Subgrid Viscosity Research Articles

Hybrid presumed pdf and flame surface density approaches for Large-Eddy Simulation of premixed turbulent combustion: Part 1: Formalism and simulation of a quasi-steady burner

Extended Coherent Flame Model for Large-Eddy Simulation (ECFM-LES) and Presumed Conditional Moments-Flame Prolongation of Intrisic Low Dimensional Manifolds (PCM-FPI) are some of the combustion models exploited for Large-Eddy Simulations (LES) of turbulent premixed flames. Combustion is then either modeled by tracking the flame surface density or by combining computations of flamelets with presumed probability density functions (pdf). The first approach enables to control the turbulent flame speed but models chemical kinetics in a simplified manner. The second directly accounts for detailed chemistry via the flamelet structure but the turbulent propagation speed cannot be easily estimated a priori. Simple one-dimensional tests are then performed in this study to evaluate flame velocities of PCM-FPI. A restricted operating range of this model, that enables to retrieve an evolution of the propagating speed similar to ECFM-LES predictions, is highlighted. This zone is limited in terms of filter width and sub-grid scale turbulent viscosity. An attractive alternative to both ECFM-LES and PCM-FPI approaches thus appears to be a model integrating their respective main strengths. For this purpose, two hybrid models are proposed in this paper and tested through LES of a lean-premixed turbulent swirling flame. Results in terms of statistics for temperature and mass fractions of chemical species are compared to experimental data and to previous results already obtained with PCM-FPI. The coupled models enable to properly locate the reaction zone via the flame surface density closures along with a correct prediction of the chemistry evolution in the flame front.

An algebraic variational multiscale-multigrid method for large-eddy simulation: generalized-α time integration, Fourier analysis and application to turbulent flow past a square-section cylinder

This article studies three aspects of the recently proposed algebraic variational multiscale-multigrid method for large-eddy simulation of turbulent flow. First, the method is integrated into a second-order-accurate generalized-? time-stepping scheme. Second, a Fourier analysis of a simplified model problem is performed to assess the impact of scale separation on the overall performance of the method. The analysis reveals that scale separation implemented by projective operators provides modeling effects very close to an ideal small-scale subgrid viscosity, that is, it preserves low frequencies, in contrast to non-projective scale separations. Third, the algebraic variational multiscale-multigrid method is applied to turbulent flow past a square-section cylinder. The computational results obtained with the method reveal, on the one hand, the good accuracy achievable for this challenging test case already at a rather coarse discretization and, on the other hand, the superior computing efficiency, e.g., compared to a traditional dynamic Smagorinsky modeling approach.

New subgrid artificial viscosity Galerkin methods for the Navier–Stokes equations

We study subgrid artificial viscosity methods for approximating solutions to the Navier–Stokes equations. Two methods are introduced that add viscous stabilization via an artificial viscosity, then remove it only on a coarse mesh. These methods can be considered as conforming, mixed methods, the first for velocity and vorticity, and the second for velocity and its gradient, the former incorporating a naturally arising grad–div stabilization term. In this paper, we rigorously study the first scheme analytically, showing that it is unconditionally stable and optimally convergent, as well as both schemes computationally. Numerical experiments demonstrate the advantages of both of these methods.

Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.

Prediction of liquid-liquid flow in an SMX static mixer using large eddy simulations

AbstractThe main purpose of the paper is to apply the large eddy simulations (LES) technique and to verify its use as a predicting tool for turbulent liquid-liquid flow in an SMX static mixer. LES modeling was carried out using the Smagorinsky-Lilly model of the turbulent subgrid viscosity for the Reynolds number of 5000 and 10000. The continuous phase was water and the dispersed phase was silicon oil. The investigation covers the effects of the density ratio between the phases. Three different cases of liquid densities were considered. The dispersed phase concentration distribution in the mixer cross-sections was compared with the corresponding time averaged results obtained formerly for the same configuration in a steady-state simulation using the standard RANS approach with the k-ɛ model. The dependency of the standard deviation of the dispersed phase concentration on the distance from the mixer inlet and the impact of the centrifugal force on the phase concentration distribution were investigated. The presented results for the SMX static mixer confirm conclusions of previous studies by Jaworski et al. (2006) obtained for a Kenics static mixer and show less a pronounced influence of the centrifugal force on the phase concentration distribution of the LES results in comparison to the RANS case.

