A fast pressure-correction method for incompressible flows over curved surfaces

In order to simulate turbulent flows over curved surfaces, the incompressible Navier-Stokes (NS) equations can be transformed to a more general formulation that allows them to be discretized in curved domains. This transformation leads to cross-derivatives in the expressions for Laplacian, advection, diffusion, and gradient operators. Further, the variable-coefficient Poisson equation for pressure is usually solved using an iterative solver at every timestep that leads to high computational costs. Thus, we have developed a new method to solve the incompressible NS equations in the orthogonal formulation. This has allowed us to develop a fast FFT-based Poisson solver which is at least forty-four times faster than the state-of-the-art multigrid-based iterative method. Provided the computational mesh satisfies the property of orthogonality, our numerical method can simulate flows over curved surfaces: surfaces of revolution (e.g., axisymmetric ramps) and surfaces of linear translation (e.g., curved ramps, bumps) with second-order accuracy. We present the results from convergence, verification and validation studies and the DNS results of flow over a smooth bump.