kim 2013) Our proposedmethod for the intermediate velocity will work well ifnabladp is subtracted For the other case the intermediatevelocity boundary condition should be modified and canbe derived by using the intermediate pressure boundarycondition in a similar way Next we will briefly describethe extension of our proposed method to an implicitscheme for the momentum equation (1)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun+1

(29)

Then we decompose Equation (29) into two steps

u minus un

t= minus(u middot nabladu)n + 1

Redu (30)

un+1 minus ut

= minusnabladpn+1 (31)

Since the boundary conditions for velocity and pres-sure gradient are known in Section 32 we can straight-forwardly compute the intermediate velocity values in the

domain boundary as

n middot u = n middot un+1 + tn middot nabladpn+1 (32)

However in that case the time step may be restrictedby numerical stability since explicit time integration forthe diffusion term is considered on the boundary Onepossible way to improve numerical stability is to replacedun bydu in Equations (22)ndash(24) and substitute theminto Equation (32) After that we can combine Equa-tions (30) and (32) and find the solutions of the inter-mediate velocity Once u is known we can straightfor-wardly compute pn+1 and un+1 by using Equations (17)and (19) respectively Note that in this paper we useEquations (14) and (15) so that we only need to solveEquation (17 ) However if the Reynolds number ismuchsmaller implicit or CrankndashNicolson schemes should beconsidered It should also be noted that the projectionmethod on staggered grids has the property that an arbi-trary boundary condition for the pressure gradient can beused first and then a corresponding boundary conditioncan be provided for the intermediate velocity throughStep 5 The corresponding boundary condition shouldcertainlymake the outflow satisfy its boundary condition

33 Mass flux correction algorithm

Undesirable mass flux loss in a global or local sense ispossible because of boundary singularities in a finite dif-ference framework The proposed outflowconditionswillleverage for unexpected mass flux loss by addressing thecoupling effect between pressure and velocity at the sin-gular point (see the circle in Figure 3(a)) The centred dif-ference approximation for the velocity in Equations (21)and (28) at the singular point is not a good choice andmay lead to flux loss because of oscillation of the valuesFurthermore the present boundary conditions require atransient time to achieve mass balance which will be dis-cussed in the next section Thus we propose a simplecorrection algorithm to maintain the global mass fluxThe idea is summarized as follows

Step 1 Update the velocity with the outlet boundarycondition u

uNx+ 12 j

= unNx+ 12 j

minus t((unNx+ 1

2 j)2

minus(unNxminus 12 j

)2)

h (33)

Step 2 Set un+1Nx+ 1

2 j= εuNx+ 1

2 jif uNx+ 1

2 jgt 0 and set

un+1Nx+ 1

2 j= uNx+ 1

2 jotherwise Here ε can be

determined from the balance of mass flux

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 75

Figure 3 Schematic illustration of channels with (a) and without (b) a singular point in the outlet

between the inlet and outlet ieNysumj=1

un+112 j

=Nysumj=1

un+1Nx+ 1

2 j (34)

The following should be pointed out

(1) Our scheme satisfies the divergence-free conditionwith the mass flux correction algorithm becauseEquations (15) and (18) are used in the projectionmethod at every time step

(2) Since the mass correction method also leads tospurious velocities our proposed outflow boundarycondition does not require it if the outflow is parallelto the wall (see Figure 3 (b)) because in this case themass flux loss is much smaller which will be shownin the next section

(3) In the case of a singular point at the outletthe correction algorithm is applied after sev-eral flow-through time periods t0 which satisfiest0intpartin

u dy = αLxLy Here α is the flow-throughtime In our approach we set α = 3 This is becausetransient flow fields may promote numerical insta-bility

(4) In the case without a singular point at the out-let boundary we can use the mass flux correc-tion algorithm only once to save transient time ieα = 1

4 Numerical results

In this section to demonstrate the performance of ourproposed scheme we perform several numerical exper-iments for parabolic flow convergence testing unsteadyflow in a channel backward-facing step flow and outflowwith a block on the outlet

41 Parabolic flow

The first numerical experiment is performed for para-bolic flow through a channel The channel ranges from 0

to 2 in the streamwise (x) direction and from 0 to 1 in thenormal (y) direction The initial velocity field consists ofrandom perturbations with a maximum amplitude of 2that is u(x y 0) = rand(x y) and v(x y 0) = rand(x y)where rand(xy) is a random number between minus1 and1 The inflow velocity is set as u(0 y t) = 8(y minus y2) andv(0 y t) = 0 A no-slip boundary condition is appliedat the top and bottom walls ie (u v) = (0 0) It iswell known that at the beginning of a random veloc-ity field the wake gradually attains a steady state for alow Reynolds number At steady state the shape of theflow is the same as that of the inflow The evolution isrun up to T=7813 with a time step of t = 4h2 Hereh = 164 and Re = 100 are used Note that althoughu0 does not satisfy the divergence-free condition it con-verges rather quickly to a divergence-free velocity field infive time steps It should be noted that although whenwe begin with u0 the projection method used can obtaina divergence-free velocity field in one time step it willtake many more iterations to ensure that the l2 normalerror of divergence of the velocity field is lower than1E-6 Therefore to reduce the computational cost wedivided each time step process into five steps with sev-eral iterations Figure 4 shows the evolution of parabolicflow The results show that our method performs wellfor simulating parabolic flow To compare the velocity atthe outlet in Figure 5(a) we show the results obtainedby our proposed models with and without a flux cor-rection algorithm along with those of the convectivemodel and the nonreflecting model Here U is set as aPoiseuille profile since the outlet flow is known in thistest Furthermore we calculate the discrete l2-norm oferror which isradicradicradicradicradic Nysum

j=1(uNx+ 1

2 jminus u 1

2 j)2Ny

The errors are 849E-3 848E-3 107E-3 and 226E-2 forour proposed models with and without a flux correctionalgorithm the convective model and the nonreflecting

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 71

Figure 1 Schematic illustration of inflowndashoutflow boundaryconditions

21 Velocity boundary conditions

At the inlet the boundary condition can be chosen basedon the physical problem In our paper we consider inflowparallel to the wall Thus for partin we use the followingboundary condition

u = u(y t) and v = 0 (3)

where u(y t) is a given x-directional inflow velocity func-tion Jin and Braza (1993) developed a nonreflectingoutlet boundary condition for incompressible unsteadyNavierndashStokes calculations

ut + uux minus 1Re

uyy = 0 and vt + uvx minus 1Re

vyy = 0

(4)

Halpern and Schatzman (1989) proposed the convectiveboundary conditions

ut + U middot nablau = 0 (5)

where U(x t) = (U(x t)V(x t)) is a known flowOlrsquoshanskii and Staroverov (2000) proposed a simple con-vective boundary condition

ut + Uux = 0 and v = 0 (6)

The computed velocityU is supplemented by a correc-tion velocity to balance the mass flux ie

intpartin

u dy =int

partout

u dy

The function U is formally chosen as constant or as aPoiseuille profile If the outlet flow is unknown the veloc-ityU can be calculated by taking the integral of both sides

of Equation (6)

U =intpartout

ut dyintpartout

ux dy=

(intpartout

u dy)tint

partoutux dy

=

(intpartin

u dy)tint

partoutux dy

(7)

However when the inflow is a steady flow the con-stant flow rate at the inlet can lead to a zero velocityU at the outlet Thus the outflow boundary conditionEquation (6) will reduce to a Dirichlet boundary condi-tion ie ut = 0 It should be noted that for unsteady flowthe outflow is clearly not constant Sohankar et al (1998)also proposed to setU as a convective velocity but to add acorrection velocity to guarantee the balance between theinlet and outlet at each time step The correction veloc-ity is a constant over the entire outlet and is zero whenthe outflow reaches a steady state However the choice ofthe convective velocity is somewhat arbitrary Note thatin this approach to compare with other boundary condi-tions we will perform several tests by using a convectiveboundary condition The chosen convective velocity Uis a constant Poiseuille profile when the outflow is inthe steady state otherwise it will be computed by usingEquation (7)

The fluid flow at low Reynolds number is gener-ally characterized by smooth and constant fluid motionbecause viscous forces are dominant Therefore we canassume that the flow is onlymoved by the convection andthat itsmain direction is normal to the outflow boundaryThe first assumption implies that the transparent bound-ary condition ut + u middot nablau = 0 is satisfied and the secondassumption implies v=0 because the outflow boundaryis perpendicular to the x-direction in our paper There-fore to simulate the fluid flow at low Reynolds numberwithout trivial boundary treatment we propose to use asimple and efficient transparent boundary condition

ut + uux = 0 and v = 0 (8)

These conditions are admissible provided that the out-let boundary is not located quite nearby the downstreamNote that in solving the incompressible NavierndashStokesequations at the inflow and outflow boundaries boththe velocity-divergence and the velocityndashpressure formsshould be considered to balance themass flux This is alsoa new aspect of this paper The form (8) may suffer fromsome numerical difficulties which will be described inSection 31 Thus by using the divergence-free conditionnabla middot u = 0 we can rewrite Equation (8) as

ut + uux = ut + uux + u(ux + vy)

= ut + (u2)x = 0 and v = 0 (9)

72 Y LI ET AL

22 Pressure boundary conditions

Taking the inner product of both sides of Equation (1)with n which is the unit normal vector to the domainboundary we obtain

n middot nablap = n middot(

minusut minus u middot nablau + 1Re

u) (10)

On partwall we get

n middot nablap = n middot uRe

(11)

Here we have used the no-slip boundary condition ieu = 0 On partin by using Equation (10) we get

px = minusut minus uux minus vuy + 1Re

(uxx + uyy)

= minusut + 1Re

uyy (12)

Here we have used the inflow boundary condition (3)For the outlet boundary by substituting Equation (8) intoEquation (10) we get

px = 1Re

(uxx + uyy) (13)

Thus the pressure boundary condition is directlyderived by using the NavierndashStokes equation and out-let flow boundary conditions The pressure conditionis mainly balanced with the viscous stress terms at theboundary All the pressure boundary conditions are ofNeumann type Thus the method used with the projec-tion method would be convenient and simple It is note-worthy that the present form of the pressure boundarycondition ismore plausible because the flows inmicroflu-idics are mostly viscous dominated For high Reynoldsnumber flows the pressure boundary condition becomesa homogeneous Neumann boundary condition This isbecause the viscous term (uxx + uyy)Re is negligible

3 Numerical methods

We use a staggered grid where the pressure field isstored at cell centres and velocities at cell interfaces (seeFigure 2) Let h be a uniform mesh spacing The centreof each cell ij is located at (xi yj) = ((i minus 05)h (j minus05)h) on the computational domain = (0 Lx) times(0 Ly) for i = 1 Nx and j = 1 Ny Nx and Nyare the numbers of cells in the x- and y-directions respec-tively The cell edges are located at (xi+ 1

