European Conference on Computational Fluid DynamicsECCOMAS CFD 2006

P. Wesseling, E. Onate and J. Periaux (Eds)c© TU Delft, The Netherlands, 2006

A THREE DIMENSIONAL, FINITE VOLUME METHODFOR INCOMPRESSIBLE NAVIER STOKES EQUATIONS

ON UNSTRUCTURED, STAGGERED GRIDS

Bojan Niceno

Paul Scherrer Institut,Nuclear Energy and Safety,

Thermal-Hydraulics Laboratory5232 Villigen PSI, Switzerlande-mail: bojan.niceno@psi.ch

Key words: Finite volume method, Staggered grids, Unstructured grids, Incompressibleflow, Projection method

Abstract. In this work, a new method was described for spatial discretization of three-dimensional Navier Stokes equations in their primitive form, on unstructured, staggeredgrids. Velocities were placed on the cell faces and pressure in cell centers and were linkedwith the projection method. Thanks to the variable arrangement, no stabilization procedurewas needed to avoid spurious pressure/velocity fields. A way around the deferred correctionwas also described and used in this work. Several laminar cases were computed to showthe validity of the method. Computation of velocities on the cell faces and the abilityto integrate in time with projection method without any stabilization procedure make theproposed method a good candidate for large eddy simulation (LES) of turbulence in complexgeometries.

1 INTRODUCTION

Collocated methods, where velocities and pressure are computed at the same placeson numerical grid, are nowadays well established. Most of the commercial CFD pack-ages are based on cell centered finite volume methods, almost invariantly stabilized withRhie & Chow techniques1 in addition to some advanced convective scheme. These typeof discretization is well suited for steady or unsteady RANS computations. However,recent advances in computer hardware and mixed success of RANS based method, ledto the interest of industry in LES as a new and emerging tool for analysis of turbulentflows. Clearly enough, industrial applications occur in complex geometry, so the highorder spectral, or finite difference codes which were most often used in the last couple ofdecades for LES, must be replaced with a more versatile methodology. Many researchgroups, realizing the industry’s interest in LES, jumped into it, ignoring the importance

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Bojan Niceno

of numerical methods2. It is generally hoped that using an existing numerical method-ology developed and tested for RANS would surely work for LES, believing that refiningthe grid and decreasing the time step, would per se provide a good platform for LES.Very little effort has been put into research of numerical methods suited for LES in com-plex geometry3. However, no matter how fine the grid is in collocated computation, onestill depends on Rhie & Chow technique, which introduces fourth order dissipative termsto governing equations4, damping turbulent fluctuations we wish to simulate with LES.Furthermore, iterative methods for coupling velocities and pressure which are almost in-variantly used in conjunction with collocated grid methods (SIMPLE, SIMPLEC, PISOto name just the few) and which are originally developed for steady flow computation,are not ideally suited for unsteady simulations, as pointed out in5. Projection methods6

7,8 are more efficient for unsteady simulations, particularly for LES. However, not allspatial discretization schemes are capable to integrate in time with projection scheme.Depending on the placement of variables, some methods are simply not capable to closethe mass conservation equation after just one solution of pressure Poisson equation.

To the authors knowledge there have been several attempts to develop unstructuredstaggered grid in 2D9,10,11,12. An interesting method, based on discretization of diver-gence and curl was proposed13. It is extended to 3D14 and named complementary volumeor covolume method. This three-dimensional extension of covolume method is seen bymany as an ideal candidate for LES in complex geometry, thanks to its conservative prop-erties15. The original method is constrained to Delaunay-Voronoi tessellations, but anextension of the method which does not have this restriction is developed16.

In present work, a new method is presented which solves the equation in their primitiveform (hence velocity components and pressure) and is yet staggered and formulated forhybrid grids. The link between velocity and pressure is achieved via projection method,having the ability to reduce the mass error to arbitrary low level after each solution ofthe pressure Poisson equation. To the authors best knowledge, it is the first primitiveformulation of staggered finite volume method on three dimensional arbitrary grids.

2 DISCRETIZATION OF GOVERNING EQUATIONS

The equations which govern the incompressible Navier Stokes equations are the massconservation equation:

∇u = 0, (1)

and momentum conservation equations:

∂u

∂t+ u∇u =

1

Re∇2u −∇p. (2)

A variety of methods exists for linking the velocity and pressure in the context of finitevolume method. A family of method, such as SIMPLE, SIMPLEC, PISO seek the pressure

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Bojan Niceno

field as an iterative process where in each iterations, by solving a Poisson-like equation,one gets a better estimate of the pressure field. These methods are very efficient for steadyflow equations and widely used, even in commercial CFD packages. On the other hand,we have projection methods, where in each time step only one solution of Poisson equationfor pseudo pressure is sufficient to close the equations and proceed with a new time step.Since the long-term objective of present author is to apply the proposed methodology toLES of turbulence, projection method is a preffered choice in this work.

