A THREE DIMENSIONAL, FINITE VOLUME METHOD FOR INCOMPRESSIBLE NAVIER STOKES EQUATIONS ON UNSTRUCTURED, STAGGERED GRIDS

European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006 P. Wesseling, E. Oñate and J. Périaux (Eds) c TU Delft, The Netherlands, 2006 A THREE DIMENSIONAL, FINITE VOLUME METHOD FOR INCOMPRESSIBLE NAVIER STOKES EQUATIONS ON UNSTRUCTURED, STAGGERED GRIDS Paul Scherrer Institut, Nuclear Energy and Safety, Thermal-Hydraulics Laboratory 5232 Villigen PSI, Switzerland e-mail: boan.niceno@psi.ch Key words: Finite volume method, Staggered grids, Unstructured grids, Incompressible flow, Proection method Abstract. In this work, a new method was described for spatial discretization of threedimensional Navier Stokes equations in their primitive form, on unstructured, staggered grids. Velocities were placed on the cell faces and pressure in cell centers and were linked with the proection method. Thanks to the variable arrangement, no stabilization procedure was needed to avoid spurious pressure/velocity fields. A way around the deferred correction was also described and used in this work. Several laminar cases were computed to show the validity of the method. Computation of velocities on the cell faces and the ability to integrate in time with proection method without any stabilization procedure make the proposed method a good candidate for large eddy simulation (LES) of turbulence in complex geometries. 1 INTRODUCTION Collocated methods, where velocities and pressure are computed at the same places on numerical grid, are nowadays well established. Most of the commercial CFD packages are based on cell centered finite volume methods, almost invariantly stabilized with Rhie & Chow techniques 1 in addition to some advanced convective scheme. These type of discretization is well suited for steady or unsteady RANS computations. However, recent advances in computer hardware and mixed success of RANS based method, led to the interest of industry in LES as a new and emerging tool for analysis of turbulent flows. Clearly enough, industrial applications occur in complex geometry, so the high order spectral, or finite difference codes which were most often used in the last couple of decades for LES, must be replaced with a more versatile methodology. Many research groups, realizing the industry s interest in LES, umped into it, ignoring the importance 1

of numerical methods 2. It is generally hoped that using an existing numerical methodology developed and tested for RANS would surely work for LES, believing that refining the 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 complex geometry 3. However, no matter how fine the grid is in collocated computation, one still depends on Rhie & Chow technique, which introduces fourth order dissipative terms to governing equations 4, damping turbulent fluctuations we wish to simulate with LES. Furthermore, iterative methods for coupling velocities and pressure which are almost invariantly used in conunction with collocated grid methods (SIMPLE, SIMPLEC, PISO to name ust the few) and which are originally developed for steady flow computation, are not ideally suited for unsteady simulations, as pointed out in 5. Proection methods 6 7, 8 are more efficient for unsteady simulations, particularly for LES. However, not all spatial discretization schemes are capable to integrate in time with proection scheme. Depending on the placement of variables, some methods are simply not capable to close the mass conservation equation after ust one solution of pressure Poisson equation. To the authors knowledge there have been several attempts to develop unstructured staggered grid in 2D 9, 10, 11, 12. An interesting method, based on discretization of divergence and curl was proposed 13. It is extended to 3D 14 and named complementary volume or covolume method. This three-dimensional extension of covolume method is seen by many as an ideal candidate for LES in complex geometry, thanks to its conservative properties 15. The original method is constrained to Delaunay-Voronoi tessellations, but an extension of the method which does not have this restriction is developed 16. In present work, a new method is presented which solves the equation in their primitive form (hence velocity components and pressure) and is yet staggered and formulated for hybrid grids. The link between velocity and pressure is achieved via proection method, having the ability to reduce the mass error to arbitrary low level after each solution of the pressure Poisson equation. To the authors best knowledge, it is the first primitive formulation 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 mass conservation equation: and momentum conservation equations: u = 0, (1) u t + u u = 1 Re 2 u p. (2) A variety of methods exists for linking the velocity and pressure in the context of finite volume method. A family of method, such as SIMPLE, SIMPLEC, PISO seek the pressure 2

