Three-dimensional mixed explicit-implicit generalized Galerkin spectral element methods for high-speed turbulent compressible flows

Abstract In high speed flows the interactions of shock waves with turbulent boundary layers are important design considerations because of the complex flowfields resulting in increased adverse pressure gradients, skin friction and temperatures. Unsteadiness and three-dimensional flowfield structure are also characteristic of shock wave turbulent boundary layer interactions. Such physical phenomena require sophisticated numerical schemes in the solution of governing equations. The purpose of this paper, therefore, is to introduce an accurate and efficient approach—the Mixed Explicit-Implicit Generalized Galerkin Spectral Element Method (MEI-GG-SEM) with Legendre polynomial spectral elements in which flowfield dependent implicitness parameters provide automatically adequate computational requirements for compressible and incompressible flows or high speed and low speed flows. This is in contrast to the traditional approach in which all-speed-regime analysis requires a separate hyperbolic-elliptic pressure equation for pressure correction if the flow becomes incompressible. In the MEI-GG-SEM scheme, mesh refinements are carried out adaptively until shock waves are resolved, followed then by the adaptive increase of Legendre polynomial degrees until turbulence microscales are resolved, in which the traditional turbulence modeling is no longer required, aimed toward direct numerical simulation. In order to demonstrate the validity of the theory and numerical procedure, two-dimensional flat plate and compression corner high speed flows are investigated, followed by a three-dimensional sharp leading edged fin for swept shock wave turbulent boundary layer interactions. Comparisons of the present study with experimental measurements and other numerical studies show favorable agreement.