The projection method, in its simplest form, can be summarized as follows8:

• Step 1: solution of the tentative velocity field (u?) from the following equation:

u? − un

δt+

3

2(u∇u)n −

1

2(u∇u)n−1 =

1

2Re

[

(∇2u)? + (∇2u)n]

, (3)

where n is the old time step, n − 1 is time step before that. Convective terms arediscretized with the Adams-Bashforth scheme for simplicity and viscous terms withCrank-Nicolson for stability. Tentative velocity field is generally not divergence-free.

• Step 2: solution of the pseudo pressure (φ) from the following equation:

∇2φ = −∇u?

δt. (4)

• Step 3: projection of the velocity to a divergence-free field:

un+1 = u? + δt∇φ, (5)

and proceeding to a new time step.

Once we have outlined the algorithm for the solution of Navier Stokes equations, weclearly see which quantities have to be discretized.

2.1 Selection of finite volumes

We define two sets of non overlapping cells: pressure and momentum cells. Each of thesets covers the entire problem domain. Pressure cells coincide with the actual cell of thenumerical grid. On the boundary, pressure cells coincide with boundary faces. Pressureis computed in the cells inside and on the boundary. Hereafter, pressure cells will bereferred to as just cells and its faces as faces for the sake of shortness.

Momentum cells are constructed around each cell face from two pyramids having thecell face as its base and spanning to the center of each of the surrounding cells. Sincethere are two possible shapes of cell faces, triangular and quadrilateral, there are alsotwo possible shapes for momentum cells inside the domain: triangular and quadrilateralbi-pyramid and two possible shapes of momentum cells at the boundary: triangular andquadrilateral pyramid. Two possible momentum cell inside the domain are shown inFigure 1. Hereafter, momentum cells will be referred to as such and its faces as facets.Facets can only have triangular shape.

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Bojan Niceno

1 2

43

(a) Pressure cells (b) Momentum cells

Figure 1: Selected finite volumes: (a) for pressure equation, (b) for momentum equations

2.2 Computation of gradients

We start the description of spatial discretization with procedures for computation ofgradients of cell and face centered variables, because it is the basic building block of themethod we propose.

2.2.1 Cell centered pressure gradients

Gradients in cell centers are computed with the least square approach17. We define agradient in a cell center with:

φi = φc + (∇φ)Tc · qi − εi, (6)

where φc denotes the value in the cell center, φi denotes the values in neighboring cellcenters. Distance vectors between neighboring cell centers and current cell center aredenoted by qi = xi − xc with qx, qy and qz as it’s components. Errors introduced bythis approximation are denoted by εi. If square of the errors is minimized with gradientcomponents, the following equation is obtained for each cell:

∑

i q2x

∑

i qxqy

∑

i qxqz∑

i qxqy

∑

i q2y

∑

i qyqz∑

i qxqz

∑

i qyqz

∑

i q2z

∂φ/∂x∂φ/∂y∂φ/∂z

=

∑

i ∆φi qx∑

i ∆φi qy∑

i ∆φi qz

, (7)

where ∆φi = φi −φc is the difference between the values in cell neighbors and cell center.Equation (7) is conveniently written in the form:

C−1c (∇φ)c = dc, (8)

to yield the expression for the gradient in the cell center:

(∇φ)c = Cc dc. (9)

This procedure could be used for all variables defined in cell centers. It should be notedthat Cc is a 3× 3 symmetric matrix dependent on geometry and dc is a vector dependenton geometry and variable φi in all cell neighbors.

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Bojan Niceno

2.2.2 Cell centered velocity gradients

Velocity gradients in cell centers are computed in the same manner as the pressuregradients, i.e. by minimizing the square of error (εj) in the expression:

uj = uc + (∇u)c rj − εj. (10)

Here uc is the velocity in the cell center, uj are the values in the faces surrounding thecell, and rj = xj−xc is the distance vector between the jth cell face and cell center and rx,ry and rz are its components. Following a similar procedure as for the pressure gradients(minimizing the square of the error by gradient components), we obtain the equation:

∑

j r2x

∑

j rxry

∑

j rxrz∑

j rxry

∑

j r2y

∑

j ryrz∑

j rxrz

∑

j ryrz

∑

j r2z

∂uT /∂x∂uT /∂y∂uT /∂z

=

∑

j ∆uTj rx

∑

j ∆uTj ry

∑

j ∆uTj rz

, (11)

with ∆uTj = uT

j − uTc . The following point deserves special attention. On the right side

of Equation (11) we have a sum of velocities at cell faces j and velocity in the cell centeruc which is not defined by the present method. If expression

