Subgrid Closure Research Articles

A hybrid DG-Monte Carlo FDF simulator

A new computational scheme is developed for large eddy simulation (LES) of turbulent reacting flows via the filtered density function (FDF) subgrid scale closure. This is a hybrid scheme, combining the discontinuous Galerkin (DG) Eulerian solver with a Lagrangian Monte Carlo FDF simulator. The methodology is shown to be suitable for LES, as a larger portion of the resolved energy is captured as the order of spectral approximation increases. The consistency and the overall performance of the DG-FDF solver and the realization of the simulated results are demonstrated via LES of a temporally developing mixing layer under both non-reacting and reacting conditions.

Assessment of scaling laws for mixing fields in inter-droplet space

Droplet evaporation and subsequent mixing of the fuel vapor with the turbulent surrounding hot oxidizer has been simulated using direct numerical simulations (DNS). The kerosene droplets are fully resolved with realistic boundary conditions at the liquid–vapor interface to ensure the correct heat and mass transfer between the two phases. The study focuses on combustible mixture preparation in the inter-droplet space at scales larger than the quasi-laminar near droplet zone. The data are analysed with respect to key quantities for mixture fraction based combustion models such as mixture fraction distribution and its dissipation. The DNS is compared with scaling relationships for the wake and for length scales of Kolmogorov size where the wake seizes to be the dominant flow structure. The scaling relationships agree reasonably well with DNS data and they may serve as sub-grid scale closures for combustion models such as conditional moment closure or flamelets. Their respective regions of validity are quantified for the different cases of turbulence intensity that are investigated here.

Subgrid-Scale Modeling of Reaction-Diffusion and Scalar Transport in Turbulent Premixed Flames

ABSTRACTA numerical study of premixed flame-turbulence interaction is performed to investigate the effects of turbulence on the structural features of the flame and the subgrid-scale (SGS) effects on vorticity dynamics, energy transfer mechanism, and turbulent transport across the flame. We consider a freely propagating methane-air turbulent premixed flame interacting with a decaying isotropic turbulence under three different initial conditions corresponding to the corrugated flamelet (CF), the thin reaction zone (TRZ), and the broken/distributed reaction zone (B/DRZ) regimes. We employ the well-established linear eddy mixing (LEM) model in large-eddy simulation (LEMLES), a new subgrid closure for reaction-diffusion occurring in the small-scales based on LEM (RRLES), and a quasi-laminar chemistry based closure in large-eddy simulation (QLLES) to simulate flame-turbulence interactions and to compare predictions with direct numerical simulation (DNS). We assess the accuracy and robustness of the closures by comparing statistical features to highlight their abilities and limitations. The newly proposed RRLES subgrid closure uses a dual-resolution grid for solving the species transport equations. Such an approach is shown to improve the existing LEMLES subgrid model especially at low Reynolds numbers. All SGS closures reveals good agreement, although there are some differences due to the closure used for convective transport of the scalar field and the reaction rate. Further analysis of the DNS dataset shows that there is a significant contribution by dilatation and baroclinic torque terms across the flame. In particular, at higher Karlovitz number, there is an abrupt change in the sign of the dilatation term, which is related to the competing effects of thermal expansion due to heat release and enhanced molecular mixing by the turbulence across the flame brush region. The enhanced mixing leads to localized pockets of cold reactants surrounded by hot products, which is only partly captured by the employed closures. The analysis of SGS kinetic energy and scalar dissipation rates indicates the presence of back-scatter of turbulent kinetic energy, and we also observe counter-gradient transport across the flame. The results suggest that further improvement of the traditional closures is needed to accurately capture the dynamics of flame-turbulence interaction.

Large eddy simulation, turbulent transport and the renormalization group

In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG), we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the model coefficients. The RNG point of view sheds light on the nonuniqueness of the infinite Reynolds number limit. For the mixing of N species, we see an N +1 parameter family of infinite Reynolds number solutions labeled by dimensionless parameters of the limiting Euler equations, in a manner intrinsic to the RNG itself. Large eddy simulations, with their Leonard stress and dynamic subgrid models, break this nonuniqueness and predict unique model coefficients on the basis of theory. In this sense large eddy simulations go beyond the RNG methodology, which does not in general predict model coefficients.

Comparison of Subgrid Closure Methods for Passive Scalar Variance at High Schmidt Number

AbstractThe subgrid closure for fluctuations of the passive scalar in liquids is investigated. Simulation results of the concentration variance based on the literature subgrid‐scale models are compared with experimental findings obtained by planar laser‐induced fluorescence measurements in a tubular reactor with T‐shaped mixing head. The new coefficient for the proposed dynamic subgrid‐scale model takes into account the spatial variation of the turbulence Reynolds number and depends on the Schmidt number and the numerical cell size. The model suggested provides excellent predictions of the concentration variance.

