Immersed boundary technique for compressible flow simulations on semi-structured meshes

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1 Center for Turbulence Research Annual Research Briefs Immersed boundary technique for compressible flow simulations on semi-structured meshes By M. de Tullio AND G. Iaccarino 1. Motivation and objectives The Immersed Boundary IB) method simplifies the grid generation process for the simulation of complex geometries by removing the constraint of the near wall grid-lines to be aligned with the surfaces of the bodies. IB methods are usually implemented using regular Cartesian grids to take advantage of simple and robust numerical algorithms; however, these grids allow only limited control on the localization of the grid points. Mesh lines propagate in the entire computational domain, introducing high concentration of grid points in regions away from the solid walls where flow gradients are usually small. Flexible local grid refinement LGR) is required to achieve high resolution near the body while saving cells in other regions of the computational domain. The IB technique was originally developed for incompressible flows Peskin 1972; Fadlun et al. 2000; Iaccarino & Verzicco 2003). Very few applications to viscous compressible flow are present in the literature, e.g. the work of De Palma et al. 2004) on structured grids. The aim of this research is to extend the method of De Palma et al. 2004) to locally refined meshes in order to describe the immersed geometry in a better way and to have higher resolution in the boundary layers and in the regions with high gradients e.g. shocks). In this paper we first introduce the details of the numerical method: the discretization scheme, the interpolation procedure at the interface cells and the semi-structured approach to handle LGR grids. The initial validation of the new solver is then reported. The applications include the steady incompressible flow past a sphere at moderate Reynolds numbers for the purpose of testing both the IB implementation and the preconditioning strategy; the low-speed unsteady flow past a heated circular cylinder for the purpose of testing the correct implementation of the energy equation and of the unsteady terms. Preliminary results of the supersonic flow past a circular cylinder and past a sphere are also presented. 2. Numerical method The Reynolds Averaged Navier Stokes RANS) equations, written in terms of Favre mass-averaged quantities and using the standard k ω turbulence model, can be written as follows: ρ u i ) t ρ t + ρ u j ) = 0, 2.1) + ρ u j u i ) = p t x i + ˆτ ji, 2.2) Dipartimento di Ingegneria Meccanica e Gestionale, DIMeG, Politecnico di Bari, Italy.

2 72 M. de Tullio & G. Iaccarino ρ H p t ) + ) ρ u j H = t ρ k) t ρ ω) t + ρ u j k) = τ ij u i β ρ ω k + + ρ u j ω) = γ ω k τ ij u i ˆτ ij + µ + σ µ t ) k q j u i β ρ ω 2 + ], 2.3) µ + σ µ t ) k ], 2.4) µ + σµ t ) ω ]. 2.5) In the equations above, H and pt are the total enthalpy and the pressure comprehensive of the turbulent kinetic energy, k; the eddy viscosity, µ t, is defined in terms of k and of the specific dissipation rate, ω, according to the k ω turbulence model of Wilcox 1998), namely: µ t = γ ρ k ω. 2.6) Moreover, ˆτ ij indicates the sum of the viscous and Reynolds τ ij ) stress tensor components. According to the Boussinesq approximation, one has: ui ˆτ ij = µ + µ t ) + u j 2 x i 3 u k δ ij x k Finally, the heat flux vector components, q j, are given as: ] 2 3 ρ k δ ij. 2.7) µ q j = P r + µ ) t h, 2.8) P r t where P r = 0.71 and P r t = 1 are the laminar and turbulent Prandtl numbers, respectively. The Sutherland law is used to compute the molecular viscosity coefficient and the turbulence models coefficients are reported in Wilcox 1998). It is useful to write the equations in the vector form: Q t + E x + F y + G z E v x F v y G v z = D, 2.9) where Q is the conservative variable vector, E, F, G and E v, F v, G v indicate the inviscid and viscous fluxes, respectively, and D is the vector of the source terms. A pseudotime derivative is added to the left-hand-side of 2.9) in order to use a time marching approach for both steady state and unsteady problems; the preconditioning matrix, Γ Venkateswaran et al. 1992); Merkle 1995)) is used to pre-multiply the pseudo-time derivative in order to improve the efficiency in low Mach number problems. The final system reads: Γ Q v τ + Q t + E x + F y + G z E v x F v y G v z = D, 2.10) where Q v = p t, u, v, w, T, k, ω) T is the primitive variable vector, which is related to Q by the Jacobian P = Q/Q v. 2.10) is discretized using an Euler implicit scheme in the pseudo-time; the physical-time derivative is approximated by a second-order-accurate