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 steady flow equations and widely used, even in commercial CFD packages. On the other hand, we have proection methods, where in each time step only one solution of Poisson equation for pseudo pressure is sufficient to close the equations and proceed with a new time step. Since the long-term obective of present author is to apply the proposed methodology to LES of turbulence, proection method is a preffered choice in this work. The proection method, in its simplest form, can be summarized as follows 8 : Step 1: solution of the tentative velocity field (u ) from the following equation: u u n + 3 δt 2 (u u)n 1 2 (u u)n 1 = 1 [ ( 2 u) + ( 2 u) n], (3) 2Re where n is the old time step, n 1 is time step before that. Convective terms are discretized with the Adams-Bashforth scheme for simplicity and viscous terms with Crank-Nicolson for stability. Tentative velocity field is generally not divergence-free. Step 2: solution of the pseudo pressure (φ) from the following equation: 2 φ = u δt. (4) Step 3: proection of the velocity to a divergence-free field: and proceeding to a new time step. u n+1 = u + δt φ, (5) Once we have outlined the algorithm for the solution of Navier Stokes equations, we clearly 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 the sets covers the entire problem domain. Pressure cells coincide with the actual cell of the numerical grid. On the boundary, pressure cells coincide with boundary faces. Pressure is computed in the cells inside and on the boundary. Hereafter, pressure cells will be referred to as ust cells and its faces as faces for the sake of shortness. Momentum cells are constructed around each cell face from two pyramids having the cell face as its base and spanning to the center of each of the surrounding cells. Since there are two possible shapes of cell faces, triangular and quadrilateral, there are also two possible shapes for momentum cells inside the domain: triangular and quadrilateral bi-pyramid and two possible shapes of momentum cells at the boundary: triangular and quadrilateral pyramid. Two possible momentum cell inside the domain are shown in Figure 1. Hereafter, momentum cells will be referred to as such and its faces as facets. Facets can only have triangular shape. 3

1 2 3 4 (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 of gradients of cell and face centered variables, because it is the basic building block of the method we propose. 2.2.1 Cell centered pressure gradients Gradients in cell centers are computed with the least square approach 17. We define a gradient in a cell center with: φ i = φ c + ( φ) T c q i ɛ i, (6) where φ c denotes the value in the cell center, φ i denotes the values in neighboring cell centers. Distance vectors between neighboring cell centers and current cell center are denoted by q i = x i x c with q x, q y and q z as it s components. Errors introduced by this approximation are denoted by ɛ i. If square of the errors is minimized with gradient components, the following equation is obtained for each cell: i q 2 x i q x q y i q x q z i q x q y i qy 2 φ/ x i q y q z φ/ y i q x q z i q y q z i qz 2 = i φ i q x i φ i q y, (7) φ/ z i φ i q z where φ i = φ i φ c is the difference between the values in cell neighbors and cell center. Equation (7) is conveniently written in the form: C 1 c ( φ) c = d c, (8) to yield the expression for the gradient in the cell center: ( φ) c = C c d c. (9) This procedure could be used for all variables defined in cell centers. It should be noted that C c is a 3 3 symmetric matrix dependent on geometry and d c is a vector dependent on geometry and variable φ i in all cell neighbors. 4

