Spectrally-accurate immersed boundary conditions method for three-dimensional flows

A three-dimensional, spectrally accurate algorithm based on the immersed boundary conditions (IBC) concept has been developed for the analysis of flows in channels bounded by rough boundaries. The algorithm is based on the velocity–vorticity formulation and uses a fixed computational domain with the flow domain immersed in its interior. The geometry of the boundaries is expressed in terms of double Fourier expansions and boundary conditions enter the algorithm in the form of constraints. The spatial discretization uses Fourier expansions in the stream-wise and span-wise directions and Chebyshev expansions in the wall-normal direction. The algorithm can use either the fixed-flow-rate constraint or the fixed-pressure-gradient constraint; a direct implementation of the former constraint is described. An efficient solver which takes advantage of the structure of the coefficient matrix has been developed. It is demonstrated that the applicability of the algorithm can be extended to more extreme geometries using the over-determined formulation. Various tests confirm the spectral accuracy of the algorithm.