Variational Multiscale Approach Research Articles

A finite-volume variational multiscale method coupled with a discrete interpolation filter for large-eddy simulation of isotropic turbulence and fully developed channel flow

The variational multiscale (VMS) approach proposed by Hughes et al. is applied to the Smagorinsky model and developed within the context of a physical-space unstructured finite-volume solver. A specific explicit filtering procedure is also developed, based on the idea of the discrete interpolation filters, originally proposed by Marsden et al. Tests on decaying isotropic turbulence at infinite Reynolds number and fully developed channel flow at friction Reynolds number Reτ=395 are performed, using the dynamic Smagorinsky model for comparison. For the VMS version of the Smagorinsky model, the conservative choice of not adapting the model’s constant is made, giving good indications on the required adjustment directions in both isotropic flows and nonisotropic flows. Promising results are obtained in general by the VMS approach with the two test cases. An overdissipative behavior is observed in the isotropic turbulence test case, which does not have a strong spectral connotation and could probably be cured adjusting the model’s constant. A very good prediction of the levels of unsteadiness is produced by VMS in the channel flow test case, showing the potential of this approach for nonisotropic flows. In both of the test cases the VMS approach gives performances of the same order of accuracy of the dynamic Smagorinsky model used for comparison, being at the same time less computationally expensive.

Large-eddy simulation of sheared interfacial flow

Large-eddy simulations (LES) of a turbulent interfacial gas-liquid flows are described in this paper. The variational multiscale approach (VMS) introduced by Hughes for single-phase flows is systematically assessed against direct numerical simulation (DNS) data obtained at a shear Reynolds number Re⋆=171, and compared to LES results obtained with the Smagorinsky model, modified by a near-interface turbulence decay treatment. The models are incorporated in the same pseudospectral DNS solver built within the boundary fitting method used by Fulgosi et al. for air-water flow. The LES are performed for physical conditions allowing low interface deformations that fall in the range of capillary waves of wave slope ak=0.01. The LES results show that both the modified Smagorinsky model and the VMS are capable to predict the boundary layer structure in the gas side, including the decay process, and to cope with the anisotropy of turbulence in the liquid blockage layer underneath the interface. Higher-order turbulence statistics, including the transfer of energy between the normal stresses is also well predicted by both approaches, but qualitatively the VMS results remain overall better than the modified Smagorinsky model. The study has demonstrated that the key to the prediction of the energy transfer mechanism is in the proper prediction of the fluctuating pressure field, which has been found out of reach of any of the LES methodologies. The superiority of the VMS is demonstrated through the analysis of the subgrid transport and exchange terms in the resolved kinetic energy, where it is indeed shown to be self-adaptive with regard to the eddy viscosity. Although VMS is shown to be sensitive to filter scale partition and model constant, the optimal setting can be easily translated in the interface tracking/finite-volume context, which makes it very useful for practical purposes. An important point is that the VMS approach yields very satisfactory results without the need for prescribing an ad-hoc damping function and the required distance to the interface.

A multiscale stabilization of the streamfunction form of the steady state Navier-Stokes equations

In this document, the derivation of a multiscale stabilized finite element method for the streamfunction formulation of the two-dimensional, incompressible Navier-Stokes equations is motivated and outlined. A linearized model problem is developed and analyzed through a variational multiscale approach to determine the form of the stabilized terms.

A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces

AbstractWe examine a bubble‐stabilized finite element method for enforcing Dirichlet constraints on embedded interfaces. By ‘embedded’ we refer to problems of general interest wherein the geometry of the interface is assumed independent of some underlying bulk mesh. As such, the robust imposition of Dirichlet constraints using a Lagrange multiplier field is not trivial. To focus issues, we consider a simple one‐sided problem that is representative of a wide class of evolving‐interface problems. The bulk field is decomposed into coarse and fine scales, giving rise to coarse‐scale and fine‐scale one‐sided sub‐problems. The fine‐scale solution is approximated with bubble functions, permitting static condensation and giving rise to a stabilized form bearing strong analogy with a classical method. Importantly, the method is simple to implement, readily extends to multiple dimensions, obviates the need to specify any free stabilization parameters, and can lead to a symmetric, positive‐definite system of equations. The performance of the method is then examined through several numerical examples. The accuracy of the Lagrange multiplier is compared to results obtained using a local version of the domain integral method. The variational multiscale approach proposed herein is shown to both stabilize the Lagrange multiplier and improve the accuracy of the post‐processed fluxes. Copyright © 2006 John Wiley & Sons, Ltd.