3 IBM for Compressible Flows 73 three-point backward differences, and the following equation in delta form is obtained: Γ + 3 τ 2 t P + τ ) A v R xx x x R xy y R xz z + τ ) B v R yx y x R yy y R yz z + τ )] 2.11) C v R zx z x R zy y R zz Q v = z 3Q r 4Q n + Q n 1 ] τ + R r, 2 t where r and τ indicate the pseudo-time level and step, n and t indicate the physicaltime level and step, A v = E/Q v, B v = F /Q v, C v = G/Q v, R ij are the viscous coefficient matrices Schwer 1999)), and the matrix Γ is evaluated as proposed in Buelow et al. 1997). The residual is given as: R r = Er E r v) x + F r F r v ) y + Gr G r v) z D r, 2.12) and Q v = Q v r+1 Q v r are the unknowns in delta form. The left-hand-side LHS) of 2.11) is modified to improve the efficiency of the method, without affecting the residual, that is, the physical solution. Firstly, the non-orthogonal viscous coefficient matrices, R xy, R xz, R yx,r yz,r zx and R zy, are neglected, and the remaining ones are approximated by the corresponding spectral radii multiplied by the identity matrix, R xx = R x I, R yy = R y I and R zz = R z I; then Schwer 1999)), the pseudo- and physical-time terms are grouped together into a new term S, S = Γ which is factored out of the LHS in 2.11), yielding S I + τs 1 A v R x I x x + τs 1 z C v R z I z )] Q v = τ τ P, 2.13) t ) + τs 1 y B v R y I ) y 3Q r 4Q n + Q n 1 + R ]. r 2 t 2.14) In order to solve the resulting linear system, the diagonalization procedure of Pulliam & Chaussee 1981) is first applied so that the matrices S 1 A v, S 1 B v and S 1 C v can be written as: S 1 A v = M x Λ x M 1 x ; S 1 B v = M y Λ y M 1 y ; S 1 C v = M z Λ z M 1 z, 2.15) where M x, M y, M z, are the right-eigenvector matrices, M 1 x, M 1 y,m 1 z are the lefteigenvector matrices; and Λ x, Λ y and Λ z are diagonal matrices containing the eigenvalues of S 1 A v, S 1 B v and S 1 C v, respectively. 2.14) is rewritten as: S I + τm x Λ x R x I ) M 1 x + τm y x x y + τm z Λ z R z I z z ) M 1 z Λ y R y I y ) M 1 y ] 3Q r 4Q n + Q n 1 Q v = τ + R ]. r 2 t 2.16)

4 74 M. de Tullio & G. Iaccarino The BiCGStab solver is employed. Therefore it is possible to solve the above equation inverting the full sparse matrix arising from the discrete LHS operator. In this work a different strategy is carried out in order to have a more compact form of the matrix. This is obtained by factorizing the LHS of 2.16): SM x I + τ x M z I + τ z Λ x R x I x Λ z R z I z )] )] M 1 x M y I + τ y Λ y R y I y )] M 1 y 3Q M 1 r 4Q n + Q n 1 z Q v = τ + R ]. r 2 t 2.17) A collocated, cell-centered finite volume space discretization is used. A first-order upwind flux-difference-splitting scheme or a second-order-accurate central differencing scheme are used to discretize the convective terms in the RHS. The viscous terms are discretized by second-order-accurate differences. The convective term in the LHS is always discretized using a first-order upwind scheme deferred-correction approach) in order to improve the convergence of the linear system solver. Finally, the boundary conditions are treated explicitly Preconditioning Following the work of Venkateswaran & Merkle 1995), the preconditioning matrix is given by: ρ p t ρ T ρ k 0 ρ p t u ρ 0 0 ρ T u ρ k u 0 ρ p t v 0 ρ 0 ρ T v ρ k v 0 Γ = ρ p t w 0 0 ρ ρ T w ρ k w 0, 2.18) ρ h pt + H ρ p t 1 ρu ρv ρw H ρ T + ρ h T hk ρ ρk + H ρ k 0 k ρ p t k ρ T ρ + k ρ k 0 ω ρ p t ω ρ T ρ k ω ρ where the subscripts indicate partial derivative operators. Notice that the derivatives of the flow variables are denoted with ), indicating that, for the general case of the RANS equations, they are functions of the primitive variables p t, T, k. The parameter ρ p t controls the scaling between relevant physical processes in order to enhance the rate of convergence of the numerical algorithm. For the case where no preconditioning is used, this parameter reverts to its non-preconditioned form, ρ p t = ρ pt. The correct asymptotic form of this parameter is determined from low Mach number and low Reynolds number scaling. The term ρ pt can be written in the form: ρ pt = 1 c 2 ρ T 1 ρ h pt ) ρ h T, 2.19) where c is the speed of sound plus a contribution from the turbulent kinetic energy, k.

