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Article

ERIK DICK

A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume…

Abstract

A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume method. In first order accurate form, a discrete set of equations is obtained which is both conservative and positive. Due to the positivity, the set of equations can be solved by collective relaxation methods in multigrid form. A full multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. Second order accuracy is obtained by the Chakravarthy‐Osher flux‐extrapolation technique, using the Roe‐Chakravarthy minmod limiter. In second order form, direct relaxation of the discrete equations is no longer possible due to the loss of positivity. A defect‐correction is used in order to solve the second order system.

Details

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

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Article

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.

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

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…

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

J.‐Y. TRÉPANIER, M. REGGIO and D. AIT‐ALI‐YAHIA

An implicit method for the solution of transonic flows modelled by the time‐dependent Euler equations is presented. The method is characterized by a robust linearization…

Abstract

An implicit method for the solution of transonic flows modelled by the time‐dependent Euler equations is presented. The method is characterized by a robust linearization for first‐ and second‐order versions of Roe's flux‐difference splitting scheme, an implicit treatment of the boundary conditions and the implementation of an adaptive grid strategy for global efficiency. The performance of the method is investigated for the GAMM test circular‐arc bump configuration and for the RAE 2822 aerofoil.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 3 no. 1
Type: Research Article
ISSN: 0961-5539

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Article

J. Steelant and E. Dick

The steady compressible Navier—Stokes equations coupled to thek—ε turbulence equations are discretized within avertex‐centered finite volume formulation. The convective…

Abstract

The steady compressible Navier—Stokes equations coupled to the k—ε turbulence equations are discretized within a vertex‐centered finite volume formulation. The convective fluxes are obtained by the polynomial flux‐difference splitting upwind method. The first order accurate part results directly from the splitting. The second order part is obtained by the flux‐extrapolation technique using the minmod limiter. The diffusive fluxes are discretized in the central way and are split into a normal and a tangential contribution. The first order accurate part of the convective fluxes together with the normal contribution of the diffusive fluxes form a positive system which allows solution by classical relaxation methods. The source terms in the low‐Reynolds k‐ε equations are grouped into positive and negative terms. The linearized negative source terms are added to the positive system to increase the diagonal dominance. The resulting positive system forms the left hand side of the equations. The remaining terms are put in the right hand side. A multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. The multigrid method itself acts on the left hand side of the equations. The right hand side is updated in a defect correction cycle.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 4 no. 2
Type: Research Article
ISSN: 0961-5539

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Article

C.P.T. GROTH and J.J. GOTTLIEB

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional…

Abstract

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 3 no. 6
Type: Research Article
ISSN: 0961-5539

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Article

M. Taha Janan and A. El Marjani

This paper aims to develop an efficient numerical method for simulating multicomponent flows by solving the system of conservative equations closed by a general two…

Abstract

Purpose

This paper aims to develop an efficient numerical method for simulating multicomponent flows by solving the system of conservative equations closed by a general two parameters equation of state.

Design/methodology/approach

A finite difference method for solving the two‐dimensional Euler or Navier‐Stokes equations for multicomponent flows in a general curvilinear coordinate system is developed. The system of conservative equations (mass, momentum and energy) is closed with a general two parameters equation of state (ρe=(p+γp)/(γ−1)), which, associated to a γ‐formulation, allows easy computation of multicomponent flows. In order to enforce the stability of the numerical scheme, the Roe's flux‐difference splitting is adopted for the numerical treatment of the inviscid fluxes. The method is adapted to treat also unsteady flows by implementing an explicit Euler scheme.

Findings

The method was applied to compute various configurations of flows, ranging from incompressible to compressible fluid, including cases of single component flows or multicomponent ones. Computations show that the use of primitive variables instead of conservative ones, especially at low Mach numbers, improves the iteration process when the resolution is performed with a relaxation procedure such as Gauss‐Seidel method. Simulations of compressible flows with a strong shock show the ability of the present method to capture shocks correctly even with the use of primitive variables. To complete numerical tests, flows involving two fluids with the presence of interactions between a shock and a discontinuity surface have been treated successfully. Also, a case of cavitating flow has been considered in this work.

