APPLICATION OF MULTIGRID TO TURBULENT COMPRESSIBLE FLOWS

           Jan Broeze, Bernard Geurts, Hans Kuerten, Martin Streng 

           Department of Applied Mathematics, University of Twente,
               P.O. Box 217, 7500 AE Enschede, The Netherlands.

                       email: j.broeze@math.utwente.nl


Algorithms for accurate and efficient numerical simulation of compressible
turbulent flows are a subject of intensive research.

Time-accurate direct numerical simulations of the Navier-Stokes equations
require a sufficiently fine grid to accurately capture the wide range of
length-scales encountered in turbulent flows.  Furthermore, a large number of
time steps is required to simulate the transition from laminar to fully
developed turbulent flow.

Numerical stability requirements impose serious restrictions on the time step,
if explicit time integration methods are used.  Therefore, we have decided to
apply an implicit second order accurate time integration method.  The system
of equations resulting from the implicit discretization is solved by means of
a pseudo-time stepping method (a five-stage Runge-Kutta method with frozen
dissipation) and accelerated by local pseudo-time stepping and a multigrid
technique.

We observed that, compared to an explicit time integration method, a speed-up
by a factor of 5 can be obtained for a wall-bounded flow.  The convergence of
the relaxation method depends strongly on the prolongation of the correction.
The high frequency components in the error which are created by the
prolongation process are very slowly damped, since no artificial dissipation
is added to the Navier-Stokes equations.  We consider the application of other
relaxation methods to improve this behaviour.