Estimation of grid-induced errors in computational fluid dynamics solutions using a discrete error transport equation

Abstract

Computational fluid dynamics (CFD) has become a widely used tool in research and engineering for the study of a wide variety of problems. However, confidence in CFD solutions is still dependent on comparisons with experimental data. In order for CFD to become a trusted resource, a quantitative measure of error must be provided for each generated solution. Although there are several sources of error, the effects of the resolution and quality of the computational grid are difficult to predict a priori. This grid-induced error is most often attenuated by performing a grid refinement study or using solution adaptive grid refinement. While these methods are effective, they can also be computationally expensive and even impractical for large, complex problems. This work presents a method for estimating the grid-induced error in CFD solutions of the Navier-Stokes and Euler equations using a single grid and solution or a series of increasingly finer grids and solutions. The method is based on the discrete error transport equation (DETE), which is derived directly from the discretized PDE and provides a value of the error at every cell in the computational grid. The DETE is developed for two-dimensional, laminar Navier-Stokes and Euler equations within a generalized unstructured finite volume scheme, such that an extension to three dimensions and turbulent flow would follow the same approach. The usefulness of the DETE depends on the accuracy with which the source term, the gridinduced residual, can be modeled. Three different models for the grid-induced residual were developed: the AME model, the PDE model, and the extrapolation model. The AME model consists of the leading terms of the remainder of a simplified modified equation. The PDE model creates a polynomial fit of the CFD solution and then uses the original PDE in differential form to calculate the residual. Both the AME and PDE are used with a single grid and solution. The extrapolation model uses a fine grid solution to calculate the grid-induced residual on the coarse grid and then extrapolates that residual back to the fine grid. The DETE and residual models were then evaluated for four flow problems: (1) steady flow past a circular cylinder; (2) steady, transonic flow past an airfoil; (3) unsteady flow of an isentropic vortex; (4) unsteady flow past a circular cylinder with vortex shedding. Results demonstrate the fidelity of the DETE with each residual model as well as usefulness of the DETE as a tool for predicting the grid-induced error in CFD solutions.