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Flowfield dependent variation method: A numerical scheme for the solution of low- to high-Mach number flow problems

Bassem R Girgis (Ansys, Inc., Lebanon, New Hampshire, United States.)
Sarma L Rani (Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, Alabama, United States.)
Abdelkader Frendi (Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, Alabama, United States.)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 6 June 2016

148

Abstract

Purpose

The purpose of this paper is to investigate the computational features of the Flowfield Dependent Variation (FDV) method, a numerical scheme built to simulate flows characterized by multiple speeds, multiple physical phenomena, and by large variations in flow variables.

Design/methodology/approach

Fundamentally, the FDV method may be regarded as a variant of the Lax-Wendroff Scheme (LWS) that is obtained by replacing the explicit time derivatives in LWS by a weighted combination of explicit and implicit time derivatives. The weighting factors – referred to as FDV parameters – may be broadly classified as convective and diffusive parameters which, for example, are determined using flow quantities such as the Mach number and Reynolds number, respectively. Hence, the reference to these parameters and the method as “flow field dependent.” A von Neumann Fourier analysis demonstrates that the increased implicitness makes FDV both more stable and less dispersive compared to LWS, a feature crucial to capturing shocks and other phenomena characterized by high gradients in variables. In the current study, the FDV scheme is implemented in a Taylor-Galerkin-based finite element method framework that supports arbitrarily high order, unstructured isoparametric elements in one-, two- and three-dimensional geometries.

Findings

At first, the spatial accuracy of the implemented FDV scheme is established using the Method of Manufactured Solutions, wherein the results show that the order of accuracy of the scheme is nearly equal to the order of the shape function polynomial plus one. The dispersion and dissipation errors of FDV, when applied to the compressible Navier-Stokes and energy equations, are investigated using a 2-D, small-amplitude acoustic pulse propagating in a quiescent medium. It is shown that FDV with third-order shape functions accurately captures both the amplitude and phase of the acoustic pulse. The method is then applied to cases ranging from low-Mach number subsonic flows (Mach number M=0.05) to high-Mach number supersonic flows (M=4) with shock-boundary layer interactions. For all cases, fair to good agreement is observed between the current results and those in the literature.

Originality/value

The spatial order of accuracy of the FDV method, its stability and dispersive properties, as well as its applicability to low- and high-Mach number flows are established.

Keywords

Acknowledgements

This work was partially supported by a NASA SBIR Phase I project (Contract Number NNX11CD05P), with Dr C.Y. Tang as the Technical Monitor. SLR gratefully acknowledges the many fruitful discussions with Shreyas Bidadi, a PhD student in the Mechanical and Aerospace Engineering Department at the University of Alabama in Huntsville.

Citation

Girgis, B.R., Rani, S.L. and Frendi, A. (2016), "Flowfield dependent variation method: A numerical scheme for the solution of low- to high-Mach number flow problems", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26 No. 5, pp. 1486-1525. https://doi.org/10.1108/HFF-04-2015-0137

Publisher

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Emerald Group Publishing Limited

Copyright © 2016, Emerald Group Publishing Limited

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