Local Preconditioning for Low Mach-number Aerodynamic Flow Simulation with a Steady Compressible Navier-Stokes Flow Solver

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1 Local Preconditioning for Low Mach-number Aerodynamic Flow Simulation with a Steady Compressible Navier-Stokes Flow Solver Kurt Sermeus and Eric Laurendeau Advanced Aerodynamics Department Bombardier Aerospace Dorval, QC, H4S 1Y9, Canada ABSTRACT This paper discusses the implementation of a local preconditioning method for accurate and efficient simulation of low Mach-number aerodynamic flows with a finite volume compressible Navier-Stokes flow solver for multiblock-structured grids. The Weiss- Smith low Mach-number preconditioner has been implemented using the formulation of Turkel. Results show that both improved accuracy and iterative convergence rate are maintained down to the incompressible limit. As an example of a complex and important aerodynamic problem for which the low Mach-number preconditioning method is an essential ingredient, the low-speed separated flow around the intake of a powered jet engine in a crosswind is computed. 1. INTRODUCTION Density-based CFD methods are well established for solving compressible flow problems. However, when applied to nearly incompressible flows, i.e. at low Mach-numbers, compressible CFD methods show severe accuracy and convergence deficiencies. These numerical difficulties originate from the increasingly large disparity in magnitude between the advection wave speeds as the Mach-number approaches zero: the acoustic waves travel with the speed of sound c, while the fluid particles (entropy and shear waves) travel with the local flow velocity u << c. While some low-speed fluid dynamics problems can effectively be solved with incompressible CFD methods using pressure-correction schemes, many problems in aircraft aerodynamics involve areas of transonic flow embedded within a mostly low-speed flow field and thus require essentially compressible CFD methods. A typical example of such problems is the flow over a transport aircraft at landing speed with the high-lift devices deployed; locally in the high-lift system, transonic flow conditions may occur. Another problem of particular interest is the flow around the intake of a jet engine at take-off power setting, while the aircraft is stationary on the runway. As a remedy to the aforementioned problems of compressible CFD methods at nearly incompressible conditions, a technique known as low Mach-number preconditioning has been discussed in the literature for over 5 years, see [1] for a review. The theoretical basis of low-mach preconditioning methods is now relatively well understood in the literature. However, the practical application of these methods to complex 3D Navier-Stokes solvers and industrial flow problems remains a challenge and involves code modifications that are mostly not so well publicized. The paper is organized as follows. In Section a brief description of Bombardier s Full-Aircraft Navier- Stokes Code (FANSC) is given. Section 3 summarizes the basics of low Mach-number preconditioning and discusses the implementation of the Weiss-Smith preconditioner in FANSC. Section 4 shows numerical results of this method with FANSC, ranging from inviscid flow over a wing to turbulent flow over a powered turbofan jet engine in a crosswind. Section 5 concludes and indicates directions of ongoing and future work.. FULL-AIRCRAFT NAVIER-STOKES CODE FANSC The Full Aircraft Navier-Stokes Code (FANSC), developed and employed by the Advanced Aerodynamics Department at Bombardier Aerospace, solves the 3D Euler and Navier-Stokes equations on multiblock structured grids [],[3]. The equations are discretized using a cell-centered finite-volume method with artificial dissipation, following Jameson et al. [4],[5]. Steady-state flow solutions are computed using an explicit multi-stage Runge-Kutta time-marching algorithm, accelerated by Implicit Residual Smoothing and the Full Approximation Storage (FAS) multigrid method [5]. For closure of the Reynolds-averaged 1

