A High-Order Spectral Difference Method for the Navier-Stokes Equations on Unstructured Moving Deformable Grids

In this paper the high-order accurate spectral difference method for the Navier-Stokes equations is applied to moving boundary problems. Boundary movements are achieved, firstly, through rigid displacement of the entire flow domain. In order to account for the dynamic rigid mesh motion, the Navier-Stokes equations are modified through an unsteady coordinate transformation. Airfoils in pitching and plunging motions are studied. In both cases, computation results are compared with existing experimental data, and favorable results have been obtained. Secondly, spectral difference method is extended to include capability for handling dynamic deforming grids. The physical boundary movement is achieved through a time dependent unsteady transformation that allows part of the flow domain to be rigidly displacing, part of it fixed, and the rest deforming smoothly in between. The time dependent transformation preserves spectral difference method’s high order accuracy by solving the governing equations in a steady reference domain where the same shape functions are used, and introducing the unsteady perturbation in the physical space only through the changes in the transformation metrics and Jacobian. The blended deforming mesh allows the far field boundary or some desirable portions of the flow domain to be unaltered. These together make the overall solver accurate, flexible, and simple to implement. The order of accuracy of the spectral difference method in highly distorted mesh has been demonstrated through simulation of euler vortex problem. Simulations for flow over a plunging cylinder with rigid displacing and dynamic deforming meshes have yielded nearly identical results.