2 yj) = (ih (j minus

05)h) and (xi yj+ 12) = ((i minus 05)h jh)

At the beginning of each time step given un wewant to find un+1 and pn+1 by solving the following

Figure 2 Velocities defined at cell boundaries and the pressurefield defined at the cell centres

discretized equations in time with a projection method(Chorin 1968)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun

(14)

nablad middot un+1 = 0 (15)

Here nablad and d denote the centred differenceapproximations for the gradient and Laplacian operatorsrespectively

nabladp =(pi+1j minus pij

hpij+1 minus pij

h

)

du = (dui+ 12 jdvij+ 1

2)

=(ui+ 3

2 j+ uiminus 1

2 j+ ui+ 1

2 j+1 + ui+ 12 jminus1 minus 4ui+ 1

2 j

h2

vij+ 32

+ vijminus 12

+ vi+1j+ 12

+ viminus1j+ 12

minus 4vij+ 12

h2

)

The outline of the main procedure in one time step isas follows

Step 1 Initialize u0 to be the divergence-free velocityfield

Step 2 Update un+1 and n middot nabladpn+1 on the entireboundaries The computational details will begiven in Section 31

Step 3 Solve an intermediate velocity field uwithout thepressure gradient term

u minus un

t= minusun middot nabladun + 1

Redun (16)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 73

Step 4 Solve the Poisson equation for the pressure

dpn+1 = 1t

nablad middot u (17)

The resulting linear system of Equation (17)is solved using a multigrid method Note thatEquation (17) is derived by applying the diver-gence operator to Equation (18)

un+1 minus ut

= minusnabladpn+1 (18)

It should be pointed out that to find the solu-tion for pressure in Equation (17) we shouldknow the boundary condition of the intermedi-ate velocity Our proposed intermediate velocityboundary condition is described in Section 32

Step 5 Update the divergence-free velocity

un+1 = u minus tnabladpn+1 (19)

These steps complete one time step For more detailsee Li Jung Lee Lee and Kim (2012) In this study wefocus on introducing efficient outflow boundary condi-tions and pressure boundary conditions

31 Boundary conditions for velocity and pressuregradient

In this section we will give detailed descriptions ofboundary conditions for the velocity and pressure gra-dient on the entire boundary First we will considerthe velocity boundary condition The velocities un+1

on partin and partwall are given as a known functionand zero respectively The outflow boundary conditionEquation (8) can be discretizedwith anupwind scheme as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus tunNx+ 1

2 junxNx+ 1

2 j (20)

Here unxi+ 12 j

is computed using the upwind procedure

unxi+ 12 j

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

uni+ 1

2 jminus un

iminus 12 j

hif un

i+ 12 j

gt 0

uni+ 3

2 jminus un

i+ 12 j

hotherwise

Note that Equation (19) yields un+1Nx+ 1

2 j= 0 when

unNx+ 1

2 j= unyNx+ 1

2 j= 0 for every j = 1 Ny Thismeans

that once u is zero at the boundary uwill always be zeroTo avoid this trivial case we use the conservation form

of the convective terms in Equation (9) The discretizedform can be written as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus t

h

[(unNx+ 1

2 j)2 minus (unNxminus 1

2 j)2]

(21)

where we used v=0 on the outflow boundary condi-tion For the pressure boundary condition we discretizeEquation (12) on partin as

pn+1x 12 j

= minusun+1

12 j

minus un12 j

t+

un12 j+1

+ un12 jminus1

minus 2un12 j

h2Re (22)

At the wall without loss of generality we only describethe pressure in the y-directionDiscretizingEquation (11)we can get

pn+1yi 12

=vniminus 1

2+ vn

i 32minus 2vn

i 12h2Re

+vni+1 12

+ vniminus1 12

minus 2vni 12

h2Re

(23)

Since the no-slip boundary condition is used at the wallwe can get vi12 = 0 Meanwhile on the staggered gridvniminus12 can be simply computed as minusvni32 to matchthe no-slip boundary condition Thus in our approachEquation (11) can be reduced to

n middot nablapn+1 = 0

For partout the pressure boundary condition can bediscretized as

pn+1xNx+ 1

2 j=

unNx+ 1

2 j+1+ un

Nx+ 12 jminus1

minus 2unNx+ 1

2 j

h2Re

+2un

Nx+ 12 j

minus 5unNxminus 1

2 j+ 4un

Nxminus 32 j

minus unNxminus 5

2 j

h2Re

(24)

Note that since there is no information for uNx+ 32 j we

use a second-order one-sided approximation to uxx

32 Boundary condition for intermediate velocity

Since we have used the projection method to discretizethe time-dependent incompressible NavierndashStokesequation in time it is necessary to define the bound-ary condition for the intermediate velocity field In ourapproach we can directly compute ulowast Since we have cal-culatedun+1 andn middot nabladp in Step 2 of Section 3 we can use

74 Y LI ET AL

them straightforwardly From Equation (19) we have

u = un+1 + tnablapn+1 (25)

On partwall we get

n middot u = n middot (un+1 + tnabladpn+1) = 0 (26)

For the inflow boundary by recalling Equation (19)we have

u 12 j

= un+112 j

+ tpn+1x 12 j

= un12 j

+ th2Re

(un1

2 j+1 + un12 jminus1 minus 2un1

2 j

) (27)

Here we have used the Dirichlet boundary conditionfor velocity and the homogeneous Neumann boundarycondition for pressure On the outflow boundary weobtain

uNx+ 12 j

= un+1Nx+ 1

2 j+ tpn+1

xNx+ 12 j

= un+1Nx+ 1

2 j+ t

(unNx+ 1

2 j+1 + unNx+ 12 jminus1

minus2unNx+ 12 j

)(h2Re) + t

(2unNx+ 1

2 j

minus5unNxminus 12 j

+ 4unNxminus 32 j

minus unNxminus 52 j

)(h2Re)

(28)

Thus the intermediate velocity boundary condition isconsistently adopted for the velocityndashpressure couplingWith different projection methods the gradient pressureterm (nabladp) is added (Kim 2005 Kim amp Moin 1985Poux Glockner ampAzaiumlez 2011 TsengampHuang 2014) orsubtracted (Chorin 1968 Li Jung Lee Lee ampKim 2012Li Yun Lee Shin Jeong amp Kim 2013) Our proposedmethod for the intermediate velocity will work well ifnabladp is subtracted For the other case the intermediatevelocity boundary condition should be modified and canbe derived by using the intermediate pressure boundarycondition in a similar way Next we will briefly describethe extension of our proposed method to an implicitscheme for the momentum equation (1)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun+1

(29)

Then we decompose Equation (29) into two steps

u minus un

t= minus(u middot nabladu)n + 1

Redu (30)

un+1 minus ut

= minusnabladpn+1 (31)

Since the boundary conditions for velocity and pres-sure gradient are known in Section 32 we can straight-forwardly compute the intermediate velocity values in the

domain boundary as

n middot u = n middot un+1 + tn middot nabladpn+1 (32)

However in that case the time step may be restrictedby numerical stability since explicit time integration forthe diffusion term is considered on the boundary Onepossible way to improve numerical stability is to replacedun bydu in Equations (22)ndash(24) and substitute theminto Equation (32) After that we can combine Equa-tions (30) and (32) and find the solutions of the inter-mediate velocity Once u is known we can straightfor-wardly compute pn+1 and un+1 by using Equations (17)and (19) respectively Note that in this paper we useEquations (14) and (15) so that we only need to solveEquation (17 ) However if the Reynolds number ismuchsmaller implicit or CrankndashNicolson schemes should beconsidered It should also be noted that the projectionmethod on staggered grids has the property that an arbi-trary boundary condition for the pressure gradient can beused first and then a corresponding boundary conditioncan be provided for the intermediate velocity throughStep 5 The corresponding boundary condition shouldcertainlymake the outflow satisfy its boundary condition

33 Mass flux correction algorithm

Undesirable mass flux loss in a global or local sense ispossible because of boundary singularities in a finite dif-ference framework The proposed outflowconditionswillleverage for unexpected mass flux loss by addressing thecoupling effect between pressure and velocity at the sin-gular point (see the circle in Figure 3(a)) The centred dif-ference approximation for the velocity in Equations (21)and (28) at the singular point is not a good choice andmay lead to flux loss because of oscillation of the valuesFurthermore the present boundary conditions require atransient time to achieve mass balance which will be dis-cussed in the next section Thus we propose a simplecorrection algorithm to maintain the global mass fluxThe idea is summarized as follows

Step 1 Update the velocity with the outlet boundarycondition u

uNx+ 12 j

= unNx+ 12 j

minus t((unNx+ 1

2 j)2

minus(unNxminus 12 j

)2)

h (33)

Step 2 Set un+1Nx+ 1

2 j= εuNx+ 1

2 jif uNx+ 1

2 jgt 0 and set

un+1Nx+ 1

2 j= uNx+ 1

2 jotherwise Here ε can be

determined from the balance of mass flux

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 75

Figure 3 Schematic illustration of channels with (a) and without (b) a singular point in the outlet

between the inlet and outlet ieNysumj=1

un+112 j

=Nysumj=1

un+1Nx+ 1

2 j (34)

The following should be pointed out

(1) Our scheme satisfies the divergence-free conditionwith the mass flux correction algorithm becauseEquations (15) and (18) are used in the projectionmethod at every time step

(2) Since the mass correction method also leads tospurious velocities our proposed outflow boundarycondition does not require it if the outflow is parallelto the wall (see Figure 3 (b)) because in this case themass flux loss is much smaller which will be shownin the next section

(3) In the case of a singular point at the outletthe correction algorithm is applied after sev-eral flow-through time periods t0 which satisfiest0intpartin

u dy = αLxLy Here α is the flow-throughtime In our approach we set α = 3 This is becausetransient flow fields may promote numerical insta-bility

(4) In the case without a singular point at the out-let boundary we can use the mass flux correc-tion algorithm only once to save transient time ieα = 1

4 Numerical results

In this section to demonstrate the performance of ourproposed scheme we perform several numerical exper-iments for parabolic flow convergence testing unsteadyflow in a channel backward-facing step flow and outflowwith a block on the outlet

