USING THE ANSYS FLUENT FOR SIMULATION OF TWO · PDF fileUSING THE ANSYS FLUENT FOR SIMULATION OF TWO-SIDED LID-DRIVEN FLOW IN A STAGGERED CAVITY ... The ANSYS FLUENT commercial software

APTEFF, 43, 1-342 (2012) UDK: 532.54:66.011:004.4 DOI: 10.2298/APT1243169M BIBLID: 1450-7188 (2012) 43, 169-178

Original scientific paper

169

USING THE ANSYS FLUENT FOR SIMULATION OF TWO-SIDED LID-DRIVEN FLOW IN A STAGGERED CAVITY

Jelena Đ. Marković*, Nataša Lj. Lukić, Jelena D. Ilić, Branislava G. Nikolovski,

Milan N. Sovilj and Ivana M. Šijački

University of Novi Sad, Faculty of Technology, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia

This paper is concerned with numerical study of the two-sided lid-driven fluid flow in a staggered cavity. The ANSYS FLUENT commercial software was used for the simulation, In one of the simulated cases the lids are moving in opposite directions (antiparallel motion) and in the other they move in the same direction (parallel motion). Calculation results for various Re numbers are presented in the form of flow patterns and velocity profiles along the central lines of the cavity. The results are compared with the existing data from the literature. In general, a good agreement is found, especially in the antiparallel motion, while in the parallel motion the same flow pattern is found, but the velocity profiles are slightly different. KEY WORDS: cavity benchmark; fluid flow; two-sided lid driven cavity; parallel mo-

tion; antiparallel motion

INTRODUCTION In the past decades, flow in a lid-driven cavity has been studied extensively as one of the most popular fluid problems in the computational fluid dynamics (CFD). This classical problem has attracted considerable attention because the flow configuration is relevant to a number of industrial applications. ANSYS FLUENT uses conventional algorithms for calculation of macroscopic variables. Computational advantages of this commercial software are simplicity of the problem setup, parallel computing and higher precision. Two-sided lid-driven staggered cavity appears to be a synthesis of two benchmark problems: a lid-driven cavity and backward facing step. Furthermore, it has all the main features of a complex geometry. Nonrectangular two-sided lid-driven cavities have been recently introduced and investigated as a potential benchmark problem by Zhou et al. (1), Nithiearasu and Liu (2) and Tekic et al. (3). Zhou et al. Presented a solution for the flow in a staggered cavity obtained by using wavelet-based discrete singular convolution. Nithiarasu and Liu solved the same problem using the artificial compressibility-based

* Corresponding author: Jelena Marković , University of Novi Sad, Faculty of Technology, Bulevar Cara Laza-

ra 1, 21000 Novi Sad, Serbia, e-mail: [email protected]



170

characteristic-based split scheme. Tekic et al. solved this problem by using the lattice-Boltzmann method. The aim of this work was to study two-sided lid-driven staggered cavity utilizing the commercial software package FLUENT. Solutions are presented in the parallel and antiparallel motion of the lid and the flow pattern which develops under these conditions.

Figure 1. Schematic diagram of two-sided lid-driven staggered cavity: (a) antiparallel;

(b) parallel motion.

MATHEMATICAL FORMULATION

General Scalar Transport Equation: Discretization and Solution - ANSYS FLU-ENT uses a control-volume-based technique to convert a general scalar transport equa-tion to an algebraic equation that can be solved numerically. This control volume techni-que consists of the integration of the transport equation about each control volume, yiel-ding a discrete equation that expresses the conservation law on a control-volume basis. Discretization of the governing equations can be illustrated most easily by considering the unsteady conservation equation for transport of a scalar quantity Φ. This is demon-strated by the following equation written in integral form for an arbitrary control volume V as follows:

VV

dVSAdAdvdVt

[1]

where ρ is the density, v - velocity vector; A - surface area vector; - diffusion coefficient for Φ, S source of Φ per unit volume. Equation [1] is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure 1 is an example of such a control volume. Discretization of Equation [1] on a given cell yield

faces facesN

f

N

f

ffffff VSAAvVt

[2]



171

where Nfaces represents the number of faces enclosing the cell, Φf is the value of con-

vected through the face f, fA is the area of the face f and V is the cell volume. The equations solved by ANSYS FLUENT take the same general form as the one given above and apply readily to multi-dimensional, unstructured meshes composed of arbitrary polyhedra.

