This paper presents the application of the structured-grid-based high-order methods on unstructured hexahedral grids by employing the embedded structured element (ESE) method. The motivation of the embedded structured element method is that, by embedding structured finite volume sub-cells in unstructured hexahedral grids, it attempts to combine favorable features of the unstructured hexahedral grid, for relatively easier generation of computational grids for complex geometries, and the current state-of-the-art structured-grid-based high-order methods with shock-capaturing capability. The embedded structured element approach enables the flexible choice of available low-dissipative high-order methods with shock-capturing capability depending on the flow problems, e.g., variants of WENO scheme, compact scheme, discontinuous high-order methods and combinations of various high-order methods, yet it allows to take advantage of unstructured hexahedral grids. It is anticipated that the ESE method provides the suitable framework for the flow problems involving turbulence/discontinuity interactions in complex geometries. To illustrate the performance of the ESE method, two structured-grid-based high-order methods are employed, namely, the 5th-order WENO-Z and the 5th-order CRWENO-Z schemes with the monotonicity preserving bound, on the unstructured base hexahedral grid. In addition, two Riemann solvers, namely, the Roe scheme and the SD-SLAU, are used to investigate the shock instabilities associated with non-alignment of shock and computational grid. The order of accuracy is examined, and three test cases are carried out with the combinations of employed high-order schemes and Riemann solvers. The simulation results by the ESE method are compared with the exact solutions and the computational results obtained on the structured grid using the hybrid spectral difference/embedded finite volume method and are discussed.
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