2 Navier-Stokes (RANS) equations, one- or twoequation turbulence models are used; these are discretized with a finite difference scheme. The turbulence model equations are solved with an implicit method, employing an approximate factorization scheme with symmetric Gauss-Seidel iteration. For computational speed-up of FANSC on massively parallel computer systems, coarse-grain parallelism is implemented using the Message Passing Interface (MPI) [6],[7]. 3. LOW MACH-NUMBER PRECONDITIONING We summarize the main ideas related to low Machnumber preconditioning, mostly following the work of Turkel and his co-authors [1],[8] [1]. Other notable references are [11] [17]. 3.1 Condition number in low Mach-number limit Consider for instance the D Euler equations in their non-conservative form: + A t where U ( ρ, ρu, ρe) T x + A x y y =, r = is the vector of conservative variables. The time-dependent Euler equations are hyperbolic, i.e. the eigenvalues of the spatial operator are real; these eigenvalues correspond to the propagation speeds of acoustic, entropy and shear waves traveling in the flow field. For waves propagating in a direction at an angle θ to the x-axis, the wave speeds are given by the eigenvalues of the matrix Ax cosθ + Ay sinθ. The wave-speeds are: λ = u c for the acoustic waves, 1, ± λ u for the shear and entropy waves, 3, 4 = where u is the velocity and c the speed of sound. The condition number of a matrix is defined as the ratio of its largest and smallest eigenvalues. For the low subsonic regime, i.e. M <. 5, the condition number κ of the Euler equations is: A κ A = λ λ max min u + c M +1 = = u M. The maximum allowable time-step of explicit timemarching schemes, given by the CFL stability limit, and the asymptotic solution change per time step are determined by respectively the fastest and slowest waves. Hence, the number of iterations to reach convergence goes up with the condition number. In the low Mach-number limit the system of Euler equations becomes ill-conditioned, i.e. lim κ M A =. As a result, exceedingly large numbers of iterations are needed to converge the numerical solution to steady-state as the Mach number approaches zero. This is also true, although to a lesser degree, for implicit solvers [17]. 3. Accuracy in low Mach-number limit In addition to the convergence slow-down, the illconditioned system of equations at low Machnumbers leads to a loss of accuracy. This occurs because the numerical dissipation introduced by the discretization method be it by artificial dissipation or upwinding does not properly scale with the Mach number to maintain more or less a balance with the advective flux terms in the Euler equations [1]. 3.3 Low Mach-number preconditioning By pre-multiplying the spatial operator of the system of partial differential equations with a preconditioning matrix P, + P A t x + A x y = y, the wave-speeds governing the time-behaviour of the system can be altered, while the steady-state solution upon convergence is retained, provided at least that P is positive definite [1]. The aim of low Mach-number preconditioning is then to define P such that the wave propagation speeds of the modified system, i.e. the eigenvalues of the matrix P(A x cosθ +A y sinθ ), are more equalized in magnitude, such that the condition number is close to one. Optimally, the preconditioner would be defined to give unity condition number, i.e. κ PA = 1, in the incompressible limit. While many low Mach-number preconditioners have been proposed in the literature, their application to complex flow problems is often plagued by robustness issues, mainly due to singularities arising at zero flow velocity (e.g. stagnation points). Based on successful results shown in the literature for other CFD codes similar to FANSC and the intended applications, the Weiss-Smith preconditioner [13] was selected for implementation in FANSC. The Weiss- Smith preconditioner has been formulated generally for both variable and constant density Newtonian

