Research on Numerical Algorithms for the Three Dimensional Navier-Stokes Equations. I. Accuracy, Convergence & Efficiency

摘要

The objective of this research is to develop a highly accurate and efficient numerical solution algorithm for the non-linear three- dimensional Navier-Stokes equations for aerodynamics applications. A candidate algorithm has been derived employing finite element interpolation theory, the non-linear extension of quasi-variational principles, and the concept of tensor product bases. The resultant solution statement is rendered soluable using an implicit integration algorithm, and the replacement of the standard Jacobian of a Newton iteration algorithm with a tensor matrix product form. A linearized stability analysis indicates the basic algorithm is spatially fourth-order accurate in its most elementary embodiment. The control of an added dissipation mechanism can elevate this to sixth order, but numerical experimentation indicates the resultant artificial diffusion is unacceptably large. By the same measure, this additional accuracy is intrinsic to the quadratic element embodiment of the algorithm with freedom from artificial diffusion. The results of several numerical experiments for single- and multi-dimensional convection-dominated test problems confirms the basic viability of this algorithm and its tensor produce formulation. The latter is of paramount importance in rendering the algorithm nominally as efficient as familiar lower-order accurate finite difference formulations. (Author)