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 tau*_{e} = ( (u+c_{x})/(Delta x) +
(v+c_{y})/(Delta y) )^{-1}
or
*
Delta tau*_{i} = ( (u+c_{x})/(Delta x) +
(v)/(Delta y) )^{-1}
for explicit and line-implicit methods respectively,
with *c*_{x} = (u^{2} + beta^{2}))^{½} and
*c*_{y} = ((v^{2} + beta^{2})^{½}
and *beta* the pseudo-acoustic speed.
Let's define the convective and acoustic *cfl* numbers in x- and y-direction as
*cfl*_{xc} = (u Delta tau)/ Delta x,
*cfl*_{yc} = (v Delta tau)/ Delta y,
*cfl*_{xa} = (u+c_{x}) Delta tau/ Delta x,
*cfl*_{ya} = (v+c_{y}) Delta tau/ Delta y.
If the flow is aligned to the x-direction, and high aspect ratios are used,
and with *Delta tau= Delta tau*_{e} and
*cfl*_{ya}=1, *cfl*_{xc}
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 tau*_{i}. If the flow is aligned to the x-direction,
the acoustic *cfl*_{xa} is equal to 1,
and the convective *cfl*_{xc} will be of the same magnitude
if *beta* is chosen appropriately. Then convergence is not deteriorated.
The acoustic *cfl*_{ya} 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.