41 Parabolic flow

The first numerical experiment is performed for para-bolic flow through a channel The channel ranges from 0

to 2 in the streamwise (x) direction and from 0 to 1 in thenormal (y) direction The initial velocity field consists ofrandom perturbations with a maximum amplitude of 2that is u(x y 0) = rand(x y) and v(x y 0) = rand(x y)where rand(xy) is a random number between minus1 and1 The inflow velocity is set as u(0 y t) = 8(y minus y2) andv(0 y t) = 0 A no-slip boundary condition is appliedat the top and bottom walls ie (u v) = (0 0) It iswell known that at the beginning of a random veloc-ity field the wake gradually attains a steady state for alow Reynolds number At steady state the shape of theflow is the same as that of the inflow The evolution isrun up to T=7813 with a time step of t = 4h2 Hereh = 164 and Re = 100 are used Note that althoughu0 does not satisfy the divergence-free condition it con-verges rather quickly to a divergence-free velocity field infive time steps It should be noted that although whenwe begin with u0 the projection method used can obtaina divergence-free velocity field in one time step it willtake many more iterations to ensure that the l2 normalerror of divergence of the velocity field is lower than1E-6 Therefore to reduce the computational cost wedivided each time step process into five steps with sev-eral iterations Figure 4 shows the evolution of parabolicflow The results show that our method performs wellfor simulating parabolic flow To compare the velocity atthe outlet in Figure 5(a) we show the results obtainedby our proposed models with and without a flux cor-rection algorithm along with those of the convectivemodel and the nonreflecting model Here U is set as aPoiseuille profile since the outlet flow is known in thistest Furthermore we calculate the discrete l2-norm oferror which isradicradicradicradicradic Nysum

j=1(uNx+ 1

2 jminus u 1

2 j)2Ny

The errors are 849E-3 848E-3 107E-3 and 226E-2 forour proposed models with and without a flux correctionalgorithm the convective model and the nonreflecting

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

72 Y LI ET AL

22 Pressure boundary conditions

Taking the inner product of both sides of Equation (1)with n which is the unit normal vector to the domainboundary we obtain

n middot nablap = n middot(

minusut minus u middot nablau + 1Re

u) (10)

On partwall we get

n middot nablap = n middot uRe

(11)

Here we have used the no-slip boundary condition ieu = 0 On partin by using Equation (10) we get

px = minusut minus uux minus vuy + 1Re

(uxx + uyy)

= minusut + 1Re

uyy (12)

Here we have used the inflow boundary condition (3)For the outlet boundary by substituting Equation (8) intoEquation (10) we get

px = 1Re

(uxx + uyy) (13)

Thus the pressure boundary condition is directlyderived by using the NavierndashStokes equation and out-let flow boundary conditions The pressure conditionis mainly balanced with the viscous stress terms at theboundary All the pressure boundary conditions are ofNeumann type Thus the method used with the projec-tion method would be convenient and simple It is note-worthy that the present form of the pressure boundarycondition ismore plausible because the flows inmicroflu-idics are mostly viscous dominated For high Reynoldsnumber flows the pressure boundary condition becomesa homogeneous Neumann boundary condition This isbecause the viscous term (uxx + uyy)Re is negligible

3 Numerical methods

We use a staggered grid where the pressure field isstored at cell centres and velocities at cell interfaces (seeFigure 2) Let h be a uniform mesh spacing The centreof each cell ij is located at (xi yj) = ((i minus 05)h (j minus05)h) on the computational domain = (0 Lx) times(0 Ly) for i = 1 Nx and j = 1 Ny Nx and Nyare the numbers of cells in the x- and y-directions respec-tively The cell edges are located at (xi+ 1

2 yj) = (ih (j minus

05)h) and (xi yj+ 12) = ((i minus 05)h jh)

At the beginning of each time step given un wewant to find un+1 and pn+1 by solving the following

Figure 2 Velocities defined at cell boundaries and the pressurefield defined at the cell centres

discretized equations in time with a projection method(Chorin 1968)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun

(14)

nablad middot un+1 = 0 (15)

Here nablad and d denote the centred differenceapproximations for the gradient and Laplacian operatorsrespectively

nabladp =(pi+1j minus pij

hpij+1 minus pij

h

)

du = (dui+ 12 jdvij+ 1

2)

=(ui+ 3

2 j+ uiminus 1

2 j+ ui+ 1

2 j+1 + ui+ 12 jminus1 minus 4ui+ 1

2 j

h2

vij+ 32

+ vijminus 12

+ vi+1j+ 12

+ viminus1j+ 12

minus 4vij+ 12

h2

)

The outline of the main procedure in one time step isas follows

Step 1 Initialize u0 to be the divergence-free velocityfield

Step 2 Update un+1 and n middot nabladpn+1 on the entireboundaries The computational details will begiven in Section 31

Step 3 Solve an intermediate velocity field uwithout thepressure gradient term

u minus un

t= minusun middot nabladun + 1

Redun (16)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 73

Step 4 Solve the Poisson equation for the pressure

dpn+1 = 1t

nablad middot u (17)

The resulting linear system of Equation (17)is solved using a multigrid method Note thatEquation (17) is derived by applying the diver-gence operator to Equation (18)

un+1 minus ut

= minusnabladpn+1 (18)

It should be pointed out that to find the solu-tion for pressure in Equation (17) we shouldknow the boundary condition of the intermedi-ate velocity Our proposed intermediate velocityboundary condition is described in Section 32

Step 5 Update the divergence-free velocity

un+1 = u minus tnabladpn+1 (19)

These steps complete one time step For more detailsee Li Jung Lee Lee and Kim (2012) In this study wefocus on introducing efficient outflow boundary condi-tions and pressure boundary conditions

31 Boundary conditions for velocity and pressuregradient

In this section we will give detailed descriptions ofboundary conditions for the velocity and pressure gra-dient on the entire boundary First we will considerthe velocity boundary condition The velocities un+1

on partin and partwall are given as a known functionand zero respectively The outflow boundary conditionEquation (8) can be discretizedwith anupwind scheme as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus tunNx+ 1

2 junxNx+ 1

2 j (20)

Here unxi+ 12 j

is computed using the upwind procedure

unxi+ 12 j

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

uni+ 1

2 jminus un

iminus 12 j

hif un

i+ 12 j

gt 0

uni+ 3

2 jminus un

i+ 12 j

hotherwise

Note that Equation (19) yields un+1Nx+ 1

2 j= 0 when

unNx+ 1

2 j= unyNx+ 1

2 j= 0 for every j = 1 Ny Thismeans

that once u is zero at the boundary uwill always be zeroTo avoid this trivial case we use the conservation form

of the convective terms in Equation (9) The discretizedform can be written as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus t

h

[(unNx+ 1

2 j)2 minus (unNxminus 1

2 j)2]

(21)

where we used v=0 on the outflow boundary condi-tion For the pressure boundary condition we discretizeEquation (12) on partin as

pn+1x 12 j

= minusun+1

12 j

minus un12 j

t+

un12 j+1

+ un12 jminus1

minus 2un12 j

h2Re (22)

At the wall without loss of generality we only describethe pressure in the y-directionDiscretizingEquation (11)we can get

pn+1yi 12

=vniminus 1

2+ vn

i 32minus 2vn

i 12h2Re

+vni+1 12

+ vniminus1 12

minus 2vni 12

h2Re

(23)

Since the no-slip boundary condition is used at the wallwe can get vi12 = 0 Meanwhile on the staggered gridvniminus12 can be simply computed as minusvni32 to matchthe no-slip boundary condition Thus in our approachEquation (11) can be reduced to

n middot nablapn+1 = 0

For partout the pressure boundary condition can bediscretized as

pn+1xNx+ 1

2 j=

unNx+ 1

2 j+1+ un

Nx+ 12 jminus1

minus 2unNx+ 1

2 j

h2Re

+2un

Nx+ 12 j

minus 5unNxminus 1

2 j+ 4un

Nxminus 32 j

minus unNxminus 5

2 j

h2Re

(24)

Note that since there is no information for uNx+ 32 j we

use a second-order one-sided approximation to uxx

32 Boundary condition for intermediate velocity

Since we have used the projection method to discretizethe time-dependent incompressible NavierndashStokesequation in time it is necessary to define the bound-ary condition for the intermediate velocity field In ourapproach we can directly compute ulowast Since we have cal-culatedun+1 andn middot nabladp in Step 2 of Section 3 we can use

74 Y LI ET AL

them straightforwardly From Equation (19) we have

u = un+1 + tnablapn+1 (25)

On partwall we get

n middot u = n middot (un+1 + tnabladpn+1) = 0 (26)

For the inflow boundary by recalling Equation (19)we have

u 12 j

= un+112 j

+ tpn+1x 12 j

= un12 j

+ th2Re

(un1

2 j+1 + un12 jminus1 minus 2un1

2 j

) (27)

Here we have used the Dirichlet boundary conditionfor velocity and the homogeneous Neumann boundarycondition for pressure On the outflow boundary weobtain

uNx+ 12 j

= un+1Nx+ 1

2 j+ tpn+1

xNx+ 12 j

= un+1Nx+ 1

2 j+ t

(unNx+ 1

2 j+1 + unNx+ 12 jminus1

minus2unNx+ 12 j

)(h2Re) + t

(2unNx+ 1

2 j

minus5unNxminus 12 j

+ 4unNxminus 32 j

minus unNxminus 52 j

)(h2Re)

(28)

Thus the intermediate velocity boundary condition isconsistently adopted for the velocityndashpressure couplingWith different projection methods the gradient pressureterm (nabladp) is added (Kim 2005 Kim amp Moin 1985Poux Glockner ampAzaiumlez 2011 TsengampHuang 2014) orsubtracted (Chorin 1968 Li Jung Lee Lee ampKim 2012Li Yun Lee Shin Jeong amp Kim 2013) Our proposedmethod for the intermediate velocity will work well ifnabladp is subtracted For the other case the intermediatevelocity boundary condition should be modified and canbe derived by using the intermediate pressure boundarycondition in a similar way Next we will briefly describethe extension of our proposed method to an implicitscheme for the momentum equation (1)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun+1

(29)

Then we decompose Equation (29) into two steps

u minus un

t= minus(u middot nabladu)n + 1

Redu (30)

un+1 minus ut

= minusnabladpn+1 (31)

Since the boundary conditions for velocity and pres-sure gradient are known in Section 32 we can straight-forwardly compute the intermediate velocity values in the

domain boundary as

n middot u = n middot un+1 + tn middot nabladpn+1 (32)

However in that case the time step may be restrictedby numerical stability since explicit time integration forthe diffusion term is considered on the boundary Onepossible way to improve numerical stability is to replacedun bydu in Equations (22)ndash(24) and substitute theminto Equation (32) After that we can combine Equa-tions (30) and (32) and find the solutions of the inter-mediate velocity Once u is known we can straightfor-wardly compute pn+1 and un+1 by using Equations (17)and (19) respectively Note that in this paper we useEquations (14) and (15) so that we only need to solveEquation (17 ) However if the Reynolds number ismuchsmaller implicit or CrankndashNicolson schemes should beconsidered It should also be noted that the projectionmethod on staggered grids has the property that an arbi-trary boundary condition for the pressure gradient can beused first and then a corresponding boundary conditioncan be provided for the intermediate velocity throughStep 5 The corresponding boundary condition shouldcertainlymake the outflow satisfy its boundary condition

