Bardina Model Research Articles

Mixed modeling for large-eddy simulation: The single-layer and two-layer minimum-dissipation-Bardina models

Predicting the behavior of turbulent flows using large-eddy simulation requires a modeling of the subgrid-scale stress tensor. This tensor can be approximated using mixed models, which combine the dissipative nature of functional models with the capability of structural models to approximate out-of-equilibrium effects. We propose a mathematical basis to mix (functional) eddy-viscosity models with the (structural) Bardina model. By taking an anisotropic minimum-dissipation (AMD) model for the eddy viscosity, we obtain the (single-layer) AMD–Bardina model. In order to also obtain a physics-conforming model for wall-bounded flows, we further develop this mixed model into a two-layer approach: the near-wall region is parameterized with the AMD–Bardina model, whereas the outer region is computed with the Bardina model. The single-layer and two-layer AMD–Bardina models are tested in turbulent channel flows at various Reynolds numbers, and improved predictions are obtained when the mixed models are applied in comparison to the computations with the AMD and Bardina models alone. The results obtained with the two-layer AMD–Bardina model are particularly remarkable: both first- and second-order statistics are extremely well predicted and even the inflection of the mean velocity in the channel center is captured. Hence, a very promising model is obtained for large-eddy simulations of wall-bounded turbulent flows at moderate and high Reynolds numbers.

Assessment of two-parameter mixed models for large eddy simulations of transitional and turbulent flows

In the present study, improved two-parameter mixed models for large eddy simulations are proposed based on previous two-parameter mixed models of Salvetti and Banerjee [1] and Horiuti [2]. The subgrid-scale (SGS) stress in our models is decomposed into the modified Leonard stress, modified cross stress and modified SGS Reynolds stress terms. Although the modified Leonard stress term is explicitly calculated based on the scale-similarity, the modified cross stress term is built using an extension of the filtered Bardina model proposed by Horiuti [3] for better predictions of the interaction between resolved and unresolved scales (i.e., energy exchange). The modified SGS Reynolds stress is modeled by the dynamic Smagorinsky model or by a dynamic global model, leading to two unknown model coefficients for the modified cross stress and the modified SGS Reynolds stress terms. In order to demonstrate the reliability of the proposed SGS models, large eddy simulations of two types of flows (i.e., a fully developed turbulent channel flow and a transitional boundary layer flow) are performed. It is shown that the modified cross stress term makes an important contribution to the accurate predictions of such flows because the emergence of negative SGS dissipation (backward scatter) by the modified cross stress term decreases the excessive positive SGS dissipation (forward scatter). A direct comparison of the turbulent statistics with those from previous SGS models shows that the proposed SGS models result in better prediction performance both in transitional and turbulent flows.

Global and exponential attractors for the 3D Kelvin-Voigt-Brinkman-Forchheimer equations

The dynamics of three dimensional Kelvin-Voigt-Brinkman- Forchheimer equations in bounded domains is considered in this work. The existence and uniqueness of strong solution to the system is obtained by exploiting the \begin{document}$ m $\end{document} -accretive quantization of the linear and nonlinear operators. The long-term behavior of solutions of such systems is also examined in this work. We first establish the existence of an absorbing ball in appropriate spaces for the semigroup associated with the solutions of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations. Then, we prove that the semigroup is asymptotically compact, which implies the existence of a global attractor for the system. Next, we show the differentiability of the semigroup with respect to the initial data and then establish that the global attractor has finite Hausdorff and fractal dimensions. Furthermore, we establish the existence of an exponential attractor and discuss about its fractal dimensions for the associated semigroup of such systems. Finally, we discuss about the inviscid limit of the 3D Kelvin-Voigt-Brinkman-Forchheimer equations to the 3D Navier-Stokes-Voigt system and then to the simplified Bardina model.

On the Bardina’s Model in the Whole Space

We consider the Bardina's model for turbulent incompressible flows in the whole space with a cut-off frequency of order 1/alpha. We show that for any alpha >0 fixed, the model has a unique regular solution defined for all time.

