Multigrid Method coupled with Large Eddy Simulation

Sandra Nägele

Interdisciplinary Centre for Scientific Computing
University of Heidelberg
Im Neuenheimer Feld 368
69120 Heidelberg


The use of multigrid methods in conjunction with a Large Eddy Simulation (LES) is very proximate since both are based on multiple scales. LES is a turbulence model which resolves large turbulent scales and models the small ones. The scale separation is performed by applying a spacial convolution operator (filter operator) to the incompressible Navier-Stokes equations. As filter operator a top hat filter is applied with a grid dependent support size. Hence the application of the filter results in a locally varying average in space. The LES model itself also depends on the grid size since the filtering process removes all subgrid scales. Some special dynamic LES models have been developed by various researchers which apply two filters with different support size at each point of the domain to the gouverning equation system. By comparison of the two different large scale resolutions the model term can be specified locally. This is similar to the multigrid cycle where the defect is restricted to the coarser grid and higher frequencies are removed. Another property of dynamic models is their ability to adjust themselves to local flow structures. This adaptivity is very useful since in some regions of the domain the flow can be laminar and a turbulence model is not necessary at all. Hence dynamic models were used in the simulations.

The simulations were carried out on the software platform UG which is based on unstructured grids. Thus the filter width varies in the solution domain. Using unstructured grids the resolution of the turbulent scales can be increased locally to decrease the modelling effort. In the neighbourhood of walls for example it is possible to use a smaller grid size. By this grid adaptation the local flow structures can be better resolved and less modelling with less modelling error is necessary.
A Krylov subspace method with linear multigrid as preconditioner is used to solve the linearized system. Within the multigrid cycle it is important to separate modelling and solving in the sense that only on the finest grid the modeled part of the equations is determined as described above. This is necessary for an appropriate modelling of the subgrid turbulent scales. Afterwards the model term is restricted to the coarse grids and a standard linear multigrid cycle can be used. This solution strategy was applied to different flow problems which will be presented.