Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

Guevremont, G.; Habashi, Wagdi G.; Hafez, M.

doi:10.1002/fld.1650110511

Cited by 14 publications

(9 citation statements)

References 11 publications

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“…It solves directly for the vorticity, and it has been argued that methods that do so are more physically accurate, particularly near boundaries [8]. Using vorticity equations for fluid dynamics solvers has a long history and has been a subject of intensive studies, see, e.g., [16,18,24,25,34,35] for a sample of results. Furthermore, it was pointed out recently in [30], see also the discussion in [14], that the discrete vorticity w h from the finite element vorticity equation is a more natural quantity than r Â u h for the discrete balance laws for vorticity, enstrophy and helicity when the forcing terms are conservative.…”

Section: ð1:3þmentioning

confidence: 99%

Assessment of a vorticity based solver for the Navier–Stokes equations

Benzi

Olshanskii

Rebholz

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tWe investigate numerically a recently proposed vorticity based formulation of the incompressible Navier-Stokes equations. The formulation couples a velocity-pressure system with a vorticity-helicity system, and is intended to provide a numerical scheme with enhanced accuracy and superior conservation properties. For a few benchmark problems, we study the performance of a finite element method for this formulation and compare it with the commonly used velocity-pressure based finite element method. It is shown that both steady and unsteady discrete problems in the new formulation admit simple decoupling strategies followed by the application of iterative solves to auxiliary subproblems. Further, we compare several iterative strategies to solve the discrete problems and study the interplay between the choice of stabilization parameters in the finite element method and the efficiency of linear algebra solvers.

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Section: ð1:3þmentioning

confidence: 99%

Assessment of a vorticity based solver for the Navier–Stokes equations

Benzi

Olshanskii

Rebholz

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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“…Initial testing of VVH has shown great promise, and there are several novel properties of the formulation that warrant further development and testing. First, VVH is a velocity-vorticity system, which can provide more accurate solutions than velocity-pressure systems [8,10,11,18,19,24,26,28,27]. Typically such an improvement in accuracy comes at a cost, but in [22] an efficient iterative scheme for VVH is devised that performed very well on initial tests.…”

mentioning

confidence: 99%

On Error Analysis for the 3D Navier–Stokes Equations in Velocity-Vorticity-Helicity Form

Lee¹,

Olshanskii²,

Rebholz³

2011

SIAM J. Numer. Anal.

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Abstract. We present a rigorous numerical analysis and computational tests for the Galerkin finite element discretization of the velocity-vorticity-helicity formulation of the equilibrium NavierStokes equations (NSEs). This formulation, recently derived by the authors, is the first NSE formulation that directly solves for helicity and the first velocity-vorticity formulation to naturally enforce incompressibility of the vorticity, and preliminary computations confirm its potential. We present a numerical scheme; prove stability, existence of solutions, uniqueness under a small data condition, and convergence; and provide numerical experiments to confirm the theory and illustrate the effectiveness of the scheme on a benchmark problem.

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“…Wong and Baker [32] introduced a second-order accurate Taylor's series expansion to compute the vorticity values at the boundaries. Davis and Carpenter [31] used an integral approach for vorticity definition at the boundary, as followed by Guevremont et al [35].…”

Section: Velocity-vorticity Formulationmentioning

confidence: 99%