Abstract
The accuracy of gradient reconstruction methods on unstructured meshes is analyzed both mathematically and numerically. Mathematical derivations reveal that, for gradient reconstruction based on the Green-Gauss theorem (the GG methods), if the summation of first-and-lower-order terms does not counterbalance in the discretized integral process, which rarely occurs, second-order accurate approximation of face midpoint value is necessary to produce at least first-order accurate gradient. However, gradient reconstruction based on the least-squares approach (the LSQ methods) is at least first-order on arbitrary unstructured grids. Verifications are performed on typical isotropic grid stencils by analyzing the relationship between the discretization error of gradient reconstruction and the discretization error of the face midpoint value approximation of a given analytic function. Meanwhile, the numerical accuracy of gradient reconstruction methods is examined with grid convergence study on typical isotropic grids. Results verify the phenomenon of accuracy degradation for the GG methods when the face midpoint value condition is not satisfied. The LSQ methods are proved to be at least first-order on all tested isotropic grids. To study gradient accuracy effects on inviscid flow simulation, solution errors are quantified using the Method of Manufactured Solutions (MMS) which was validated before adoption by comparing with an exact solution case, i.e., the 2-dimensional (2D) inviscid isentropic vortex. Numerical results demonstrate that the order of accuracy (OOA) of gradient reconstruction is crucial in determining the OOA of numerical solutions. Solution accuracy deteriorates seriously if gradient reconstruction does not reach first-order.
Highlights
In the last several decades, research and applications of unstructured grids in Computational Fluid Dynamics (CFD) numerical simulations had drawn much attention
7 Conclusions and future work Gradient reconstruction based on the Green-Gauss theorem and the least-squares approaches are analyzed both mathematically and numerically
Mathematical derivations reveal that, for gradient reconstruction based on the Green-Gauss theorem, if the summation of first-and-lower-order terms does not counterbalance in the discretized integral process, which rarely occurs, second-order accurate approximation of face midpoint value is necessary to produce at least first-order accurate gradient
Summary
In the last several decades, research and applications of unstructured grids in Computational Fluid Dynamics (CFD) numerical simulations had drawn much attention. There are mainly two types of gradient reconstruction methods which can be readily implemented on unstructured second-order FV discretization of inviscid and viscous fluxes. The focus of this study is analyzing gradient reconstruction methods both mathematically and numerically for cell-centered FV schemes, evaluating the gradient effects on solution accuracy of inviscid flow simulations. A second-order spatial discretization can be obtained by assuming a linear distribution of flow variables in each cell With this assumption, the left and the right states are reconstructed through a piecewise linear interpolation as Eq (4) [6]. These two types of methods are introduced and analyzed
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