∑

j ∆uTj rx, is expanded, we

get:

∑

j

(uTj − uT

c )(xj − xc) =∑

j

uTj xj − xc

∑

j

uTj − uT

c

=0︷ ︸︸ ︷

(∑

j

xj − N xc), (12)

where N is the number of faces around the cell. The last term in Equation (12) is equal tozero if cell center is defined as an average of the face centers encompassing it. In presentwork, cell centers have been defined in such a way to satisfy this property. 1 Hence, theright hand side of Equation (11) can be rewritten as:

∑

j ∆uTj rx

∑

j ∆uTj ry

∑

j ∆uTj rz

=

∑

j uTj rx

∑

j uTj ry

∑

j uTj rz

, (13)

where the dependence on the velocity in the cell center uc is lost. Equation (11) can bewritten in short form as:

F−1c (∇u)T

c = Gc. (14)

The gradients can finally be computed from:

(∇u)Tc = Fc Gc, (15)

where both F and Gc are a 3 × 3 matrices, former depending on geometry only andlater on geometry and velocity components defined on the faces encompassing the cell.It is worth noting that although velocities are defined at the faces, velocity gradients aredefined in the cell centers.

1A reader can check that it is automatically satisfied for all but pyramid cell shape.

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Bojan Niceno

2.3 Pseudo pressure equation

The first step in discretization any differential equation by the finite volume family ofmethods is to re-write it in integral form and apply Gauss theorem to lower the orderof differentiation. 2 Pseudo pressure Equation (4) in integral form, and after applyingGaussian theorem, reads:

∫

S

(∇φ)dS = −1

δt

∫

S

u?dS, (16)

where S is the closed surface encompassing the control volume. Once the integral form ispresent, we may readily write its discretized version:

∑

j

(∇φ)jSj = −1

∆t

∑

j

u?Sj, (17)

where j is a face index, Sj is the face surface vector and (∇φ)j is the gradient at the cellface. This is the discretized form of the pseudo pressure equation. The difficulty with itlies in the fact that we somehow need to compute the gradient at each cell face. The mostnatural choice, which is followed by many authors18,19 is to compute it as an arithmeticaverage:

(∇φ)j = (∇φ)j =1

2[(∇φ)a + (∇φ)b], (18)

where subscripts a and b denote gradients computed in cell centers surrounding the facej as illustrated in Figure 2. There is a problem with this approach, hindering its straight-

jB

A b

a

PSfrag

replac

ement

s

δa

δb

Figure 2: Computation of the flux on a cell face.

forward application. Namely, the left hand side of (17) does not depend at all on the

2A reader may argue that we are going two steps forward and one step back, i.e. form physical laws tointegral formulation, then from integral to differential form just to go one step back to integral formulation.We are aware of that, but yet we find that many expressions are shorter to write in differential form.

6

Bojan Niceno

value of φ in its first neighbors. A small proof follows. Direct discretization of the lefthand side of (17), using the approximation (18), for each cell c we get:

∑

j

(∇φ)jSj =1

2(∇φ)c

=0︷ ︸︸ ︷∑

j

Sj +1

2

∑

j

(∇φ)ijSj, (19)

where ij denotes cell neighbor to cell c on the opposite side of face j, for each of the facessurrounding the cell c. The first term on right hand side vanishes, because sum of thesurface vectors encompassing a closed volume is zero, and as a consequence, we get:

∑

j

(∇φ)jSj =1

2

∑

j

(∇φ)ijSj. (20)

If we recall that gradients computed in the neighboring cells ij do not depend on thevalues in themselves (12), Equation (20) has a very undesirable property that it links thevalue in the cell center only with the values in its second neighbors, which is illustrated inFigure 3. This issue is addressed by the introduction of the so-called deferred correction

c i,j

j

c

i,jj

Figure 3: Illustration of the numerical molecule obtained on regular grids after discretization by Equa-tion (20). Non-zero terms are shaded. Triangular grid features the same problem as quadrilateral.

approach18,20,19 where the authors introduce a first order term connecting the values ineach cell center with its neighbors and putting it in the system matrix and the differencebetween the first order term and the second order is placed on the right hand side oflinear system of equations and iterated within the solution algorithm. If this correction israther high (which happened if the cell shapes are very irregular) right hand side of thelinear system grows relative to the left making the solution unstable or even lead to non-physical solutions. This issue was studied in19 who, among other things, examined threedifferent approaches to deferred correction and analyzed their influence on the stabilityof the solution procedure.