Figure 2. Control volume used to illustrate discretization of a scalar transport equation.

For relatively uncomplicated problems (laminar flows with no additional models acti-vated) in which convergence is limited by the pressure-velocity coupling, a converged so-lution can often be obtained more quickly using SIMPLEC. With SIMPLEC, the pres-sure-correction under-relaxation factor is generally set to 1.0, which aids in convergence speedup. In the present study, a slightly more conservative under-relaxation value was used, and it is equal to 0.7 .Special practices related to the discretization of the momen-tum and continuity equations and their solution by means of the pressure-based solver is most easily described by considering the steady-state continuity and momentum equati-ons in the integral form:

0 Adv [3]

V

dVFAdAdpIAdvv [4]

where I is the identity matrix, is the stress tensor, and F is the force vector. Discretization of the Momentum Equation - previously described a discretization scheme for a scalar transport equation is also used to discretize the momentum equations. For example, the x-momentum equation can be obtained by setting u :

SiApuauanb

fnbnbP ^

[5]

If the pressure field and face mass fluxes are known, Equation [5] can be solved in the previously outlined manner, and a velocity field can be obtained. However, the pressure field and face mass fluxes are not known a priori and have to be obtained as a part of the solution. There are important issues with respect to the storage of pressure and the discretization of the pressure gradient term. ANSYS FLUENT uses a co-located scheme, whereby pressure and velocity are both stored at cell centers. However, Equation [5] requires the value of the pressure at the face between cells c0 and c1, shown in Figure 2.



172

Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values. Discretization of continuity equation- Equation [1] may be integrated over the control volume to yield the following discrete equation

facesN

fff AJ 0 [6]

where Jf is the mass flux through the face nv . In order to proceed further, it is necessary

to relate the face values of the velocity, nv , to the stored values of velocity at the cell

centers. Linear interpolation of cell-centered velocities to the face results in an unphysical checker-boarding of pressure. ANSYS FLUENT uses a procedure similar to that outlined by Rhie and Chow (4) to prevent checkerboarding. The face value of velocity is not ave-raged linearly; instead, momentum-weighted averaging, using weighting factors based on the aP coefficient from the equation [5], is performed. Using this procedure, the face flux, Jf, may be written as:

)()))(())((( 1

^

10

,,

,,,,

01100

10

1100

ccffccccf

cpcp

cncpcncp

ff ppdJrpprppdaa

vavaJ

[7]

where 0cp ,

1cp and 0,cnv ,

1,cnv , are the pressures and normal velocities, respectively,

within the two cells on either side of the face, and ^

fJ contains the influence of velocities

in these cells (Figure 2). The term fd is a function of Pa , the average of the momentum

equation of the Pa coefficients for the cells on either side of the face f.

Spatial Discretization - By default, FLUENT stores discrete values of the scalar at

the cell centers (c0 and c1 in Figure 2). However, the face values f are required for the

convection terms in Equation [2] and they have to be interpolated from the cell center values. This is accomplished using an upwind scheme. Upwinding means that the face value f is derived from quantities in the cell upstream, or „upwind“, relative to the di-rection of the normal velocity vn in Equation [2]. The diffusion terms are central-differenced and are always second-order accurate. When second-order accuracy is desired, the quantities at cell faces are computed using a multidimensional linear reconstruction approach (5,6). In this approach, higher-order accuracy is achieved at cell faces through a Taylor series expansion of the cell-centered solution about the cell centroid. Thus, when second-order upwinding is selected, the face value f is computed using the following expression:

rSOUf , [8]

where and are the cell-centered value and its gradient in the upstream cell, and r

is the displacement vector from the upstream cell centroid to the face centroid. This for-mulation requires the determination of the gradient in each cell. Finally, the gradient

is limited so that no new maxima or minima are introduced.

