Variational Multiscale Methods in Computational Fluid Dynamics

Codina, Ramon; Badia, Santiago; Baiges, Joan; Principe, Javier

doi:10.1002/9781119176817.ecm2117

Cited by 70 publications

(83 citation statements)

References 129 publications

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“…To circumvent the restrictions imposed by the inf‐sup condition and convection dominated flows, a VMS stabilization is applied, originally proposed in and later further developed in (see also for a review). When applied to the Navier‐Stokes problem, the time discrete version of Equation is replaced by:

\begin{align} false(ρ_{fl} δ_{2, t} {bold-italicu}_{h}^{n + 1}, {bold-italicv}_{h} false) + B false({bold-italicU}_{h}^{n + 1}, {bold-italicV}_{h} false) + \sum_{K} {(⟨⟩, {\overset{bold-italicu}{true}}^{n + 1}, ρ_{fl} {bold-italicu}_{h}^{n + 1} \cdot \nabla {bold-italicv}_{h} + μ_{fl} normalΔ {bold-italicv}_{h} + \nabla q_{h})}_{K} \\ prefix- \sum_{K} {(⟨⟩, {\overset{p}{true}}^{n + 1}, \nabla \cdot {bold-italicv}_{h})}_{K} = L false({bold-italicV}_{h} false), \end{align}

where

{\tilde{u}}^{n + 1}

and

{\tilde{p}}^{n + 1}

are the solution of

\begin{align} \frac{ρ_{fl}}{normalΔ t} ((), {\overset{bold-italicu}{true}}^{n + 1} prefix- {\overset{bold-italicu}{true}}^{n}) \end{align}

…”

Section: Incompressible Navier‐stokes Equationsmentioning

confidence: 99%

Fluid structure interaction by means of variational multiscale reduced order models

Tello

Codina

Baiges

2020

Numerical Meth Engineering

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A reduced order model designed by means of a variational multiscale stabilized formulation has been applied successfully to fluid-structure interaction problems in a strongly coupled partitioned solution scheme. Details of the formulation and the implementation both for the interaction problem and for the reduced models, for both the off-line and on-line phases, are shown. Results are obtained for cases in which both domains are reduced at the same time.Numerical results are presented for a semistationary and a fully transient case. K E Y W O R D Sfluid structure Interaction, nonlinear solid elastodynamics, reduced order model, variational multiscale method 1 Int J Numer Methods Eng. 2020;121:2601-2625.wileyonlinelibrary.com/journal/nme

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\begin{align} false(ρ_{fl} δ_{2, t} {bold-italicu}_{h}^{n + 1}, {bold-italicv}_{h} false) + B false({bold-italicU}_{h}^{n + 1}, {bold-italicV}_{h} false) + \sum_{K} {(⟨⟩, {\overset{bold-italicu}{true}}^{n + 1}, ρ_{fl} {bold-italicu}_{h}^{n + 1} \cdot \nabla {bold-italicv}_{h} + μ_{fl} normalΔ {bold-italicv}_{h} + \nabla q_{h})}_{K} \\ prefix- \sum_{K} {(⟨⟩, {\overset{p}{true}}^{n + 1}, \nabla \cdot {bold-italicv}_{h})}_{K} = L false({bold-italicV}_{h} false), \end{align}

where

{\tilde{u}}^{n + 1}

and

{\tilde{p}}^{n + 1}

are the solution of

\begin{align} \frac{ρ_{fl}}{normalΔ t} ((), {\overset{bold-italicu}{true}}^{n + 1} prefix- {\overset{bold-italicu}{true}}^{n}) \end{align}

…”

Section: Incompressible Navier‐stokes Equationsmentioning

confidence: 99%

Fluid structure interaction by means of variational multiscale reduced order models

Tello

Codina

Baiges

2020

Numerical Meth Engineering

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“…A residual based VMS method has been used in CFD applications for a large variety of problems (see e.g. [7,17,21,29,42]), and as a result such VMS method has proven to be an efficient and accurate turbulence model for applications characterized by moderately high Reynolds number flows. The method is a two-scales method, even though other VMS variants with a larger number of scales are available in literature (e.g.…”

Section: The Full Order Modelmentioning

confidence: 99%

A reduced order variational multiscale approach for turbulent flows

et al. 2019

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The purpose of this work is to present different reduced order model strategies starting from full order simulations stabilized using a residual based Variational MultiScale (VMS) approach. The focus is on flows with moderately high Reynolds numbers. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremizers in ROMs based on a VMS formulation. Finally, the developed methods are tested on a numerical benchmark.

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“…The VMS methodology has enjoyed a lot of progress since then. For an overview of the development consult the review paper [9].…”

Section: Contextmentioning

confidence: 99%

Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier–Stokes equations

Eikelder

Akkerman

2018

Computer Methods in Applied Mechanics and Engineering

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This paper presents the construction of a correct-energy stabilized finite element method for the incompressible Navier-Stokes equations. The framework of the methodology and the correct-energy concept have been developed in the convective-diffusive context in the preceding paper [M.F.P. ten Eikelder, I. Akkerman, Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. I: The convective-diffusive context, Comput. Methods Appl. Mech. Engrg. 331 (2018) 259-280]. The current work extends ideas of the preceding paper to build a stabilized method within the variational multiscale (VMS) setting which displays correct-energy behavior. Similar to the convection-diffusion case, a key ingredient is the proper dynamic and orthogonal behavior of the small-scales. This is demanded for correct energy behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin (SUPG) and the Galerkin/least-squares method (GLS).The presented method is a Galerkin/least-squares formulation with dynamic divergence-free small-scales (GLSDD). It is locally mass-conservative for both the large-and small-scales separately. In addition, it locally conserves linear and angular momentum. The computations require and employ NURBS-based isogeometric analysis for the spatial discretization. The resulting formulation numerically shows improved energy behavior for turbulent flows comparing with the original VMS method.

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Variational Multiscale Methods in Computational Fluid Dynamics

Cited by 70 publications

References 129 publications

Fluid structure interaction by means of variational multiscale reduced order models

Fluid structure interaction by means of variational multiscale reduced order models

A reduced order variational multiscale approach for turbulent flows

Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier–Stokes equations

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