2.2.2 Cell centered velocity gradients Velocity gradients in cell centers are computed in the same manner as the pressure gradients, i.e. by minimizing the square of error (ɛ ) in the expression: u = u c + ( u) c r ɛ. (10) Here u c is the velocity in the cell center, u are the values in the faces surrounding the cell, and r = x x c is the distance vector between the th cell face and cell center and r x, r y and r z 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: rx 2 r x r y r x r z r x r y ry 2 r y r z r x r z r y r z rz 2 u T / x u T / y u T / z = u T r x u T r y u T r z, (11) with u T = u T ut c. The following point deserves special attention. On the right side of Equation (11) we have a sum of velocities at cell faces and velocity in the cell center u c which is not defined by the present method. If expression u T r x, is expanded, we get: (u T ut c )(x x c ) = u T x x c u T ut c =0 {}}{ ( x N x c ), (12) where N is the number of faces around the cell. The last term in Equation (12) is equal to zero if cell center is defined as an average of the face centers encompassing it. In present work, cell centers have been defined in such a way to satisfy this property. 1 Hence, the right hand side of Equation (11) can be rewritten as: u T r x u T u T r x r y = u T u T r y, (13) r z u T r z where the dependence on the velocity in the cell center u c is lost. Equation (11) can be written in short form as: The gradients can finally be computed from: F 1 c ( u) T c = G c. (14) ( u) T c = F c G c, (15) where both F and G c are a 3 3 matrices, former depending on geometry only and later 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 are defined in the cell centers. 1 A reader can check that it is automatically satisfied for all but pyramid cell shape. 5

2.3 Pseudo pressure equation The first step in discretization any differential equation by the finite volume family of methods is to re-write it in integral form and apply Gauss theorem to lower the order of differentiation. 2 Pseudo pressure Equation (4) in integral form, and after applying Gaussian 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 is present, we may readily write its discretized version: ( φ) S = 1 u S, (17) t where is a face index, S is the face surface vector and ( φ) is the gradient at the cell face. This is the discretized form of the pseudo pressure equation. The difficulty with it lies in the fact that we somehow need to compute the gradient at each cell face. The most natural choice, which is followed by many authors 18, 19 is to compute it as an arithmetic average: ( φ) = ( φ) = 1 2 [( φ) a + ( φ) b ], (18) where subscripts a and b denote gradients computed in cell centers surrounding the face as illustrated in Figure 2. There is a problem with this approach, hindering its straight- a B δ b δ a A b 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 PSfrag replacements 2 A reader may argue that we are going two steps forward and one step back, i.e. form physical laws to integral formulation, then from integral to differential form ust 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

value of φ in its first neighbors. A small proof follows. Direct discretization of the left hand side of (17), using the approximation (18), for each cell c we get: =0 ( φ) S = 1 2 ( φ) {}}{ c S + 1 ( φ) i S, (19) 2 where i denotes cell neighbor to cell c on the opposite side of face, for each of the faces surrounding the cell c. The first term on right hand side vanishes, because sum of the surface vectors encompassing a closed volume is zero, and as a consequence, we get: ( φ) S = 1 ( φ) i S. (20) 2 If we recall that gradients computed in the neighboring cells i do not depend on the values in themselves (12), Equation (20) has a very undesirable property that it links the value in the cell center only with the values in its second neighbors, which is illustrated in Figure 3. This issue is addressed by the introduction of the so-called deferred correction c i, c i, Figure 3: Illustration of the numerical molecule obtained on regular grids after discretization by Equation (20). Non-zero terms are shaded. Triangular grid features the same problem as quadrilateral. approach 18, 20, 19 where the authors introduce a first order term connecting the values in each cell center with its neighbors and putting it in the system matrix and the difference between the first order term and the second order is placed on the right hand side of linear system of equations and iterated within the solution algorithm. If this correction is rather high (which happened if the cell shapes are very irregular) right hand side of the linear system grows relative to the left making the solution unstable or even lead to nonphysical solutions. This issue was studied in 19 who, among other things, examined three different approaches to deferred correction and analyzed their influence on the stability of the solution procedure. 7