An entropy-variable-based VMS/GLS method for the simulation of compressible flows on unstructured grids

The variational multiscale approach for large-eddy simulation is investigated within a finite element framework. A filtering analog of this method, providing a model, which shares basic structures with hyperviscosity models, is used and evaluated on a freely decaying isotropic turbulence in the limit of infinite Reynolds numbers. The method used is based on a symmetric form of the Navier–Stokes equations stabilized with a Galerkin/least-squares approach. The proposed procedure provides appealing results. It is as satisfactory as the use of Germano and Lilly’s dynamic algorithm, without stabilization flaws, and at a lower cost. Comparisons between numerical dissipation and the proper subgrid closure show that the variational multiscale models tend to compute a turbulent viscosity that accounts for the numerical sources of dissipation, which reveals an important feature for an intensive and robust use of VMS-large-eddy simulation.

A variational multiscale higher-order finite element formulation for turbomachinery flow computations

The Variational MultiScale approach for finite elements addresses the inclusion of the effect of fine scales of the solution in the coarse problem. In this framework advective–diffusive–reactive equations modelling turbomachinery flows feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method called V-SGS ( Variable-SubGrid Scale), designed for quadratic elements, with a variable ‘intrinsic time’ parameter. Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, including the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on cruder meshes.

Variational multiscale stabilized FEM formulations for transport equations: stochastic advection–diffusion and incompressible stochastic Navier–Stokes equations

An extension of the deterministic variational multiscale (VMS) approach with algebraic subgrid scale (SGS) modeling is considered for developing stabilized finite element formulations for the stochastic advection and the incompressible stochastic Navier–Stokes equations. The stabilized formulations are numerically implemented using the spectral stochastic formulation of the finite element method (SSFEM). Generalized polynomial chaos and Karhunen–Loève expansion techniques are used for representation of uncertain quantities. The proposed stabilized method is then applied to various standard advection–diffusion and fluid-flow examples with uncertainty in essential boundary conditions. Comparisons are drawn between the numerical solutions and Monte-Carlo/analytical solutions wherever possible.

Multiscale analysis and modeling of multiphase chemical reactors

Abstract Multiphase gas–solids fluidized bed reactors are of multiscale structure, i.e. single particles, particle clusters/bubbles and reactor vessel, and of multiple physics, i.e. hydrodynamics, heat and mass transfer, and reaction kinetics. The formation of complex structures/patterns in each regime is a result of a compromise between dominant mechanisms at multiple scales. Coupling of hydrodynamics, heat and mass transfer, and reaction kinetics takes place at molecular and particle levels where conductive and convective transfer and diffusion within the internal pores of the catalyst are accompanied by the adsorption, surface reaction and desorption of reactant and product on the surface. This work first presents an overview of the general multiscale approaches, including the sequential approach, iterative/concurrent approach and the variational multiscale approach, applicable to multiphase flow and reaction systems, followed by a review on multiscale modeling of gas–solids fluidized bed reactors including bubbling/turbulent and fast fluidized bed reactors. Finally, a multiscale circulating fluidized bed reactor model is outlined in which hydrodynamics are modeled by the combined energy-minimization multiscale/two-fluid computational fluid dynamics coupled with multiscale heat and mass transfer sub-models and reaction kinetics. The interaction and compromise among dominant mechanisms are represented by the variational criterion.

A variational multiscale approach to strain localization – formulation for multidimensional problems

The multiscale approach to strain localization problems developed earlier, is generalized to higher dimensions. The key idea is the identification of the fine scale field with a component of the displacement that has a large gradient. The difference between the total displacement and the fine scale is the coarse scale field. The formulation is variationally based, with the weak form as the point of departure. With suitable assumptions on the space of fine scale fields, the weak form is separable into coarse and fine scale parts. The weak form of the fine scale problem is used to eliminate the fine scales from the formulation. This is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone. In this framework, fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh. It is demonstrated that by correctly accounting for the “microstructural” fine scale behavior, the formulation results in solutions devoid of pathological mesh sensitivity. For problems wherein the detailed structure of the localized displacement is of interest, recovery of the fine scale components can be performed to reconstruct the entire displacement field. Several numerical examples are presented that demonstrate the robustness of the formulation.