33 Mass flux correction algorithm

Undesirable mass flux loss in a global or local sense ispossible because of boundary singularities in a finite dif-ference framework The proposed outflowconditionswillleverage for unexpected mass flux loss by addressing thecoupling effect between pressure and velocity at the sin-gular point (see the circle in Figure 3(a)) The centred dif-ference approximation for the velocity in Equations (21)and (28) at the singular point is not a good choice andmay lead to flux loss because of oscillation of the valuesFurthermore the present boundary conditions require atransient time to achieve mass balance which will be dis-cussed in the next section Thus we propose a simplecorrection algorithm to maintain the global mass fluxThe idea is summarized as follows

Step 1 Update the velocity with the outlet boundarycondition u

uNx+ 12 j

= unNx+ 12 j

minus t((unNx+ 1

2 j)2

minus(unNxminus 12 j

)2)

h (33)

Step 2 Set un+1Nx+ 1

2 j= εuNx+ 1

2 jif uNx+ 1

2 jgt 0 and set

un+1Nx+ 1

2 j= uNx+ 1

2 jotherwise Here ε can be

determined from the balance of mass flux

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 75

Figure 3 Schematic illustration of channels with (a) and without (b) a singular point in the outlet

between the inlet and outlet ieNysumj=1

un+112 j

=Nysumj=1

un+1Nx+ 1

2 j (34)

The following should be pointed out

(1) Our scheme satisfies the divergence-free conditionwith the mass flux correction algorithm becauseEquations (15) and (18) are used in the projectionmethod at every time step

(2) Since the mass correction method also leads tospurious velocities our proposed outflow boundarycondition does not require it if the outflow is parallelto the wall (see Figure 3 (b)) because in this case themass flux loss is much smaller which will be shownin the next section

(3) In the case of a singular point at the outletthe correction algorithm is applied after sev-eral flow-through time periods t0 which satisfiest0intpartin

u dy = αLxLy Here α is the flow-throughtime In our approach we set α = 3 This is becausetransient flow fields may promote numerical insta-bility

(4) In the case without a singular point at the out-let boundary we can use the mass flux correc-tion algorithm only once to save transient time ieα = 1

4 Numerical results

In this section to demonstrate the performance of ourproposed scheme we perform several numerical exper-iments for parabolic flow convergence testing unsteadyflow in a channel backward-facing step flow and outflowwith a block on the outlet

41 Parabolic flow

The first numerical experiment is performed for para-bolic flow through a channel The channel ranges from 0

to 2 in the streamwise (x) direction and from 0 to 1 in thenormal (y) direction The initial velocity field consists ofrandom perturbations with a maximum amplitude of 2that is u(x y 0) = rand(x y) and v(x y 0) = rand(x y)where rand(xy) is a random number between minus1 and1 The inflow velocity is set as u(0 y t) = 8(y minus y2) andv(0 y t) = 0 A no-slip boundary condition is appliedat the top and bottom walls ie (u v) = (0 0) It iswell known that at the beginning of a random veloc-ity field the wake gradually attains a steady state for alow Reynolds number At steady state the shape of theflow is the same as that of the inflow The evolution isrun up to T=7813 with a time step of t = 4h2 Hereh = 164 and Re = 100 are used Note that althoughu0 does not satisfy the divergence-free condition it con-verges rather quickly to a divergence-free velocity field infive time steps It should be noted that although whenwe begin with u0 the projection method used can obtaina divergence-free velocity field in one time step it willtake many more iterations to ensure that the l2 normalerror of divergence of the velocity field is lower than1E-6 Therefore to reduce the computational cost wedivided each time step process into five steps with sev-eral iterations Figure 4 shows the evolution of parabolicflow The results show that our method performs wellfor simulating parabolic flow To compare the velocity atthe outlet in Figure 5(a) we show the results obtainedby our proposed models with and without a flux cor-rection algorithm along with those of the convectivemodel and the nonreflecting model Here U is set as aPoiseuille profile since the outlet flow is known in thistest Furthermore we calculate the discrete l2-norm oferror which isradicradicradicradicradic Nysum

j=1(uNx+ 1

2 jminus u 1

2 j)2Ny

The errors are 849E-3 848E-3 107E-3 and 226E-2 forour proposed models with and without a flux correctionalgorithm the convective model and the nonreflecting

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 73

Step 4 Solve the Poisson equation for the pressure

dpn+1 = 1t

nablad middot u (17)

The resulting linear system of Equation (17)is solved using a multigrid method Note thatEquation (17) is derived by applying the diver-gence operator to Equation (18)

un+1 minus ut

= minusnabladpn+1 (18)

It should be pointed out that to find the solu-tion for pressure in Equation (17) we shouldknow the boundary condition of the intermedi-ate velocity Our proposed intermediate velocityboundary condition is described in Section 32

Step 5 Update the divergence-free velocity

un+1 = u minus tnabladpn+1 (19)

These steps complete one time step For more detailsee Li Jung Lee Lee and Kim (2012) In this study wefocus on introducing efficient outflow boundary condi-tions and pressure boundary conditions

31 Boundary conditions for velocity and pressuregradient

In this section we will give detailed descriptions ofboundary conditions for the velocity and pressure gra-dient on the entire boundary First we will considerthe velocity boundary condition The velocities un+1

on partin and partwall are given as a known functionand zero respectively The outflow boundary conditionEquation (8) can be discretizedwith anupwind scheme as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus tunNx+ 1

2 junxNx+ 1

2 j (20)

Here unxi+ 12 j

is computed using the upwind procedure

unxi+ 12 j

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

uni+ 1

2 jminus un

iminus 12 j

hif un

i+ 12 j

gt 0

uni+ 3

2 jminus un

i+ 12 j

hotherwise

Note that Equation (19) yields un+1Nx+ 1

2 j= 0 when

unNx+ 1

2 j= unyNx+ 1

2 j= 0 for every j = 1 Ny Thismeans

that once u is zero at the boundary uwill always be zeroTo avoid this trivial case we use the conservation form

of the convective terms in Equation (9) The discretizedform can be written as

un+1Nx+ 1

2 j= unNx+ 1

2 jminus t

h

[(unNx+ 1

2 j)2 minus (unNxminus 1

2 j)2]

(21)

where we used v=0 on the outflow boundary condi-tion For the pressure boundary condition we discretizeEquation (12) on partin as

pn+1x 12 j

= minusun+1

12 j

minus un12 j

t+

un12 j+1

+ un12 jminus1

minus 2un12 j

h2Re (22)

At the wall without loss of generality we only describethe pressure in the y-directionDiscretizingEquation (11)we can get

pn+1yi 12

=vniminus 1

2+ vn

i 32minus 2vn

i 12h2Re

+vni+1 12

+ vniminus1 12

minus 2vni 12

h2Re

(23)

Since the no-slip boundary condition is used at the wallwe can get vi12 = 0 Meanwhile on the staggered gridvniminus12 can be simply computed as minusvni32 to matchthe no-slip boundary condition Thus in our approachEquation (11) can be reduced to

n middot nablapn+1 = 0

For partout the pressure boundary condition can bediscretized as

pn+1xNx+ 1

2 j=

unNx+ 1

2 j+1+ un

Nx+ 12 jminus1

minus 2unNx+ 1

2 j

h2Re

+2un

Nx+ 12 j

minus 5unNxminus 1

2 j+ 4un

Nxminus 32 j

minus unNxminus 5

2 j

h2Re

(24)

Note that since there is no information for uNx+ 32 j we

use a second-order one-sided approximation to uxx

32 Boundary condition for intermediate velocity

Since we have used the projection method to discretizethe time-dependent incompressible NavierndashStokesequation in time it is necessary to define the bound-ary condition for the intermediate velocity field In ourapproach we can directly compute ulowast Since we have cal-culatedun+1 andn middot nabladp in Step 2 of Section 3 we can use

74 Y LI ET AL

them straightforwardly From Equation (19) we have

u = un+1 + tnablapn+1 (25)

On partwall we get

n middot u = n middot (un+1 + tnabladpn+1) = 0 (26)

For the inflow boundary by recalling Equation (19)we have

u 12 j

= un+112 j

+ tpn+1x 12 j

= un12 j

+ th2Re

(un1

2 j+1 + un12 jminus1 minus 2un1

2 j

) (27)

Here we have used the Dirichlet boundary conditionfor velocity and the homogeneous Neumann boundarycondition for pressure On the outflow boundary weobtain

uNx+ 12 j

= un+1Nx+ 1

2 j+ tpn+1

xNx+ 12 j

= un+1Nx+ 1

2 j+ t

(unNx+ 1

2 j+1 + unNx+ 12 jminus1

minus2unNx+ 12 j

)(h2Re) + t

(2unNx+ 1

2 j

minus5unNxminus 12 j

+ 4unNxminus 32 j

minus unNxminus 52 j

)(h2Re)

(28)

Thus the intermediate velocity boundary condition isconsistently adopted for the velocityndashpressure couplingWith different projection methods the gradient pressureterm (nabladp) is added (Kim 2005 Kim amp Moin 1985Poux Glockner ampAzaiumlez 2011 TsengampHuang 2014) orsubtracted (Chorin 1968 Li Jung Lee Lee ampKim 2012Li Yun Lee Shin Jeong amp Kim 2013) Our proposedmethod for the intermediate velocity will work well ifnabladp is subtracted For the other case the intermediatevelocity boundary condition should be modified and canbe derived by using the intermediate pressure boundarycondition in a similar way Next we will briefly describethe extension of our proposed method to an implicitscheme for the momentum equation (1)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun+1

(29)

Then we decompose Equation (29) into two steps

u minus un

t= minus(u middot nabladu)n + 1

Redu (30)

un+1 minus ut

= minusnabladpn+1 (31)

Since the boundary conditions for velocity and pres-sure gradient are known in Section 32 we can straight-forwardly compute the intermediate velocity values in the

domain boundary as

n middot u = n middot un+1 + tn middot nabladpn+1 (32)

However in that case the time step may be restrictedby numerical stability since explicit time integration forthe diffusion term is considered on the boundary Onepossible way to improve numerical stability is to replacedun bydu in Equations (22)ndash(24) and substitute theminto Equation (32) After that we can combine Equa-tions (30) and (32) and find the solutions of the inter-mediate velocity Once u is known we can straightfor-wardly compute pn+1 and un+1 by using Equations (17)and (19) respectively Note that in this paper we useEquations (14) and (15) so that we only need to solveEquation (17 ) However if the Reynolds number ismuchsmaller implicit or CrankndashNicolson schemes should beconsidered It should also be noted that the projectionmethod on staggered grids has the property that an arbi-trary boundary condition for the pressure gradient can beused first and then a corresponding boundary conditioncan be provided for the intermediate velocity throughStep 5 The corresponding boundary condition shouldcertainlymake the outflow satisfy its boundary condition