Continuous data assimilation algorithm for simplified Bardina model

We present a continuous data assimilation algorithm for three-dimensional viscous simplified Bardina turbulence model, based on the fact that dissipative dynamical systems possess finite degrees of freedom. We construct an approximating solution of simplified Barbina model through an interpolant operator which is obtained using observational data of the system. This interpolant is inserted to theoric model coupled to a relaxation parameter, and our main result provides conditions on the finite-dimensional spatial resolution of collected measurements sufficient to ensure that the approximating solution converges to the theoric solution of the model. Global well-posedness of approximating solutions and related results with degrees of freedom are also presented.

The Cauchy problem for a Bardina–Oldroyd model to the incompressible viscoelastic flow

In this paper, we study the Cauchy problem for a regularized viscoelastic fluid model in space dimension two, the Bardina–Oldroyd model, which is inspired by the simplified Bardina model for the turbulent flows of fluids, introduced by Cao et al. (2006). In particular, we obtain the local existence of smooth solutions to this model via the contraction mapping principle. Furthermore, we prove the global existence of smooth solutions to this system.

On a regularized family of models for the full Ericksen–Leslie system

We consider a general family of regularized systems for the full Ericksen–Leslie model for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-α-like equation, the Leray-α equation, the Modified Leray-α equation, the Simplified Bardina model, the Navier–Stokes–Voigt model and the Navier–Stokes equation) for the fluid velocity u suitably coupled with a parabolic equation for the director field d. We establish existence, stability and regularity results for this family. We also show the existence of a finite dimensional global attractor for our general model, and then establish sufficiently general conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.

Numerical study of a regularized barotropic vorticity model of geophysical flow

We study a Crank–Nicolson in time, finite element in space, numerical scheme for a Bardina regularization of the barotropic vorticity (BV) model. We derive the regularized model from the simplified Bardina model in primitive variables, present a numerical algorithm for it, and prove the algorithm is unconditionally stable with respect to the timestep size and optimally convergent in both space and time. Numerical experiments are provided that verify the theoretical convergence rates, and also that test the model/scheme on a benchmark double-gyre wind forcing experiment. For the latter test, we find the proposed model/scheme gives a good coarse mesh approximation to the highly resolved direct numerical simulation of the BV model, and compares favorably to related regularization model results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1492–1514, 2015

Inertial Manifolds for Certain Subgrid-Scale $\alpha$-Models of Turbulence

In this paper we prove the existence of an inertial manifold, i.e., a globally invariant, exponentially attracting, finite-dimensional smooth manifold, for two different subgrid-scale $\alpha$-models of turbulence, the simplified Bardina model and the modified Leray-$\alpha$ model, in two-dimensional space. That is, we show the existence of an exact rule that parameterizes the dynamics of small spatial scales in terms of the dynamics of large ones. In particular, this implies that the long-time dynamics of these turbulence models is equivalent to that of a finite-dimensional system of ordinary differential equations.

Regularized family of models for incompressible Cahn–Hilliard two-phase flows

We consider a general family of regularized models for incompressible two-phase flows based on the Cahn–Hilliard formulation in n-dimensional compact Riemannian manifolds (with or without boundary) for n=2,3. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes model, and many others) for the fluid velocity u suitably coupled with a convective Cahn–Hilliard equation for the order (phase) parameter ϕ. We give a unified analysis of the entire three-parameter family of two-phase models. We first establish existence, stability and regularity results. Then, we show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory (u,ϕ) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.

On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in $$n$$ -dimensional compact Riemannian manifolds for $$n=2,3$$ . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- $$\alpha $$ -like model, the Leray- $$\alpha $$ model, the modified Leray- $$\alpha $$ model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity $$u$$ suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter $$\phi $$ . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the $$\alpha \rightarrow 0$$ limit in $$\alpha $$ models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory $$\left( u,\phi \right) $$ converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in $$n$$ -dimensional compact Riemannian manifolds.

Stochastic simplified bardina turbulent model: existence of weak solution

In this paper, we consider the stochastic version of the 3D Bardina model arising from the turbulent flows of fluids. We obtain the existence of probabilistic weak solution for the model with the non-Lipschitz condition.

