Variational Multiscale Formulation Research Articles

A mixed large eddy simulation model based on the residual-based variational multiscale formulation

A mixed model based on the residual-based variational multiscale (RBVM) formulation is proposed for the large eddy simulation of turbulent flows. In this model the cross stresses are modeled by the RBVM term and the Reynolds stresses are represented by a Smagorinsky eddy viscosity. The mixed model is motivated by an a priori analysis of a turbulent flow field that indicates that the RBVM term correctly models the dissipation induced by the cross-stress term but underpredicts the contribution from the Reynolds stress term. The dynamic version of the mixed model is implemented in a Fourier-spectral code in order to predict the decay of incompressible, three-dimensional homogeneous isotropic turbulence. It is found that the mixed model yields better agreement with direct numerical simulation solution than the dynamic Smagorinsky and the RBVM models, which are its building blocks.

Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients. In this paper, we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques. These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart–Thomas spaces, and the second non-negative formulation is based on the variational multiscale formulation. For the former formulation we comment on the effect of adding the non-negative constraint on the local mass balance property of the Raviart–Thomas formulation. We perform numerical convergence analysis of the proposed optimization-based non-negative mixed formulations. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

AbstractIn the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.

On a two-level element-free Galerkin method for incompressible fluid flow

In this paper, a new element-free Galerkin method called the two-level element-free Galerkin method is presented for incompressible fluid flow. This method consists of two parts: One at the global level and the other at the local level. The new method is based on the Hughes' variational multiscale formulation, and arises from a decomposition of the unknown variables into coarse/resolved scale and fine/unresolved scale. In order to find the unresolved part of the solution, the momentum equation is used to define the unresolved governing equation on the local level. Finally the application of the proposed method to incompressible fluid flow is displayed. Numerical solutions obtained from the two-level element-free Galerkin method are present for three benchmark problems, and the results indicate that the proposed method is a robust and accurate method. This is due to the addition of the unresolved scale which adds stability to the discrete problem when equal order basis is used for pressure and velocity.

A stochastic multiscale framework for modeling flow through random heterogeneous porous media

Flow through porous media is ubiquitous, occurring from large geological scales down to the microscopic scales. Several critical engineering phenomena like contaminant spread, nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena. Such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system. Any realistic modeling of these transport phenomena has to resolve two key issues: (i) the multi-length scale variations in permeability that these systems exhibit, and (ii) the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes. A stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features. A stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process. Recent developments in linear and non-linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model. An adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations (SPDEs). The framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given.

A two-parameter variational multiscale method for large eddy simulation

A two-parameter version of the variational multiscale (VMS) formulation with two Smagorinsky-type terms is proposed for the large eddy simulation of incompressible turbulent flows. One of the terms is the standard Smagorinsky eddy viscosity model that is applied to all resolved scales. The other is a fine scale addition that is constructed from the fine scale velocity field and is applied only to the fine scale equations. The new model is a generalization of the original variational multiscale model, which employs an eddy viscosity that acts only in the fine scales. The ability of the new two-parameter model to accurately predict sustained homogeneous isotropic turbulence is assessed by implementing it in a spectral code. Its performance is compared to the Smagorinsky model and the single-parameter VMS model. It is found that the single-parameter VMS model performs well at low to moderate Reynolds numbers, that the Smagorinsky model is more accurate at high Reynolds numbers, and that the new two-parameter VMS model performs well over the entire range. This performance is explained on the basis of the effective wavenumber dependent eddy viscosity for each model.

On a multi-scale element-free Galerkin method for the Stokes problem

In this paper, a multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy.

A multi-scale formulation for compressible turbulent flows suitable for general variational discretization techniques

Based on the recently introduced variational multi-scale (VMS) approach to large-eddy simulation (LES) as introduced in [T.J.R. Hughes, L. Mazzei, K.E. Jansen, Large eddy simulation and the variational multiscale method, Comput. Visual. Sci. 3 (2001) 47–59; S.S. Collis, Monitoring unresolved scales in multiscale turbulence modeling, Phys. Fluids 13 (6) (2001) 1800–1806], we present a VMS formulation which can be used in the simulation of compressible flows. Special attention is given to obtain a VMS formulation which is suitable for complex flow domains and general variational discretization techniques. A generalization of the Favre-averaging procedure is introduced such that the formulation resembles the Favre-filtered Navier–Stokes equations traditionally used in LES of compressible flow, and no explicit subgrid terms arise in the continuity equation. Also, we show that with the use of discretization methods other than Fourier-spectral methods the VMS-projection no longer commutes with differentiation. This results in additional subgrid scale terms which resemble the commutator error as encountered in the traditional filtering approach to LES.

