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Validation of a high-order numerical discretization scheme for the solution of the 3-D Euler equations

Angelopoulos Dimitrios

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Summary

In this study, the application and evaluation of a high-order spatial and time discretization method for the numerical solution of 2-dimensional Euler equations is reported. An alternative high-order approach enhances the in-house academic solver, named EU2, employing the dimensionless Euler equations, discretized with a node-centered finite volume method on triangular unstructured girds, to simulate inviscid compressible flows. Most methodologies that have been developed during the past years, e.g. the discontinuous Galerkin and K-exact scheme, necessitate a non-trivial increase of the DoFs (Degrees of Freedom) and consequently a considerable increase of computational resources. Moreover, major modifications to existing CFD codes are required for their implementation. The adopted high-order scheme is based on the incorporation of additional high order terms to the reconstructed nodal values, used for the computation of the inviscid fluxes. The required higher-order derivatives are computed with the corresponding lower-order ones on the existing DoFs via a successive differentiation technique. As a result, the connectivity requirements are restricted to the first neighbouring points, overcoming the inherent constraint of the unstructured solvers to retrieve information from a wider computational stencil. The aforementioned technique was incorporated with a variable extrapolation numerical scheme, named U-MUSCL, which closely resembles the traditional MUSCL one, and was coupled with a high-order time discretization that employs a Strong Stability Preserving Runge-Kutta method (SSPRK). To assess the effectiveness of the aforementioned numerical scheme, the EU2 solver is used against a benchmark problem having analytic solution. A satisfactory agreement is obtained, demonstrating the proposed scheme’s potential to increase the solution’s accuracy for a given grid density. Furthermore, a corresponding high-order formulation is extended to a 3-dimensional numerical fluid model. An elaborate construction method of 3-d computational meshes for various grid types is presented in detail for future exploitation on the numerical evaluation of equivalent 3-d high-order schemes.

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