APTEFDOI: 1

Sifinem0.001densilid vesired for thof mfirst-oused

Inlid drresulsgeomresult

Figu

Ttours velocFigur

FF, 43, 1-342 (2012)10.2298/APT1243169

imulation setup -ment adjacent to th1 Pas. Reynolds nity; μ is dynamic velocity in the x dirRe number. Boun

he upper and bottomoving depending

order upwind scheas starting conditi

Validatio

n order to validateriven square cavist in the available

metric center of thts of Chen et al. (6

ure 3. Velocity pro

he results for antat various Re nu

city u – and v-prore 5. For comparis

M

Mesh was createdhe walls. Density

number was calcuvisocity of the flurection. The velocndary conditions wom moving lid as

on the case (pareme were taken aions for the secon

RESULTS

on or results of o

e the simulation mity is simulated fliterature. Figure

he cavity. The o6) and Ghia et al (

ofiles u – and v- asq

Antiparall

tiparallel motion oumbers are presentofiles through the son sake, the resul

d with 140x140 ny of the fluid was lated as Re = uL

uid; L is the characcity of the movingwere set as no-slimoving walls wit

rallel or antiparals 0.5 velocity of t

nd-order upwind sc

AND DISCUSSI

one-sided lid –dri

mehod, a popular bfor different Re n

3 shows the u- anobtained results a(7).

along the vertical aquare cavity.

lel motion of the l

of lids are listed ited in Figure 4, wmid-section of th

lts obtained by Te

UDK: 53BIBLID: 1450-7188 (

Origin

number of elemenset to 1 kg/m3, a

L/. where ρ reprecteristical length og lid was calculateip for the left and th defined velocitllel). Starting conthe moving lids, acheme.

ION

iven square cavit

benchmark problenumbers and comnd v-velocity profare in good agree

and horizontal cen

lids

in Table 1. Streamwhile the results ohe staggered caviekic et al. (3) are a

32.54:66.011:004.4 (2012) 43, 169-178 nal scientific paper

173

nts with grid re-and viscosity to esents the fluid of cavity, and u ed based on de-right wall, and

ty and direction nditions for the nd results were

ty

em of one-sided mpared with the file, through the ement with the

nterlines of the

mfunction con-obtained for the ity are given in also presented.

APTEFDOI: 1

174

Itcompcompstron Abehavpattermoreferredcally locate

Wthe excreasmarycorne

Figu

VThe m

FF, 43, 1-342 (2012)10.2298/APT1243169

t is evident that wponents also increpared to the viscoger for higher Re.

As previously menvior for Re numbrns are achieved e

e precisely there ard to as secondary aligned along the

ed in the left and r

Figure 4. Streamf

With the increase oxpense of the prim

se of the Re numby vortices along ters next to the mov

ure 5. Velocity prostaggered c

Velocity profiles almost notable diff

M

with the increase iease in magnitudous ones. As a re. ntioned, three studbers above 1000. even at Re numbere three primary vfor easier compa

e mid-section of thright bottom corne

function contours

of Re, the primarymary vortex locat

ber, the bottom lefhe long diagonalving lid.

ofiles u – and v- acavity –antiparalle

long the vertical cference is for Re

in the Re numberde. Furthermore, tesult, the gradien

dies on staggered In the present stu

ers lower than 100vortices, although arison of the resulthe cavity. Opposeer of the cavity.

at various Re num

y vortex located inted in the upper rft corner vortex dl of the cavity, se

along the vertical ael motion (Rea – T

centerline of the c=100. While the

UDK: 53BIBLID: 1450-7188 (

Origin

r, extreme values the inertial forcesnts close to the m

cavity, (1)-(3) shudy, symmetric a00. Multiple vorticin Table 1 the thi

ts. Primary vorticed to this, seconda

mbers – antiparall

n the left bottom cright corner. Withisappears, and theecondary vortices

and horizontal cenTekic et al. results

cavity differ for soresults of Tekic


of the velocity s are dominant

moving lids are

howed unsteady and asymmetric ces are formed, ird vortex is re-es are all verti-ary vortices are

el motion.