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Bojan Niceno

In this work we propose a different approach. We do not approximate gradient on thecell face and multiply it with surface vector, as done by equations (17) and (18), but werather approximate the flux itself, i.e. cell face gradient multiplied with surface vector:

(∇φ · S)j =|Sj|

|dj|(φB − φA), (21)

where A and B are virtual cell centers orthogonal to the face j, and |dj| is the distancebetween them. The situation is shown in Figure 2. The values in these, orthogonal, cellcenters are obtained from the following equations:

φA = φa + (∇φ)a ·−→δA, (22)

and:

φB = φb + (∇φ)b ·−→δB, (23)

with a and b being the cells around the face j. The flux on the cell face can finally bewritten as:

(∇φ · S)j =|Sj|

|dj|[φb − φa + (∇φ)b ·

−→δB − (∇φ)a ·

−→δA]. (24)

In order to discretize the entire system of equations, we browse through cell faces, and foreach of them we discretize Equation (24) using the cell gradient matrices defined by (7).We get the following expression:

(∇φ · S)j =|Sj|

|dj|[φb − φa + (Cb db) ·

−→δB − (Ca da) ·

−→δA], (25)

which is used to compute the coefficients of the system matrix connecting the values in cellcenters a and b encompassing face j and all of their neighbors. Paraphrasing finite elementmethodology, we call Equation (25) face stiffness matrix. Non-zero terms from the facestiffness matrix are illustrated in Figure 4. No term resulting from a discretization of thepseudo pressure equation is treated explicitly by the solution algorithm. Everything isplaced in the system matrix. The reasons for that are threefold:

• Since the pseudo pressure equation poses a difficulty for linear solvers due to itsnon-singularity, we did not want to make things even harder on it by loading theright hand side.

• Our final goal is to use the proposed methodology for unsteady flow and we wishto link velocity and pressure via the projection method. Thus, any type of outeriterations (which are needed if you treat part of the equation explicitly) would meanfailing to achieve our objective.

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Bojan Niceno

j

a

b

j

ba

Figure 4: Illustration of the numerical molecule obtained on regular grid after discretization by Equa-tion (25). Non-zero terms are shaded.

• Implicit treatment of entire pseudo pressure equation makes the implementationof the vanishing gradient boundary condition a straightforward task, as will beexplained bellow.

2.3.1 Implementation of boundary conditions

The only type of boundary condition used for the pseudo pressure equation is a zerogradient, namely:

∂φ

∂n= 0, (26)

for all boundaries where the velocity is specified. In present work, it is on all boundaries.When dealing with boundary values of the pressure, two approaches are possible:

• Define boundary values as unknowns, create an equation for each of them and solvethem in a linear system of discretized equations together with values in the interior.This approach is almost invariantly used with finite element discretization and nodecentered finite volumes.

• Do not treat boundary values as unknowns (do not place them in the linear system ofequations), but rather calculate them from the values in the interior and specifiedboundary conditions. This is how it is done in most cell centered finite volumeapproaches20.

Since we are defining a cell centered method for the pressure, we have first considered thelater approach. Implementing a zero gradient conditions (26), which is needed for pseudopressure equation, leaves some ambiguity in computing the boundary values: one mayeither just copy (or mirror) the values from the inside to the boundary cell, or one mayuse more sophisticated extrapolation using the gradient from inside. None of these twoapproaches, from present authors experience, is satisfactory for the projection method,i.e. none of them results in a pressure field which is able to project the velocities into thedivergence-free field. Moreover, they both require iterations for computation of gradients,

9

Bojan Niceno

deteriorating the efficiency of the solution algorithm. Therefore, we re-write the cell facestiffness matrix (25) for the boundary cell:

(∇φ · S)j =|Sj|

|dj|[φA − φb + (Ca da) ·

−→δA]. (27)

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j

B = b

a

A

PSfrag

replac

ement

s

−→δA

Figure 5: Implementation of theboundary condition.

In (27), it has been acknowledged that δB is zero forthe boundary cell (see Figure 5) and thus we got rid of theentire part multiplying non-existent gradient matrix forthe boundary cell. Boundary face stiffness matrices (27)are assembled in the same linear system as the values inthe interior are. In Equation (27) each boundary valuesdepends on the value of the first neighbor inside (a) andall of its neighbors (via da). By doing so we have removedany ambiguity in defining the values in the boundarycells and we have also obtained the pressure field whichprojects the velocity into the zero (to the machine accu-racy) divergence field.

