Flowfield-Dependent Variation (FDV) method for compressible, incompressible, viscous, and inviscid flow interactions with FDV adaptive mesh refinements and parallel processing

A new approach to solution-adaptive grid refinement using the finite element method and Flowfield-Dependent Variation (FDV) theory applied to the Navier-Stokes system of equations is discussed. Flowfield-Dependent Variation (FDV) parameters are introduced into a modified Taylor series expansion of the conservation variables, with the Navier-Stokes system of equations substituted into the Taylor series. The FDV parameters are calculated from the current Fowfield conditions, and automatically adjust the resulting equations from elliptic to parabolic to hyperbolic in type to assure solution accuracy in evolving fluid flowfields that may consist of interactions between regions of compressible and incompressible flow, viscous and inviscid flow, and turbulent and laminar flow. The system of equations is solved using an element-by-element iterative GMRES solver with the elements grouped together to allow the element operations to be performed in parallel. The FDV parameters play many roles in the numerical scheme. One of these roles is to control formations of shock wave discontinuities in high speeds and pressure oscillations in low speeds. To demonstrate these abilities, various example problems are shown, including supersonic flows over a flat plate and a compression corner, and flows involving triple shock waves generated on fin geometries for high speed compressible flows. Furthermore, analysis of low speed incompressible flows is presented in the form of flow in a lid-driven cavity at various Reynolds numbers. Another role of the FDV parameters is their use as error indicators for a solution-adaptive mesh. The finite element grid is refined as dictated by the magnitude of the FDV parameters. Examples of adaptive grids generated using the FDV parameters as error indicators are presented for supersonic flow over flat plate/compression ramp combinations in both two and three dimensions. Grids refined using the FDV parameters as error indicators are comparable to ones refined using primitive variable error indicators, and require less computational time to generate the grids. The use of parallel processing in performing some element operations is shown to reduce the wall clock time approximately forty-five percent in going from one to eight processors. Finally, the algorithm's ability to solve a flowfield containing interactions and transitions between regions of incompressible and compressible, viscous and inviscid, and laminar and turbulent flow is demonstrated by modeling the flowfield generated by supersonic flow over a compression ramp located between two fins. The structure of the resulting systems of shock waves are analyzed and compared with planar laser scattering images obtained experimentally for similar flow structures.