corner grows at h the further in-ere are two pri-s appear in the

nterlines of the (3))

ome Re values. et al. (3) show

APTEFDOI: 1

moremore(Re=Consof Ghprevifoundtions TTablevortic Tabl

Acompdiffer

Itsheargular

Re

50a

50b

50c

100a

100b

100c

FF, 43, 1-342 (2012)10.2298/APT1243169

e flattened profilese similar to the p1000). The veloc

sidering that therehia et al. (6) andiously mentioned d in different Re caused by the difo summarize the e 1 and comparedces are in good ag

le 1. Locations an

As expected, paralpared to the antiprent Re numbers.

Figure 6. Stream

t can be noticed thr layer forms betwr cavities, where t

Primary vortex

(xc1,yc1)

(0.9781, 1.1600) (0.4219, 0.2518)

(0.9637, 1.1551) (0.4494, 0.2543)

(1.10383,1.17135) (0.41911,0.343648)

(1.0172, 1.1091) (0.3828, 0.2889)

(1.0031, 1.1382) (0.4082, 0.2693)

(1.20251, 1.25965) (0.452649,0.360417)

M

s, ANSYS FLUENprofiles which Tcity profiles alon is a very good ag

d Chen et al. (7), authors, the reasonumber definition

fferent numerical aresults, the loca

d with (1) and (3greement with the

nd secondary vortiet al.(3

Par

llel motion of theparallel motion. F

mfunction contou

hat two primary coween them. Compathe free shear laye

First secondary vortex R(xc2,yc2)

(1.3556, 0.4405) (0.0444, 0.9595)

40

(1.3484, 0.4476) (0.0460, 0.9543)

40

(1.27995, 0.69013) (0.032408, 0.8355)

40

(1.3556, 0.4486) (0.0444, 0.9514)

10

(1.3502, 0.4457) (0.0460, 0.9543)

10

(0.90263, 0.89427) (0.04669, 0.84673)

10

NT results show thTekic et al. (3) s

g the horizontal greement betweenand also between

ons for disagreemn and different boapproach.

ations of the cent3). It can be notiresults in the avai

ices – antiparallel 3), c present study

rallel motion

e opposite lids deFigure 6 shows t

urs at various Re n

ounter-rotating voared to the previoer is formed along

Re Primary vortex

(xc1,yc1)

00a (0.7000, 0.7000)

00b (0.6822, 0.6859)

00c (0.472214,0.445062)

000a (0.7000, 0.7000)

000b (0.7000, 0.7000)

000c (0.6934, 0.6972)

UDK: 53BIBLID: 1450-7188 (

Origin

he existence of a sshowed for highecenterline are al

n the present studyn results of Tekicents with present oundary condition

ers of the vorticeced that the resuilable literature.

motion, aZhou et

evelops a differenthe streamfunctio

numbers – parallel

ortices are presentous studies of flowg a horizontal cen

First secondary vortex (xc2,yc2)

(1.3500, 0.4656) (0.0500, 0.9344)

(1.3522, 0.4607) (0.0554, 0.9506)

) (1.25479, 0.60078) (0.05631, 0.60016)

(1.3250, 0.4844) (0.0750, 0.9063)

(1.3250, 0.4844) (0.0750, 0.9063)

(1.3371, 0.4851) (0.0722, 0.9280)


175

sine-like curve, er Re numbers most identical. y and the work c et al. (3) and study could be

ns implementa-

es are listed in lts for primary

al. (1), b Tekic

nt flow pattern on contours for

motion.

t and that a free w inside rectan-nterline (6), (8),

Second secondary vortex (xc3,yc1)

(0.4703, 1.625) (0.9219, 0.2375)

(0.4382, 1.1345) (0.9824, 0.2862)

(0.42548, 1.07007)

(0.7256,0.2000) (0.5339, 1.1907)

(0.8811,0.2167) (0.5301, 1.1962)