On the model coefficients for the standard and the variational multi-scale Smagorinsky model

A theoretical analysis is presented on the behaviour of the model coefficients for the well-known Smagorinsky model and two variational multi-scale (VMS) variants of the Smagorinsky model. The dependency on two important parameters is addressed, i.e. the ratio of the LES-filter width ∆ and the Kolmogorov scale η on the one hand, and the ratio of the integral length scale L and the LES-filter width ∆ on the other hand. First of all, it is demonstrated that the model coefficients vary strongly with ∆/η. By evaluating the model coefficients as functions of the subgrid activity s (which expresses the relative contribution of the subgrid-scale model in the total dissipation, and corresponds to a nonlinear transformation of ∆/η), we show that a classical Lilly–Smagorinsky model overestimates the dissipation, even in cases where the dissipation of the subgrid-scale model is dominant. Therefore, generic and easy-to-use modifications to the different models are proposed, which provide close approximations to the models employing ‘exact’ coefficients. For the standard Smagorinsky model, this modified model corresponds to approximating the eddy viscosity νt as νt =( ν 2 Lilly + ν 2 ) 1/2 − ν, with νLilly the turbulent viscosity obtained by employing Lilly’s classical Smagorinsky constant and ν the laminar viscosity. Similar easy-to-use relations are presented for the variational multi-scale Smagorinsky models. Next to the ∆/η dependence of the model coefficients, the L/∆ behaviour is also elaborated. Although a strong dependence on L/∆ is observed for low values of the ratio, we do not advocate the use of L/∆-dependent model coefficients. Rather, the asymptotic L/∆ independence and the speed of asymptotic convergence are used as a tool to compare the quality of subgrid-scale models (e.g. L/∆ > 10 is a minimum order of magnitude for the small–small VMS model), and differences are observed between the standard Smagorinsky model and its two VMS variants. Finally, for the VMS models, the influence of the shape of the high-pass filter, used in the variational multi-scale formulation, is investigated. We observed that smooth high-pass filters result in more robust VMS Smagorinsky models.

A stabilized finite element method based on SGS models for compressible flows

We present an appropriate extension of the stabilized finite element formulation, introduced in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823] for the prediction of incompressible flows, aimed at compressible flows. The stabilized formulation is the so-called variable subgrid scale method (V-SGS) based on an approximation of the class of subgrid scale models (SGS) derived from the Hughes variational multiscale method (Hughes-VMS) introduced in [T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401]. It is characterized by a variable stabilization parameter within the domain of element. We also propose an innovative procedure for computing the stabilization parameter by using the one-dimensional element Green’s functions corresponding to an advective–diffusive differential operator. The stabilization parameter is defined as the sum of two components, the first providing its mean value and the second its element-wise space dependency. On this basis the present work proposes the stabilization matrix for the multidimensional advective–diffusive system of equations governing compressible flows. The procedure developed for computing the stabilization parameter can be applied to other differential operators as it has been shown in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823]. The proposed stabilization device is validated by simulation of inviscid and viscous supersonic flow configurations.

Improved Discontinuity-capturing Finite Element Techniques for Reaction Effects in Turbulence Computation

Recent advances in turbulence modeling brought more and more sophisticated turbulence closures (e.g. k-ɛ, k-ɛ -v2-f, Second Moment Closures), where the governing equations for the model parameters involve advection, diffusion and reaction terms. Numerical instabilities can be generated by the dominant advection or reaction terms. Classical stabilized formulations such as the Streamline–Upwind/Petrov–Galerkin (SUPG) formulation (Brook and Hughes, comput methods Appl Mech Eng 32:199–255, 1982; Hughes and Tezduyar, comput methods Appl Mech Eng 45: 217–284, 1984) are very well suited for preventing the numerical instabilities generated by the dominant advection terms. A different stabilization however is needed for instabilities due to the dominant reaction terms. An additional stabilization term, called the diffusion for reaction-dominated (DRD) term, was introduced by Tezduyar and Park (comput methods Appl Mech Eng 59:307–325, 1986) for that purpose and improves the SUPG performance. In recent years a new class of variational multi-scale (VMS) stabilization (Hughes, comput methods Appl Mech Eng 127:387–401, 1995) has been introduced, and this approach, in principle, can deal with advection–diffusion–reaction equations. However, it was pointed out in Hanke (comput methods Appl Mech Eng 191:2925–2947) that this class of methods also need some improvement in the presence of high reaction rates. In this work we show the benefits of using the DRD operator to enhance the core stabilization techniques such as the SUPG and VMS formulations. We also propose a new operator called the DRDJ (DRD with the local variation jump) term, targeting the reduction of numerical oscillations in the presence of both high reaction rates and sharp solution gradients. The methods are evaluated in the context of two stabilized methods: the classical SUPG formulation and a recently-developed VMS formulation called the V-SGS (Corsini et al. comput methods Appl Mech Eng 194:4797–4823, 2005). Model problems and industrial test cases are computed to show the potential of the proposed methods in simulation of turbulent flows.