33 Mass flux correction algorithm

Undesirable mass flux loss in a global or local sense ispossible because of boundary singularities in a finite dif-ference framework The proposed outflowconditionswillleverage for unexpected mass flux loss by addressing thecoupling effect between pressure and velocity at the sin-gular point (see the circle in Figure 3(a)) The centred dif-ference approximation for the velocity in Equations (21)and (28) at the singular point is not a good choice andmay lead to flux loss because of oscillation of the valuesFurthermore the present boundary conditions require atransient time to achieve mass balance which will be dis-cussed in the next section Thus we propose a simplecorrection algorithm to maintain the global mass fluxThe idea is summarized as follows

Step 1 Update the velocity with the outlet boundarycondition u

uNx+ 12 j

= unNx+ 12 j

minus t((unNx+ 1

2 j)2

minus(unNxminus 12 j

)2)

h (33)

Step 2 Set un+1Nx+ 1

2 j= εuNx+ 1

2 jif uNx+ 1

2 jgt 0 and set

un+1Nx+ 1

2 j= uNx+ 1

2 jotherwise Here ε can be

determined from the balance of mass flux

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 75

Figure 3 Schematic illustration of channels with (a) and without (b) a singular point in the outlet

between the inlet and outlet ieNysumj=1

un+112 j

=Nysumj=1

un+1Nx+ 1

2 j (34)

The following should be pointed out

(1) Our scheme satisfies the divergence-free conditionwith the mass flux correction algorithm becauseEquations (15) and (18) are used in the projectionmethod at every time step

(2) Since the mass correction method also leads tospurious velocities our proposed outflow boundarycondition does not require it if the outflow is parallelto the wall (see Figure 3 (b)) because in this case themass flux loss is much smaller which will be shownin the next section

(3) In the case of a singular point at the outletthe correction algorithm is applied after sev-eral flow-through time periods t0 which satisfiest0intpartin

u dy = αLxLy Here α is the flow-throughtime In our approach we set α = 3 This is becausetransient flow fields may promote numerical insta-bility

(4) In the case without a singular point at the out-let boundary we can use the mass flux correc-tion algorithm only once to save transient time ieα = 1

4 Numerical results

In this section to demonstrate the performance of ourproposed scheme we perform several numerical exper-iments for parabolic flow convergence testing unsteadyflow in a channel backward-facing step flow and outflowwith a block on the outlet

41 Parabolic flow

The first numerical experiment is performed for para-bolic flow through a channel The channel ranges from 0

to 2 in the streamwise (x) direction and from 0 to 1 in thenormal (y) direction The initial velocity field consists ofrandom perturbations with a maximum amplitude of 2that is u(x y 0) = rand(x y) and v(x y 0) = rand(x y)where rand(xy) is a random number between minus1 and1 The inflow velocity is set as u(0 y t) = 8(y minus y2) andv(0 y t) = 0 A no-slip boundary condition is appliedat the top and bottom walls ie (u v) = (0 0) It iswell known that at the beginning of a random veloc-ity field the wake gradually attains a steady state for alow Reynolds number At steady state the shape of theflow is the same as that of the inflow The evolution isrun up to T=7813 with a time step of t = 4h2 Hereh = 164 and Re = 100 are used Note that althoughu0 does not satisfy the divergence-free condition it con-verges rather quickly to a divergence-free velocity field infive time steps It should be noted that although whenwe begin with u0 the projection method used can obtaina divergence-free velocity field in one time step it willtake many more iterations to ensure that the l2 normalerror of divergence of the velocity field is lower than1E-6 Therefore to reduce the computational cost wedivided each time step process into five steps with sev-eral iterations Figure 4 shows the evolution of parabolicflow The results show that our method performs wellfor simulating parabolic flow To compare the velocity atthe outlet in Figure 5(a) we show the results obtainedby our proposed models with and without a flux cor-rection algorithm along with those of the convectivemodel and the nonreflecting model Here U is set as aPoiseuille profile since the outlet flow is known in thistest Furthermore we calculate the discrete l2-norm oferror which isradicradicradicradicradic Nysum

j=1(uNx+ 1

2 jminus u 1

2 j)2Ny

The errors are 849E-3 848E-3 107E-3 and 226E-2 forour proposed models with and without a flux correctionalgorithm the convective model and the nonreflecting

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

74 Y LI ET AL

them straightforwardly From Equation (19) we have

u = un+1 + tnablapn+1 (25)

On partwall we get

n middot u = n middot (un+1 + tnabladpn+1) = 0 (26)

For the inflow boundary by recalling Equation (19)we have

u 12 j

= un+112 j

+ tpn+1x 12 j

= un12 j

+ th2Re

(un1

2 j+1 + un12 jminus1 minus 2un1

2 j

) (27)

Here we have used the Dirichlet boundary conditionfor velocity and the homogeneous Neumann boundarycondition for pressure On the outflow boundary weobtain

uNx+ 12 j

= un+1Nx+ 1

2 j+ tpn+1

xNx+ 12 j

= un+1Nx+ 1

2 j+ t

(unNx+ 1

2 j+1 + unNx+ 12 jminus1

minus2unNx+ 12 j

)(h2Re) + t

(2unNx+ 1

2 j

minus5unNxminus 12 j

+ 4unNxminus 32 j

minus unNxminus 52 j

)(h2Re)

(28)

Thus the intermediate velocity boundary condition isconsistently adopted for the velocityndashpressure couplingWith different projection methods the gradient pressureterm (nabladp) is added (Kim 2005 Kim amp Moin 1985Poux Glockner ampAzaiumlez 2011 TsengampHuang 2014) orsubtracted (Chorin 1968 Li Jung Lee Lee ampKim 2012Li Yun Lee Shin Jeong amp Kim 2013) Our proposedmethod for the intermediate velocity will work well ifnabladp is subtracted For the other case the intermediatevelocity boundary condition should be modified and canbe derived by using the intermediate pressure boundarycondition in a similar way Next we will briefly describethe extension of our proposed method to an implicitscheme for the momentum equation (1)

un+1 minus un

t= minus(u middot nabladu)n minus nabladpn+1 + 1

Redun+1

(29)

Then we decompose Equation (29) into two steps

u minus un

t= minus(u middot nabladu)n + 1

Redu (30)

un+1 minus ut

= minusnabladpn+1 (31)

Since the boundary conditions for velocity and pres-sure gradient are known in Section 32 we can straight-forwardly compute the intermediate velocity values in the

domain boundary as

n middot u = n middot un+1 + tn middot nabladpn+1 (32)

However in that case the time step may be restrictedby numerical stability since explicit time integration forthe diffusion term is considered on the boundary Onepossible way to improve numerical stability is to replacedun bydu in Equations (22)ndash(24) and substitute theminto Equation (32) After that we can combine Equa-tions (30) and (32) and find the solutions of the inter-mediate velocity Once u is known we can straightfor-wardly compute pn+1 and un+1 by using Equations (17)and (19) respectively Note that in this paper we useEquations (14) and (15) so that we only need to solveEquation (17 ) However if the Reynolds number ismuchsmaller implicit or CrankndashNicolson schemes should beconsidered It should also be noted that the projectionmethod on staggered grids has the property that an arbi-trary boundary condition for the pressure gradient can beused first and then a corresponding boundary conditioncan be provided for the intermediate velocity throughStep 5 The corresponding boundary condition shouldcertainlymake the outflow satisfy its boundary condition

33 Mass flux correction algorithm

Undesirable mass flux loss in a global or local sense ispossible because of boundary singularities in a finite dif-ference framework The proposed outflowconditionswillleverage for unexpected mass flux loss by addressing thecoupling effect between pressure and velocity at the sin-gular point (see the circle in Figure 3(a)) The centred dif-ference approximation for the velocity in Equations (21)and (28) at the singular point is not a good choice andmay lead to flux loss because of oscillation of the valuesFurthermore the present boundary conditions require atransient time to achieve mass balance which will be dis-cussed in the next section Thus we propose a simplecorrection algorithm to maintain the global mass fluxThe idea is summarized as follows

Step 1 Update the velocity with the outlet boundarycondition u

uNx+ 12 j

= unNx+ 12 j

minus t((unNx+ 1

2 j)2

minus(unNxminus 12 j

)2)

h (33)

Step 2 Set un+1Nx+ 1

2 j= εuNx+ 1

2 jif uNx+ 1

2 jgt 0 and set

un+1Nx+ 1

2 j= uNx+ 1

2 jotherwise Here ε can be

determined from the balance of mass flux

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 75

Figure 3 Schematic illustration of channels with (a) and without (b) a singular point in the outlet

between the inlet and outlet ieNysumj=1

un+112 j

=Nysumj=1

un+1Nx+ 1

2 j (34)

The following should be pointed out

(1) Our scheme satisfies the divergence-free conditionwith the mass flux correction algorithm becauseEquations (15) and (18) are used in the projectionmethod at every time step

(2) Since the mass correction method also leads tospurious velocities our proposed outflow boundarycondition does not require it if the outflow is parallelto the wall (see Figure 3 (b)) because in this case themass flux loss is much smaller which will be shownin the next section

(3) In the case of a singular point at the outletthe correction algorithm is applied after sev-eral flow-through time periods t0 which satisfiest0intpartin

u dy = αLxLy Here α is the flow-throughtime In our approach we set α = 3 This is becausetransient flow fields may promote numerical insta-bility

(4) In the case without a singular point at the out-let boundary we can use the mass flux correc-tion algorithm only once to save transient time ieα = 1

4 Numerical results

In this section to demonstrate the performance of ourproposed scheme we perform several numerical exper-iments for parabolic flow convergence testing unsteadyflow in a channel backward-facing step flow and outflowwith a block on the outlet

41 Parabolic flow

The first numerical experiment is performed for para-bolic flow through a channel The channel ranges from 0

to 2 in the streamwise (x) direction and from 0 to 1 in thenormal (y) direction The initial velocity field consists ofrandom perturbations with a maximum amplitude of 2that is u(x y 0) = rand(x y) and v(x y 0) = rand(x y)where rand(xy) is a random number between minus1 and1 The inflow velocity is set as u(0 y t) = 8(y minus y2) andv(0 y t) = 0 A no-slip boundary condition is appliedat the top and bottom walls ie (u v) = (0 0) It iswell known that at the beginning of a random veloc-ity field the wake gradually attains a steady state for alow Reynolds number At steady state the shape of theflow is the same as that of the inflow The evolution isrun up to T=7813 with a time step of t = 4h2 Hereh = 164 and Re = 100 are used Note that althoughu0 does not satisfy the divergence-free condition it con-verges rather quickly to a divergence-free velocity field infive time steps It should be noted that although whenwe begin with u0 the projection method used can obtaina divergence-free velocity field in one time step it willtake many more iterations to ensure that the l2 normalerror of divergence of the velocity field is lower than1E-6 Therefore to reduce the computational cost wedivided each time step process into five steps with sev-eral iterations Figure 4 shows the evolution of parabolicflow The results show that our method performs wellfor simulating parabolic flow To compare the velocity atthe outlet in Figure 5(a) we show the results obtainedby our proposed models with and without a flux cor-rection algorithm along with those of the convectivemodel and the nonreflecting model Here U is set as aPoiseuille profile since the outlet flow is known in thistest Furthermore we calculate the discrete l2-norm oferror which isradicradicradicradicradic Nysum

j=1(uNx+ 1

2 jminus u 1

2 j)2Ny

The errors are 849E-3 848E-3 107E-3 and 226E-2 forour proposed models with and without a flux correctionalgorithm the convective model and the nonreflecting

