Keywords: computational fluid dynamics, adaptive grid refinement, finite volume, block-structured grids, multigrid.

Computational fluid dynamics is becoming increasingly important in computational engineering. A major problem when dealing with complex flows is the occurence of largely different scales, e.g. in case of turbulent flows. When generating the computational grid, an orientation on the smallest scales leads to a very high number of grid cells and thus to a high memory requirement and significant CPU time.

This paper presents an algorithm which adapts the grid spacing locally to the needs in different parts of the computational domain, thus saving a large number of grid cells while maintaining the desired accuracy. Since the first research in this area, done by Brandt with his multi-level adaptive technique [1] and by Berger and Collela with their AMR method [2], grid refinement methods have been advanced and applied to different flow configurations.

The presented local adaptive grid refinement algorithm is implemented in a three-dimensional finite volume flow solver using block-structured grids and combined with a multigrid method. Unlike a block-wise grid refinement, which does not show enough flexibility and enhancement in efficiency in the case of three-dimensional flows [3], the refinement in this approach is performed on the basis of single grid cells. After the calculation of an initial solution on the original grid, the grid cells are evaluated using an error estimator. If marked for refinement, the cells are clustered to new, refined blocks, to keep the blockstructure of the grid.

The new blocks are treated detached from the underlying original domain. To obtain the information of the original coarse grids, i.e. velocity and pressure values, the coarse grid values are interpolated to the refined grids. On coarse/fine block boundaries the interpolated values are used as boundary conditions, whereas on fine/fine block boundaries communication is performed via copying the according values into a layer of additional grid cells on the neighbouring blocks.

To retain the mass conservation of the newly generated blocks, which may be disturbed by numerical errors, a flux correction scheme is applied. The algorithm is embedded into a multigrid method. Multigrid algorithms are well developed and have proven to be a powerful tool to accelerate flow computations [4]. In combining those two highly efficient strategies, a full-multigrid-like scheme is created.

For verifying and illustrating the functionality of the presented algorithm, the paper shows two test cases of different complexity. Further studies will investigate the efficiency of the approach involving test cases of varying complexity.

1

A. Brandt, "Multi-level adaptive techniques", IBM I.J. Watson Research Center, Yorktown Heigths, New York, IBM Research Report RC6026, 1976.

2

M.J. Berger, P. Collela, "Local adaptive mesh refinement for shock hydrodynamids", Journal of Computational Physics 82, 67-84, 1989. doi:10.1016/0021-9991(89)90035-1

3

T. Lehnhäuser, "Eine effiziente numerische Methode zur Gestaltoptimierung von Ströumgsgebieten", Dissertation, TU Darmstadt, 2003.

4

O. Iliev, D. Stoyanov, "Multigrid - adaptive local refinement solver for incompressible flows", Berichte des Fraunhofer ITWM, 54, 2003.

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