Progress with Multigrid Schemes for Hypersonic Flow Problems (original) (raw)

Multigrid for hypersonic viscous two- and three-dimensional flows

10th Computational Fluid Dynamics Conference, 1991

We consider the use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows. The time-dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that remove the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock-capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are giver for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional viscous flow over a blunt biconic.

Extension of multigrid methodology to supersonic/hypersonic 3-D viscous flows

International Journal for Numerical Methods in Fluids, 1993

A multigrid acceleration technique developed for solving the three-dimensional Navier-Stokes equations for subsonicjtransonic flows has been extended to supersonic/hypersonic flows. An explicit multistage Runge-Kutta type of time-stepping scheme is used as the basic algorithm in conjunction with the multigrid scheme. Solutions have been obtained for a blunt conical frustum at Mach 6 to demonstrate the applicability of the multigrid scheme to high-speed flows. Computations have also been performed for a generic High-speed Civil Transport configuration designed to cruise at Mach 3. These solutions demonstrate both the efficiency and accuracy of the present scheme for computing high-speed viscous flows over configurations of practical interest.

An effective multigrid method for high-speed flows

Communications in Applied Numerical Methods, 1992

We consider the use of a multigrid method with central differencing to solve the Navier-Stokes equations for high-speed flows. The time-dependent form of the equations is integrated with a Runge-Kutta scheme accelerated by local time stepping and variable coefficient implicit residual smoothing. Of particular importance are the details of the numerical dissipation formulation, especially the switch between the second and fourth difference terms. Solutions are given for two-dimensional lamipar flow over a circular cylinder and a 15 degree compression ramp.

Analysis of Robust Multigrid Methods for Steady Viscous Low Mach Number Flows

Journal of Computational Physics, 1997

up with a well-conditioned system matrix. This manipulation changes the time behaviour of the system such that A theoretical and numerical analysis of different implicit methods in multigrid form is carried out in order to study their behaviour at certain invariants are transported at nearly identical velocivarious flow conditions. The Fourier analysis for scalar convectionties. Nevertheless, these analytic preconditioners do not diffusion equations is extended to the coupled set of laminar Navierguarantee a well-conditioned system for high aspect ratios Stokes equations. For first-and second-order discretized Navier-[2, 3] or for viscosity dominated flows [1].

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.

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.

A multigrid semi-implicit line-method for viscous incompressible flows

1998

Discretization of viscous incompressible and viscous low-Mach-number flows often leads to a system of equations, which is very difficult to solve. There are two reasons. First, the use of high aspect ratio grids results in a very numerically anisotropic behaviour of the diffusive and acoustic terms and second, in low-Machnumber flow, the ratio of the convective and acoustic eigenvalues of the inviscid system becomes very high. We implemented an AUSM based discretization method, using an explicit third-order discretization for the convective part and a line-implicit central discretization for the acoustic part and for the diffusive part. The lines are chosen in the direction of the grid points with shortest connection. The preconditioned semi-implicit line method is used in multistage form because of the explicit third-order discretization of the convective part. Multigrid is used as an acceleration technique. It is shown that the convergence is very good, independent of grid aspect ratio and Mach number.

On multigrid methods for the solution of least-squares finite element models for viscous flows

International Journal of Computational Fluid Dynamics, 2012

There is a vast literature on least-squares finite element models (LSFEM) applied to fluid dynamics problems. The hp version of the least-squares models is computationally expensive, which necessitates the usage of elegant methods for solving resulting systems of equations. Amongst some of the schemes used for solving large systems of equations is the element-by-element (EBE) solution technique, which has found widespread use in least-squares applications. However, the use of EBE techniques with Jacobi preconditioning leads to very little performance gains as compared to solving a non-preconditioned system. Because of such considerations, the hp version LSFEM solutions are computationally intensive. In this study, we propose to solve the LSFEM systems using the multigrid method, which offers superior convergence rates compared to the EBE-JCG. We demonstrate the superior convergence of the Multigrid solver compared to Jacobi preconditioning for the wall-driven cavity and backward facing step problems using the full Navier-Stokes equations. Load balancing issues encountered with multigrid solvers in a parallel environment are resolved elegantly with an element-by-element solution of the coarse grid problem with Jacobi preconditioning.

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.