5 The term ρ p t IBM for Compressible Flows 75 is given by the following relation: ρ p t = 1 Vr 2 ρ T 1 ρ h pt ). 2.20) ρ h T The term V r is a reference velocity that is chosen to appropriately precondition the relevant time-scales. One possible way of choosing V r is V 2 r = min c 2, max V 2 inv, V 2 vis, V 2 unsteady)]. 2.21) The inviscid velocity scale can be defined as Vinv 2 M 2 c 2, where M is the Mach number. The viscous velocity scale is given by Vvis 2 V inv 2 /Re2 x, where Re x is a local cell Reynolds number. The unsteady velocity scale can be computed as Vunsteady 2 l/ tπ)2, where l is a characteristic length scale of the problem, and t is the physical-time step Venkateswaran & Merkle 1995)) Local grid refinement algorithm Local grid refinement LGR) allows for efficient clustering of cells in the vicinity of the immersed boundary. The present implementation is an extension of the classical adaptive mesh refinement AMR) technique for non-isotropic refinement. It can also be interpreted as a generalization of the procedure used for building coarse grids for geometric multigrid on structured meshes. The basic idea was introduced in Durbin & Iaccarino 2002) for a finite difference discretization and extended for finite volume formulations by Iaccarino et al. 2004). The LGR grid is considered as a coarsened version of a fine, structured grid Fig. 1); the following discussion of the algorithm refers to two dimensions, although the extension to three-dimensions is straightforward. The cells are defined as usual by a couple of vertices with indices i, j) and i + 1, j + 1). On the LGR grid, each element is bounded by the grid lines passing through the vertices i, j) and i + i, j + j), see Fig. 1. The effective element size in this case is not constant since i, j depend on i, j). Therefore, the cells are not organized in a structured way with one-to-one neighbors in each Cartesian direction. This requires a modification of the algorithm to handle with hanging nodes. In the present finite-volume approach, the fluxes for each face e.g. E f in the x-direction) are evaluated using the first-order upwind Flux Difference Splitting using the two adjacent cells on the left L and on the right R: E f = 1 2 E L + E R ) 1 ΓMx Λ x M 1 ) x 2 Q f v,r Q v,l ), 2.22) where the matrices Γ, M x, Λ x and M 1 x are calculated using the Roe-averaged quantities. For each direction e.g. the x-direction) the balance of the fluxes in each computational cell is evaluated using all the contiguous cells for each side Nf E cells on the right side and Nf W cells on the left side). Each face flux is computed using the two adjacent cells and all the contributions are collected to build the corresponding convective operators for the cell: V N E N E f f W x δv = E E E W )S x = Ef,nS E x,n E Ef,mS W x,m. W 2.23) n m The number of adjacent cells per face Nf E, N f W ) is limited to two four in threedimensions). A similar approach is employed to build the diffusive operator. The major advantage of the present LGR approach with respect to classical OCTREE based Berger & Aftosmis 1998)) and fully-unstructured Ham et al. 2002)) algorithms lies