3 fluids. For an ideal gas, the Weiss-Smith preconditioner boils down to the simplest form of the Turkel preconditioner [1]. As we are only concerned with the ideal gas case in the present work, we followed Turkel s formulation of the preconditioner. 3.4 Turkel preconditioner For the D Euler equations transformed to the entropy variables W, W = ( p,u,v,s), with S denoting entropy, a generalized form of the Turkel preconditioner [1] (or rather its inverse) is, 1 δ β α u 1 1 P = T cβ α v, 1 cβ 1 with α, β and δ free parameters. Taking different choices for the parameters, this definition covers a whole family of preconditioners. For the choice β = 1 and α = δ = there is no preconditioning. The wave-speeds of the preconditioned Euler equations are given by the eigenvalues of PA: 1 λ = ± + 1, z u z u 4β c u, λ u, where 3, 4 = z = 1+ β α. For the condition number κ PA of the preconditioned system PA to remain bounded in the M limit, β needs to be of the order of the Mach-number, i.e. β = k1m, where k1 1. The preconditioner thus effectively slows down the acoustic waves to be of the same magnitude as the flow velocity. Note that δ does not affect the inviscid eigenvalues. Setting δ = defines the family of preconditioners proposed by Turkel [1], while δ = 1, α = recovers the preconditioner of Choi and Merkle [11]. Setting α = δ = leads to the simplest form of the Turkel preconditioner, which is in fact identical to the Weiss-Smith preconditioner [13] for ideal gases. The choice α = 1 is the optimal preconditioner in the Turkel family, as it leads to a condition number of one in the incompressible limit. The choice α =, i.e. the Weiss-Smith preconditioner, gives a higher condition number, which has an M limit value of ( 1 + 5) ( 1 5). 6. Figure 1 illustrates the effect of these different choices of the preconditioner on the condition number for the D Euler equations. The graph also lists the van Leer- Lee-Roe preconditioner [1], which can be shown to achieve the lowest possible condition number throughout the Mach-number range. The condition number is however not the only criterion to consider. Because the van Leer-Lee-Roe preconditioner is formulated in a stream-aligned coordinate system, which is ill-defined in stagnation regions, there are robustness issues with this preconditioner. 3.5 Implementation of Weiss-Smith preconditioner In FANSC, the Euler or Navier-Stokes equations are discretized using a cell-centered finite-volume method with artificial dissipation and solved by means of an explicit multi-stage Runge-Kutta time marching algorithm, following Jameson et al. [4],[5]. The implementation of the Weiss-Smith preconditioner in FANSC follows mostly the approach outlined in [8] Time-marching scheme The conservative variables U were chosen to update the solution in every stage of the Runge-Kutta timemarching scheme: k k ( ) 1 k Ui = Ui αk ti R ~ i U 1, with κ R ~ i ( U ) = Pi Ri ( U ), 1 Euler equations Turkel preconditioner, α = (Weiss-Smith) Turkel preconditioner, α = 1 Turkel preconditioner, α = β van Leer-Lee-Roe preconditioner M Figure 1. Comparison of characterisctic condition number for the D Euler equations with and without low Mach-number preconditioner. 3

4 where R i is the cell residual obtained from the flux balance in conservative variables and P i is the preconditioner for that cell. The preconditioner P is applied in conservative variables, after transforming from the entropy variables W, in which the Turkel preconditioner P T is defined: W P = PT W U Since the density is constant in the incompressible limit (except for thermal effects), the ( p,u,t r ) set of variables would in principle make a more appropriate choice for the update variable than the conservative variables. However, for our aerodynamic applications the interest is in low Mach-number flows and not in totally incompressible flows. Numerical evidence also shows that the variables used in the artificial dissipation have a much larger impact on the accuracy of the steady-state solution than the update variables [1]. Using low Mach-number preconditioning, the local time-step estimate t i for stability has to be based on the spectral radius (largest eigenvalue) of PA instead of A, i.e. the reduced acoustic wave speed instead of u + c. As suggested by Turkel [1], the