In this work we propose a different approach. We do not approximate gradient on the cell face and multiply it with surface vector, as done by equations (17) and (18), but we rather approximate the flux itself, i.e. cell face gradient multiplied with surface vector: ( φ S) = S d (φ B φ A ), (21) where A and B are virtual cell centers orthogonal to the face, and d is the distance between them. The situation is shown in Figure 2. The values in these, orthogonal, cell centers are obtained from the following equations: and: φ A = φ a + ( φ) a δ A, (22) φ B = φ b + ( φ) b δ B, (23) with a and b being the cells around the face. The flux on the cell face can finally be written as: ( φ S) = S d [φ b φ a + ( φ) b δ B ( φ) a δ A ]. (24) In order to discretize the entire system of equations, we browse through cell faces, and for each of them we discretize Equation (24) using the cell gradient matrices defined by (7). We get the following expression: ( φ S) = S d [φ b φ a + (C b d b ) δ B (C a d a ) δ A ], (25) which is used to compute the coefficients of the system matrix connecting the values in cell centers a and b encompassing face and all of their neighbors. Paraphrasing finite element methodology, we call Equation (25) face stiffness matrix. Non-zero terms from the face stiffness matrix are illustrated in Figure 4. No term resulting from a discretization of the pseudo pressure equation is treated explicitly by the solution algorithm. Everything is placed in the system matrix. The reasons for that are threefold: Since the pseudo pressure equation poses a difficulty for linear solvers due to its non-singularity, we did not want to make things even harder on it by loading the right hand side. Our final goal is to use the proposed methodology for unsteady flow and we wish to link velocity and pressure via the proection method. Thus, any type of outer iterations (which are needed if you treat part of the equation explicitly) would mean failing to achieve our obective. 8

b a b a Figure 4: Illustration of the numerical molecule obtained on regular grid after discretization by Equation (25). Non-zero terms are shaded. Implicit treatment of entire pseudo pressure equation makes the implementation of the vanishing gradient boundary condition a straightforward task, as will be explained bellow. 2.3.1 Implementation of boundary conditions The only type of boundary condition used for the pseudo pressure equation is a zero gradient, 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 solve them in a linear system of discretized equations together with values in the interior. This approach is almost invariantly used with finite element discretization and node centered finite volumes. Do not treat boundary values as unknowns (do not place them in the linear system of equations), but rather calculate them from the values in the interior and specified boundary conditions. This is how it is done in most cell centered finite volume approaches 20. Since we are defining a cell centered method for the pressure, we have first considered the later approach. Implementing a zero gradient conditions (26), which is needed for pseudo pressure equation, leaves some ambiguity in computing the boundary values: one may either ust copy (or mirror) the values from the inside to the boundary cell, or one may use more sophisticated extrapolation using the gradient from inside. None of these two approaches, from present authors experience, is satisfactory for the proection method, i.e. none of them results in a pressure field which is able to proect the velocities into the divergence-free field. Moreover, they both require iterations for computation of gradients, 9

deteriorating the efficiency of the solution algorithm. Therefore, we re-write the cell face stiffness matrix (25) for the boundary cell: PSfrag replacements a δa B = b A Figure 5: Implementation of the boundary condition. ( φ S) = S d [φ A φ b + (C a d a ) δ A ]. (27) 2.3.2 Discretization of the source term In (27), it has been acknowledged that δ B is zero for the boundary cell (see Figure 5) and thus we got rid of the entire part multiplying non-existent gradient matrix for the boundary cell. Boundary face stiffness matrices (27) are assembled in the same linear system as the values in the interior are. In Equation (27) each boundary values depends on the value of the first neighbor inside (a) and all of its neighbors (via d a ). By doing so we have removed any ambiguity in defining the values in the boundary cells and we have also obtained the pressure field which proects the velocity into the zero (to the machine accuracy) divergence field. The discretization of the source term for the pseudo pressure equation is straightforward. Rewriting it in integral form and applying Gauss theorem yields: 1 u dv = 1 u ds 1 δt V δt S t u S. (28) We have merely replaced the tentative velocity u field by its discretized counterpart u, which is already defined on cell faces. The very fact that no interpolation on velocity is done 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, with omitted pressure gradient term, reads: V u t S dv + uuds = uds. (29) S Terms on the left are inertial and convective terms, whereas on the right we have the viscous term. Pressure gradient term is omitted on purpose, since it was not used in the present proection method. Discretization of the gradient of the pseudo pressure is outlined in the next section. 10

