A Finite Element Variational Multiscale Method for the Navier--Stokes Equations

John, Volker; Kaya, Songül

doi:10.1137/030601533

Cited by 162 publications

(179 citation statements)

References 24 publications

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“…In [10], the method is extended to compressible Navier-Stokes equations coupled with further transport equations to describe chemically reacting flows. Finally, turbulent flows including the 3D mixing layer problem are numerically computed by the local projection method in [29,31].…”

mentioning

confidence: 99%

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

172

183

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Abstract. The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard two-level version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach.Mathematics Subject Classification. 65N12, 65N30, 76D05.

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mentioning

confidence: 99%

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

172

183

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“…We will use in our studies the Crank-Nicolson scheme, θ 1 = θ 2 = θ 3 = θ 4 = 0.5, since this scheme has been proven to be a good compromise between accuracy and efficiency [16,24]. For the main features of an efficient implementation of the fully implicit approach we refer to [18].…”

Section: Projection-based Finite Element Variational Multiscale Methodsmentioning

confidence: 99%

“…In [18] it has been shown that an efficient implementation of the one-level method requires this space to consist of discontinuous functions. This is also understandable from the fact that the resolved small scales are the projection of an already discontinuous function, namely the deformation tensor of the finite element velocity.…”

Section: The Projection-based Finite Element Variational Multiscale Mmentioning

confidence: 99%

“…This is also understandable from the fact that the resolved small scales are the projection of an already discontinuous function, namely the deformation tensor of the finite element velocity. In [18,25,20,21] numerical studies were performed for L H = P 0 and…”

Section: The Projection-based Finite Element Variational Multiscale Mmentioning

confidence: 99%

“…The projection for the definition of the scales is contained explicitly in the set of equations. This projection-based FEVMS method was proposed in [18]. Its parameters are the finite element spaces used to define the scale decomposition and the turbulence model acting directly only on the resolved small scales.…”

Section: Introductionmentioning

confidence: 99%

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A variational multiscale method for turbulent flow simulation with adaptive large scale space

John¹,

Kindl²

2010

Journal of Computational Physics

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In turbulent flows it is only feasible to simulate large flow structures. Variational multiscale (VMS) methods define these flow structures by projections into appropriate function spaces. This paper presents a finite element VMS method which defines the large scale projection space adaptively. The adaption controls the influence of an eddy viscosity model and it is based on the size of the so-called resolved small scales. The adaptive procedure is described in detail. Numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented. It is shown that the method selects the large scale space in a correct way and that appropriately chosen parameters improve the results compared to the basic method, which uses the same local large scale space in the whole domain and for all times.

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Sensitivity computations of eddy viscosity models with an application in drag computation

Pahlevani

2006

Numerical Methods in Fluids

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SUMMARYThis paper presents a numerical study of the sensitivity of an eddy viscosity model with respect to the variation of the eddy viscosity parameter for the two-dimensional driven cavity problem and ow around a cylinder. The main objective is to provide a comparison between computing the sensitivity using sensitivity equation and computing the sensitivity using ÿnite di erence methods and also numerically illustrate the application of the sensitivity computations in improving drag ow functional.

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A Finite Element Variational Multiscale Method for the Navier--Stokes Equations

Cited by 162 publications

References 24 publications

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

A variational multiscale method for turbulent flow simulation with adaptive large scale space

Sensitivity computations of eddy viscosity models with an application in drag computation

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