2.3.2 Discretization of the source term

The discretization of the source term for the pseudo pressure equation is straightfor-ward. Rewriting it in integral form and applying Gauss theorem yields:

1

δt

∫

V∇u?dV =

1

δt

∫

S

u?dSj ≈1

∆t

∑

j

u?jSj. (28)

We have merely replaced the tentative velocity u? field by its discretized counterpart u?j ,

which is already defined on cell faces j. The very fact that no interpolation on velocity isdone to close the continuity equation should yield the method energy conserving 3.

2.4 Momentum equations

Momentum conservation equation for incompressible flow, in its integral form, withomitted pressure gradient term, reads:

∫

V

∂u

∂tdV +

∫

S

uudS = −∫

S

∇udS. (29)

Terms on the left are inertial and convective terms, whereas on the right we have theviscous term. Pressure gradient term is omitted on purpose, since it was not used inthe present projection method. Discretization of the gradient of the pseudo pressure isoutlined in the next section.

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Bojan Niceno

2.4.1 Inertial term

Inertial term is discretized by approximating the volume integral in each momentumcell with:

∫

V

∂u

∂tdV ≈

u? − un

∆tVf , (30)

where Vf is a volume of each momentum cell around face f .

2.4.2 Convective terms

Convective term is discretized by approximating the surface integral of the convectiveterm by:

−∫

S

uudS ≈ −∑

k

ukukSk, (31)

where k stands for summation over facets of each momentum cell and uk is the velocityat the centroid of each facet k. Sk is the facet surface vector. The velocity at each facetis obtained from the following relation:

uk = w[ua + (∇u)a−→γa] + (1 − w)[ub + (∇u)b

−→γb ], (32)

B

b

a

c

k

PSfrag

replacements

−→γA−→γB

Figure 6: Computation of convective flux on thefacet k. Momentum cells are denoted by a and b

where w is a weight factor in the range 0.5−1 determining type of convective schemewhich is used. For w = 0.5 we have a centralconvective scheme, whence for w = 1 we havea linear upwind differencing scheme (LUDS).Subscripts a and b denote faces (momentumcells) around facet k and −→γa and −→γb are vec-tors connecting centers of momentum cells aand b with facet center (see Figure 6).

If the definitions for velocity gradient (15)is introduced into (32), we get the final expression for velocity at the facet:

uk = wua + (1 − w)ub + Fc Gc [w−→γa + (1 − w)−→γb ], (33)

which is used to compute the discretized convective terms. Since in the projection methodwe are using to link velocities and pressure, convective terms are treated with Adams-Bashforth scheme in time, we only compute terms (33) from known velocities and put itinto the right hand side of the linear system of discretized equations.

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Bojan Niceno

2.4.3 Viscous terms

Since the method presented in this work is used only for incompressible flow prob-lems with constant physical properties, the viscous stress tensor reduces to the velocitygradient:

∇u + (∇u)T = ∇u. (34)

After integration of (34) over momentum cell, we get:∫

S

∇u · dS =∑

k

∇u · Sk, (35)

where k denotes facets of momentum cells. This term is discretized following a proceduresimilar to the one for pseudo pressure equation, i.e. by discretizing the viscous force ateach facet as:

(∇u · S)k =|Sk|

|dk|(uB − uA), (36)

where uA and uB are velocities in orthogonal face centers A and B, and |dk| is the distancebetween the orthogonal face centres (see Figure 7). Velocities uA and uB are computed

k B

bA

a

c

PSfrag

replacements

−→δA

−→δB

Figure 7: Computation of the flux on a facet. Cell center is denoted by c. Two of the cell faces are shownand their centers denoted by a and b. Two momentum cells around the faces a and b are also shown anda facet between them is shaded and denoted by k

by using (15) as:

uA = ua + (∇u)c−→δA = ua + (Fc Gc) ·

−→δA, (37)

and:

uB = ub + (∇u)c−→δB = ub + (Fc Gc) ·

−→δB. (38)

Because each facet can be contained by only one cell, the same face gradient matrix Fc

is used for computation of uA and uB.As a result of the discretization, we get the following expression for each facet:

(∇u)kSk =|Sk|

|dk|[ub − ua + F GT

j (−→δB −

−→δA)], (39)

which represents the facet stiffness matrix. As in the case of pseudo pressure equation,all the terms from the face stiffness matrices are treated implicitly by a linear solver.