Implicit Residual Smoothing and the Full Approximation Storage (FAS) multigrid algorithms are applied on the preconditioned residuals, i.e. the residual smoothing and coarse-grid restriction operations are done on PR, rather than R Artificial dissipation In order to ensure good solution accuracy a proper scaling of the dissipation flux terms at all flow conditions is needed. This is not the case in the incompressible limit if the dissipation is based on the original, i.e. non-preconditioned, equations. It can be shown that applying the Turkel low Mach-number preconditioner to the artificial dissipation terms of the central discretization scheme, whether they are scalar or matrix-valued, ensures the proper scaling as M [8]. The same applies to upwind schemes as well. For the Jameson-Schmidt-Turkel (JST) scalar dissipation scheme [4] for instance, the second and fourth-order dissipation flux are scaled with the spectral radius σ of the preconditioned system PA, instead of A: F F where ( ) σ ( PA) ( U + 1 U ) S 1 D 1 i + 1 = Pi + 1 ε i + 1 i i i + ( δ U δ U ) S 1 ( 4) ( ) ( 3) ( 3) D4 1 i + 1 = Pi + 1 ε i + 1 σ PA i + 1 i i + ( 3 δ ) U i = Ui + 1 U i + U i 1 and ( ε ) ν η 1, ( 4) ( ε max ν ε ) i + 1 = i + ( ) i + 1 =, 4 i + 1, with the ν and ν 4 the second and fourth order dissipation coefficients, and η i +1 represents the JST pressure sensor [4]. Including the inverse of the preconditioner in the dissipation flux maintains the telescopingproperty of the finite-volume scheme so that conservation is ensured [8]. Indeed, the above artificial dissipation fluxes are added to the advective and diffusive fluxes of the Navier-Stokes equations, forming in each cell i the un-preconditioned residual R i, before multiplying with the preconditioner P i to form the preconditioned residual R ~ i ( U ) = Pi Ri ( U ). Note that such conservative formulation is essential since we consider applications where shocks may be embedded in a mostly low Mach-number flow Boundary conditions As the low Mach-number preconditioner modifies the wave speeds of the time-dependent equations, the treatment of those boundary conditions that are based on characteristic relations (Riemann invariants) needs to be adapted accordingly for the preconditioned equations. In practice however, as suggested by Turkel et al. [8], a simplified set of non-characteristic relations can be used for the far-field boundary conditions instead of the characteristic relations of the preconditioned equations. At an inflow boundary, the freestream velocity and temperature are imposed, while the pressure is extrapolated from the interior domain. At an outflow boundary, the freestream pressure is imposed, while the velocity and temperature are extrapolated from the interior Definition of parameter β To prevent β from becoming zero and causing a singularity of the preconditioner at stagnation points (M = ), a cut-off is added, proportional to M ref [1]: ( k M,k ) β = max, 1 M ref where for external flow problems M ref can be taken equal to the free-stream Mach number. Finally a cutoff is added to switch-off preconditioning at supersonic flow conditions: β ( max( k M,k M ), 1) = min ref 1 Following computational experience and suggestions from the literature, the cut-off parameter k is typically set to.4 for inviscid cases and. for viscous cases. Generally, the value of k needs to be higher for tough cases, which means that the preconditioner will be constant, i.e. β = k flow field. M ref, in a larger portion of the 4

5 4. RESULTS To illustrate the implementation of the Turkel low Mach-number preconditioner in FANSC, the following sections discuss results ranging from inviscid flow over a wing to the turbulent separated flow over a powered turbofan engine in crosswind. Unless indicated differently, all solutions have been obtained using the JST scheme, with second-order and fourth-order dissipation coefficients set to ν = 1/ and ν 4 = 1/3, 5-stage Runge-Kutta time-stepping, augmented with a 3-level W-cycle FAS multigrid scheme and Implicit Residual Smoothing. 