Multigrid Methods for Compressible Navier-Stokes Equations (original) (raw)
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Multigrid methods for compressible Navier-Stokes equations in low-speed flows
Journal of Computational and Applied Mathematics, 1997
The multigrid performance of pointwise, linewise and blockwise Gauss-Seidel relaxations for compressible laminar and turbulent Navier-Stokes equations is illustrated on two low-speed test problems: a flat plate and a backward facing step. The line method is an Alternating Symmetric Line Gauss-Seidel relaxation. In the block methods, the grid is subdivided into geometric blocks of n x n points with one point overlap. With in the blocks, the solution is obtained by a direct method or with an alternating modified incomplete lower-upper decomposition. The analysis is focused on flows typical for boundary layers, stagnation and recirculation regions. These are characterized by very small Mach numbers, high Reynolds numbers and high mesh aspect ratios.
12th Computational Fluid Dynamics Conference, 1995
A relaxation method for the steady turbulent compressible Navier-Stokes equations is developed. The flow equations are fully coupled to the turbulence equations. The principles of the method are illustrated for three different k-E models. The relaxation method can be used in multigrid form. Multigrid results are given for one turbulence model on a flat plate test case. Two variants of the multigrid formulation are considered. In the first, the turbulence equations are not used on the coarse grids. In the second, the turbulence equations are used on all grids. In this last variant, without damping of the coarse grid corrections for the turbulence quantities, negative values for k and E are always encountered leading to a breakdown of the calculation. Therefore, damping of the coarse grid corrections for k and E is introduced.
A multigrid method for the Navier Stokes equations
24th Aerospace Sciences Meeting, 1986
A multigrid method for solving the compressible Navier Stokes equations is presented. The dimensionless conservation equations are discretized by a finite volume technique and time integration is performed by using a mltistage explicit algorithm. Convergence to a steady state. is enhanced by local time stepping, implicit smoothing of the residuals and the use of m l t i p l e grids. The raethod has been implemented in two different ways: firstly a cell centered and secondly a corner point formulation (i. e. the unknown variables are defined either at the center of a computational cell or at its vertices). laminar and turbulent two dimensional flows over airfoils.
2008
Implicit time-integration techniques are envisioned to be the methods of choice for direct numerical simulations (DNS) for flows at high Reynolds numbers. Therefore, the computational efficiency of implicit flow solvers becomes critically important. The textbook multigrid efficiency (TME), which is the optimal efficiency of a multigrid method, is achieved if accurate solutions of the governing equations are obtained with the total computational work that is a small (less than 10) multiple of the operation count in one residual evaluation. In this paper, we present a TME solver for unsteady subsonic compressible Navier–Stokes equations in three dimensions discretized with an implicit, second-order accurate in both space and time, unconditionally stable, and non-conservative scheme. A semi-Lagrangian approach is used to discretize the time-dependent convection part of the equations; viscous terms and the pressure gradient are discretized on a staggered grid. The TME solver for the imp...
Journal of Computational Physics, 2008
Implicit time-integration techniques are envisioned to be the methods of choice for direct numerical simulations (DNS) for flows at high Reynolds numbers. Therefore, the computational efficiency of implicit flow solvers becomes critically important. The textbook multigrid efficiency (TME), which is the optimal efficiency of a multigrid method, is achieved if accurate solutions of the governing equations are obtained with the total computational work that is a small (less than 10) multiple of the operation count in one residual evaluation. In this paper, we present a TME solver for unsteady subsonic compressible Navier-Stokes equations in three dimensions discretized with an implicit, second-order accurate in both space and time, unconditionally stable, and non-conservative scheme. A semi-Lagrangian approach is used to discretize the time-dependent convection part of the equations; viscous terms and the pressure gradient are discretized on a staggered grid. The TME solver for the implicit equations is applied at each time level. The computational efficiency of the solver is designed to be independent of the Reynolds number. Our tests show that the proposed solver maintains its optimal efficiency at high Reynolds numbers and for large time steps.
2004
A fast multigrid solver for the steady, incompressible Navier-Stokes equations is presented. The multigrid solver is based upon a factorizable discrete scheme for the velocity-pressure form of the Navier-Stokes equations. This scheme correctly distinguishes between the advection-diffusion and elliptic parts of the operator, allowing efficient smoothers to be constructed. To evaluate the multigrid algorithm, solutions are computed for flow over a flat plate, parabola, and a Kármán-Trefftz airfoil. Both nonlifting and lifting airfoil flows are considered, with a Reynolds number range of 200 to 800. Convergence and accuracy of the algorithm are discussed. Using Gauss-Seidel line relaxation in alternating directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of Runge-Kutta and implicit upwind based multigrid methods.
International Journal for Numerical Methods in Fluids, 2004
The paper's leitmotiv is condensed in one word: robustness. This is a real hindrance for the successful implementation of any multigrid scheme for solving the Navier-Stokes set of equations. In this paper, many hints are given to improve this issue. Instead of looking for the best possible speed-up rate for a particular set of problems, at a given regime and in a given condition, the authors propose some ideas pursuing reasonable speed-up rates in any situation. In (Vázquez et al., 2001), a multigrid method for solving the incompressible turbulent RANS equations is introduced, with particular care in the robustness and flexibility of the solution scheme. Here, these concepts are further developed and extended to compressible laminar and turbulent flows. This goal is achieved by introducing a non-linear multigrid scheme for compressible laminar (NS equations) and turbulent flow (RANS equations), taking benefit of a convenient master-slave implementation strategy, originally proposed in
Development of pressure-based composite multigrid methods for complex fluid flows
Progress in Aerospace Sciences, 1996
Progress in the development of a multiblock, multigrid algorithm, and a critical assessment of the k-e two-equation turbulent model for solving fluid flows in complex geometries is presented. The basic methodology employed is a unified pressure-based method for both incompressible and compressible flows, along with a TVD-based controlled variation scheme (CVS), which uses a second-order flux estimation bounded by flux limiters. Performance of the CVS is assessed in terms of its accuracy and convergence properties for laminar and turbulent recirculating flows as well as compressible flows containing shocks. Several other conventional schemes are also employed, including the first-order upwind, central difference, hybrid, second-order upwind and QUICK schemes. For better control over grid quality and to obtain accurate solutions for complex flow domains, a multiblock procedure is desirable and often a must. Here, a composite grid algorithm utilizing patched (abutting) grids is discussed and a conservative flux treatment for interfaces between blocks is presented. A full approximation storage-full multigrid (FAS-FMG) algorithm that is incorporated in the flow solver for increasing the efficiency of the computation is also described. For T:urbulent flows, implementation of the k-e two-equation model and in particular the wall functions at solid boundaries is also detailed. In addition, different modifications to the basic k-e model, which take the non-equilibrium between the production and dissipation of k and e and rotational effects into account, have also been assessed. Selected test cases are used to demonstrate the robustness of the solver in terms of the convection schemes, the multiblock interface treatment, the multigrid speedup and the turbulence models.