6 76 M. de Tullio & G. Iaccarino N1 N1 N1 j+ j=const i+2,j+2) W1 W1 P P E1 P E1 j=const W2 i,j) W2 P P E1 S1 S2 S1 S2 i=const a) i+ i=const Figure 1. a) LGR grid showing a cell P and its neighbors; b) Cell identification array, ID, on the fine underlying grid showing one-to-one connectivity. b) in the economy and flexibility of storing and retrieving connectivity information due to the presence of the underlying notional grid. In particular, on a N i N j underlying grid, only N cells are effectively defined using the two sets of indices i, j) and i, j), with a total storage cost of 4 N integers. In addition, an array of integers, ID i,j) is defined on the fine grid to store the correspondence between the underlying cell and the actual LGR element Fig. 1). In other words, all the underlying cells that are not present in the actual mesh, namely, those included in the range i : i + i 1] and j : j + j 1], are tagged using the same LGR cell number. The total storage required for allocating ID i,j) is, therefore, N i N j. The connectivity information for each cell is retrieved consistently to a structured framework by indirectly querying the array ID i,j). The neighbors of an LGR cell are ID i 1,k) and ID i+1,k) for k ranging in j : j + j 1] in the positive and negative i-direction, respectively Refinement criteria The generation of LGR grids is carried out by creating the underlying fine) grid and coarsening it in the regions away from the immersed boundary. The advantage of this approach is that all the cell tagging ray tracing) can be performed on a structured grid, taking full advantage of the alignment of the cell centers and the grid nodes. The coarsening and the generation of the connectivity information is the last step of the grid generation process. Another important aspect of the application of the LGR is the selection of the refinement/coarsening criteria. In the present implementation, LGR is used to increase the resolution in the surroundings of the immersed boundary and, therefore, the only criteria used is the geometrical distance between each cell and the boundary itself. The cells are tagged as fluid or solid if the cell center is outside or inside the immersed boundary, respectively. Interface elements are not important at this stage and, therefore, a cell is tagged by considering only the position of its cell center. An integer value ±1 is assigned to each cell. The gradient of this function is non-zero only at the immersed boundary, and it is dependent on the local grid size. This gradient is used to select the cells to be refined. The grid is refined until a user specified resolution is achieved at the boundary. A smoothing function can be applied on the ±1 tagging function to obtain a smeared interface that will allow a smoother transition between coarse and refined regions. In addition to the automatic refinement, the user can define

7 IBM for Compressible Flows 77 a) b) c) Figure 2. Different steps of refinement: a) automatic on the immersed boundary; b) specified window wake); c) external surface shock). regions of the computational domains to be refined windows), selecting the resolution of the refinement. Finally it is possible to refine on void surfaces, that is surfaces without solid or interface points like the bow-shock shown in Fig Interpolation method at the immersed surfaces After the pre-processor detects the faces of the cells that are cut by the body, the cells are divided in those that are inside and outside of the body. The cells that have at least one of their neighbors inside the body are labeled as interface cells. The imposition of the boundary conditions on the immersed surface is treated in an explicit way. In the solid cells, the velocity components are set to zero, the temperature in the case of isothermal surface is set to the wall value, the pressure and the temperature in the case of adiabatic surface are not imposed. At the interface cells, the nearby wall is modeled with an off wall boundary condition that consists of an interpolation of the flow variables, using the computed values of the surrounding fluid cells and the imposed values of the wall. For each interface cell it is possible to find N nbr contiguous fluid cells and N ib intersections of the faces of the cell with the immersed boundary. For the velocity components and the temperature in the case of isothermal surface Dirichlet boundary condition), the interpolation formula used is the following: φ int = N nbr i α i q φ N ib β j i + q φ j,wall, 2.24) j where φ j,wall are the values of the flow variable to be imposed on the immersed surface, q = N nbr i N ib α i + β j, 2.25) and α i and β j are the inverse distances between the surrounding cell centers and the interface cell center and between the wall intersections and the interface cell center, respectively. It can be shown that in the one-dimensional case, this interpolation procedure results in a scheme which coincides with the linear procedure used in Fadlun et al. 2000) and De Palma et al. 2004). For the temperature on an adiabatic surface and for the j

8 78 M. de Tullio & G. Iaccarino pressure) it is necessary to impose the value of the normal gradient at the immersed surface. It is possible to write the interpolation formula 2.24) for the gradient as φ int = N nbr ) N φ nbr α i = n int q i i α i q φ N ib β j i + q φ j,wall, 2.26) j ) φ n i N ib + j β j q ) φ. 2.27) n j,wall Finally, we can relate φ j,wall with φ int assuming a linear variation and approximating the normal direction with the direction along which the distance 1/β j is calculated: ) φ 1 φ j,wall = φ int. 2.28) n β j Substituting 2.28) in 2.26), we can obtain the final equation for the flow variable value to be imposed in the interface cell: ) Nnbr α i i q φ i φ 1 n q N ib α i φ int = 1 N ib j β j q int int Nnbr i 1 N ib j q φ i β j q where the last expression is obtained assuming φ/n) wall = φ/n) int., 2.29) 3. Results 3.1. Incompressible flow around a sphere The low Mach number flow past a sphere has been considered to evaluate the accuracy of the present flow solver, the preconditioning strategy, and the immersed boundary method. A single value of the free-stream Mach number, M = 0.03, and four values of the Reynolds number, 40, 60, 80 and 100 have been considered. The Reynolds number is based on the sphere diameter, D, the free-stream velocity, U, and kinematic viscosity, ν. The computational domain is a box and, to avoid errors due to the proximity of the sphere to the boundaries, the inlet and outlet boundary planes are located at x i = 10D and x o = 15D, and the others boundaries are located at y w = ±10D and z w = ±10D, the origin of the box coinciding with the center of the sphere. Standard characteristic boundary conditions have been imposed at inlet and outlet surfaces, whereas free-shear wall boundary conditions are imposed at the other surfaces. Computations have been performed using a semi-structured, non-uniform mesh with 159, 484 fluid cells, 1, 248, 315 faces and 760 halo cells. The grid is refined on the sphere surface, in order to have a good resolution of the boundary layer, and in a box surrounding the sphere and the wake, in order to capture well the separation region. The equivalent underlying structured fine mesh is with 2, 007, 236 cells. Figure 3 shows the influence of the preconditioning on the convergence history of the x-momentum equation in the case of Re = 100. It is known from experimental evidence that the flow around a sphere does not separate up to Re 24, and for increasing Reynolds number, the axial length of the separation bubble grows linearly up to Re 100. This result is recovered by the present numerical simulations. Figure 4 shows the length of the separation bubble compared to the interpolation line of experimental data.

