Extension of multigrid methodology to supersonic/hypersonic 3-D viscous flows (original) (raw)
Related papers
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.
Progress with multigrid schemes for hypersonic flow problems
Journal of Computational Physics, 1995
JOURNAL OF COMPUTATIONAL PHYSICS 1X6, 103-122 (1995) Progress with Multigrid Schemes for Hypersonic Flow Problems R. Radespiel DLR, Braunschweig, Germany AND R, C. SWANSON NASA Langley Research Center, Hampton, Virginia 23681 Received March 2, Î992; ...
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.
International Journal for Numerical Methods in Fluids, 2005
In a companion paper, the authors used a hierarchical formulation based on a Helmholtz velocity decomposition to simulate transonic ows over airfoils. The potential ow formulation is augmented with entropy and vorticity corrections and the numerical results are compared to standard Euler and Navier-Stokes calculations. For many aerodynamic applications, the corrections are limited to relatively small regions; the ow in the near and far ÿelds is irrotational and isentropic. The entropy and vorticity corrections are governed by convection=di usion equations while the non-homogeneous potential equation is of mixed type; elliptic in the subsonic domain and hyperbolic in the supersonic one. The forcing function represents the necessary correction for mass conservation. Upwind schemes are used for the convection terms of the scalar equations of the corrections and for the potential equation in the supersonic region. The formulation can be viewed as an implementation of a viscous=inviscid interaction procedure which is equivalent to Navier-Stokes equations in the inner ÿeld.
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...
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.
On the development of an agglomeration multigrid solver for turbulent flows
Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2003
The paper describes the implementation details and validation results for an agglomeration multigrid procedure developed in the context of hybrid, unstructured grid solutions of aerodynamic flows. The governing equations are discretized using an unstructured grid finite volume method, which is capable of handling hybrid unstructured grids. A centered scheme as well as a second order version of Liou's AUSM+ upwind scheme are used for the spatial discretization. The time march uses an explicit 5-stage Runge-Kutta time-stepping scheme. Convergence acceleration to steady state is achieved through the implementation of an agglomeration multigrid procedure, which retains all the flexibility previously available in the unstructured grid code. The calculation capability created is validated considering 2-D laminar and turbulent viscous flows over a flat plate. Studies of the various parameters affecting the multigrid acceleration performance are undertaken with the objective of determining optimal numerical parameter combinations.
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.
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.