A model for two coupled turbulent fluids Part III: Numerical approximation by finite elements

Bernardi, Christine; Rebollo, Tomás Chacón; Mármol, Macarena Gómez; Lewandowski, Roger; Murat, François

doi:10.1007/s00211-003-0490-9

Cited by 15 publications

(21 citation statements)

References 13 publications

(25 reference statements)

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“…We refer to [7] for the mathematical theory in the spirit of Theorem 1.1, for further features related to this system and references regarding the analysis, numerical computations and some applications (further details and more references can be found in a more recent book [13]). From the point of view of mathematical analysis of initial and/or boundary-value problems relevant to the Navier-Stokes system with the viscosity depending on other scalar quantity/quantities, we recall several works on analysis of problems related to or motivated by (1.59) that were established prior to [13], see [27,28,29,4,5,3,18]. 1.7.…”

Section: Main Resultmentioning

confidence: 99%

“…In the same manner, by using (4.40) and (4.39) (or (4.31)), we conclude that v k E k ⇀ vE weakly in L For the last term appearing in (4.61) for which we have not identified the weak limit yet, we use (4.35), (4.24) and (4.40) to conclude that π k 3 bv k ⇀ p 3 v weakly in L βg (0, T ; L 2 loc (Ω) 3 ). (4.67) Hence, recalling also (4.55) and (4.52), we can deduce from (4.61) that, for arbitrary Ω ′ ⊂ Ω ′ ⊂ Ω, where β min is defined in Theorem 1.1.…”

Section: 31mentioning

confidence: 95%

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Large data analysis for Kolmogorov’s two-equation model of turbulence

Bulíček

Málek

2019

Nonlinear Analysis: Real World Applications

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Kolmogorov seems to have been the first to recognize that a twoequation model of turbulence might be used as the basis of turbulent flow prediction. Nowadays, a whole hierarchy of phenomenological two-equation models of turbulence is in place. The structure of their governing equations is similar to the Navier-Stokes equations for incompressible fluids, the difference is that the viscosity is not constant but depends on the fraction of the scalar quantities that measure the effect of turbulence: the average of the kinetic energy of velocity fluctuations (i.e. the turbulent energy) and the measure related to the length scales of turbulence. For these two scalar quantities two additional evolutionary convection-diffusion equations are augmented to the generalized Navier-Stokes system. Although Kolmogorov's model has so far been almost unnoticed it exhibits interesting features. First of all, in contrast to other two-equation models of turbulence there is no source term in the equation for the frequency. Consequently, nonhomogeneous Dirichlet boundary conditions for the quantities measuring the effect of turbulence are assigned to a part of the boundary. Second, the structure of the governing equations is such that one can find an "equivalent" reformulation of the equation for turbulent energy that eliminates the presence of the energy dissipation acting as the source in the original equation for turbulent energy and which is merely an L 1 quantity. Third, the material coefficients such as the viscosity and turbulent diffusivities may degenerate, and thus the a priori control of the derivatives of the quantities involved is unclear.We establish long-time and large-data existence of a suitable weak solution to three-dimensional internal unsteady flows described by Kolmogorov's two-equation model of turbulence. The governing system of equations is completed by initial and boundary conditions; concerning the velocity we consider generalized stick-slip boundary conditions. The fact that the admissible class of boundary conditions includes various types of slipping mechanisms on the boundary makes the result robust from the point of view of possible applications.

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Section: Main Resultmentioning

confidence: 99%

Section: 31mentioning

confidence: 95%

Large data analysis for Kolmogorov’s two-equation model of turbulence

Bulíček

Málek

2019

Nonlinear Analysis: Real World Applications

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“…Following the recent works [5] and [30], we are interested in the finite element discretization of the still rather complex system (1.2). Our approach relies on the following remark: Both experiments of measurements and numerical simulations indicate that, in a large part of the domain Ω, the turbulent energy k is negligible.…”

Section: Introductionmentioning

confidence: 99%

Automatic Insertion of a Turbulence Model in the Finite Element Discretization of the Navier–stokes Equations

Bernardi

Rebollo

Hecht

et al. 2009

Math. Models Methods Appl. Sci.

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We consider the finite element discretization of the Navier-Stokes equations locally coupled with the equation for the turbulent kinetic energy through an eddy viscosity. We prove a posteriori error estimates which allow to automatically determine the zone where the turbulent kinetic energy must be inserted in the Navier-Stokes equations and also to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.Résumé: Nous considérons une discrétisation paréléments finis deséquations de NavierStokes couplées localement avec l'équation de l'énergie cinétique turbulente par une viscosité turbulente. Nous prouvons des estimations d'erreur a posteriori qui permettent de déterminer automatiquement la zône où l'énergie cinétique turbulente doitêtre insérée dans leséquations de Navier-Stokes et simultanément d'adapter le maillage pour optimiser la discrétisation de ceséquations. Des résultats numériques confirment l'intérêt d'une telle approche.

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“…To perform a somewhat realistic computation, we consider turbulent viscosities α i and γ i with the structure ν t + √ k. We consider the data reported in [5]: • Friction coefficients (coming also from [5]):…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Spectral and Finite Element discretizations were studied in subsequent papers by the same authors and coworkers (see [4,5]). In these papers, the ability of these discretization techniques to approach the solution of model (1.1) was proved.…”

Section: Introductionmentioning

confidence: 99%

An iterative procedure to solve a coupled two-fluids turbulence model

2010

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Abstract. This paper introduces a scheme for the numerical approximation of a model for two turbulent flows with coupling at an interface. We consider the variational formulation of the coupled model, where the turbulent kinetic energy equation is formulated by transposition. We prove the convergence of the approximation to this formulation for 3D flows for large turbulent viscosities and smooth enough flows, whenever bounded in W 1,p Sobolev norms for p large enough. Under the same assumptions, we show that the limit is a solution of the initial problem. Finally, we give some numerical experiments to enlighten the theoretical work.Mathematics Subject Classification. 63N30, 76M10.

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A model for two coupled turbulent fluids Part III: Numerical approximation by finite elements

Cited by 15 publications

References 13 publications

Large data analysis for Kolmogorov’s two-equation model of turbulence

Large data analysis for Kolmogorov’s two-equation model of turbulence

Automatic Insertion of a Turbulence Model in the Finite Element Discretization of the Navier–stokes Equations

An iterative procedure to solve a coupled two-fluids turbulence model

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