Abstract
In this article, a quasi-linear semi-discrete analysis of shock capturing schemes in two dimensional wavenumber space is proposed. Using the dispersion relation of the two dimensional advection and linearized Euler equations, the spectral properties of a spatial scheme can be quantified in two dimensional wavenumber space. A hybrid scheme (HYB-MDCD-TENO6) which combines the merits of the minimum dispersion and controllable dissipation (MDCD) scheme with the targeted essentially non-oscillatory (TENO) scheme was developed and tested. Using the two dimensional analysis framework, the scheme was spectrally optimized in such a way that the linear part of the scheme can be separately optimized for its dispersion and dissipation properties. In order to compare its performance against existing schemes, the proposed scheme as well as the baseline schemes were tested against a series of benchmark test cases. It was found that the HYB-MDCD-TENO6 scheme provides similar or better resolution as compared to the baseline TENO6 schemes for the same grid size.
Highlights
Turbulent flows are usually characterised by a large range of length and time scales
We present a framework for which the non-linear scheme can be optimized in two dimensional wavenumber space based on the dispersion relation of the two dimensional advection equations
The free parameters to be optimized in this context is the values of γdisp and γdiss used in the weighting of the sub scheme of the TENO6 scheme
Summary
Turbulent flows are usually characterised by a large range of length and time scales. In order to obtain accurate results, a direct numerical simulation (DNS) of such flows must resolve all these range of scales, especially the finest scales with accuracy in both amplitude and phase. The dissipation and dispersion properties of a numerical scheme has direct relevance to its scale resolving abilities. Due to their superior spectral properties, spectral methods
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