A multigrid semi-implicit line method for Navier-Stokes equations

J. Vierendeels
Department of Mechanical and Thermal Engineering
Universiteit Gent, Belgium

K. Riemslagh
Department of Mechanical and Thermal Engineering
Universiteit Gent, Belgium

E. Dick
Department of Mechanical and Thermal Engineering
Universiteit Gent, Belgium

Abstract

Preconditioning of the incompressible and compressible Navier-Stokes equations is used by many authors in order to accelerate convergence, especially for low Mach number flow. However this technique does not always provide good results on high aspect ratio grids, because of the stiffness due to the numerically-anisotropic behaviour of the diffusive and acoustic terms.

The stiffness on high aspect ratio grids is due to two reasons: first, the discretized diffusion terms are numerically very anisotropic on such grids and second, the ratio of the cfl numbers in both directions is proportional to the aspect ratio. Assume a control volume with Delta y much smaller than Delta x. For incompressible flow, using an artificial compressibility approach, the timestep Delta tau can be computed as

Delta taue = ( (u+cx)/(Delta x) + (v+cy)/(Delta y) )-1
or
Delta taui = ( (u+cx)/(Delta x) + (v)/(Delta y) )-1
for explicit and line-implicit methods respectively, with cx = (u2 + beta2))½ and cy = ((v2 + beta2)½ and beta the pseudo-acoustic speed. Let's define the convective and acoustic cfl numbers in x- and y-direction as cflxc = (u Delta tau)/ Delta x, cflyc = (v Delta tau)/ Delta y, cflxa = (u+cx) Delta tau/ Delta x, cflya = (v+cy) Delta tau/ Delta y. If the flow is aligned to the x-direction, and high aspect ratios are used, and with Delta tau= Delta taue and cflya=1, cflxc becomes very small, which means that the propagation of the convective wave is degraded in the x-direction. Therefore convergence will be deteriorated. In the incompressible case, the acoustic eigenvalues are formed by the divergence of the velocity field in the continuity equation and by the pressure in the momentum equation. If these components of the inviscid system are discretized impliciteyearly in the y-direction, the determination of the timestep is changed into Delta tau= Delta taui. If the flow is aligned to the x-direction, the acoustic cflxa is equal to 1, and the convective cflxc will be of the same magnitude if beta is chosen appropriately. Then convergence is not deteriorated. The acoustic cflya is now bigger than 1 but this is allowed since the acoustic terms are treated impliciteyearly. The numerical anisotropy of the viscous system on the same high aspect ratio grid can also be accounted for, if also the viscous terms are discretized line impliciteyearly in the y-direction. The viscous terms in the x-direction are discretized point-implicit. The discretization of the non-linear convective part of the inviscid system is done expliciteyearly. Velocity upwinding is used and higher order is obtained with the MUSCLE aproach. Due to the preconditioning, this part is stepped with a cfl number in the order of unity. A similar approach is used for low Mach number compressible flows.

The preconditioned semi-implicit line method is used in a multistage scheme because of the stability restrictions on the explicit part. Multigrid is then used as acceleration technique. The convergence is very fast, independent of the aspect ratio and Mach number.