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 75

Figure 3 Schematic illustration of channels with (a) and without (b) a singular point in the outlet

between the inlet and outlet ieNysumj=1

un+112 j

=Nysumj=1

un+1Nx+ 1

2 j (34)

The following should be pointed out

(1) Our scheme satisfies the divergence-free conditionwith the mass flux correction algorithm becauseEquations (15) and (18) are used in the projectionmethod at every time step

(2) Since the mass correction method also leads tospurious velocities our proposed outflow boundarycondition does not require it if the outflow is parallelto the wall (see Figure 3 (b)) because in this case themass flux loss is much smaller which will be shownin the next section

(3) In the case of a singular point at the outletthe correction algorithm is applied after sev-eral flow-through time periods t0 which satisfiest0intpartin

u dy = αLxLy Here α is the flow-throughtime In our approach we set α = 3 This is becausetransient flow fields may promote numerical insta-bility

(4) In the case without a singular point at the out-let boundary we can use the mass flux correc-tion algorithm only once to save transient time ieα = 1

4 Numerical results

In this section to demonstrate the performance of ourproposed scheme we perform several numerical exper-iments for parabolic flow convergence testing unsteadyflow in a channel backward-facing step flow and outflowwith a block on the outlet

41 Parabolic flow

The first numerical experiment is performed for para-bolic flow through a channel The channel ranges from 0

to 2 in the streamwise (x) direction and from 0 to 1 in thenormal (y) direction The initial velocity field consists ofrandom perturbations with a maximum amplitude of 2that is u(x y 0) = rand(x y) and v(x y 0) = rand(x y)where rand(xy) is a random number between minus1 and1 The inflow velocity is set as u(0 y t) = 8(y minus y2) andv(0 y t) = 0 A no-slip boundary condition is appliedat the top and bottom walls ie (u v) = (0 0) It iswell known that at the beginning of a random veloc-ity field the wake gradually attains a steady state for alow Reynolds number At steady state the shape of theflow is the same as that of the inflow The evolution isrun up to T=7813 with a time step of t = 4h2 Hereh = 164 and Re = 100 are used Note that althoughu0 does not satisfy the divergence-free condition it con-verges rather quickly to a divergence-free velocity field infive time steps It should be noted that although whenwe begin with u0 the projection method used can obtaina divergence-free velocity field in one time step it willtake many more iterations to ensure that the l2 normalerror of divergence of the velocity field is lower than1E-6 Therefore to reduce the computational cost wedivided each time step process into five steps with sev-eral iterations Figure 4 shows the evolution of parabolicflow The results show that our method performs wellfor simulating parabolic flow To compare the velocity atthe outlet in Figure 5(a) we show the results obtainedby our proposed models with and without a flux cor-rection algorithm along with those of the convectivemodel and the nonreflecting model Here U is set as aPoiseuille profile since the outlet flow is known in thistest Furthermore we calculate the discrete l2-norm oferror which isradicradicradicradicradic Nysum

j=1(uNx+ 1

2 jminus u 1

2 j)2Ny

The errors are 849E-3 848E-3 107E-3 and 226E-2 forour proposed models with and without a flux correctionalgorithm the convective model and the nonreflecting

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

76 Y LI ET AL

Figure 4 Snapshots of velocity field at times t= 0 (a) t= 0391 (b) t= 1563 (c) and t= 7813 (d)

Figure 5 Comparisons of our proposed models with and without a flux correction algorithm to the convective model and the nonre-flecting model (a) Velocity profiles at the outflow boundary position x= 2 (b) Flux loss rate which is computed as the ratio of inlet tooutlet Note the reason for the jump in flux rate after three flow-through time period intervals the fluid retains mass flux conservationin our approach Without a mass flux correction method the mass flux will be conservative after more flow-through time periods

model respectively These results indicate that the firstthree models can efficiently emulate parabolic flow Thereason for the failure of the nonreflecting model is thatit is derived from a wave equation and not for parabolicflow In Figure 5(b) we illustrate the flux loss rate whichis computed by the ratio of inlet and outlet fluxes As canbe seen the convective model and our model satisfy fluxconservation whereas the nonreflecting model does not

Note that our correction algorithm is performed after

t0 = αLxLy⎛⎝ Nysum

j=1u12jh

⎞⎠ = 45

before α = 3 is introduced The jump in flux rate asshown in Figure 5(b) is due to the correction algorithmafter three time periods The numerical jump disappears

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 77

later It should be pointed out that without the masscorrection algorithm our model also maintains flux con-servation after more fluid exits the outlet boundarybecause our outflow boundary condition and pressureboundary conditions are defined by matching with theNavierndashStokes equationUnless otherwisementionedwewill not use our correction algorithm for the channelswhich have no singular points

42 Convergence test

We next perform spatial and temporal convergence testsof the proposed method To obtain the spatial conver-gence rate we perform a number of simulations withincreasingly fine grids h = 12n for n=5 6 7 and 8 onthe domain = (0 2) times (0 1) The initial condition andparameters are the same as in Section 41 It is easy tofind that at equilibrium the constant pressure gradientequals uyyRe ie 16Re Thus the equilibrium solutionof parabolic flow is given by

u(x y) = 8y(1 minus y) v(x y) = 0 and

p(x y) = 16(1 minus x)Re

Observing the evolution which is described in Section41 we can see that the outflow reaches a steady stateafter t=7813 Thus in this section we also run thesimulations up to T=7813 We define the error of thenumerical solution on a grid as the discrete l2-norm ofthe difference between the numerical solution uh and theexact solution uexact ehij = uhij minus uexactij The other termsare defined similarly The rates of convergence are definedas the ratio of successive errors log2(eh2eh22) andlog2(ehinfineh2infin) Here e22 is a discrete l2 norm

and is defined as

e22 =Nxsumi=1

Nysumj=1

e2ij(NxNy)

and einfin is a discrete linfin norm which is defined aseinfin = max(|eij|) Since we refined the spatial and tem-poral grids by factors of 4 and 2 respectively the ratioof successive errors should increase by a factor of 2 Theerrors and rates of convergence obtained using these def-initions are given in Table 1 The results suggest thatthe scheme is almost second-order accurate in space asexpected from the discretization

To obtain the convergence rate for temporal dis-cretization we fix the space step size as h = 164 andchoose a set of decreasing time steps t = 1953E-39766E-4 4883E-4 and 2441E-4 The errors and ratesof convergence obtained using these definitions are givenin Table 2 First-order accuracy with respect to time isobserved as expected from the discretization

43 Backward-facing step flow

To examine the robustness and accuracy of the pro-posed method we consider a backward-facing step flowWe then compare our results with the previous experi-mental (Armaly Durst amp Pereira 1983) and numerical(Barton 1997 Kim ampMoin 1985) results

The backward-facing step geometry with channeldimensions is shown in Figure 6 To verify the robust-ness of the proposed numerical scheme we consider aninitial condition of a random velocity field on the compu-tational domain = (0 12) times (0 1) The inflow velocityisu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =0 These are shown in Figure 7(a) The computationsare performed with Re = 500 h = 164 and t = 5h2

Table 1 Error and convergence results with various mesh gridst = 2h2 is used

Grid

Case 64 times 32 128 times 64 256 times 128 512 times 256

u l2-error 275E-2 7747E-3 275E-3 5607E-4Rate 188 186 187

linfin-error 5900E-2 1562E-2 4167E-3 1150E-3Rate 192 191 186

v l2-error 1311E-3 3147E-4 7585E-5 2162E-5Rate 206 205 181

linfin-error 2292E-3 5623E-4 1399E-4 3915E-5Rate 203 200 184

p l2-error 2424E-2 6888E-3 1898E-3 5163E-4Rate 181 186 187

linfin-error 4438E-2 1273E-3 3445E-3 9438E-4Rate 180 189 187

nablad middot u l2-error 4449E-3 1212E-3 3335E-4 9132E-5Rate 188 186 187

linfin-error 5090E-3 1389E-3 3827E-4 1048E-4Rate 187 186 187

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

78 Y LI ET AL

Table 2 Error and convergence results with various time steps 64times128 is fixed

t

Case 1953E-3 9766E-4 4883E-4 2441E-4

u l2-error 7744E-2 3161E-2 1598E-2 8203E-3Rate 129 098 096

linfin-error 1700E-1 6622E-2 3262E-2 1667E-2Rate 136 102 097

v l2-error 3547E-3 1254E-3 5982E-4 3034E-4Rate 150 107 098

linfin-error 6265E-3 2260E-3 1094E-3 5602E-4Rate 147 105 097

p l2-error 5003E-2 2678E-2 1459E-2 7568E-3Rate 090 088 095

linfin-error 9491E-2 4829E-2 2660E-2 1378E-3Rate 097 086 095

nablad middot u l2-error 6273E-2 2571E-2 1301E-2 6670E-3Rate 129 098 096

linfin-error 7197E-2 2950E-2 1493E-2 7654E-3Rate 129 098 096

Figure 6 Schematic of a flow over a backward-facing step

Figures 7(a)ndash7(d) show snapshots of the velocity field attimes t = 0 1465 2441 and 48828 respectively As canbe seen our model can simulate backward-facing stepflow well

Plots of reattachment length at steady states with vari-ousReynolds numbers Re = 100 200 800 are shown

in Figure 8(b) Typically wall shear stress can be usedto identify the reattachment location however in ourcomputations the wall shear stress is calculated verynearly but not exactly at the wall To obtain a more accu-rate result we numerically compute the reattachmentlength Xh

r by using the quadratic polynomial α = aβ2 +bβ + c with three given points (xm y3) (xp y2) and(xq y1) (see Figure 8(a)) Herem isin [i i + 1] which satis-fies uiminus123 le 0 and ui+123 gt 0 for some integer i Thenxm is computed by the linear interpolation of nearby twopoints ie xm = (xi+1uiminus123 minus xiui+123)(uiminus123 minusui+123) Note that xp and xq are defined in a similarfashion Finally approximating the quadratic polynomialα = aβ2 + bβ + c we can get Xh

r = α∣∣β=0 To compare

the proposed algorithm to the previous results we put

Figure 7 Snapshots of the velocity field at times t= 0 (a) t= 1465 (b) t= 2441 (c) and t= 48828 (d)

