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Article
Publication date: 1 January 2006

Mahmood K. Mawlood, Shahnor Basri, Waqar Asrar, Ashraf A. Omar, Ahmad S. Mokhtar and Megat M.H.M. Ahmad

To develop a high‐order compact finite‐difference method for solving flow problems containing shock waves.

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Abstract

Purpose

To develop a high‐order compact finite‐difference method for solving flow problems containing shock waves.

Design/methodology/approach

A numerical algorithm based on high‐order compact finite‐difference schemes is developed for solving Navier‐Stokes equations in two‐dimensional space. The convective flux terms are discretized by using advection upstream splitting method (AUSM). The developed method is then used to compute some example laminar flow problems. The problems considered have a range of Mach number that corresponds to subsonic incompressible flow to hypersonic compressible flows that contain shock waves and shock/boundary‐layer interaction.

Findings

The paper shows that the AUSM flux splitting and high‐order compact finite‐difference methods can be used accurately and robustly in resolving shear layers and capturing shock waves. The highly diffusive nature of conventional flux splitting especially on coarse grids makes them inaccurate for boundary layers even with high‐order discretization.

Originality/value

This paper presents a high‐order numerical method that can accurately and robustly capture shock waves without deteriorating oscillations and resolve boundary layers and shock/boundary layer interaction.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 16 no. 1
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 1 June 2004

Mahmood K. Mawlood, ShahNor Basri, Waqar Asrar, Ashraf A. Omar, Ahmad S. Mokhtar and Megat M.H.M. Ahmad

A high‐order compact upwind algorithm is developed for solving Navier‐Stokes equations in two‐space dimensions. The method is based on advection upstream splitting method and…

1318

Abstract

A high‐order compact upwind algorithm is developed for solving Navier‐Stokes equations in two‐space dimensions. The method is based on advection upstream splitting method and fourth‐order compact finite‐difference schemes. The convection flux terms of the Navier‐Stokes equations are discretized by a compact cell‐centered differencing scheme while the diffusion flux terms are discretized by a central fourth‐order compact scheme. The midpoint values of the flux functions required by the cell‐centered compact scheme are determined by a fourth‐order MUSCL approach. For steady‐state solutions; first‐order implicit time integration, with LU decomposition, is employed. Computed results for a laminar flow past a flat plate and the problem of shock‐wave boundary layer interaction are presented.

Details

Aircraft Engineering and Aerospace Technology, vol. 76 no. 3
Type: Research Article
ISSN: 0002-2667

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Article
Publication date: 18 May 2021

J.I. Ramos

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of…

117

Abstract

Purpose

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

Design/methodology/approach

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

Findings

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

Originality/value

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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