Mathematical Results for Some α Models of Turbulence with Critical and Subcritical Regularizations

In this paper, we establish the existence of a unique "regular" weak solution to turbulent flows governed by a general family of $\alpha$ models with critical regularizations. In particular this family contains the simplified Bardina model and the modified Leray-$\alpha$ model. When the regularizations are subcritical, we prove the existence of weak solutions and we establish an upper bound on the Hausdorff dimension of the time singular set of those weak solutions. The result is an interpolation between the bound proved by Scheffer for the Navier-Stokes equations and the regularity result in the critical case.

Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier–Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier–Stokes-α equation in the nonmagnetic case). However, the Navier–Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model.

Global well-posedness of a Bardina model

We prove the existence of a global smooth solution to a viscous simplified Bardina turbulence model when the spacial dimension n satisfies 3≤n≤8.

Analysis of a General Family of Regularized Navier–Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.

Global existence and finite dimensional global attractor for a 3D double viscous MHD-α model

We consider a magnetohydrodynamic-α model with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. We prove the existence of a global solution and a global attractor. Moreover, we provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the system. In some sense, this result provides an intermediate bound between the number of degrees of freedom for the simplified Bardina model and the Navier–Stokes-α equation.

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

The authors investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, upper bounds for the number of determining modes are derived for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view the authors consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, it is also shown that the weak solutions of the Navier-Stokes-Voight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient ν → 0.

Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence

Inviscid regularization modeling of turbulent flow is investigated. Homogeneous, isotropic, decaying turbulence is simulated at a range of filter widths. A coarse-graining of turbulent flow arises from the direct regularization of the convective nonlinearity in the Navier–Stokes equations. The regularization is translated into its corresponding sub-filter model to close the equations for large-eddy simulation (LES). The accuracy with which primary turbulent flow features are captured by this modeling is investigated for the Leray regularization, the Navier–Stokes-α formulation (NS-α), the simplified Bardina model and a modified Leray approach. On a PDE level, each regularization principle is known to possess a unique, strong solution with known regularity properties. When used as turbulence closure for numerical simulations, significant differences between these models are observed. Through a comparison with direct numerical simulation (DNS) results, a detailed assessment of these regularization principles is made. The regularization models retain much of the small-scale variability in the solution. The smaller resolved scales are dominated by the specific sub-filter model adopted. We find that the Leray model is in general closest to the filtered DNS results, the modified Leray model is found least accurate and the simplified Bardina and NS-α models are in between, as far as accuracy is concerned. This rough ordering is based on the energy decay, the Taylor Reynolds number and the velocity skewness, and on detailed characteristics of the energy dynamics, including spectra of the energy, the energy transfer and the transfer power. At filter widths up to about 10% of the computational domain-size, the Leray and NS-α predictions were found to correlate well with the filtered DNS data. Each of the regularization models underestimates the energy decay rate and overestimates the tail of the energy spectrum. The correspondence with unfiltered DNS spectra was observed often to be closer than with filtered DNS for several of the regularization models.

?Perfect? modeling framework for dynamic SGS model testing in large eddy simulation

A perfect modeling framework for the systematic study of the effect of filter shape on the resolved scales of motion in large eddy simulation is developed. The effects of the explicit and implicit filtering approaches in large eddy simulation are considered. A simple model for smooth filtering is proposed and the related effects are analyzed. The proposed approach provides an effective research tool for assessing the behavior of sub-grid scale models in a dynamic fashion. The performances of various classical models are examined by using the perfect modeling formalism for simulating the large and/or the small residual scales effect. Numerical experiments are performed for decaying isotropic turbulence. The consistency of the sub-grid scale models with the effective composite filter employed in real simulations is discussed. The necessity of using mixed models when solving doubly-filtered Navier–Stokes equations is verified. It is found that time evolution of large scale velocity field is more sensitive to sub-grid large scale models like Bardina model, while the grid-filtered sub-filter scale model is necessary to ensure the proper energy dissipation.