4.1 ONERA M6 wing Inviscid flow To verify the convergence of the preconditioned Euler equations in the limit of zero Mach-number, the 3D inviscid flow over the ONERA M6 wing at º incidence is computed at several free-stream Machnumbers approaching zero, starting from Mach.3. This computation uses an 8-block C-H type grid with a total of 1.1 million cells. The cut-off parameter in the preconditioner is set at k =.4. (a) Mach number Figure shows for the case of a free-stream Machnumber M =., the distribution of local Mach number and entropy on the wing s upper surface and compares the original non-preconditioned Euler flow solution with the preconditioned Euler solution. It can be seen that the low Mach-number preconditioner eliminates the large spurious entropy sources, which are indicative of numerical error, near the leading-edge and wing-tip. The iterative convergence histories shown in Figure 3 prove that with the addition of the low Mach-number preconditioner, the convergence rate of the solver is made independent of the free-stream Mach number. Note that the value at which the residual norm levels out (machine zero) goes up as the Mach number goes down. This is because the residual norm is defined as relative to the residual of the initial solution and because the latter is proportional to γm. Since the ONERA M6 wing is entirely symmetric and because the flow is subsonic and inviscid, the exact solution of the Euler equations at zero lift is known to have zero drag. Any non-zero drag in the numerical solution of the Euler equations is therefore spurious drag and a direct measure of discretization error. The drag coefficient shown in Figure 4 as a function of free-stream Mach-number therefore corresponds entirely to discretization error. It is seen that the low Mach-number preconditioner ensures that the numerical solution asymptotes to an incompressible limit with a spurious drag value of 1 counts, instead of diverging with increasingly large spurious drag in the non-preconditioned solution. Even at M =.3, the spurious drag is reduced by half, from counts to 11 counts. ONERA M6 - Euler α = o M =.3, JST M =.3, JST + P T M =.1, JST - M =.1, JST + P T M =., JST M =., JST + P T -4 log(res L ) (b) Entropy Figure. Inviscid flow over ONERA M6 wing at M =., α = º. Original Euler solution (left) vs. preconditioned Euler solution (right) iterations Figure 3. ONERA M6 wing, inviscid flow, α = º. Iterative convergence of density residual (L -norm) for M =.3,.1 and.. Dashed lines: nonpreconditioned. Solid lines: preconditioned. 5

6 ONERA M6 - Euler α = o 5 JST JST + P T 1.8 Horizontal velocity through vertical centreline C D x y Mach Figure 4. ONERA M6 wing, inviscid flow, α = º. Drag coefficient as a function of Mach number. Dashed line: non-preconditioned. Solid line: preconditioned. 4. Lid-Driven Cavity Laminar flow The D lid-driven cavity flow is a classical problem of fluid dynamics that has been investigated by many authors [18]. The vertical walls and the bottom wall of the square cavity are fixed. The upper wall moves horizontally, from left to right, at a velocity u w and forces the fluid motion in the cavity through shear stresses. Figure 5 shows the velocity contours and streamlines of the steady Navier-Stokes flow solution for the liddriven cavity problem at Re = 1, which was computed with the preconditioned method for a wall velocity u w corresponding to M =.1. The cut-off parameter in the preconditioner was set to k =.. Figure 5. D lid-driven cavity. Steady laminar flow solution at Re = 1, M =.1. Velocity contours and streamlines u / u lid Figure 6. D lid-driven cavity. Steady laminar flow solution at Re = 1. Solid line: present study with preconditioning at M =.1. Symbols: incompressible flow solution from [19]. This computation used a rectangular grid of points, with a cosine-function point distribution for grid refinement near the walls. The solution exhibits a large primary vortex and two smaller secondary vortices in the bottom corners of the cavity. To verify the accuracy of the present method, the M =.1 solution has been compared with highly accurate incompressible flow results from a spectral method with 16 Chebyshev polynomials [19]. Figure 6 shows the comparison of the horizontal velocity component u along the vertical centre line of the cavity. An excellent agreement with the reference numerical solution can be observed. 