2.4.1 Inertial term Inertial term is discretized by approximating the volume integral in each momentum cell with: where V f is a volume of each momentum cell around face f. 2.4.2 Convective terms V u t dv u u n V f, (30) t Convective term is discretized by approximating the surface integral of the convective term by: uuds S k u k u k S k, (31) where k stands for summation over facets of each momentum cell and u k is the velocity at the centroid of each facet k. S k is the facet surface vector. The velocity at each facet is obtained from the following relation: PSfrag replacements u k = w[u a + ( u) a γa ] + (1 w)[u b + ( u) b γb ], (32) where w is a weight factor in the range 0.5 1 determining type of convective scheme which is used. For w = 0.5 we have a central k B convective scheme, whence for w = 1 we have a γ A γb a linear upwind differencing scheme (LUDS). b Subscripts a and b denote faces (momentum cells) around facet k and γ a and γ b are vectors connecting centers of momentum cells a c Figure 6: Computation of convective flux on the and b with facet center (see Figure 6). facet k. Momentum cells are denoted by a and b If the definitions for velocity gradient (15) is introduced into (32), we get the final expression for velocity at the facet: u k = wu a + (1 w)u b + F c G c [w γ a + (1 w) γ b ], (33) which is used to compute the discretized convective terms. Since in the proection method we 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 it into the right hand side of the linear system of discretized equations. 11

2.4.3 Viscous terms Since the method presented in this work is used only for incompressible flow problems with constant physical properties, the viscous stress tensor reduces to the velocity gradient: u + ( u) T = u. (34) After integration of (34) over momentum cell, we get: u ds = u S k, (35) S k where k denotes facets of momentum cells. This term is discretized following a procedure similar to the one for pseudo pressure equation, i.e. by discretizing the viscous force at each facet as: ( u S) k = S k d k (u B u A ), (36) PSfrag replacements where u A and u B are velocities in orthogonal face centers A and B, and d k is the distance between the orthogonal face centres (see Figure 7). Velocities u A and u B are computed a δ A A k B δb b Figure 7: Computation of the flux on a facet. Cell center is denoted by c. Two of the cell faces are shown and their centers denoted by a and b. Two momentum cells around the faces a and b are also shown and a facet between them is shaded and denoted by k c by using (15) as: and: u A = u a + ( u) c δa = u a + (F c G c ) δ A, (37) u B = u b + ( u) c δb = u b + (F c G c ) δ B. (38) Because each facet can be contained by only one cell, the same face gradient matrix F c is used for computation of u A and u B. As a result of the discretization, we get the following expression for each facet: ( u) k S k = S k d k [u b u a + F G T ( δ B δ A )], (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. 12

2.4.4 Pseudo pressure gradient At each momentum cell, we have to discretize the pressure gradient term to compute the last step of the proection method (5). There are several possibilities and we will briefly describe some of them. We might have used the volume integral of pressure gradient: φ dv φ V f, (40) V where φ is a gradient in momentum cell center computed from (18), and V f is a volume of momentum cell. Alternatively, we might have applied Gauss theorem to get the surface integral of the pressure gradient term: φ dv = φ ds S k φ k S k, (41) where k implies summation over facets and φ k is the pressure interpolated on the facet. None of the above two approaches gave satisfactory discretization of the pressure gradient, i.e. none of them could proect velocity into divergence-free field. The probable reason for that lies in the fact that in both (40) and (41) we have interpolations of pressure gradient (or pressure) which do introduce some numerical errors. If we recall that the discretized PSfrag replacements a w p t v u b p n p Figure 8: Proection of velocity on the cell face pseudo pressure equation was derived by balancing fluxes through faces with values in the virtually orthogonal cell centers (21), it becomes apparent that we should use these, orthogonal, values to proect velocity into the divergence-free field. Hence, the pressure gradient should be computed as: φ n = φ A φ B d n = φ a + φ δ A φ b φ δ B d n (42) where ( φ) a and ( φ) b are evaluated using expression (9) and n is normal on the cell face computed as n = S/ S. Equation (42) gives ust one component of the pressure gradient normal on the cell face. Although it proects velocity into the divergence-free field, it also introduces large numerical errors in momentum equations, because the local threedimensionality of the proection is lost. Therefore we also must introduce the tangential pressure gradient as: φ t = φ φ n. (43) 13

