Hybrid Spectral Difference/Embedded Finite Volume Method for Conservation Laws

A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field. The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.Comment: 27 pages, 17 figures, 2 tables, Accepted for publication in the Journal of Computational Physics, April 201

Solution of cavitating compressible flows using Discontinuous Galerkin discretisation

A methodology for modelling cavitating flows using a high-order Adaptive Mesh Refinement (AMR) approach based on the Discontinuous Galerkin method (DG) is presented. The AMR implementation used features on-the-fly adaptive mesh refinement for unstructured hybrid meshes. The specific implementation has been developed for the resolution of complex multi-scale phenomena where high accuracy p-adaptive discretisations are combined with an h-adaptive data structure. This approach accommodates the fine spatial resolution for the interface discontinuities and the shock waves observed in compressible cavitating flows. The Tait equation of state is used for the modelling of the liquid phase while an isentropic path is assumed for the liquid/vapour mixture. Second order spatial and a third order non-oscillatory temporal discretisation are used for the integration of the mass and momentum conservation equations, in order to resolve the flow structures responsible for the formation of cavitation bubbles and the resulting compression waves. Assessment of the developed methodology is performed for the one-dimensional advancement of a compressible liquid-vapour interface and the symmetric collapse of a spherical vapour bubble. Following, results obtained with the developed multi-scale modelling AMR approach has revealed a complex bubble collapse mechanism near a rigid wall, providing evidence of processes that have been unknown before due to reduced resolution and dissipative nature of past simulations. The impinging jet accompanying the collapse of a bubble near a wall, was found to induce vortical structures, which result to the formation of a secondary cavitation of a wall-attached bubble at the vicinity of the impingement jet shear layer. At the final stages of the initial bubble collapse, the impinging jet was found to penetrate the centre-line of the wall bubble inducing its partial collapse. This secondary collapse results to a rich spatial structure of shock waves, interacting with the secondary bubbles. Moreover, the calculated pressure level are found to be much higher than those reported from previous methodologies

Issures in Discontinuous High-Order Methods: Broadband Wave Computation and Viscous Boundary Layer Resolution

A new discontinuous formulation named Correction Procedure via Reconstruction (CPR) was developed for conservation laws. CPR is an efficient nodal differential formulation unifying the discontinuous Galerkin (DG), spectral volume (SV) and spectral difference (SD) methods, is easy to implement. In this thesis, we deal with two issues: the efficient computation of broadband waves, and the proper resolution of a viscous boundary layer with the high-order CPR method. A hybrid discontinuous space including polynomial and Fourier bases is employed in the CPR formulation in order to compute broad-band waves. The polynomial bases are used to achieve a certain order of accuracy, while the Fourier bases are able to exactly resolve waves at a certain frequency. Free-parameters introduced in the Fourier bases are optimized in order to minimize both dispersion and dissipation errors by mimicking the dispersion-relation-preserving (DRP) method for a one-dimensional wave problem. For the one-dimensional wave problem, the dispersion and dissipation properties and the optimization procedure are investigated through a wave propagation analysis. The optimization procedure is verified with a wave propagation analysis and several numerical tests. The two-dimensional wave behavior is investigated through a wave propagation analysis and the wave propagation properties are verified with a numerical test of the two-dimensional acoustic wave equation. In order to understand the mesh size requirement to resolve a viscous boundary layer using CPR method, grid resolution studies are performed. . It is well known that the mesh size, which is defined from non-dimensional wall distance y^+=1, gives accepted results to simulate viscous boundary layer problem for 2nd order finite volume method. For high-order CPR formulation, what grid size y^+ is required to match the results from the 2nd order finite volume method with y^+=1. 1D and 2D burger\u27s equation are used as the viscous boundary layer model problem. Skin friction is used as the indicator of accuracy for the resolution of a boundary layer. Keywords: (Correction Procedure via Reconstruction), A Hybrid Discontinuous Space, Wave Propagation Analysis, Grid Resolution Study, Method of Manufactured Solution