A variational multiscale stabilized finite element method for the Stokes flow problem

A new stabilized method is proposed for the 2-D Stokes flow problem. The new approach is based on the Variational Multiscale(VM) formulation. The Green's functions for an infinite domain are adopted to compute the two-scale interaction. The method is shown to be able to stabilized low order Finite Element intepolation pairs and has a formulation similar to the one of Galerkin/Least-squares(GLS) formulation. The stabilization term obtained in this method has a physical interpretation. A couple of numerical examples show the method has a good convergence rate.

Smart element method II. An element based on the finite Eshelby tensor

In this study, we apply the newly derived finite Eshelby tensor in a variational multiscale formulation to construct a smart element through a more accurate homogenization procedure. The so-called Neumann–Eshelby tensor for an inclusion in a finite domain is used in the fine scale feedback procedure to take into account the interactions among different scales and elements. Numerical experiments have been conducted to compare the performance and robustness of the new element to earlier formulations. The results showed that the smart element constructed via the Neumann–Eshelby tensor of a finite domain provides better numerical accuracy than that constructed via the Eshelby tensor of an infinite domain. Moreover, it can relieve volumetric locking. Copyright © 2005 John Wiley & Sons, Ltd.

Multiscale-stabilized solutions to one-dimensional systems of conservation laws

We present a variational multiscale formulation for the numerical solution of one-dimensional systems of conservation laws. The key idea of the proposed formulation, originally presented by Hughes [Comput. Methods Appl. Mech. Engrg., 127 (1995) 387–401], is a multiple-scale decomposition into resolved grid scales and unresolved subgrid scales. Incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties, capable of accurately representing the sharp features of the solution. In the formulation developed herein, the multiscale split is invoked prior to any linearization of the equations. Special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity-capturing diffusion. The methodology is applied to the one-dimensional simulation of three-phase flow in porous media, and the shallow water equations. These numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear, nearly hyperbolic systems on very coarse grids. Application of the numerical formulation to multidimensional problems is presented in a forthcoming paper.

Variational eigenstrain multiscale finite element method

A new subgrid (subscale) finite element method is proposed, which is termed as the variational eigenstrain multiscale method. It combines the essential ideas of Hughes' variational multiscale formulation [Comput. Methods Appl. Mech. Engrg. 127 (1995) 387] with Eshelby's inclusion theory [Proc. R. Soc. Lond. A 252 (1957) 561] and Mura's equivalent eigenstrain principle [Micromechanics of Defects in Solids, Martinus Nijhoff Publisher, 1987] to homogenize numerical error due to finite element discretization. By synthesizing variational multiscale method with the equivalent eigenstrain principle, we have developed a new finite element formulation that can automatically homogenize its own discretization errors so that it may attain better accuracy in a coarse scale finite element computation than that of the original coarse scale finite element computation. The paper provides the theoretical foundation of the method as well as two numerical examples that illustrate and validate the proposed method.

A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes––application to vortex shedding

The variational multiscale formulation of large eddy simulations developed by Hughes and his co-workers for incompressible turbulent flows is first extended to the case of compressible turbulent flows. Next, a finite volume/element discretization on tetrahedral meshes of this method is proposed. This discretization features an economical procedure based on agglomeration for separating a priori the scales, a corresponding projector for eliminating the small scales from the system of equations, and an effective control of the numerical dissipation induced by upwinding. The resulting LES method is validated with the three-dimensional numerical simulation of a low-speed flow past a square cylinder at M ∞=0.1 and Re=22,000, and various comparisons with experimental data.

Large eddy simulation of turbulent channel flows by the variational multiscale method

The variational multiscale formulation of LES is applied to two-dimensional equilibrium and three-dimensional nonequilibrium channel flows. Simple, constant-coefficient Smagorinsky-type eddy viscosities, without wall damping functions, are used to model the decay of small scales, an approach which is not viable for wall-bounded flows within the traditional LES framework. Nevertheless, very good results are obtained.