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Bojan Niceno

2.4.4 Pseudo pressure gradient

At each momentum cell, we have to discretize the pressure gradient term to computethe last step of the projection method (5). There are several possibilities and we will brieflydescribe some of them. We might have used the volume integral of pressure gradient:

∫

V∇φ dV ≈ ∇φ Vf , (40)

where ∇φ is a gradient in momentum cell center computed from (18), and Vf is a volumeof momentum cell. Alternatively, we might have applied Gauss theorem to get the surfaceintegral of the pressure gradient term:

∫

∇φ dV =∫

S

φ dS ≈∑

k

φk Sk, (41)

u

w

ja

b

v

PSfrag

replac

ement

s

∇p

∂p

∂n

∂p

∂t

Figure 8: Projection of velocity on the cell face

where k implies summation over facetsand φk is the pressure interpolated on thefacet. None of the above two approachesgave satisfactory discretization of the pres-sure gradient, i.e. none of them couldproject velocity into divergence-free field.The probable reason for that lies in thefact that in both (40) and (41) we have in-terpolations of pressure gradient (or pres-sure) which do introduce some numericalerrors. If we recall that the discretizedpseudo pressure equation was derived bybalancing fluxes through faces with values in the virtually orthogonal cell centers (21),it becomes apparent that we should use these, orthogonal, values to project velocity intothe divergence-free field. Hence, the pressure gradient should be computed as:

−→∂φ

∂n=

φA − φB

|d|· n =

φa + ∇φ · δA − φb −∇φ · δB

|d|· n (42)

where (∇φ)a and (∇φ)b are evaluated using expression (9) and n is normal on the cell facecomputed as n = S/|S|. Equation (42) gives just one component of the pressure gradientnormal on the cell face. Although it projects velocity into the divergence-free field, italso introduces large numerical errors in momentum equations, because the local three-dimensionality of the projection is lost. Therefore we also must introduce the tangentialpressure gradient as:

−→∂φ

∂t= ∇φ −

−→∂φ

∂n. (43)

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Bojan Niceno

Although ∇φ is unable to project velocity in the divergence-free field, as discussed above,it does not matter for tangential component since it does not contribute to the massconservation. The final form of a pseudo-pressure gradient term in momentum cell is:

∫

V∇φ dV ≈ (

−→∂φ

∂n+

−→∂φ

∂t) Vf . (44)

3 RESULTS

3.1 Flow over a step

In this section, a flow over a step placed in a square channel, was analyzed. This caseis performed to check whether the staggered algorithm we propose gives the solution freeof check-board velocity and/or pressure fields. Furthermore, it was used as a first testto check whether the pseudo pressure equation is able to project the velocities into adivergence-free field.

The geometry is illustrated in Figure 9. On the inlet, a fully developed parabolicvelocity profile was prescribed, whereas at the outlet, a convective outflow condition wasspecified. Reynolds number, based on bulk velocity and the channel height was 200. Sincethe projection method is used to link velocity and pressure, the simulations were run asunsteady and the results which are shown here are after 10 non-dimensional time units.Although this case is two-dimensional, the solver which was developed for the presentwork is only three-dimensional and the grids shown in Figures 9 and 10 are essentiallycut-planes. The results are plotted for the values computed in cells and no interpolationis used for plotting. This was done on purpose, to illustrate that no spurious fields occurin the solution.

Although the case considered here was quite simple, it proved that the method fulfilledtwo important requirements. The computations clearly show that no spurious fields occurneither in the pressure nor in the velocity fields. The author has also observed thatpressure was able to project the velocities into a divergence-free field after each timestep and reduce the mass error to the level of machine accuracy. Hence it is clear thatwe do have defined a staggered method on unstructured grids which did not need anystabilization procedure in linking velocity and pressure and we were able to use projectionmethod with it.

3.2 Lid-driven cavity flow

To check the validity of the spatial discretization, the well-know lid-driven cavity flowwas chosen at Re value of 1000, which is known to yield a steady solution.

Three different grid topologies were used for the computation of the lid-driven cavity:hexahedral orthogonal grid, three-side prismatic grid and hybrid grid. Orthogonal gridwas uniform with 80 × 80 in non-homogeneous direction. In the homogeneous direction,five layers of uniform cells were used. Three-side prismatic grid was created with the

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Bojan Niceno

-0.641034-0.435446-0.229858-0.02427030.1813180.3869060.5924940.7980821.003671.20926

0.00.1825120.3650240.5475370.7300490.9125611.095071.277591.46011.64261

Figure 9: Results for hexahedral grid: non-dimensional velocity vectors (top), velocity magnitude field(middle) and pressure field (bottom)

-0.734437-0.522556-0.310675-0.09879450.1130860.3249670.5368480.7487290.960611.17249

0.00.1731780.3463560.5195340.6927120.8658891.039071.212251.385421.5586

Figure 10: Results for prismatic grid: non-dimensional velocity vectors (top), velocity magnitude field(middle) and pressure field (bottom)

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Bojan Niceno

Gambit 3 mesh generator by forcing the number of cells to be equal to the number of cellsof the hexahedral grid and by trying to keep the triangle edges as uniform as possible.

