Comparison of Implicit Multigrid Schemes for Three-Dimensional Incompressible Flows (original) (raw)
Related papers
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.
Multigrid Methods for Compressible Navier-Stokes Equations
1996
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.
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.
Efficient Multigrid Techniques for the Solution of Fluid Dynamics Problems
Ph.D. Dissertation, 2013
The multigrid technique (MG) is one of the most efficient methods for solving a large class of problems very efficiently. One of these multigrid techniques is the algebraic multigrid (AMG) approach which is developed to solve matrix equations using the principles of usual multigrid methods. In this work, various algebraic multigrid methods are proposed to solve different problems including: general linear elliptic partial differential equations (PDEs), as anisotropic Poisson equation, problems with steep boundary layers, as convectional dominant convection-diffusion equations, and nonlinear system of equations as Navier-stokes equations. In addition, a new technique is introduced for solving convection-diffusion equation by predicting a modified diffusion coefficient (MDC) such that the discretization process applies on the modified equation rather than the original one. For a class of one-dimensional convection-diffusion equation, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. Extending the same technique to obtain analytic MDC for other classes of convection-diffusion equations is not always straight forward especially for higher dimensions. However, we have extended the derived analytic formula of MDC (of the studied class) to general convection-diffusion problems. The analytic formula is computed locally within each element according to the mesh size and the values of the associated coefficients in each direction. The numerical results for two-dimensional, variable coefficients, convection-dominated problems show that although the discrete solution does not coincide with the exact one, it provides stable and accurate solution even on coarse grids. As a result, multigrid-based solvers benefit from these accurate coarse grid solutions and retained its efficiency when applied for convection–diffusion equations. Many numerical results are presented to investigate the convergence of classical algebraic and geometric multigrid solvers as well as Krylov-subspace methods preconditioned by multigrid. Also, in this thesis, we were concerned with the channel flow, which is an interesting problem in fluid dynamics. This type of flow is found in many real-life applications such as irrigation systems, pharmacological and chemical operations, oil- v refinery industries, etc. In the present work, the channel flow with one and two obstacles are considered. The methodology is based on the numerical solution of the Navier-Stokes equations by using a suitable computational domain with appropriate grid and correct boundary conditions. Large-eddy simulation (LES) was used to handle the turbulent flow with Smagorinsky modeling. Finite- element method (FEM) was used for the discretization of the governing equations. Adaptive time stepping is used and the resulting linear algebraic systems are solved by different methods including preconditioned minimum residual method, geometric and algebraic multigrid methods. The investigation was carried out for a range of Reynolds number (Re) from 1 to 300 with a fixed blockage ratio β = 0.25 and an artificial source of turbulence is introduced in the inflow velocity profile to ensure the turbulent nature of the flow. The finite element method is used in the present work to discretize many CFD problems and we have developed algebraic multigrid (AMG) approaches for anisotropic elliptic equation, problems with steep boundary layers, as convectional dominant convection-diffusion equations, and nonlinear system of equations as Navier-stokes equations. The conclusions which are obtained in the present work can be stated as: (i) AMG can be used for many kinds of problems where the application of standard multigrid methods is difficult or impossible. (ii) Implementation of the proposed MDC technique produces the exact nodal solutions for the 1-D singularly-perturbed convection diffusion problems even on coarse grids with uniform or non-uniform mesh sizes. (iii) Numerical results show that extension of MDC to 2-D eliminates the oscillations and produces more accurate solutions compared with other existing methods. (iv) As a result, multigrid-based solvers retain its efficient convergence rates for singularly-perturbed convection diffusion problems. (v) Excellent convergence behavior is obtained for numerical solution of Navier-Stokes system for different values of Re in two cases, 1- and 2- obstacles, when we used the proposed AMG algorithm as a solver or a preconditioner of GMRES.
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...
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.
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.
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.
Multigrid Applied to a Fully Implicit Fem Solver for Turbulent Incompressible Flows
2001
A methodology for convergence speed-up of a fully implicit solver for the Ran- dom Averaged Navier-Stokes (RANS) equations for incompressible flows using multigrid (MG) techniques is here presented. The RANS set, comprising the mean flow Navier- Stokes equations and a 2-equation k-" turbulence model, is discretized in space by applying the finite element method onto a hierarchy of meshes of dierent element sizes. To solve the system in the finest discretization, a non-linear multigrid scheme is applied to the hi- erarchy. A second objective of this work is to make more robust multigrids, trying to keep good speed-up rates. Considering this paper as a first part of a larger work, we leave aside momentarily the quest for the best possible speed-up rates, to develop some ideas that can make our multigrid more reliable, less dependent to the kind of problem we deal with or to the way the grid hierarchies are constructed: no particular grid-coarsening strategy is studied here. Among...
International Journal for Numerical Methods in Fluids - INT J NUMER METHOD FLUID, 1998
The application of standard multigrid methods for the solution of the Navier-Stokes equations in complicated domains causes problems in two ways. First, coarsening is not possible to full extent since the geometry must be resolved by the coarsest grid used. Second, for semi-implicit time-stepping schemes, robustness of the convergence rates is usually not obtained for convection-diffusion problems, especially for higher Reynolds numbers. We show that both problems can be overcome by the use of algebraic multigrid (AMG), which we apply for the solution of the pressure and momentum equations in explicit and semi-implicit time-stepping schemes. We consider the convergence rates of AMG for several model problems and demonstrate the robustiness of the proposed scheme.