A Subgrid Model for the Time-Dependent Navier-Stokes Equations

We propose a stabilized subgrid finite-element method for the two-dimensional (2D) nonstationary incompressible Naver-Stokes equation (NSE). This method yields a subgrid eddy viscosity which does not act on the large flow structures. The proposed eddy viscous term is constructed by a fluctuation operator based on an L2-projection. The fluctuation operator can be implemented by the L2-projection from high-order interpolation finite-element spaces to the low-order interpolation finite-element spaces. In this paper, P2/P1 mixed finite-element spaces are adopted to implement the calculation and the analysis. The error analysis is given based on some regular assumptions. Finally, in the part of numerical tests, the numerical computations show that the numerical results agree with theoretical analysis very well. Meanwhile, the numerical investigations demonstrate that the proposed subgrid is very effective for high Reynolds number fluid flows and superior to other referred subgrid models.

Reduction of the Germano-identity error in the dynamic Smagorinsky model

We revisit the Germano-identity error in the dynamic modeling procedure in the sense that the current modeling procedure to obtain the dynamic coefficient may not truly minimize the error in the mean and global sense. A “corrector step” to the conventional dynamic Smagorinsky model is proposed to obtain a corrected eddy viscosity which further reduces the error. The change in resolved velocity due to the coefficient variation as well as nonlocal nature of the filter and flow unsteadiness is accounted for by a simplified suboptimal control formalism without resorting to the adjoint equations. The objective function chosen is the Germano-identity error integrated over the entire computational volume and pathline. In order to determine corrected eddy viscosity, the Fréchet derivative of the objective function is directly evaluated by a finite-differencing formula in an efficient predictor-corrector-type framework. The proposed model is applied to decaying isotropic turbulence and turbulent channel flow at various Reynolds numbers and resolutions to obtain noticeable reduction in the Germano-identity error and significantly improved flow statistics. From channel flow large-eddy simulation, it is shown that conventional dynamic model underestimates subgrid scale eddy viscosity when the resolution gets coarse, and this underestimation is responsible for increased anisotropy of predicted Reynolds stress. The proposed model raises both the overall and near-wall subgrid scale eddy viscosity to reduce exaggerated Reynolds stress anisotropy and yield significantly improved flow statistics.

A posteriori error estimates for subgrid viscosity stabilized approximations of convection–diffusion equations

We derive a posteriori error estimates for subgrid viscosity stabilized finite element approximations of convection–diffusion equations in the high Péclet number regime. Two estimators are analyzed: an asymptotically robust one and a fully robust one with respect to the Péclet number. Numerical results on test cases with boundary layers or internal layers show that the asymptotically robust estimator can be used to construct adaptive meshes.

Numerical analysis of a method for high Peclet number transport in porous media

A variationally consistent eddy viscosity discretization is presented in [W.J. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002) 147–157] for the stationary convection diffusion problem. This discretization is extended to the evolutionary problem in [N. Heitmann, Subgridscale stabilization of time-dependent convection dominated diffusive transport, J. Math. Anal. Appl. 331 (2007) 38–50] with a near optimal error bound. In the following, we couple this discretization with the porous media problem. We present a comprehensive analysis of stability and error for the velocity field derived from the porous media problem. Next, using a backward Euler approximation for the time derivative we follow the inherited error in velocity through the coupling with the convection diffusion problem. The method is shown to be stable and the error near optimal and independent of the diffusion coefficient, ϵ.