U0

H

H

x

y

hexahedral prismatic hybrid

Figure 11: Schematic representation of the computational domain (left) and cuts through computationalgrids in homogeneous planes

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

u

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

v

QuadrilateralTriangularHybridGhia et al.

PSfrag replacements−→δA

Figure 12: Results for the cavity flow. Com-parison of non-dimensional velocity profileswith reference results

The hybrid grid was also generated with Gam-bit. Circular region inside the domain, wherethe strongest vortex occurs, was covered witharbitrary hexahedral cells, whereas the rest ofdomain, between the circular region and cavitywalls, is covered with three-sided prisms. Thenumber of cells was forced to be equal to theprevious grids. The cuts through all the grids inhomogeneous direction are depicted in Figure 11.

The comparison of computed velocity profileswith benchmark solutions is given in Figure 12for all the considered grids. It can be observedthat the computations on all types of grids com-pare very good with benchmark solutions.

3.3 Flow in a square duct with 90o bend

The cases computed so far proved that the new method works and that results comparegood to benchmark solutions, but they were essentially two-dimensional. To validate thenew method on a three-dimensional flow, we have chosen a flow through a square duct with90o bend. The results are compared to experimental results21 and numerical results22.

Figure 13 shows the computational domain and characteristic flow pattern. The innerradius of the bend is r = 0.072 and outer radius is R = 0.112. Reynolds number basedon the bulk velocity and hydraulic diameter is 790. The grid is created by extruding an

3Gambit is a trademark of Fluent corporation.

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Bojan Niceno

tetrahedral grid at the inlet plane in the flow direction, resulting in a grid made up fromthree-sided prism throughout the domain. The surface grid at the inlet and the side ofthe domain are plotted in Figure 13.

R

rθ

INLET

OUTLET

SIDE VIEW

FRONT VIEW

Figure 13: Computational domain (left) and grid (right) for the flow in a square duct with 90o bend

Topology of the flow is depicted in Figure 13. A fully developed profile enters thedomain, hits the bend, creating secondary structures in the flow. The cuts of secondaryflow structures at four stream-wise cross-planes at: 0, 30, 60 and 90 degrees shown inFigure 14.

θ = 0o θ = 30o θ = 60o θ = 90o

Figure 14: Secondary flow structures at four stream-wise cross-planes, illustrated by the non-dimensionalvelocity vectors projected on the cross-planes

Figure 15 shows computed stream-wise velocity profiles for four stream-wise planes,compared to experiments21 and numerical results22. The latter are obtained with a nodecentered finite volume method. Computed velocity profiles compare well with experimen-tal results in 0, 30 and 90 degree planes. There are some discrepancies at 60 degree plane,

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Bojan Niceno

but the same behavior is observed for other numerical simulations22 as well. It might beattributed to insufficient resolution of both computations.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.4

0.8

1.2

1.6

2.0

v

PresentShinHumphrey

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.4

0.8

1.2

1.6

2.0

v

PresentShinHumphrey

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.4

0.8

1.2

1.6

2.0

v

Present ShinHumphrey

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.4

0.8

1.2

1.6

2.0

v

PresentShinHumphrey

θ = 0o θ = 30o

θ = 60o θ = 90o

Figure 15: Comparison of non-dimensional stream-wise velocity profiles at mid-line of four stream-wisecross-planes

4 CONCLUSIONS

In this work, a new method for spatial discretization of Navier-Stokes equations wasproposed. The method works on three-dimensional grids and puts no restriction on cellshape, meaning that existing grids can be used with the new method. The new spatialdiscretization can be used with projection method for time discretization, and is ableto project the velocities into a divergence-free field after each solution of the pseudopressure equation. An important feature of the proposed method is that it discretizesNavier-Stokes equations in their primitive form, as opposed to16, making implementationof physical models and boundary conditions a straightforward task. To present authorsbest knowledge, the method proposed here is the first formulation of three-dimensionalstaggered spatial discretization method for Navier-Stokes equations in primitive form, onunstructured hybrid grids. The most important feature of the new method is the fact thatit does not need any stabilization procedure (such as Rhie & Chow, Arakawa, additionof 4th order dissipative terms, pressure reconstruction) to couple velocity and pressure.Moreover, the presented discretization also avoids the need for deferred correction, making

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Bojan Niceno

the computations more robust.If compared to the usual collocated cell-centered approach, the proposed method has

disadvantages. In the usual three-dimensional grid, there are roughly three times as manyfaces than cells. Hence, the proposed method has to store roughly three times as manyvelocity unknowns than the cell-centered method. The increase in number of unknownvelocities lead to a small increase in computing time, since most time is spent in thepressure solver, which remains cell-centered. Having two grids (pressure and momentumcells) means keeping two connectivity sets and more geometrical data, which increases thememory usage even further. However, this increase in required memory is also featuredby the covolume method14,15,16. The advantage of the present method over covolumesand it variants is in the fact it uses non-transformed formulation of governing equations.

