Aaron Katz

High-order flux correction/finite difference schemes for strand grids

A novel high-order method combining unstructured flux correction along body surfaces and high-order finite differences normal to surfaces is formulated for unsteady viscous flows on strand grids. The flux correction algorithm is applied in each unstructured layer of the strand grid, and the layers are then coupled together via a source term containing derivatives in the strand direction. Strand-direction derivatives are approximated to high-order via summation-by-parts operators for first derivatives and second derivatives with variable coefficients. We show how this procedure allows for the proper truncation error canceling properties required for the flux correction scheme. The resulting scheme possesses third-order design accuracy, but often exhibits fourth-order accuracy when higher-order derivatives are employed in the strand direction, especially for highly viscous flows. We prove discrete conservation for the new scheme and time stability in the absence of the flux correction terms. Results in two dimensions are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.

High-order strand grid methods for shock turbulence interaction

ABSTRACT In this work, we examine the flux correction method for three-dimensional transonic turbulent flows on strand grids. Building upon previous work, we treat flux derivatives along strands with high-order summation-by-parts operators and penalty-based boundary conditions. A finite-volume like limiting strategy is implemented in the flux correction algorithm in order to sharply capture shocks. To achieve turbulence closure in the Reynolds-Averaged Navier–Stokes equations, a robust version of the Spalart–Allmaras turbulence model is employed that accommodates negative values of the turbulence working variable. Validation studies are considered which demonstrate the flux correction method achieves a high degree of accuracy for turbulent shock interaction flows.

Source Term Discretization Effects on the Steady-State Accuracy of Finite Volume Schemes

Source terms often arise in Computational Fluid Dynamics to describe a variety of physical phenomena, including turbulence, chemical reactions, and certain methods used for code verification, such as the method of manufactured solutions. While much has already been published on the treatment of source terms, here we follow an uncommon approach, designing compatible source term discretizations in terms of spatial truncation error for finite volume schemes in multiple dimensions. In this work we examine the effect of source term discretization on three finite volume flux schemes applied to steady flows: constant reconstruction, linear reconstruction, and a recently published third-order flux correction method. Three source term discretization schemes are considered, referred to as point, Galerkin, and corrected. The corrected source discretization is a new method that extends our previous work on flux correction to equations with source terms. In all cases, computational grid refinement studies confirm the compatibility (or lack thereof) of various flux-source combinations predicted through detailed truncation error analysis.

Development and Application of an Incompressible Strand Solver

Computational fluid dynamics (CFD) is an ever-increasing component of the naval design and analysis process. Most current high-fidelity CFD methods used by the US Navy require body-fitted volume grids, which are time-consuming and depend on seasoned engineers with significant grid generation experience. Automated grid generation using a strand gridding approach is a relatively new concept that could transform the role of higher fidelity CFD within the Navys design and analysis process. The HPCMP CREATE-SH team seeks to expand the compressible strand solver under development as part of the HPCMP CREATE-AV effort to incompressible flow problems for applications of interest to the Navy. The completed incompressible strand solver would allow quicker turnaround time for experienced CFD engineers while allowing more naval architects to use higher fidelity CFD as a part of the overall design process. The incompressible strand solver is a new development of the HPCMP CREATE-SH Hydro team, and this work will demonstrate the solver on validation cases of naval interest.

Source Term Discretization Effects on the Accuracy of Finite Volume Schemes

Source terms often appear in Computational Fluid Dynamics problems. They are used to help describe the physics of turbulence, chemical reactions, and certain methods used for code verification, such as the Method of Manufactured Solutions. Proper discretization of source terms is essential to provide accurate results. If source term discretization is insufficient it can spoil the solution accuracy. For this work we examine the effect of source terms on three finite volume flux schemes: constant reconstruction, linear reconstruction, and third-order flux correction schemes. We also develop three source term discretization schemes: point source, Galerkin, and corrected. Results indicate that source term discretization is essential to providing correct order of accuracy for finite volume schemes.