(0.82157, 0.13123) (0.61360, 0.96692)

APTEFDOI: 1

176

in theis no movecornestrengcauseshowze, sothe cabody velocwith Furthincreaboutresult

Figu

Aprofildifferhorizwhileles obnounned, the di

FF, 43, 1-342 (2012)10.2298/APT1243169

e staggered cavitylonger symmetric

es towards the offer of the right wagth at the cost of es splitting of the

wn in Figure 6. Boo that viscous effavity (9). As menwith a constant a

city profiles alongthe increase in th

her, the free shearase in the Re nut the horizontal cets of Tekic et al., i

ure 7. Velocity prstaggered

As in the case of anles, while the presrences occur at lo

zontal centerline she for the Re=100 tbtained by simulaced minimum andcould be a result ifferent boundary

M

y the free shear laycal due to the uppfset. At low Re nuall and offset. Athe upper primary

e primary vortex oth primary vorticfects are confined ntioned by Sahin aangular velocity ag mid sections of he Re number, extr layer formed be

umber due to turbenterline of the cavit can be seen that

rofiles u – and v- acavity – parallel m

ntiparallel motionsent study shows twer values of Re how relatively gothere is a more sig

ation in the presend maximum veloc

of the different Rconditions.

yer is formed alonper lid moving froumbers, a secondas the Re numbery vortex. At higеand formation o

ces have become mto the thin bound

and Owens (10), fat high Re numbe

the cavity. As ditreme velocity vaetween the two prbulence. The profvity, as previouslyt the obtained prof

along the vertical motion (Rea – Tek

n, the results of Tethe existence of anumber (50 and od agreement for gnificant differencnt study are more sity pitch. These diRe calculation pr

UDK: 53BIBLID: 1450-7188 (

Origin

ng the shorter diagom the offset, whilary vortex is preser increases, this vеr Re numbers, sef a second seconmore prominent adary layers close fluid begins to rotrs. Figure 7 showiscussed in the pralues also increaserimary vortices shfiles confirm asymy mentioned. Comfiles are quite sim

an horizontal cenkic et al. results (3

ekic et al. (3) givea minimum veloci100). Velocity prothe Re values 50,

ce. In general, thesymmetrical and hifferences, as prev

rocedure, and imp


gonal. The flow le the lower lid ent close to the vortex gains in condary vortex

ndary vortex as and larger in si-

to the walls of tate like a rigid

ws the u- and v-revious section, e in magnitude. hrinks with the mmetrical flow mpared with the

milar.

nterlines of the 3)).

more flattened ity pitch. These ofiles along the , 400 and 1000, e velocity profi-have more pro-viously mentio-plementation of



177

CONCLUSION

Results of the ANSYS FLUENT commercial software simulation of two-sided lid-driven flow inside a staggered cavity are presented in this article. Both antiparallel and parallel motions of two facing lids are investigated. The benchmark results obtained with ANSYS FLUENT are in good agreement with the results available in the literature. For antiparallel motion of lids in a staggered cavity results show symmetrical and asy-mmetrical flow patterns. Velocity profiles along the horizontal centerline are in a good agreement with existing data from the literature, while the profiles along the vertical cen-terline are slightly different from those used for comparison, especially for Re=50 and Re=100. These differences could be explained by the different Re calculation procedures and different boundary conditions implementation methods, considering the different numerical approach. The situation is quite similar in case of parallel motion of lids. Un-like for antiparallel motion, steady-state asymmetric patterns are obtained for all investi-gated Re numbers. It can be noticed that a free shear layer is formed along the short dia-gonal of the staggered cavity. All the main features of the flow are shown, streamline contours, horizontal and vertical velocity components along the mid sections of the cavity are visually presented, while the location of vortices is presented in Table 1.

Acknowledgement This research was financially supported by the Ministry of Science and Technological Development of the Republic of Serbia (Project No. 46010)

REFERENCES

1. Zhou, Y.C., Patnaik, B.S.V., Wan, D.C., and Wei, G.W.: Dsc Solution for Fow in a Staggered Double Lid Driven Cavity. Int. J. Num. Meth. Eng. 57 (2003) 211-234.