4.3 DLR-F6 Wing-Body Turbulent flow To verify the convergence of the preconditioned Navier-Stokes equations in the limit of zero Machnumber, the 3D turbulent flow over the DRL-F6 wingbody configuration at 5º angle of attack is computed at several Mach-numbers approaching zero, starting from Mach.3. The Reynolds number, based on the mean aerodynamic chord, is maintained at a constant value, Re = 3. million. The DRL-F6 wing-body configuration is a wellknown test case, which has been used by many authors, especially since the Second AIAA Drag Prediction Workshop (DPW); see for example []. Validation of FANSC solutions with wind tunnel data for transonic flow over the DLR-F6 wing-body configuration has been presented in [1] and []. 6

7 preconditioned method at M =.1, where the cut-off parameter was set at k =.. The effect of the low Mach-number preconditioner can be observed in the surface pressure and entropy distribution on the nose of the body. The addition of the low Mach-number preconditioner completely resolves the spurious errors seen in the wing leadingedge and the body nose areas of the nonpreconditioned solution. The improvement of the iterative convergence due to the preconditioner is shown in Figure 9. Similar to what was shown for the Euler flow over a wing, the low Mach-number preconditioner ensures that the numerical Navier-Stokes solution asymptotes to the incompressible limit, as indicated by the drag coefficient shown in Figure 1. Figure 7. Turbulent flow over DLR-F6 wing-body configuration at M =.1, α = 5º, Re = 3. 16; Preconditioned RANS solution; pressure contours. DLR-F6 WB - fully turbulent (S-A) o 6 α = 5, Re = 3x1, M = CL M =.1, JST M =.1, JST + P T 1 (a) Pressure contours iterations Figure 9. Turbulent flow over DLR-F6 wing-body configuration at M =.1, α = 5º, Re = Iterative convergence of lift coefficient. DLR-F6 wing-body, RANS, S-A model o 6 α = 5, Re = 3x1, M = JST JST + P T CD x (b) Entropy contours. Figure 8. Turbulent flow over DLR-F6 wing-body configuration at M =.1, α = 5º, Re = Non-preconditioned RANS solution (left) vs. preconditioned RANS solution (right). 6 4 The present study uses a 54-block O-O type grid with a total of 3.7 million cells, which has been generated using the MBGRID grid generation package [3]. The solution was run using the Spalart-Allmaras oneequation turbulence model. Figure 7 shows the surface pressure distribution computed with the Mach Figure 1. Turbulent flow over DLR-F6 wing-body configuration at α = 5º, Re = 3. 16, M. Convergence of the drag coefficient to the incompressible limit. 7

8 4.4 Turbofan engine at take-off in crosswind One of the key requirements for the aerodynamic design of the nacelle for a jet engine is to ensure acceptable levels of total pressure distortion in the flow entering the engine at the fan face. Too large flow distortion will result in loss of engine performance and may further lead to compressor stall, engine surge and increased compressor fatigue. A case of particular concern occurs when the engine is running at take-off power, i.e. high mass flow rate, while the aircraft is stationary on the ground under crosswind conditions. The co-existence of regions of transonic flow (and possibly shocks on the inside of the nacelle lip) with mostly low-speed flow of the crosswind necessitates the use of a compressible flow solver with low Mach-number preconditioning. As a preliminary study of this problem, we have computed the flow over a generic turbofan engine at high mass flow, in a uniform 35 kts crosswind at 8º angle of incidence from the right-hand side and at standard sea-level atmospheric conditions. The ground plane is omitted in this preliminary study. The engine inlet and outlet boundary conditions are set to simulate a reduced take-off power setting at standard sea-level conditions. The preconditioned RANS computation uses an O-O type grid with.15 million cells. The grid has