Although φ is unable to proect velocity in the divergence-free field, as discussed above, it does not matter for tangential component since it does not contribute to the mass conservation. The final form of a pseudo-pressure gradient term in momentum cell is: 3 RESULTS 3.1 Flow over a step V φ φ φ dv ( n + t ) V f. (44) In this section, a flow over a step placed in a square channel, was analyzed. This case is performed to check whether the staggered algorithm we propose gives the solution free of check-board velocity and/or pressure fields. Furthermore, it was used as a first test to check whether the pseudo pressure equation is able to proect the velocities into a divergence-free field. The geometry is illustrated in Figure 9. On the inlet, a fully developed parabolic velocity profile was prescribed, whereas at the outlet, a convective outflow condition was specified. Reynolds number, based on bulk velocity and the channel height was 200. Since the proection method is used to link velocity and pressure, the simulations were run as unsteady 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 present work is only three-dimensional and the grids shown in Figures 9 and 10 are essentially cut-planes. The results are plotted for the values computed in cells and no interpolation is used for plotting. This was done on purpose, to illustrate that no spurious fields occur in the solution. Although the case considered here was quite simple, it proved that the method fulfilled two important requirements. The computations clearly show that no spurious fields occur neither in the pressure nor in the velocity fields. The author has also observed that pressure was able to proect the velocities into a divergence-free field after each time step and reduce the mass error to the level of machine accuracy. Hence it is clear that we do have defined a staggered method on unstructured grids which did not need any stabilization procedure in linking velocity and pressure and we were able to use proection method with it. 3.2 Lid-driven cavity flow To check the validity of the spatial discretization, the well-know lid-driven cavity flow was 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 grid was 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 14

1.64261 1.4601 1.27759 1.09507 0.912561 0.730049 0.547537 0.365024 0.182512 0.0 1.20926 1.00367 0.798082 0.592494 0.386906 0.181318-0.0242703-0.229858-0.435446-0.641034 Figure 9: Results for hexahedral grid: non-dimensional velocity vectors (top), velocity magnitude field (middle) and pressure field (bottom) 1.5586 1.38542 1.21225 1.03907 0.865889 0.692712 0.519534 0.346356 0.173178 0.0 1.17249 0.96061 0.748729 0.536848 0.324967 0.113086-0.0987945-0.310675-0.522556-0.734437 Figure 10: Results for prismatic grid: non-dimensional velocity vectors (top), velocity magnitude field (middle) and pressure field (bottom) 15

Gambit 3 mesh generator by forcing the number of cells to be equal to the number of cells of the hexahedral grid and by trying to keep the triangle edges as uniform as possible. U 0 H y x H hexahedral prismatic hybrid Figure 11: Schematic representation of the computational domain (left) and cuts through computational grids in homogeneous planes ag replacements δ A v 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6-0.8-1.0-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 u Quadrilateral Triangular Hybrid Ghia et al. Figure 12: Results for the cavity flow. Comparison of non-dimensional velocity profiles with reference results The hybrid grid was also generated with Gambit. Circular region inside the domain, where the strongest vortex occurs, was covered with arbitrary hexahedral cells, whereas the rest of domain, between the circular region and cavity walls, is covered with three-sided prisms. The number of cells was forced to be equal to the previous grids. The cuts through all the grids in homogeneous direction are depicted in Figure 11. The comparison of computed velocity profiles with benchmark solutions is given in Figure 12 for all the considered grids. It can be observed that the computations on all types of grids compare very good with benchmark solutions. 3.3 Flow in a square duct with 90 o bend The cases computed so far proved that the new method works and that results compare good to benchmark solutions, but they were essentially two-dimensional. To validate the new method on a three-dimensional flow, we have chosen a flow through a square duct with 90 o bend. The results are compared to experimental results 21 and numerical results 22. Figure 13 shows the computational domain and characteristic flow pattern. The inner radius of the bend is r = 0.072 and outer radius is R = 0.112. Reynolds number based on the bulk velocity and hydraulic diameter is 790. The grid is created by extruding an 3 Gambit is a trademark of Fluent corporation. 16

