A case-study in open-source CFD code verification. Part II: Boundary condition non-orthogonal correction
Abstract This investigation constitutes the follow-up to a previous work in which we applied the manufactured solution procedure to verify the OpenFOAM diffusion operator on different types of meshes. Theoretical convergence orders were observed for Poisson’s equation solver on orthogonal hexahedral grids using several boundary conditions. We noticed that on non-orthogonal grids even slight mesh distortions reduce the theoretical second order convergence rate to first order. Our investigation showed that this loss in convergence order takes place when Dirichlet and/or Neumann boundary conditions are used on non-orthogonal meshes. In this paper, we introduce ways to achieve theoretical second order convergence accuracy, mainly based on applying non-orthogonal correction to Dirichlet and/or Neumann boundary condition schemes. A peculiarity was discovered for the Least-Squares gradient scheme that slightly affects second-order convergence, and which we attribute to the non-orthogonal correction at boundaries in the Least-Squares vector reconstruction library. The verification procedure is performed using the method of manufactured solutions for the Poisson equation and an analytical solution, the SIMPLE solver, for the Navier–Stokes equations. The average and maximum error norms were used to calculate the convergence rate. Comparative results are presented.
Verification of Euler/Navier–Stokes codes using the method of manufactured solutions
The method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.
Verication of Calculations: an Overview of the Lisbon Workshop
This paper presents an overview of a Workshop dedicated to the Verification of Calculations held in Lisbon in October 2004. The Workshop aimed at evaluation of Computational Fluid Dynamics Uncertainty Estimators by comparing results of six groups on two twodimensional incompressible turbulent flows involving separated turbulent flows. Various numerical methods were represented, and calculations were done on smooth grids as well as deliberately poor grids in order to burden the Uncertainty Estimators. Even though local grid convergence was often non-monotone, the overall consensus evaluation was generally favorable for the Grid Convergence Index and the Least Squares version of it, as well as a monogrid Error Transport Equation method. Some residual doubts remain on the inter-group comparisons because of conceptual modelling differences and the possibility of undetected coding errors. These could be eliminated by the use of a realistic exact analytical solution produced by the Method of Manufactured Solutions, and the consensus favored this approach for a second Workshop on this theme.