9 IBM for Compressible Flows 79-3 preconditioning ON preconditioning OFF -4 Log error) iterations Figure 3. Convergence history of the x-momentum equation for incompressible, steady flow around a sphere Unsteady flow past a heated circular cylinder The unsteady two-dimensional low-mach number flow past a heated circular cylinder has been chosen in order to validate both the unsteady terms and the correct implementation of the energy equations, since experimental Wang et al. 2000)) and numerical Sabanca & Durst 2003)) investigations indicate that the temperature fields have a significant influence on the flow pattern, especially when the ratio between the cylinder wall temperature T w and the free-stream one T, T = T w /T exceeds 1.1. In particular, Wang et al. 2000) showed that, for a given Re, the vortex shedding frequency, f, and consequently the Strouhal number St = fd/u, decreases for increasing values of T. The computational domain has the inlet and outlet boundary planes located at x i = 10D and x o = 40D, and the other boundaries are located at y w = ±15D and z w = ±10D, the origin of the box coinciding with the center of the cylinder. Standard characteristic boundary conditions have been imposed at inlet and outlet surfaces, free-shear wall boundary conditions are imposed at the y surfaces, and periodic boundary conditions are imposed at the z surfaces in order to have two-dimensional simulations. Computations have been performed using a semi-structured, non uniform mesh with 83, 018 fluid cells, 587, 294 faces and 83, 254 halo cells. The grid is highly refined on the cylinder surface, in order to solve the thermal boundary layer, and in a box surrounding the cylinder and the wake, in order to capture well the wake region. The equivalent underlying structured fine mesh is with 1, 211, 440 cells. The physical time has been set in order to have about 500 steps per shedding period, and about 250 inner iterations are needed to reduce the residual to 10 6 at every physical time step. Figure 5 provides the computed

10 80 M. de Tullio & G. Iaccarino L/D Re Figure 4. Length of the separation bubble as a function of Re for incompressible, steady flow around a sphere. : fit of experimental data, : present simulations. values of the Strouhal number, St, for Re = 100, 120, 140 and T = 1.0, 1.1, 1.5, 1.8, compared with the experimental results found in Wang et al. 2000) and Sabanca & Durst 2003); a very good agreement is obtained, and the results are comparable to the numerical results obtained with the structured version of the code De Palma et al. 2004)) Supersonic flows As preliminary test cases of fully compressible flows, the supersonic flow past a circular cylinder and the three-dimensional flow past a sphere have been considered. The Mach and Reynolds numbers are M = 2.0 and Re = 7000 and M = 1.7, and Re = 5000 for the cylinder and the sphere respectively. The domain boundaries are located 10D away from the obstacle, and standard characteristic boundary conditions are imposed at inlet and outlet surfaces. Initially, simulations on a coarse grid were performed to find the position of the bow-shock. After that, the surface of the shock was exported, and a new fine grid was generated using the LGR on the bow-shock surface; afterward, the grid was refined in the vicinity of the shock position. For the cylinder a final grid consisting of 146, 200 fluid cells, 1, 037, 060 faces and 146, 396 halo cells, and the equivalent underlying structured fine mesh of 837 1, with 4, 399, 272 cells was used. Pressure contours are given in Fig. 6. showing that a clear description of the bow-shock in front of the cylinder is obtained, thanks to the resolution of the grid in that zone. For the sphere calculations, the grid used consisted of 306, 690 fluid cells, 2, 625, 077 faces and 512 halo cells, and the equivalent underlying structured fine mesh is