Computation of velocities on the cell faces and the ability to integrate in time withprojection method without any stabilization procedure make the proposed method a goodcandidate for LES of turbulent flows in complex geometries.

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[2] B. Niceno. An unstructured parallel algorithm for large eddy and conjugate heattransfer simulations. PhD thesis, Delft University of Technology, 2001.

[3] P. Moin. Advances in large eddy simulation for complex flows. Int. J. Heat and FluidFlow, 23:710–720, 2002.

[4] P. Wesseling. Principles of computational fluid dynamics. Springer, 2001.

[5] J.R. Manson, G. Pender, and S.G. Wallis. Limitations of traditional finite volumediscretizations for unsteady computational fluid dynamics. AIAA J., 34 (5):1074–1076, 1996.

[6] A. J. Chorin. Numerical solution method for the Navier-Stokes equations. Math.Comput., 22:745–762, 1968.

[7] J. Kim and P. Moin. Application of a fractional step method to incompressibleNavier-Stokes equations. J. Comput. Phys., 59:308–323, 1985.

[8] P. M. Gresho. On the theory of semi-implicit projection method for viscous incom-pressible flow and its implementation via a finite element method that also introducesa nearly consistent mass matrix. Part 1: Theory. Int. J. Num. Meth. Fluids, 11:587–620, 1990.

[9] Y.H. Hwang. Calculations of incompressible flow on a staggered triangular grids.part i: Mathematical formulation. Numer. Heat Transfer, Part B, 27:323–336, 1995.

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[10] M. Thomadakis and M. Leschziner. A pressure correction method for the solutionof incompressible viscous flows on unstructured grids. Int. J. Numer. Meth. Fluids,22:581–601, 1996.

[11] M.H. Kobayashi, J.M.C. Pereira, and J.C.F. Pereira. A conservative finite volumesecond order accurate projection method on hybrid unstructured grids. J. Comput.Phys., 150:40–75, 1999.

[12] I. Wenneker, A. Segal, and P. Wesseling. Conservation properties of a new unstruc-tured staggered scheme. Computers and Fluids, 32:139–147, 2003.

[13] R.A. Nicolaides. Direct discretization of planar div-curl problems. SIAM J. Numer.Anal., 29:32–56, 1992.

[14] J.C. Cavendish, C.A. Hall, and T.A. Porsching. A complementary volume approachfor modelling three-dimensional navier-stokes equations using dual delaunay/voronoitesselations. J. Numer. Meth. Heat Fluid Flow, 4:329–345, 1994.

[15] B. Perot. Conservative properties of unstructured staggered mesh schemes. J. Comp.Phys., 158:58–89, 2000.

[16] K. Mahesh, G. Constantinescu, and P. Moin. Large-eddy simulation of gas-trubinecombustors. In Annual Research Briefs - 2001. Center for Turbulence Research,Stanford University, 2000.

[17] T. J. Barth. Aspects of unstructured grids and finite volume solvers for the Eulerand Navier-Stokes equations. In von Karman Institute for Fluid Dynamics LectureSeries 1994-05, 1994.

[18] I. Demirdzic, S. Muzaferija, and M. Peric. Advances in computation of heat transfer,fluid flow and solid body deformation using finite volume approaches. In W. J.Minkowicz and E. M. Sparrow, editors, Advances in numerical heat transfer 1, pages59–96, 1997.

[19] H. Jasak. Error analysis and estimation in the Finite Volume method with applica-tions to fluid flows. PhD thesis, Imperial College, University of London, 1996.

[20] J. H. Ferziger and M. Peric. Computational methods for fluid flow. Springer-Verlag,1996.

[21] J. A. C. Humphrey, Taylor A. M., and Whitelaw J. H. Laminar flow in a square ductof strong curvature. J. Fluid Mech., 83 (part 3):509–527, 1977.

[22] S. Shin. Reynolds-averaged, Navier-Stokes computations of tip clearance flow in acompressor cascade using an unstructured grid. PhD thesis, Virginia Tech, 2001.

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