Boan Nic eno tetrahedral grid at the inlet plane in the flow direction, resulting in a grid made up from three-sided prism throughout the domain. The surface grid at the inlet and the side of the domain are plotted in Figure 13. OUTLET FRONT VIEW θ r SIDE VIEW INLET R Figure 13: Computational domain (left) and grid (right) for the flow in a square duct with 90 o bend Topology of the flow is depicted in Figure 13. A fully developed profile enters the domain, hits the bend, creating secondary structures in the flow. The cuts of secondary flow structures at four stream-wise cross-planes at: 0, 30, 60 and 90 degrees shown in Figure 14. θ = 0o θ = 30o θ = 60o θ = 90o Figure 14: Secondary flow structures at four stream-wise cross-planes, illustrated by the non-dimensional velocity vectors proected on the cross-planes Figure 15 shows computed stream-wise velocity profiles for four stream-wise planes, compared to experiments 21 and numerical results 22. The latter are obtained with a node centered finite volume method. Computed velocity profiles compare well with experimental results in 0, 30 and 90 degree planes. There are some discrepancies at 60 degree plane, 17

but the same behavior is observed for other numerical simulations 22 as well. It might be attributed to insufficient resolution of both computations. v 2.0 1.6 1.2 0.8 0.4 Present Shin Humphrey 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 θ = 0 o v 2.0 1.6 1.2 0.8 0.4 Present Shin Humphrey 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 θ = 30 o 2.0 1.6 Present Shin Humphrey 2.0 1.6 Present Shin Humphrey v 1.2 v 1.2 0.8 0.8 0.4 0.4 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 θ = 60 o 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 θ = 90 o Figure 15: Comparison of non-dimensional stream-wise velocity profiles at mid-line of four stream-wise cross-planes 4 CONCLUSIONS In this work, a new method for spatial discretization of Navier-Stokes equations was proposed. The method works on three-dimensional grids and puts no restriction on cell shape, meaning that existing grids can be used with the new method. The new spatial discretization can be used with proection method for time discretization, and is able to proect the velocities into a divergence-free field after each solution of the pseudo pressure equation. An important feature of the proposed method is that it discretizes Navier-Stokes equations in their primitive form, as opposed to 16, making implementation of physical models and boundary conditions a straightforward task. To present authors best knowledge, the method proposed here is the first formulation of three-dimensional staggered spatial discretization method for Navier-Stokes equations in primitive form, on unstructured hybrid grids. The most important feature of the new method is the fact that it does not need any stabilization procedure (such as Rhie & Chow, Arakawa, addition of 4 th order dissipative terms, pressure reconstruction) to couple velocity and pressure. Moreover, the presented discretization also avoids the need for deferred correction, making 18

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 many faces than cells. Hence, the proposed method has to store roughly three times as many velocity unknowns than the cell-centered method. The increase in number of unknown velocities lead to a small increase in computing time, since most time is spent in the pressure solver, which remains cell-centered. Having two grids (pressure and momentum cells) means keeping two connectivity sets and more geometrical data, which increases the memory usage even further. However, this increase in required memory is also featured by the covolume method 14, 15, 16. The advantage of the present method over covolumes and 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 with proection method without any stabilization procedure make the proposed method a good candidate for LES of turbulent flows in complex geometries. References [1] C. M. Rhie and W. L. Chow. A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA J., 21:1525 1532, 1983. [2] B. Ničeno. An unstructured parallel algorithm for large eddy and conugate heat transfer simulations. PhD thesis, Delft University of Technology, 2001. [3] P. Moin. Advances in large eddy simulation for complex flows. Int. J. Heat and Fluid Flow, 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 volume discretizations 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 incompressible Navier-Stokes equations. J. Comput. Phys., 59:308 323, 1985. [8] P. M. Gresho. On the theory of semi-implicit proection method for viscous incompressible flow and its implementation via a finite element method that also introduces a 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. 19

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