Block-spectral mapping for multi-scale solution

In this paper, an efficient multi-scale method is presented. Discrete block domains with high mesh resolution are used to resolve fine scale features and the number of the blocks to be solved is kept as small as being sufficient for constructing a block–block spatial spectrum. A pointwise spectrum for the blockwise variation is generated for each mesh point on the corresponding block boundaries. These block-to-block spectra provide the required formation for all the block interfaces, and hence enable the fine scale solutions in these solved blocks to be mapped to a large domain simultaneously. The structured blocking also provides a convenient domain division for parallel processing. The method has been implemented in a 3-D time-marching finite volume solver for the Reynolds-averaged compressible Navier–Stokes equations. The double Fourier series is adopted to construct the pointwise spatial spectrum, which is well suited for wall bounded problems where a domain layer with fine scale resolution is required. Several test cases of mass and heat transfer problems have been examined and the results show consistently that the accurate solutions for fine scale domains can be obtained very efficiently with a reduction in degrees of freedom by two orders of magnitude, illustrating the potential of solving complex problems which might currently be intractable.