Chaos, transport and mesh convergence for fluid mixing

Chaotic mixing of distinct fluids produces a convoluted structure to the interface separating these fluids. For miscible fluids (as considered here), this interface is defined as a 50% mass concentration isosurface. For shock wave induced (Richtmyer-Meshkov) instabilities, we find the interface to be increasingly complex as the computational mesh is refined. This interfacial chaos is cut off by viscosity, or by the computational mesh if the Kolmogorov scale is small relative to the mesh. In a regime of converged interface statistics, we then examine mixing, i.e. concentration statistics, regularized by mass diffusion. For Schmidt numbers significantly larger than unity, typical of a liquid or dense plasma, additional mesh refinement is normally needed to overcome numerical mass diffusion and to achieve a converged solution of the mixing problem. However, with the benefit of front tracking and with an algorithm that allows limited interface diffusion, we can assure convergence uniformly in the Schmidt number. We show that different solutions result from variation of the Schmidt number. We propose subgrid viscosity and mass diffusion parameterizations which might allow converged solutions at realistic grid levels.

Analysis of wind-induced inflow and outflow through a single opening using LES & DES

In order to investigate induced flow properties through a single opening, large-eddy simulations (LES) are performed on ventilated air through the single opening of a room. In this flowfield, the outdoor air flows parallel to the wall and to the opening, and the room airflow is induced by the outdoor flow through the opening. Both the outdoor and indoor airflows are simultaneously simulated. Firstly, the dependency of the induced room airflow to the shape of the opening is investigated (by changing the shape of the opening). Secondly, the distributions of mean and turbulent variables at the boundary plane of the opening are investigated. It is clarified that the shape of the opening and the resultant mixing layer thickness developed along the opening boundary plane affect the characteristics of the air exchange rate and the feature of the room's airflow. In the case of a longer opening, the change in air exchange characteristics is apparent. Furthermore, detached-eddy simulation (DES) was performed on the same flowfields in order to examine the applicability of DES to building ventilation analysis. Two types of calculation (LES calculation and DES calculation with the same grid) were performed. It is concluded that DES over-estimates the sub-grid scale viscosity in the LES region, but the difference in room mean property between both cases is low.

Large-eddy simulation of two-phase spray combustion for gas turbine combustors

In three-dimensional arbitrary curvilinear coordinates, an Eulerian–Lagrangian formulation is applied to large-eddy simulation (LES) of instantaneous gas–liquid two-phase turbulent combustion flows in gas turbine combustors. Three dimensional block-structured grids are generated by zone method and solving a system of elliptic partial differential equations. The k-equation sub-grid scale model is used to simulate the sub-grid eddy viscosity and the EBU combustion sub-grid scale model is employed to predict the chemical reaction rate. The gas-phase governing equations are solved with SIMPLE algorithm and hybrid scheme in non-staggered grid system. A stochastic separated flow formulation is used to track the droplet trajectories velocities, size and temperature history by Lagrangian equations of motion and thermal balance. Multi-zone coupling method is employed to transport data between interfaces. The influences of two different primary hole positions and three different fuel–air ratios on turbulent two-phase reacting flows are calculated. Predictions are in reasonable agreement with the measured velocity using PIV system and temperature, species concentration measurements at the exit. It is shown that the present approach may be used to study spray combustion flow fields for guiding the design of advanced gas turbine combustors.

A nonlinear subgrid method for advection–diffusion problems

This work presents a general framework for approximating advection–diffusion equations based on principles of scale separation. A two-level decomposition of the discrete approximation space is performed and the local problem is modified to capture both local and nonlocal discontinuities. The new feature is the local control resulting from decomposing the velocity field into the resolved and unresolved scales and requiring the satisfaction of the discrete model problem at the element level for a minimum kinetic energy associated to the unresolved scales. This procedure leads to a nonlinear subgrid model that acts only on the unresolved scales but does not require any tuned-up parameter. It can be considered a self-adaptive method such that the amount of the subgrid viscosity is automatically introduced according to the residual of the resolved scale at element level.

