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Article
Publication date: 6 July 2015

Di Zhao

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special…

Abstract

Purpose

The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES.

Design/methodology/approach

In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

Findings

Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers.

Originality/value

Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

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Article
Publication date: 1 April 2004

S.Ø. Wille, Ø. Staff, A.F.D. Loula and G.F. Carey

The indefinite nature of the mixed finite element formulation of the Navier‐Stokes equations is treated by segregation of the variables. The segregation algorithm…

Abstract

The indefinite nature of the mixed finite element formulation of the Navier‐Stokes equations is treated by segregation of the variables. The segregation algorithm assembles the coefficients which correspond to the velocity variables in the upper part of the equation matrix and the coefficients which corresponds to the pressure variables in the lower part of the equation matrix. During the incomplete; elimination of the velocity matrix, fill‐in will occur in the pressure matrix, hence, divisions with zero are avoided. The fill‐in rule applied here is related to the location of the node in the finite element mesh, rather than the magnitude of the fill‐in or the magnitude of the coefficient at the location of the fill‐in. Different orders of fill‐in are explored for ILU preconditioning of the mixed finite element formulation of the Navier‐Stokes equations in two dimensions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 14 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 1 April 1991

C.T. SHAW

To‐date, several segregated finite element algorithms have been proposed that solve the Navier—Stokes equations. These have considered only steady‐state cases. This paper…

Abstract

To‐date, several segregated finite element algorithms have been proposed that solve the Navier—Stokes equations. These have considered only steady‐state cases. This paper describes the addition of the time‐dependent terms to one such segregated solution scheme. Several laminar flow examples have been computed and comparisons made to predictions obtained with both finite difference and finite volume solution schemes. The finite element results compare very well with the results from the other schemes, both in terms of accuracy and the qualitative behaviour of the iterative schemes.

Details

Engineering Computations, vol. 8 no. 4
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 February 1994

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
Publication date: 1 April 1994

J.C. Cavendish, C.A. Hall and T.A. Porsching

We describe a new mathematical approach for deriving and solvingcovolume models of the three‐dimensional, incompressibleNavier—Stokes flow equations. The approach…

Abstract

We describe a new mathematical approach for deriving and solving covolume models of the three‐dimensional, incompressible Navier—Stokes flow equations. The approach integrates three technical components into a single modelling algorothm: automatic grid generation; covolume equation generation; dual variable reduction.

Details

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

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Article
Publication date: 11 October 2011

V.P. Vallala, J.N. Reddy and K.S. Surana

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations

Abstract

Purpose

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

Design/methodology/approach

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

Findings

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Originality/value

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

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Article
Publication date: 20 November 2007

J.M.F. Trindade and J.C.F. Pereira

This paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured…

Abstract

Purpose

This paper aims to focus on the temporal and spatial fourth‐order finite volume discretization of the incompressible form of the Navier‐Stokes equations on structured uniform grids. The main purpose of the paper is to assess the accuracy enhancement with the inclusion of a high‐order reconstruction of the point‐wise velocity field on a fourth‐order accurate numerical scheme for the solution of the unsteady incompressible Navier‐Stokes equations.

Design/methodology/approach

The present finite volume method uses a fractional time‐step for decoupling velocity and pressure. A Runge‐Kutta integration scheme is implemented for integrating the momentum equation along with a polynomial interpolation and Simpson formula for space‐integration. The formulation is based on step‐by‐step de‐averaging process applied to the velocity field.

Findings

The reconstruction of the point‐wise velocity field on a higher‐order basis is essential to obtain solutions that effectively stand for a fourth‐order approximation of the point‐wise one. Results are provided for the Taylor vortex decay problem and for co‐ and counter‐rotating vortices to assess the increase in accuracy promoted by the inclusion of the high‐order de‐averaging procedure.

Research limitations/implications

High‐order reconstruction of the point‐wise velocity field should be considered in high‐order finite volume methods for the solution of the unsteady incompressible form of the Navier‐Stokes equations on structured grids.

Practical implications

The inclusion of a high‐order reconstruction of the point‐wise velocity field is a simple and effective method of enhancing the accuracy of a finite volume code for the computational fluid dynamics analysis.

Originality/value

The paper develops an improved version of a fourth‐order accurate finite volume projection method with the inclusion of a high‐order reconstruction step.

Details

Engineering Computations, vol. 24 no. 8
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 1 June 1993

A. KANIEL, M. MOND and G. BEN‐DOR

Isotropic artificial dissipation is added to the Navier‐Stokes equations along with a correction term which cancels the artificial dissipation term in the limit when the…

Abstract

Isotropic artificial dissipation is added to the Navier‐Stokes equations along with a correction term which cancels the artificial dissipation term in the limit when the mesh size is zero. For a finite mesh size, the correction term replaces the artificial viscosity terms with hyperviscosity terms, i.e., with an artificial dissipation which depends on the fourth derivatives of the velocity. Hyperviscosity more effectively suppresses the higher wave number modes and has a smaller effect on the inertial modes of the flow field than does artificial viscosity. This scheme is implemented using the finite element method and therefore the required amount of dissipation is determined by analysing the discretization on a finite element. The scheme is used to simulate the flow in a driven cavity and over a backward facing step and the results are compared to existing results for these cases.

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
Publication date: 1 October 1995

E. Dick and J. Steelant

A comparison of the accuracy of the central discretization scheme withartificial dissipation and the upwind flux‐difference TVD scheme has beenmade for the compressible…

Abstract

A comparison of the accuracy of the central discretization scheme with artificial dissipation and the upwind flux‐difference TVD scheme has been made for the compressible Navier‐Stokes equations for high Reynolds number flows. First, a comparison is made on two one‐dimensional model problems. Then the schemes are compared on flat plate boundary layer flow. It is shown that a central scheme basically has poor accuracy due to the isotropic nature of the artificial dissipation. An upwind scheme decomposes the flow into different components and adapts the dissipation to the velocity of the components. The associated ansitropic dissipation results in a good accuracy. It is further discussed how a central discretization scheme with artificial dissipation can be improved at the expense of the same complexity of an upwind scheme.

Details

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

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Article
Publication date: 1 February 1995

Nick Foster and Dimitri Metaxas

A software package is developed for the modelling and animation of viscous incompressible fluids. The full time‐dependent Navier‐Stokes equations are used to simulate 2D…

Abstract

A software package is developed for the modelling and animation of viscous incompressible fluids. The full time‐dependent Navier‐Stokes equations are used to simulate 2D and 3D incompressible fluid phenomena which include shallow and deep fluid flow, transient dynamic flow, vorticity and splashing in simulated physical environments. The package also allows the inclusion of variously shaped and spaced static or moving obstacles that are fully submerged or penetrate the fluid surface. Stable numerical analysis techniques based on finite‐differences are used for the solution of the Navier‐Stokes equations. To model free‐surface fluids, a technique based on the Marker‐and‐Cell method is presented. Based on the fluid’s pressure and velocities obtained from the solution of the Navier‐Stokes equations this technique allows modelling of the fluid’s free surface either by solving a surface equation of by tracking the motion of marker particles. The latter technique is suitable for visualization of splashing and vorticity. Furthermore, an editing tool is developed for easy definition of a physical‐world which includes obstacles, boundaries and fluid properties such as viscosity, initial velocity and pressure. Using the editor, complex fluid simulations can be performed without prior knowledge of the underlying fluid dynamics equations. Finally, depending on the application fluid rendering techniques are developed using standard Silicon Graphics workstation hardware routines.

Details

Engineering Computations, vol. 12 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

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