Higher Order Variable Extrapolation For Unstructured Finite Volume RANS Flow Solvers

A new variable extrapolation formulation for unstructured finite volume codes is developed which closely resembles the MUSCL-scheme used within structured flow solvers. This new formulation is based on information currently available to the unstructured flow solvers, namely the variable information and the gradient information, and as such, it is trivial to implement within most finite volume flow solvers. This new variable extrapolation formulation represents a one-parameter family of equations and under certain circumstances, it is fully equivalent to the MUSCL-scheme, which is also a one-parameter family. A wide variety of results are presented, including theoretical analysis of the truncation error and numerical analysis of the truncation error for a one-dimensional problem, using the method of manufactured solutions, as well as for a two-dimensional problem. Inviscid three-dimensional results are presented which demonstrate that the numerical viscosity can be greatly decreased via this formulation, and the viscous results for a variety of cases indicate that the drag is better predicted and that the vortical structures within the flow field are captured more accurately and further downstream of their origination. Finally, the observed improved convergence levels and stability of the new formulation are discussed, as well as the issues involved with extending this approach to achieve third-order spatial accuracy for 2D and 3D grids.