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 79

Figure 8 (a) Schematic of computing the reattachment length Xhr in the numerical solution Here open circle filled square and starindicate the positive zero and negative values of u respectively (b) Reattachment length at steady statewith various Reynolds numbers

Figure 9 Snapshots of streamlines with four domain sizes = (0 4) times (0 1) (a) = (0 5) times (0 1) (b) = (0 6) times (0 1) (c) and = (0 8) times (0 1) (d) Comparison of velocity with the three cases (e)

them together From these results we observe that theresults obtained by using our proposed method are ingood agreement with those from the previous numer-ical methods (Barton 1997 Kim amp Moin 1985) andexperiment (Armaly Durst amp Pereira 1983)

Figures 9(a)ndash9(e) are snapshots of streamlines forvarious domain sizes = (0 4) times (0 1) (0 5) times (0 1)(0 6) times (0 1) and (0 8) times (0 1) respectivelyHereRe =400 is used Figure 9(f) shows a comparison of velocity atthe outflow boundary for different domain sizes

The agreement in our computational results obtainedfrom a larger domain and a smaller domain suggeststhat our proposed outflow boundary condition is effi-cient It should be pointed out that if the computational

domain is smaller then the numerical error increases(see Figure 9(a)) This is because our proposed methodis simple we have assumed that the flow only moves byconvection and that its main direction is normal to theoutflow boundary

44 Comparisonwith Neumann boundarycondition

In this section we will compare the results obtained byour proposed and Neumann outflow boundary condi-tions to show the efficiency and accuracy of our pro-posed method The backward-facing step flow drawn

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

80 Y LI ET AL

Figure 10 Snapshots of streamlines for the proposed outflow boundary (a) and a Neumann outflow boundary (b) From left to rightthey are results for = (0 4) times (0 1) and = (0 7) times (0 1) respectively

Table 3 The l2-errors with various domain sizes for the proposedoutflow boundary and the Neumann outflow boundary

Domain ()

Case (0 4) times (0 1) (0 5) times (0 1) (0 6) times (0 1) (0 7) times (0 1)

u Neumann 9871E-2 2566E-2 1599E-2 1299E-2Proposed 3494E-2 1012E-2 6010E-3 2314E-3

v Neumann 2315E-2 5442E-3 3387E-3 2796E-3Proposed 1745E-2 4147E-3 2188E-3 1051E-3

in Section 43 is considered The parameters and initialcondition are chosen to be the same as in Section 43

The comparisons with the two boundary conditionsare drawn in Figure 10 From left to right they are resultsfor = (0 4) times (0 1) and = (0 7) times (0 1) respec-tively As can be observed from Figure 10(a) the agree-ment between the results computed from the larger andsmaller domains is good This suggests that our proposedoutflow boundary condition is more efficient than theNeumann boundary condition

Furthermore we perform a number of simulationswith increasing domains = (0 Lx) times (0 1) for Lx =4 5 6 and 7 Since there is no closed-form analyticalsolution for this problem we consider a reference numer-ical solution which is obtained in a large domain =(0 8) times (0 1) Therefore we can compute the l2 errorsof velocities directly for each case These l2 errors aregiven in Table 3 The results suggest that as the domainsize increases the l2 errors converge These results alsosuggest that our proposed outflow boundary conditionis more accurate than the Neumann outflow boundarybecause the l2 errors obtained by our proposed out-flow boundary condition are much smaller than thoseobtained by the Neumann outflow boundary

45 Efficiency of the proposedmethod

To investigate the efficiency of our proposed method wemeasure CPU times needed to solve the following prob-lem u(x y 0) = rand(x y) v(x y 0) = rand(x y) onthe domain = (0 4) times (0 1) with 2n+2 times 2n meshesfor n = 4 5 6 7 and 8 The inflow velocities areu(0 y t) = max(24(1 minus y)(y minus 05) 0) and v(0 y t) =

103

104

105

101

102

103

Number of mesh grid

CP

Utim

e(s

)

Numerical dataFitting plot

Figure 11 Logarithm of the total CPU time versus number ofmesh grid (NxNy) in multigrid solver

0 Here Re = 500 is fixed All simulations are run up to1000t with the time step t = h2 Note that each sim-ulation is run until the maximum error of the residual isless than 10minus7 Since we refined the spatial grids by a fac-tor of four computation times should be increased ideallyby a factor of four owing to the discretization (17) whichis solved by using a multigrid method with no specialcomputing operators for the inflow and outflow bound-ary conditions added Figure 11 shows the total CPUtime versus number of mesh grid (NxNy) The straightfitting plot implies that the multigrid solver achievesO(NxNy) efficiency Furthermore the computing time forthe boundary conditions can be negligible because ourmethod updates the boundary velocity explicitly If animplicit update for the boundary condition is chosenthen the computational time is not negligible

46 Three-dimensional flow

Our method can also be straightforwardly extended tothree-dimensional flows Here we consider a backward-facing step flow with an initial condition of a randomvelocity field on the computational domain = (0 3) times(0 1) times (0 1) The inflow velocities are u(0 y z t) =

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 81

Figure 12 Snapshots of the velocity field inhorizontal (a) andvertical (b) views From left to right the computational times are t = 00980293 0586 and 2441

Figure 13 Schematic illustration of inflowndashoutflow boundary conditions (a) Schematic illustration of inflowndashoutflow boundary condi-tions and computing domain for finding the exact solution (b)

max(24(1minus y)(yminus 05)(1minus z)(zminus 05) 0) v(0 y z t) =0 and w(0 y z t) = 0 The computations are performedwith Re = 100 h = 164 and t = h2 Figures 12(a)and 12(b) show snapshots of the velocity field in hor-izontal and vertical views respectively From left toright the computational times are t = 0098 02930586 and 2441 As can be seen from the figureour proposed model can simulate three-dimensionalflows well

47 Block on the outlet

We now consider the flow geometry shown inFigure 13(a) A fluid enters the upper half of a rectilinearchannel at the left domain boundary and exits the lowerhalf of the channel at the right boundary The inflowvelocity is set as u(0 y 0) = max(24(1 minus y)(y minus 05) 0)and v(0 y 0) = 0 on = (0 Lx) times (0 1)with h = 164t = 4h2 Lx = 4 and Re = 500 A block is set at ygt05at the outflow boundary Note that in this case thereis a singularity point on the outflow boundary Thus weshould use the proposed mass correction method Sincethere is no closed-form analytical solution for this prob-lem we consider a reference numerical solution which isfirst obtained in a large domain = (0 Lx + 2) times (0 1)as shown in Figure 13(b) and then in the domain =(0 Lx) times (0 1) The other parameters are chosen to be

the same as in the above initial condition Figure 14shows contour plots of the stream functions at t=3052and 42724 for convective nonreflecting our proposedmodel and a reference numerical solution respectivelyHere U is computed by using Equation (7) since theoutlet flow is unknown in this test As can be seencompared with the convective and proposed models flu-ids cannot completely go across the outlet boundary byusing the nonreflecting model Since its outflow bound-ary condition is derived from a wave equation with-out flux correction unphysical flow appears for lowerReynolds numbers Also the convective model can sim-ulate inflow and outflow when a block is set on the outletboundary whereas the outlet flow boundary is reducedto a Dirichlet boundary though the outflow does notreach a steady flow This is not a naturally occurringphysical phenomenon Compared with them in our pro-posed model the outflow seems to be a parabolic-typeflow which is closer to the reference numerical solutionThe reason for this is that our model allows the out-flow to be adjusted on the basis of the current inflowand inside fluid whereas the convective model workswith a given artificial velocity and mass flux correctionmethod

Figures 15(a)ndash15(d) show the contour plot of thestream function with three domain sizes Lx = 2 4 and6 for convective nonreflecting our proposed models

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

82 Y LI ET AL

Figure 14 Evolution of outflow with a block obtained by using our proposed model From top to bottom they are for convective non-reflecting our proposed models and the reference numerical solution which is obtained in a larger domain as shown in Figure 13(b)respectively The computational times are shown for the left and right sides of the figure

Figure 15 Comparison of three models with different domain sizes at T = 61035 (a) Convective model (b) nonreflecting model (c)our proposedmodel and (d) the reference numerical solution From left to right the domain sizes are (0 2) times (0 1) (0 4) times (0 1) and(0 6) times (0 1)

Figure 16 Comparison of the outlet velocity for three models with different domain sizes at T = 61035 To compare them we put thereference numerical solutions together (a) Lx = 2 (b) Lx = 4 and (c) Lx = 6

and the reference numerical solutions respectively InFigure 16 we compare the outlet velocity for the fourcases with the three domain sizes

As can be seen the outflows obtained by using thenonreflecting model lead to flux loss with the increase inthe domain size Meanwhile with the convective model

the outflow reaches a similar constant outflow profile fordifferent domain sizes Compared with the nonreflect-ing and convective models our proposed model reachesa natural and efficient outflow which seems to be aparabolic-type flow and is closer to the reference numer-ical solution

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 83

5 Conclusions

In this paper we presented a simple and efficient outflowboundary condition for an incompressibleNavierndashStokesequation The proposed outflow boundary condition isbased on the transparent equation where a weak for-mulation is used By matching with the NavierndashStokesequation adopted inside the domain we exactly pre-sented the pressure boundary condition Furthermorewe proposed the pressure boundary for convective andnonreflecting models In the numerical algorithm astaggered marker-and-cell grid was used and tempo-ral discretization was based on a projection methodThe intermediate velocity boundary condition is con-sistently adopted to handle the velocityndashpressure cou-pling Although our numerical scheme is an explicitscheme in time and has first-order accuracy in time wefocus on proposing efficient outflow boundary condi-tions and pressure boundary conditions in this study Theextensions to implicit and CrankndashNicolson schemes arestraightforward Characteristic numerical experimentswere presented to demonstrate the accuracy of our pro-posed model Although a first-order equation is usedon the outlet boundary almost second-order accuracy isobserved for our model in the convergence test becausewe use a full second-order scheme to discretize theNavierndashStokes equation inside the domain Comparedwith the convective model and our proposed modelthe nonreflecting model is not a good choice for lowReynolds number flow and the outflow when a block ison the outlet boundary because the nonreflecting modelcannot maintain mass flux conservation and parabolicflow is not its solution The convective model is simpleand effective satisfying mass flux conservation althoughonce the mass flux of the inflow becomes constant theconvectivemodel will be reduced to aDirichlet boundarycondition Compared with these two methods our pro-posedmethod allows the outlet flow to be adjusted on thebasis of the current inflow and fluids inside the domainwhen the Reynolds number is low Our proposedmethodis simple compared to that of Dong et al (2014) How-ever this comparison is in some ways unfair because themethod presented in Dong et al (2014) can maximizethe domain truncation without adversely affecting theflow physics even at higher Reynolds numbers whereasour approach aims to propose an efficient and simpletransparent boundary condition to simulate channel-type flows efficiently without trivial boundary treatmentat lower Reynolds numbers When the Reynolds numberis high our method requires a much larger computa-tional domain to reduce numerical errors compared tothat of Dong et al (2014) owing to the simplicity ofour proposed transparent boundary condition In futurewe will extend our work to simulate multiphase fluids