11 IBM for Compressible Flows St T*=1.0 T*=1.1 T*=1.5 T*=1.8 T*=1.0 T*=1.1 T=*1.5 T*= exp. exp. exp. exp. present present present present Re Figure 5. Strouhal number vs Reynolds number for the unsteady flow past a heated circular cylinder. p: Figure 6. Pressure contours for supersonic, steady flow around a cylinder. with 7, 935, 532 cells. Figure 7 shows the plane z = 0 of the computational mesh and an iso-velocity surface, showing that the flow is axisymmetric.

12 82 M. de Tullio & G. Iaccarino Figure 7. Velocity isosurface V /V = for supersonic, steady flow around a sphere. 4. Conclusions and future work This paper presents the details of a Cartesian immersed boundary IB) solver for compressible RANS flow simulations. It focuses on the numerical method and on the IB implementation on semistructured grids. The methodology has been applied to compute steady and unsteady flows past circular cylinders and a sphere in a wide range of the Mach numbers, thanks to the preconditioning technique, thereby demonstrating its versatility as well as its accuracy for moderate values of the Reynolds number. These first numerical results also demonstrate the advantages of the LGR technique in increasing the resolution where needed. In particular, in the case of the supersonic flows, the LGR technique allows for a highly refined region on the bow-shock which is captured very well. Work is in progress to implement the k ω turbulence model, thereby extending the capability of the code to compute turbulent flows with a RANS approach and to extend the spatial accuracy to the 2nd-order upwind in order to have high-order solutions for compressible flows as well. Work is required to improve the interpolation procedure at the interface cells, to reduce the numerical errors, and to impose the boundary conditions accurately. The next objective is to render the code more efficient in terms of computational time on a single CPU. Finally, the parallelization with MPI is necessary for complex simulations of industrial interest. REFERENCES Berger, M. & Aftosmis, M Aspects and aspect ratios) of Cartesian Mesh Methods, 16th International Conference on Numerical Methods in Fluid Dynamics. Buelow, P. E. O., Schwer, D. A., Feng, J. Z., Merkle, C. L. & Choi, D A preconditioned dual time diagonalized ADI scheme for unsteady computations. AIAA

13 IBM for Compressible Flows 83 De Palma, P., de Tullio, M., Pascazio, G. & Napolitano, M An immersed boundary method for compressible viscous flows. Euro. Cong. Comp. Methods App. Sci. Eng., Jyvaskyla. Durbin P.A. & Iaccarino G Adaptive grid refinement for structured grids, J. Comp. Phys. 128, Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd-Yosuf, J Combined immersed boundary finite-difference methods for three-dimensional complex flow simulations. J. Comp. Phys. 161, Ham, F. E., Lien, F. S. & Strong, A. B. A Cartesian grid method with transient anisotropic adaptation J. Comp. Phys Iaccarino, G. & Verzicco, R Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56, Iaccarino, G., Kalitzin, G., Moin, P. & Khalighi, B Local grid refinement for an immersed boundary RANS solver. AIAA Merklecw, C. L Preconditioning methods for viscous flow calculations. Computational Fluid Dynamics, M. Hafez and K. Oshima eds., John Wiley & Sons, Venkateswaran S. & Merkle, C. L Dual time stepping and preconditioning for unsteady computations. AIAA Peskin, C.S Flow patterns around heart valves: A numerical method. J. Comp. Phys. 10, Pulliam, T. H. & Chaussee, D. S A diagonal form of an implicit factorization algorithm. J. Comp. Phys. 39, Sabanca, M. and Durst, F Flows past a tiny circular cylinder at high temperature ratios and slight compressible effects on the vortex shedding. Phys. Fluids., 15, Schwer, D. A Numerical study of unsteadiness in non-reacting and reacting mixing layers. Ph.D. Thesis, Department of Mechanical Engineering, Penn-state University. Venkateswaran, S., Weiss, S., Merkle, C. L. & Choi, Y. H Propulsion related flow fields using the preconditioned Navier Stokes equations. AIAA Wang, A.B., Trávníĉek, Z. & Chia, C.H On the relationship of effective Reynolds number and Strouhal number for the laminar vortex shedding of a heated circular cylinder. Phys. Fluids, 12, Wilcox, D. C Turbulence Models for CFD. 2nd edn., DCW Industries, Inc.