CFD Modelling of Liquid Homogenization in Stirred Tanks with One and Two Impellers Using Large Eddy Simulation

A prediction of liquid homogenization in stirred tanks using a CFD code FLUENT is presented. The study was performed for tanks agitated by one and two impellers on a centrally-located shaft. Two types of impellers were used: a six-blade 45° pitched blade turbine and a standard Rushton turbine. Different methods were employed for simulations of fluid flow in the stirred tanks—the multiple reference frames technique together with the standard k– ɛ turbulence model, the sliding mesh technique with the standard k– ɛ turbulence model, and the sliding mesh technique with the large eddy simulation (LES) model and a non-iterative time-advancement algorithm. The dynamic Smagorinsky–Lilly model using a locally calculated subgrid scale dynamic viscosity constant was used in the LES calculations. Tracer mass fractions in the tanks were recorded during time-dependent species transport simulations. The calculated flow field results—velocity profiles, power and pumping numbers were compared with the experimental results from literature. Time traces of normalized concentrations obtained from the simulations were compared with our experiments and the resulting mixing times were compared to literature correlations. The LES approach was the most time-consuming method; however, it described the real flow in stirred tanks better and we obtained more realistic courses of the liquid homogenization and the best agreement of the computed and experimental homogenization times of all the used models.

On the justification and extension of mixed models in LES

A reformulation of the large eddy simulation (LES) equations is presented, based on an alternative decomposition of the subgrid stress tensor leading to modified Leonard, cross, and Reynolds terms, which are all individually frame indifferent. The new Leonard tensor, identical to the scale similarity model proposed by Bardina et al(J. Bardina, J. H. Ferziger and W. C. Reynolds, 1980, AIAA Paper 80-1357), is computable and thus becomes an integral part of the LES equations, so this formulation clearly emphasizes the difference between Reynolds averaged Navier–Stokes (RANS) and LES. The remaining modified cross and Reynolds terms can be regrouped further into a reconstructable part and a true subgrid component, where we here use an approximate deconvolution model and a subgrid viscosity model for the respective parts. This structure justifies the use of a dissipative model term in mixed models based on, e.g., the scale similarity model. The reformulated LES model is tested on two basic flows, the Taylor Green flow and a fully developed turbulent channel flow at Reynolds numbers (Reτ) between 395 and 1800. The results are compared with direct numerical simulation and experimental data as well as other LES using other subgrid models. In both cases the predictions are improved by the more detailed modeling. Including the resolved, or scale similarity, tensor has larger effect than including the reconstructable tensor, but the latter is by no means negligible. The effects are larger for cases with lower resolution, where the subgrid model has to account for a larger amount of flow scales.

Subgridscale stabilization of time-dependent convection dominated diffusive transport

In [W.J. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002) 147–157], a variationally consistent eddy viscosity discretization is given for the stationary convection diffusion equation. We further develop this discretization to include the time-dependent problem. We give comprehensive stability and error analysis of the semi-discrete case. We also state the stability and error results for the fully discrete algorithm with a Crank–Nicholson time discretization. The error bound is near optimal and independent of the diffusion coefficient, ϵ. Finally, we give guidance on optimal parameter selection for some common finite element spaces.

Joint RANS/LES Approach to Premixed Flame Modelling in the Context of the TFC Combustion Model

We present an original timesaving joint RANS/LES approach to simulate turbulent premixed combustion. It is intended mainly for industrial applications where LES may not be practical. It is based on successive RANS/LES numerical modelling, where turbulent characteristics determined from RANS simulations are used in LES equations for estimation of the subgrid chemical source and viscosity. This approach has been developed using our TFC premixed combustion model, which is based on a generalization of the Kolmogorov’s ideas. We assume existence of small-scale statistically equilibrium structures not only of turbulence but also of the reaction zones. At the same time, non-equilibrium large-scale structures of reaction sheets and turbulent eddies are described statistically by model combustion and turbulence equations in RANS simulations or follow directly without modelling in LES. Assumption of small-scale equilibrium gives an opportunity to express the mean combustion rate (controlled by small-scale coupling of turbulence and chemistry) in the RANS and LES sub-problems in terms of integral or subgrid parameters of turbulence and the chemical time, i.e. the definition of the reaction rate is similar to that of the mean dissipation rate in turbulence models where it is expressed in terms of integral or subgrid turbulent parameters. Our approach therefore renders compatible the combustion and turbulent parts of the RANS and LES sub-problems and yields reasonable agreement between the RANS and averaged LES results. Combining RANS simulations of averaged fields with LES method (and especially coupled and acoustic codes) for simulation of corresponding nonstationary process (and unsteady combustion regimes) is a promising strategy for industrial applications. In this work we present results of simulations carried out employing the joint RANS/LES approach for three examples: High velocity premixed combustion in a channel, combustion in the shear flow behind an obstacle and the impinging flame (a premixed flame attached to an obstacle).