Hybrid Large-Eddy/Reynolds-Averaged Navier–Stokes Simulations of Flow Through a Model Scramjet

The reactive flow through a model scramjet combustor is simulated using a hybrid large-eddy/Reynolds-averaged Navier–Stokes technique. The scramjet configuration considered is similar to the one investigated at the Institute for Chemical Propulsion of the DLR, German Aerospace Center. The model scramjet consists of 15 fuel-injecting holes, located on the base of a wedge-shaped fuel injector, through which hydrogen is injected at sonic conditions. In the present study, only five of the 15 fuel-injecting holes are considered, and periodicity is assumed in the spanwise direction. Several parametric studies are conducted with a view toward determining the sensitivities of the predictions to modeling and algorithmic variations. Different grids (two different topologies), flux reconstruction methods (total variation diminishing and piecewise parabolic method), reaction mechanisms, and inflow boundary conditions (uniform and nonuniform) are used. To enhance fuel–air mixing, a synthetic eddy method is used to generate turbulence in the injector boundary layers and the hydrogen jets. Finally, a partially stirred reactor-type subgrid combustion model is used as a subgrid closure for the species production rates. The results show that, in all the cases, a lifted flame is predicted with varying standoff distances, heat releases, and shapes.

A filtered-laminar-flame PDF sub-grid scale closure for LES of premixed turbulent flames. Part I: Formalism and application to a bluff-body burner with differential diffusion

A sub-grid scale closure for Large Eddy Simulation (LES) of turbulent combustion based on physical-space filtering of laminar flames is discussed. Applied to an unstructured grid, the combustion LES filter size is not fixed in this novel approach devoted to LES with refined meshes, but calibrated depending on the local level of unresolved scalar fluctuations. The context is premixed or stratified flames, the derived model relies on four balance equations for mixture fraction and its variance, and a progress variable and its variance. The proposed formalism is based on a presumed probability density function (PDF) derived from the filtered flames. Closures for the terms of the equations that are unresolved over LES grids are achieved through the PDF. The method uses flamelet tabulated detailed chemistry and is first applied to the simulation of laminar flames (1D and 2D) over various grids for validation, before simulating a turbulent burner studied experimentally by Sweeney et al. (2012). Since this burner also features differential diffusion effects, the numerical model is modified to account for accumulation of carbon in the recirculation zone behind the bluff-body. A differential diffusion number based on the gradient of residence times is proposed, in an attempt to globally quantify differential diffusion effects in burners.

Vortex shedding in a highly porous structure

Highly porous media (with a porosity ≥85%), like metal foams, fins or designed porous inserts are widely used in industrial flow applications. Motivated by the use of highly porous inserts in continuous chemical reactors, the time dependent behavior of flows through a highly porous structure is investigated experimentally and numerically for Reynolds numbers ReD=33 and 330 (based on the ligament diameter and the bulk velocity). The experimental studies are based on the Particle Image Velocimetry (PIV), whereas the numerical studies are conducted by using the Large Eddy Simulation (LES) approach with the dynamic Lagrangian subgrid closure. The vortex shedding phenomena is observed for both values of Reynolds number. It is demonstrated that the length of the wake and the orientation of the shedded vortices are dependent on the position within the highly porous structure.

Convergence towards weak solutions of the Navier-Stokes equations for a finite element approximation with numerical subgrid-scale modelling

Residual-based stabilized nite element techniques for the Navier-Stokes equations lead to numerical discretizations that provide convection stabilization as well as pressure stability without the need to satisfy an inf-sup condition. They can be motivated by using a variational multiscale framework, based on the decomposition of the uid velocity into a resolvable nite element component plus a modeled subgrid scale component. The subgrid closure acts as a large eddy simulation turbulence model, leading to accurate under-resolved simulations. However, even though variational multiscale formulations are increasingly used in the applied nite element community, their numerical analysis has been restricted to a priori estimates and convergence to smooth solutions only, via a priori error estimates. In this work we prove that some versions of these methods (based on dynamic and orthogonal closures) also converge to weak (turbulent) solutions of the Navier-Stokes equations. These results are obtained by using compactness results in Bochner-Lebesgue spaces. Navier-Stokes equations; stability; convergence; stabilized nite element methods; subgrid scales; variational multiscale methods.