been partitioned in 95 blocks for parallel performance [6]. Due to the severity of the flow gradients in this problem, a higher dissipation coefficient of ν 4 = 1/16 was necessary for numerical stability. Also the cut-off parameter of the preconditioner needed to be set to a higher value, k = 3., and a reduced time-step at CFL =. was necessary. Figure 11 shows the pressure distribution and skin-friction lines on the nacelle, computed with the Spalart-Allmaras one-equation turbulence model and the preconditioned scheme. Figure 1 shows the iterative convergence history of the integrated pressure force coefficient on the engine nacelle. The computation is started with the unpreconditioned method, which does not converge due to the low Mach-number issues. Upon activating the low Mach-number preconditioner after 4 iterations, the RANS computation is seen to converge quickly to a quasi-steady solution. The remaining unsteadiness is of small amplitude and likely related to the flow separation on the outside of the nacelle. While the present results illustrate the importance of the low Mach-number preconditioner for nacelle design problems, predicting the actual engine inlet flow distortion requires the introduction of a ground plane or aircraft body to the grid, respectively for wing-mounted or tail-mounted engines. Figure 11. Engine at take-off in 35 kts cross-wind: β = 8º, M =.53, Re/m = m -1. Pressure distribution and skin-friction lines on the nacelle; view from the leeward side. C Dp Engine at take-off in 35 kts crosswind β = 8 o, M =.59-1 JST JST + P T iterations Figure 1. Engine at take-off in 35 kts cross-wind: β = 8º, M =.53, Re/m = m -1. Iterative convergence history of pressure force coefficient. Furthermore, extensive validation and investigation of the turbulence modeling effects will be needed, as was demonstrated in a similar study by Airbus [4]. 5. CONCLUDING REMARKS This paper describes the implementation of the Weiss- Smith low Mach-number preconditioning method in Bombardier s 3D multi-block compressible Euler and Navier-Stokes code FANSC, using the formulation of Turkel et al. [8]. The basics of this preconditioning method have been briefly summarized and the various choices involved in the implementation for the JST artificial dissipation scheme have been discussed. The computational results presented for the Euler and Navier-Stokes equations demonstrate that with the Weiss-Smith preconditioner, the solution accuracy and 8

9 iterative convergence rate are improved at subsonic speeds and are maintained down to the incompressible limit, which confirms with known results from the literature. In addition to making the low Mach-number preconditioner available with other discretization schemes in FANSC than the JST scheme, especially the matrix dissipation scheme [5],[6], future work will be devoted to the implementation of the so-called preconditioning-squared method [7],[1], which combines the low Mach-number preconditioner presented here with the recently implemented block- Jacobi preconditioner [6]. For the simulation of problems with unsteady separated flow, the low Machnumber preconditioner will be implemented in a recently developed time-accurate version of FANSC using a dual time-stepping scheme. The improvements obtained for the steady solver are expected to translate over to the unsteady solver, as shown in [1]. ACKNOWLEDGEMENTS The work presented in this paper was partly funded by the Department of National Defence of Canada under the Defence Industrial Research (DIR) program. The authors like to acknowledge the contribution of their colleagues Josée Boudreau and Reza Sadri in the area of grid generation. REFERENCES [1] Turkel, E., Preconditioning techniques in computational fluid dynamics, Annual Rev. Fluid Mech., Vol 31, pp , [] Laurendeau, E., Zhu, Z., and Mokhtarian, F., Development of the FANSC Full Aircraft Navier-Stokes Code, Proceedings of the 46 th Annual Conference of the Canadian Aeronautics and Space Institute, Montreal, May -5, [3] Zhu, Z., Laurendeau, E., and Mokhtarian, F., Cell-centered and cell vertex algorithms for complex flow