2. Nithiarasu, P. and Liu, C.-B.: Steady and Unsteady Incompressible Flow in a Double Driven Cavity Using the Artificial Compressibility (Ac)-Based Characteristic-Based Split (Cbs) Scheme. Int. J. Num. Meth. Eng. 63 (2005) 380-397.

3. Tekić, P., Rađenović, J., Lukić, N., and Popovic, S.: Lattice Boltzmann Simulation of Two-Sided Lid-Driven Flow in a Staggered Cavity. Int. J. Comp. Fluid Dyn. 24 (2010) 383-390.

4. Rhie, C.M. and Chow, W.L.: Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation, AIAA 21 (1983) 1525-1532.

5. Barth, J. and Jespersen, D.: The Design and Application of Upwind Schemes on Un-structured Meshes, AIAA-89-0366, AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, (1989) 10-13

6. Chen, S., Tolke, J., and Krafczyk, M.: A New Method for the Numerical Solution of Vorticity-Streamfunction Formulations. Comp. Meth. Applied Mech. Eng., 198 (2008) 367-376.



178

7. Ghia, U., Ghia, K.N., and Shin, C.T.: High-Re Solutions for Incompressible Flow Using the Navier–Stokes Equations and a Multigrid Method. J. Comput. Phys. 48 (1982) 387-411.

8. Perumal, D.A. and Dass, A.K.: Simulation of Flow in Two-Sided Lid-Driven Square Cavities by the Lattice Boltzmann Method, Advances in fluid mechanics VII. Boston, MA: WIT Press (2008) 45-54.

9. Patil, D.V., Lakshmisha, K.N., and Rogg, B.: Lattice Boltzmann Simulation of Lid-Driven Flow in Deep Cavities, Comput. Fluids 35 (2006) 1116-1125.

10. Sahin, M. and Owens, R.G.: A Novel Fully Implicit Finite Volume Method Applied to the Lid-Driven Cavity Problem – Part I: High Reynolds Number Flow Calculati-ons., Int. J. Numer. Methods Fluids 42 (2003) 57-77.

СИМУЛАЦИЈА ТОКА У ДВОСТРАНО ВОЂЕНОМ ПОКРЕТНОМ КАНАЛУ ПОМОЋУ ANSYS FLUENT ПРОГРАМСКОГ ПАКЕТА

Јелена Ђ. Марковић, Наташа Љ. Лукић, Јелена Д. Илић, Бранислава Г. Николовски,

Милан Н.Совиљ и Ивана М. Шијачки

Универзитет у Новом Саду, Технолошки факултет, Булевар цара Лазара 1, 21000 Нови Сад, Србија

Рад се бави проблематиком нумеричке анализе струјања флуида у каналима у којима струјање флуида настаје услед кретања горње и доње странице канала. Ко-мерцијални софтвер ANSYS FLUENT је коришћен за симулацију двострано вође-ног струјања флуида. Симулација је урађена за два случаја, први – када се горња и доња страна крећу у супротним смеровима (антипаралелно струјање) и други – када се горња и доња страна крећу у истом смеру. Резултати прорачуна за низ вредности Рејнолдсовог броја приказани су у виду путања струјања флуида и профила брзина дуж хоризонталне и вертикалне централне линије канала. Добијени резултати су упоређени са потојећим подацима у литератури. Генерално уочено је добро слагање са резултатим претходних истраживања, нарочито када се ради о антипаралелном струјању. У случају паралелног струјања, визуелно ток флуида је исти, али потоји мала разлика у профилима брзина. Кључне речи: симулација, Ansys Fluent, струјање флуида, двострано вођени

покретни канали

Received: 6 July 2012 Accepted: 14 September 2012

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USING THE ANSYS FLUENT FOR SIMULATION OF TWO · PDF fileUSING THE ANSYS FLUENT FOR SIMULATION OF TWO-SIDED LID-DRIVEN FLOW IN A STAGGERED CAVITY ... The ANSYS FLUENT commercial software