Mass conservation should be satisfied for each fluid sep-arately if the mass flux is to be conserved Therefore theinflow and outflow boundaries should be coupled withthe phases in that case In future work we will investigateefficient inflow and outflow boundaries to conserve thetotal mass of each fluid

Acknowledgments

The authors are grateful to the anonymous referees whosevaluable suggestions and comments significantly improved thequality of this paper

Disclosure statement

No potential conflict of interest was reported by the authors

Funding

YB Li is supported by the Fundamental Research Funds forthe Central Universities China [No XJJ2015068] and by theNational Natural Science Foundation of China [No 11601416]J-I Choi was supported by the National Research Foundation(NRF) of Korea grant funded by theMinistry of Science Infor-mation and Communications Technology and Future Plan-ning (MSIP) of the Korea government [No NRF-2014R1A2A1A11053140] JS Kimwas supported by the National ResearchFoundation of Korea grant funded by the Korea government(MSIP) [NRF-2014R1A2A2A01003683]

References

Armaly B F Durst F Pereira J C F amp Schonung B (1983)Experimental and theoretical investigation of backward-facing flow Journal of Fluid Mechanics 127 473ndash496doi101017S0022112083002839

Barton I E (1997) The entrance effect of laminar flow overa backward-facing step geometry International Journal forNumerical Methods in Fluids 25 633ndash644 doi101002(SICI)1097-0363(19970930)2563C633AID-FLD5513E30CO2-H

Chau W K amp Jiang Y W (2001) 3D numerical model forPearl river estuary Journal of Hydraulic Engineering 12772ndash82 doi101061(ASCE)0733-9429(2001)1271(72)

Chau K W amp Jiang Y W (2004) A three-dimensionalpollutant transport model in orthogonal curvilinear andsigma coordinate system for Pearl river estuary Interna-tional Journal of Environment and Pollution 21 188ndash198doi101504IJEP2004004185

Chorin J A (1968) Numerical solution of the NavierndashStokesequations Mathematics of Computation 22 745ndash762doi101090S0025-5718-1968-0242392-2

De Mello A J amp Edel J B (2007) Hydrodynamic focusingin microstructures Improved detection efficiencies in sub-femtoliter probe volumes Journal of Applied Physics 101084903 doi10106312721752

Dong S Karniadakis E G amp ssostomidis C (2014) Arobust and accurate outflow boundary condition for incom-pressible flow simulations on severely-truncated unboundeddomains Journal of Computational Physics 261 83ndash105doi101016jjcp201312042

84 Y LI ET AL

Epely-Chauvin G De Cesare G amp Schwindt S (2014)Numerical modelling of plunge pool scour evolution in non-cohesive sediments Engineering Applications of Computa-tional Fluid Mechanics 8 477ndash487 doi10108019942060201411083301

Gambin Y Simonnet C VanDelinder V Deniz A amp Gro-isman A (2010) Ultrafast microfluidic mixer with three-dimensional flow focusing for studies of biochemical kinet-ics Lab on a Chip 10 598ndash609 doi101039B914174J

Gholami A Akbar A A Minatour Y Bonakdari H ampJavadi A A (2014) Experimental and numerical studyon velocity fields and water surface profile in a strongly-curved 90 open channel bend Engineering Applications ofComputational Fluid Mechanics 8 447ndash461 doi10108019942060201411015528

Guermond J L amp Shen J (2003) Velocity-correction pro-jection methods for incompressible flows SIAM Journal onNumerical Analysis 41 112ndash134 doi101137S0036142901395400

Hagstrom T (1991) Conditions at the downstream bound-ary for simulations of viscous incompressible flow SIAMJournal on Scientific and Statistical Computing 12 843ndash858doi1011370912045

Halpern L (1986) Artificial boundary conditions for the linearadvectionndashdiffusion equations Mathematics of Computa-tion 46 25ndash438 doi101090S0025-5718-1986-0829617-8

Halpern L amp Schatzman M (1989) Artificial boundaryconditions for incompressible viscous flows SIAM Jour-nal on Mathematical Analysis 20 308ndash353 doi1011370520021

Han H Lu J amp Bao W (1994) A discrete artificial bound-ary condition for steady incompressible viscous flows ina no-slip channel using a fast iterative method Journalof Computational Physics 114 201ndash208 doi101006jcph19941160

Hasan N Anwer S F amp Sanghi S (2005) On the out-flow boundary condition for external incompressible flowsA new approach Journal of Computational Physics 206661ndash683 doi101016jjcp200412025

Hedstrom G W (1979) Nonreflecting boundary conditionsfor nonlinear hyperbolic systems Journal of ComputationalPhysics 30 222ndash237 doi1010160021-9991(79)90100-1

Jin G amp Braza M (1993) A nonreflecting outlet boundarycondition for incompressible unsteady NavierndashStokes cal-culations Journal of Computational Physics 107 239ndash253doi101006jcph19931140

Johansson V (1993) Boundary conditions for open bound-aries for the incompressible NavierndashStokes equations Jour-nal of Computational Physics 105 233ndash251 doi101006jcph19931071

Jordan S A (2014) On the axisymmetric turbulent bound-ary layer growth along long thin circular cylinders Journalof Fluids Engineering 36 051202 doi10111514026419

Kim J (2005) A continuous surface tension force formula-tion for diffuse-interface models Journal of ComputationalPhysics 204 784ndash804 doi10111514026419

Kim J amp Moin P (1985) Application of a fractional-stepmethod to incompressible NavierndashStokes equations Jour-nal of Computational Physics 59 308ndash323 doi1010160021-9991(85)90148-2

Kirkpatrick M amp Armfield S (2008) Open boundary con-ditions in numerical simulations of unsteady incompressible

flow ANZIAM Journal 50 5760ndash5773 doi1021914anziamjv50i01457

Lei H Fedosov D A amp Karniadakis G E (2011) Time-dependent and outflow boundary conditions for dissipativeparticle dynamics Journal of Computational Physics 2303765ndash3779 doi101016jjcp201102003

Li Y Jung E Lee W Lee H G amp Kim J (2012) Volumepreserving immersed boundarymethods for two-phase fluidflows International Journal for Numerical Methods in Fluids69 842ndash858 doi101002fld2616

Li Y Yun A Lee D Shin J Jeong D amp Kim J (2013)Three-dimensional volume-conserving immersed bound-ary model for two-phase fluid flows Computer Methods inAppliedMechanics and Engineering 257 36ndash46 doi101016jcma201301009

Liu J (2009) Open and traction boundary conditions for theincompressible NavierndashStokes equations Journal of Com-putational Physics 228 7250ndash7267 doi101016jjcp200906021

Liu T amp Yang J (2014) Three-dimensional computations ofwaterndashair flow in a bottom spillway during gate openingEngineering Applications of Computational Fluid Mechanics8 104ndash115 doi10108019942060201411015501

Mao X Lin S C S Dong C amp Huang T J (2009) Tun-able liquid gradient refractive index (L-GRIN) lens withtwo degrees of freedom Lab on a Chip 9 1583ndash1589doi101039b822982a

Mateescu D Muntildeoz M amp Scholz O (2010) Analysis ofunsteady confined viscous flows with variable inflow veloc-ity and oscillating wallsASME Journal of Fluids Engineering132 041105 doi10111514001184

Olrsquoshanskii M A amp Staroverov V M (2000) On simula-tion of outflow boundary conditions in finite differencecalculations for incompressible fluid International Journalfor Numerical Methods in Fluids 33 499ndash534 doi1010021097-0363(20000630)334lt 499AID-FLD1930CO2-7

Pabit S A amp Hagen S J (2002) Laminar-flow fluid mixerfor fast fluorescence kinetics studies Biophysical Journal83 2872ndash2878 Retrieved from httpdxdoiorg101016S0006-3495(02)75296-X

Poux A Glockner S Ahusborde E amp Azaiumlez M (2012)Open boundary conditions for the velocity-correctionscheme of the NavierndashStokes equationsComputers amp Fluids70 29ndash43 doi101016jcompfluid201208028

Poux A Glockner S amp Azaiumlez M (2011) Improvements onopen and traction boundary conditions for NavierndashStokestime-splitting methods Journal of Computational Physics10 4011ndash4027 doi101016jjcp201102024

Regulagadda P Naterer G F amp Dincer I (2011) Transientheat exchanger response with pressure regulated outflowJournal of Thermal Science and Engineering Applications 3021008 doi10111514004009

Rudy D H amp Strikwerda J C (1980) A nonreflectingoutflow boundary condition for subsonic NavierndashStokescalculations Journal of Computational Physics 36 55ndash70doi1010160021-9991(80)90174-6

Sani R L amp Gresho P M (1994) Reacutesumeacute and remarks onthe open boundary condition minisymposium Journal forNumerical Methods in Fluids 18 983ndash1008 doi101002fld1650181006

Sani R L Shen J Pironneau O amp Gresho P M(2006) Pressure boundary condition for the time-dependent

ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 85

incompressible NavierndashStokes equations International Jour-nal forNumericalMethods in Fluids 50 673ndash682 doi101002fld1062

Sohankar A Norberg C amp Davidson L (1998) Low-Reynolds number flow around a square cylinder at inci-dence Study of blockage onset of vortex shedding and openboundary conditions International Journal for NumericalMethods in Fluids 26 39ndash56 doi101002(SICI)1097-0363(19980115)261$$3C39AID-FLD623$$3E30CO2-P

Thompson K W (1987) Time dependent boundary con-ditions for hyperbolic systems Journal of ComputationalPhysics 68 1ndash24 doi1010160021-9991(87)90041-6

Tseng Y H amp Huang H (2014) An immersed boundarymethod for endocytosis Journal of Computational Physics273 143ndash159 doi101016jjcp201405009

Wang M M Tu E Raymond D E Yang J M ZhangH Hagen N amp Butler W F (2005) Microfluidic sort-ing of mammalian cells by optical force switching NatureBiotechnology 23 83ndash87 doi101038nbt1050

Whitesides G M (2006) The origins and the future ofmicrofluidicsNature 442 368ndash373 doi101038nature05058

Wu C L amp Chau K W (2006) Mathematical model ofwater quality rehabilitation with rainwater utilisation A casestudy at Haigang International Journal of Environment andPollution 28 534ndash545

Yun H Bang H Min J Chung C Chang J K amp Han DC (2010) Simultaneous counting of two subsets of leuko-cytes using fluorescent silica nanoparticles in a sheathlessmicrochip flow cytometer Lab on a Chip 10 3243ndash3254doi101039c0lc00041h