Implicit high-order flux reconstruction solver for high-speed compressible flows

Abstract

The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The resulting solver is fully implicit and able to simulate compressible flow problems governed by either the Euler or the Navier-Stokes equations in two and three dimensions. Furthermore, it can run in parallel on multiple CPU-cores and is designed to handle unstructured grids consisting of both straight and curved edged quadrilateral or hexahedral elements. While most of the implementation relies on state-of-the-art FR algorithms, an improved and more case-independent shock capturing scheme has been developed in order to tackle the first viscous hypersonic simulations using the FR method. Extensive verification of the FR solver has been performed through the use of reproducible benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 17.6. The obtained results have been favorably compared to those available in literature. Furthermore, so-called "super-accuracy" is retrieved for certain cases when solving the Euler equations. The strengths of the FR solver in terms of computational accuracy per degree of freedom are also illustrated. Finally, the influence of the characterizing parameters of the FR method as well as the the influence of the novel shock capturing scheme on the accuracy of the developed solver is discussed. order of accuracy per degree of freedom, they also provide a lower numerical dissipation [1]. However, high-order methods are more complicated to implement as compared to low-order methods and are often less robust. The fact that robust high-order mesh generators are not readily available has also limited their use for industrial applications [2]. Nonetheless, high-order methods have drawn considerable attention among researchers over the past couple of years due to their higher accuracy and their capability to deal with complex geometries [3]. Furthermore, the generation of high-order curved meshes has become increasingly supported by mesh generators such as the open-source Gmsh [4]. This has resulted in several successful attempts to develop high-order extensions of the existing FV, FD and FE schemes.FD methods have been extended to higher orders in a straightforward manner by widening the stencil. In this regard, Compact Finite Difference [5] schemes have proven to provide higher orders of accuracy. Nonetheless, all FD methods exhibit the same weakness: they are incapable to readily deal with complex geometries, unless utilized within an overset framework which complicates mesh generation [6,7,8]. FV methods, on the other hand, do not have this shortcoming. As a consequence, several high-order extensions have been developed, such as the k-Exact method [9, 10, 11], the Essential Non-Oscillatory (ENO) [12,13,14], and Weighted Essential Non-Oscillatory (WENO) [15,16,17] schemes. These methods do not generally use a compact ste...

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