simulations, Proceedings of the 8 th Annual Conference of the CFD Society of Canada, Montreal,. [4] Jameson, A., Schmidt, W., and Turkel, E., Numerical Solutions of the Euler Equations by Finite Volume Methods using Runge-Kutta Time-Stepping Schemes, AIAA Paper , [5] Jameson, A., Multigrid Algorithms for Compressible Flow Calculations, Proceedings of the Second European Conference on Multigrid Methods, Cologne, October 1985, Lecture Notes in Mathematics, Vol. 18, pp , Springer-Verlag, [6] Sermeus, K., Laurendeau, E., and Parpia, F., Parallelization and performance optimization of Bombardier multiblock structured Navier-Stokes solver on IBM eserver Cluster 16, AIAA Paper 7-119, 7. [7] Laurendeau, E., Sermeus, K., Piperni, P., Leblond, D., and Kafyeke, F., CFD aerodynamic design and analysis at Bombardier Aerospace using massively parallel computing, Proceedings of the 15 th Annual Conference of the CFD Society of Canada, Toronto, 7. [8] Turkel, E., Radespiel, R., and Kroll, N., Assessment of preconditioning methods for multidimensional aerodynamics, Computers & Fluids, Vol. 6, No. 6, pp , [9] Turkel, E., Robust low speed preconditioning for viscous high lift flows, AIAA Paper - 96,. [1] Turkel, E., and Vatsa, V.N., Local preconditioners for steady and unsteady flow applications, in Mathematical Modelling and Numerical Analysis, MAN, Vol. 39, No. 3, pp , 5. [11] Choi, Y.-H., and Merkle, C.L., The application of preconditioning in viscous flows, J. Comp. Physics, Vol. 15, pp. 3 3, [1] van Leer, B., Lee, W.-T., and Roe, P., Characteristic time-stepping or local preconditioning of the Euler equations, AIAA Paper , [13] Weiss, J.M., and Smith, W.A., Preconditioning applied to variable and constant density flows, AIAA Journal, Vol. 33, No. 11, [14] Lee, D., van Leer, B., and Lynn, J.F., A local Navier-Stokes preconditioner for all Mach and cell Reynolds numbers, AIAA Paper 97-4, [15] Unrau, D., and Zingg, D.W., Viscous airfoil computations using local preconditioning, AIAA Paper 97-7, [16] Darmofal, D., and van Leer, B., Local preconditioning: manipulating mother nature to fool father time, Computing the Future II: Advances and Prospects in Computational Aerodynamics, M. Hafez and D. Caughey, Eds., John Wiley and Sons, [17] Vigneron, D., Deliége, G., and Essers, J.-A., Low Mach number local preconditioning for unsteady viscous finite volumes simulations on 3D unstructured grids, Proc. of ECCOMAS 6 Conference, 6. [18] Bruneau, C.-H., and Saad, M., The D liddriven cavity problem revisited, Computers & Fluids, Vol. 35, No. 3, pp , 6. 9

10 [19] Botella, O., and Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Computers & Fluids, Vol. 7, No. 4, pp , [] Brodersen, O., Rakowitz, M., Amant, S., Larrieu, P., Destarac, D., and Sutcliffe, M., Airbus, ONERA, and DLR results from the Second AIAA Drag Prediction Workshop, Journal of Aircraft, Vol. 4, No. 4, pp , 5. [1] Laurendeau, E., and Mokhtarian, F., Recent improvements in the Bombardier Full-Aircraft Navier-Stokes Code FANSC, Proceedings of the 5 th Annual Conference of the Canadian Aeronautics and Space Institute, April 3. [] Laurendeau, E., and Boudreau, J., Drag prediction methods using Bombardier Full- Aircraft Navier-Stokes Code FANSC, Proceedings of the 5 nd Annual Conference of the Caniadian Aeronautics and Space Institute, April 5. [3] Piperni, P., and Boudreau, J., The evolution of structured grid generation at Bombardier Aerospace, Proceedings of the 5 th Annual Conference of the Caniadian Aeronautics and Space Institute, April 3. [4] Tourette, L., Navier-Stokes simulations of airintakes in crosswind using local preconditioning, AIAA Paper -739,. [5] Swanson, R.C., and Turkel, E., On central difference and upwind schemes, J. Comp. Physics, Vol. 11, pp. 9 36, 199. [6] Laurendeau, E., and Sermeus, K., Implementation of matrix dissipation and Jacobi preconditioner within Bombardier s Full-Aircraft Navier-Stokes Code FANSC, Proceedings of the 54 th Annual Conference of the Canadian Aeronautics and Space Institute, Toronto, 7. [7] Turkel, E., Preconditioning-squared methods for multidimensional aerodynamics, AIAA Paper 97-5,