Finite element, stream function–vorticity solution of steady laminar natural convection

Stevens, W.N.R.

doi:10.1002/fld.1650020404

Cited by 29 publications

(7 citation statements)

References 15 publications

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“…[14]. See [7,12] for the case of the biharmonic equation and [2,1,5,18,25] for the steady Navier-Stokes equation. The main contribution of this paper is an efficient time stepping procedure of (1.5) which will be described in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Simple finite element method in vorticity formulation for incompressible flows

Liu

Weinan

2000

Math. Comp.

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Abstract. A very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation. This method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work. Optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements. The method is convectively stable and is particularly suited for moderate to high Reynolds number flows.

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Section: Introductionmentioning

confidence: 99%

Simple finite element method in vorticity formulation for incompressible flows

Liu

Weinan

2000

Math. Comp.

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“…To solve this problem in the same form as given by others, 25 Results are presented in terms of the Rayleigh number, which is Ra = GrPr. All results are given for Pr =0.71.…”

Section: Natural Convection Problemmentioning

confidence: 99%

Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations

Macarthur

Patankar

1989

Numerical Methods in Fluids

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SUMMARYSemidirect solution techniques can be an effective alternative to the more conventional iterative approaches used in many finite difference methods. This paper summarizes several semidirect techniques which generally have not been applied to the Navier-Stokes and energy equations in finite difference form. The methods presented use both successive substitution and Jacobian-based updates as well as two variations of Broyden's full matrix update. A hybrid method is also presented, as is a norm-reducing search technique that can be used to enhance the convergence characteristics of Any semidirect approach. These methods have been compared with the well known iterative methods SIMPLE and SIMPLER. The comparison was performed on the natural convection and driven cavity problems. The semidirect methods proved to be reliably convergent without the need for a priori specification of variable under-relaxation factors, which was necessary with the iterative methods. Natural convection and driven cavity solutions have been readily obtained with the proposed methods for Rayleigh and Reynolds numbers up to 10" and 10' respectively. Of the semidirect techniques, the hybrid approach was the most robust. From an arbitrary zero initial guess this method was able to obtain a solution to the natural convection problem for Rayleigh numbers three orders of magnitude larger than was possible with the Newton--Raphson update. The computational effort required by the semidirect methods is comparable to that required by the iterative methods; however, the memory requirements can be significantly greater.

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“…In this paper the theoretical aspects is explained only for natural convection but in the illustrations both advection and natural convection problems were discussed. If the internal heating by viscous dissipation is neglected, :t/L chosen as u* and the Rayleigh and Prandtl numbers employed as dimensionless groups, eqns (12) and (13) may be rewritten as:…”

Section: The Density Satisfies An Equation Of State Of the Form P = Pmentioning

confidence: 99%

Finite element solution for advection and natural convection flows

Ramaswamy

1988

Computers & Fluids

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A new equal order velocity-pressure finite element procedure is presented for the calculation of 2-D viscous, incompressible flows of a recirculating nature. As in the finite difference procedures, velocity and pressure are uncoupled and the equations are solved one after the other. In this splitting-up method, an auxilary velocity field is computed first, which accounts for all contributions to the acceleration, except pressure, and satisfies the velocity boundary conditions. Then, the final velocities are evaluated by adding to the auxilary velocities pressure contributions which are computed to satisfy the continuity equation. The effectiveness is illustrated via example problems of 2-D advection and natural convection flows. 69~ s~o U UO!lOOAUOO [l~.myeu pue uo!:looAp~ .loj uognlos luotuolo ol[u!A

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Finite element, stream function–vorticity solution of steady laminar natural convection

Cited by 29 publications

References 15 publications

Simple finite element method in vorticity formulation for incompressible flows

Simple finite element method in vorticity formulation for incompressible flows

Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations

Finite element solution for advection and natural convection flows

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