Large Eddy Simulation of the ventilated wave boundary layer

A Large Eddy Simulation (LES) of (1) a fully developed turbulent wave boundary layer and (2) case 1 subject to ventilation (i.e., suction and injection varying alternately in phase) has been performed, using the Smagorinsky subgrid‐scale model to express the subgrid viscosity. The model was found to reproduce experimental results well. However, in case 1, the near‐bed ensemble averaged velocity is underestimated during the acceleration stage, probably due to the Smagorinsky subgrid‐scale model not being able to capture the physics well in that region. Also, there is a general overestimation of the streamwise turbulence intensity, while an underestimation of the intensities in the two other directions. This may be an effect from the stretched computational mesh in the streamwise direction, since the Smagorinsky subgrid viscosity assumes proportionality to one scalar expressing the overall (local) grid size. The results indicate that the large eddies develop in the resolved scale, corresponding to fluid with an effective viscosity decided by the sum of the kinematic and subgrid viscosity. Regarding case 2, the results are qualitatively in accordance with experimental findings. Injection generally slows down the flow in the full vertical extent of the boundary layer, destabilizes the flow and decreases the mean bed shear stress significantly; whereas suction generally speeds up the flow in the full vertical extent of the boundary layer, stabilizes the flow and increases the mean bed shear stress significantly. Ventilation therefore results in a net current, even in symmetric waves.

The variational multiscale method for laminar and turbulent flow

The present article reviews the variational multiscale method as a framework for the development of computational methods for the simulation of laminar and turbulent flows, with the emphasis placed on incompressible flows. Starting with a variational formulation of the Navier-Stokes equations, a separation of the scales of the flow problem into two and three different scale groups, respectively, is shown. The approaches resulting from these two different separations are interpreted against the background of two traditional concepts for the numerical simulation of turbulent flows, namely direct numerical simulation (DNS) and large eddy simulation (LES). It is then focused on a three-scale separation, which explicitly distinguishes large resolved scales, small resolved scales, and unresolved scales. In view of turbulent flow simulations as a LES, the variational multiscale method with three separated scale groups is refered to as a “variational multiscale LES”. The two distinguishing features of the variational multiscale LES in comparison to the traditional LES are the replacement of the traditional filter by a variational projection and the restriction of the effect of the unresolved scales to the smaller of the resolved scales. Existing solution strategies for the variational multiscale LES are presented and categorized for various numerical methods. The main focus is on the finite element method (FEM) and the finite volume method (FVM). The inclusion of the effect of the unresolved scales within the multiscale environment via constant-coefficient and dynamic subgrid-scale modeling based on the subgrid viscosity concept is also addressed. Selected numerical examples, a laminar and two turbulent flow situations, illustrate the suitability of the variational multiscale method for the numerical simulation of both states of flow. This article concludes with a view on potential future research directions for the variational multiscale method with respect to problems of fluid mechanics.

Large eddy simulation in a fully developed turbulent flow in a channel and comparison of subgrid eddy viscosity models

The accuracy and computational efficiency are compared for a number of models of subgrid eddy viscosity (Smagorinsky model, renormalization group model, and dynamic and one-parameter models). Space-filtered Navier-Stokes equations are solved numerically by the control-volume approach on a nonuniform grid with the use of high-resolution schemes in time and space. The numerical data are compared with the results of a physical experiment and direct numerical simulation.