A modified conservation principles theory leading to an optimal Galerkin CFD algorithm

摘要

A modified conservation principles theory in one, then multi-dimensions, admits the prediction of an optimally accurate algorithm construction for the unsteady incompressible Navier-Stokes (INS) equations. Via a time Taylor series (TS) operation, followed by a pseudo-limit process, the theory generates a modified, but still analytical, INS system parameterized by a set of coefficients constrained only by a convexity requirement. A spatially discretized finite element implementation of a Galerkin weak statement on this modified INS system, termed the 'Taylor weak statement (TWS),' generates a parameterized CFD algorithm for analysis. TWS algorithm phase velocity and amplification factor error functions are derived for linear and bilinear basis implementations assembled at the generic node. A subsequent TS expansion in wave number space admits analytical identification of parameter set options affecting lowest order error terms. The results of definitive verification- and validation-class computational experiments for a range of published CFD algorithms belonging to the TWS class, reported herein, clearly confirm theoretical prediction of the optimal TWS algorithm for INS thermal/fluid transport applications.