Originality/value

The present method permits the simulation of a large variety of multicomponent complexes flows with an efficient numerical taking advantage of Roe's flux‐difference splitting in curvilinear coordinate system.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 17 no. 8
Type: Research Article
ISSN: 0961-5539

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Article

Masoud Mirzaei, Seyedeh Nasrin Hosseini and Jafar Roshanian

This paper's purpose is to deal with single point and multipoint optimization of an airfoil. The aim of the paper is to discuss optimization in several design points…

Abstract

Purpose

This paper's purpose is to deal with single point and multipoint optimization of an airfoil. The aim of the paper is to discuss optimization in several design points (multipoint optimization) and compare the results with those of optimization at a specified design point.

Design/methodology/approach

A gradient‐based method is adopted for optimization and the flow is governed by two dimensional, compressible Euler equations. A finite volume code based on unstructured grid is developed to solve the equations.

Findings

Two test cases are studied for an airfoil with initial profile of NACA0012, with two types of design variables. And at the end a multi‐point case is presented.

Originality/value

The advantage of this technique over the other gradient‐based methods is its high‐convergence rate.

Details

Aircraft Engineering and Aerospace Technology, vol. 79 no. 6
Type: Research Article
ISSN: 0002-2667

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Article

P.R. Ess and C.B. Allen

A computational fluid dynamics code for the calculation of laminar hypersonic multi‐species gas flows in chemical non‐equilibrium in axisymmetric or two‐dimensional…

Abstract

Purpose

A computational fluid dynamics code for the calculation of laminar hypersonic multi‐species gas flows in chemical non‐equilibrium in axisymmetric or two‐dimensional configuration on shared and distributed memory parallel computers is presented and validated. The code is designed to work efficiently in combination with an automatic domain decompositioning method developed to facilitate efficient parallel computations of various flow problems.

Design/methodology/approach

The baseline implicit numerical method developed is the lower‐upper symmetric Gauss‐Seidel scheme, which is combined with a sub‐iteration scheme to achieve time‐accuracy up to third‐order. The spatial discretisation is based on Roe's flux‐difference splitting and various non‐linear flux limiters maintaining total‐variation diminishing properties and up to third‐order spatial accuracy in continuous regions of flow. The domain subdivision procedure is designed to work for single‐ and multi‐block domains without being constrained by the block boundaries, and an arbitrary number of processors used for the computation.

Findings

The code developed reproduces accurately various types of flows, e.g. flow over a flat plate, diffusive mixing and oscillating shock induced combustion around a projectile fired into premixed gas, and demonstrates close to linear scalability within limits of load imbalance.

Research limitations/implications

The cases considered are axisymmetric or two‐dimensional, and assume laminar flow. An extension to three‐dimensional turbulent flows is left for future work.

Originality/value

Results of a parallel computation, utilising a newly developed automatic domain subdivision procedure, for oscillating shock‐induced combustion around a projectile and various other cases are presented. The influence of entropy correction in Roe's flux‐difference splitting algorithm on diffusive mixing of multi‐species flows was examined.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 15 no. 3
Type: Research Article
ISSN: 0961-5539

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Article

P. Glaister

An algorithm based on flux difference splitting is presented for thesolution of two‐dimensional, steady, supercritical open channel flows. Atransformation maps a…

Abstract

An algorithm based on flux difference splitting is presented for the solution of two‐dimensional, steady, supercritical open channel flows. A transformation maps a non‐rectangular, physical domain into a rectangular one. The governing equations are then the shallow water equations, including terms of slope and friction, in a generalised coordinate system. A regular mesh on a rectangular computational domain can then be employed. The resulting scheme has good jump capturing properties and the advantage of using boundary/body‐fitted meshes. The scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 10
Type: Research Article
ISSN: 0961-5539

Keywords

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