Evaluation of Richardson extrapolation in computational fluid dynamics

The concept of Richardson extrapolation is evaluated for improving the solution accuracy of two well-investigated two-dimensional flow problems: (1) laminar cavity flows with Re = 100 and 1,000; and (2) the Reynolds-averaged backward-facing step turbulent flow with Re = 10⁶, aided by the widely used k-ε two-equation model with wall function. Uniform grid systems are employed in all cases to facilitate unambiguous assessment. By systematically refining the grid, computational fluid dynamics (CFD) solutions with different resolutions are first obtained, then extrapolated from a finer to a coarser grid using Lagrangian interpolation with either a 9-point or a 16-point formula. For laminar flows, Richardson extrapolation does not exhibit consistent trends in order of accuracy. Furthermore, the relative performance of Richardson extrapolation, in comparison with solutions obtained directly from mesh refinement, is not sensitive to the level of residuals contained in each computation, the detailed interpolation formula between grids, the choice between second-order central difference and upwind convection schemes, and the selection of the error norms. For turbulent flow computations, large jumps in velocity profiles between the wall and the first node cause difficulties in interpolation, and Richardson extrapolation performs unsatisfactorily under such situations. The present study indicates that Richardson extrapolation does not work consistently in approaches typically employed for engineering CFD applications.