Multiscale block spectral solution for unsteady flows

Many problems of interest are characterized by 2 distinctive and disparate scales and a huge multiplicity of similar small‐scale elements. The corresponding scale‐dependent solvability manifests itself in the high gradient flow around each element needing a fine mesh locally and the similar flow patterns among all elements globally. In a block spectral approach making use of the scale‐dependent solvability, the global domain is decomposed into a large number of similar small blocks. The mesh‐pointwise block spectra will establish the block‐block variation, for which only a small set of blocks need to be solved with a fine mesh resolution. The solution can then be very efficiently obtained by coupling the local fine mesh solution and the global coarse mesh solution through a block spectral mapping. Previously, the block spectral method has only been developed for steady flows. The present work extends the methodology to unsteady flows of short temporal and spatial scales (eg, those due to self‐excited unsteady vortices and turbulence disturbances). A source term–based approach is adopted to facilitate a two‐way coupling in terms of time‐averaged flow solutions. The global coarse base mesh solution provides an appropriate environment and boundary condition to the local fine mesh blocks, while the local fine mesh solution provides the source terms (propagated through the block spectral mapping) to the global coarse mesh domain. The computational method will be presented with several numerical examples and sensitivity studies. The results consistently demonstrate the validity and potential of the proposed approach.