Large Eddy Simulation of a Free-Burning Arc Discharge in Argon with a Turbulent Cross Flow and External Field

In most of the numerical approaches proposed for modeling high-intensity plasma-arcs, the effects of turbulence on the arc structure are often excluded because of the intricate physics originating from the interaction of turbulent scales, high-temperature gas dynamics, magnetohydrodynamics (MHD) and chemical kinetics. The goal of this study is threefold: to develop a generic turbulent MHD model to simulate free-burning arc discharges, to validate the code with available experimental data, and to investigate the effect of an external field and turbulent cross flow on the free-burning arc configuration. The governing equations are solved in conservative form using a hybrid scheme that combines a high-order monotonic upwind scheme with a second-order central scheme. The fluid and MHD turbulence are resolved using a large eddy simulation (LES) approach with a recently developed sub-grid closure model. An implicit scheme is used to compute the magnetic diffusion term appearing in the magnetic induction equation to alleviate the severe time-step constraint. The comparison of the model prediction with experimental data for Argon arcs at different current intensities shows generally good agreement. When an external field is applied, the overall shape of the free-burning arc drastically changes. The straightening of the arc indicates the potential for stabilization of a free-burning arc by magnetic forces. Even though the turbulence is significantly attenuated as a result of the thermal expansion near the cathode, it adds an unsteady characteristic to the arc and, in general, has a negative impact on the stabilization of the electrical discharge.

Large-Eddy Atmosphere–Land-Surface Modelling over Heterogeneous Surfaces: Model Development and Comparison with Measurements

A model is developed for the large-eddy simulation (LES) of heterogeneous atmosphere and land-surface processes. This couples a LES model with a land-surface scheme. New developments are made to the land-surface scheme to ensure the adequate representation of atmosphere–land-surface transfers on the large-eddy scale. These include, (1) a multi-layer canopy scheme; (2) a method for flux estimates consistent with the large-eddy subgrid closure; and (3) an appropriate soil-layer configuration. The model is then applied to a heterogeneous region with 60-m horizontal resolution and the results are compared with ground-based and airborne measurements. The simulated sensible and latent heat fluxes are found to agree well with the eddy-correlation measurements. Good agreement is also found in the modelled and observed net radiation, ground heat flux, soil temperature and moisture. Based on the model results, we study the patterns of the sensible and latent heat fluxes, how such patterns come into existence, and how large eddies propagate and destroy land-surface signals in the atmosphere. Near the surface, the flux and land-use patterns are found to be closely correlated. In the lower boundary layer, small eddies bearing land-surface signals organize and develop into larger eddies, which carry the signals to considerably higher levels. As a result, the instantaneous flux patterns appear to be unrelated to the land-use patterns, but on average, the correlation between them is significant and persistent up to about 650 m. For a given land-surface type, the scatter of the fluxes amounts to several hundred W $$\text{ m }^{-2}$$ , due to (1) large-eddy randomness; (2) rapid large-eddy and surface feedback; and (3) local advection related to surface heterogeneity.

A thermodynamic and dynamic subgrid closure model for two‐material cells

SUMMARYA general and robust subgrid closure model for two‐material cells is proposed. The conservative quantities of the entire cell are apportioned between two materials, and then, pressure and velocity are fully or partially equilibrated by modeling subgrid wave interactions. An unconditionally stable and entropy‐satisfying solution of the processes has been successfully found. The solution is valid for arbitrary level of relaxation. The model is numerically designed with care for general materials and is computationally efficient without recourse to subgrid iterations or subcycling in time. The model is implemented and tested in the Lagrange‐remap framework. Two interesting results are observed in 1D tests. First, on the basis of the closure model without any pressure and velocity relaxation, a material interface can be resolved without creating numerical oscillations and/or large nonphysical jumps in the problem of the modified Sod shock tube. Second, the overheating problem seen near the wall surface can be solved by the present entropy‐satisfying closure model. The generality, robustness, and efficiency of the model make it useful in principle in algorithms, such as ALE methods, volume of fluid methods, and even some mixture models, for compressible two‐phase flow computations. Copyright © 2013 John Wiley & Sons, Ltd.

Efficiency of Scale-Similarity Model for Study of Forced Compressible Magnetohydrodynamic Turbulence

Efficiency of scale-similarity model for study of forced compressible magnetohydrodynamic turbulence is studied. The scale-similarity model has several important advantages in contrast to the eddy-viscosity subgrid closures: good reproduction of the correlation between actual and model turbulent stress tensor even when the flow is highly anisotropic, and absence of special model constants. These advantages may be very essential for study of forced magnetohydrodynamic turbulence. Numerical computations under various similarity parameters are carried out and the obtained results are analyzed by means of comparison with results of direct numerical simulation and Smagorinsky closure for magnetohydrodynamics. Linear forcing algorithm is applied to keep the characteristics of turbulence stationary in time. Influence of discrete filter shapes on the scale-similarity model is studied as well. It is shown that the scale-similarity model provides good accuracy and the results agree well with the direct numerical simulation results. The present results show that the scale-similarity model might be a useful subgrid closure for study of scale-invariance properties of forced compressible magnetohydrodynamic turbulence in the inertial range and in contrast to decaying case the scale-similarity model can serve as a stand alone subgrid model.

Stochastic closure for local averages in the finite-difference discretization of the forced Burgers equation

We present a new approach for the construction of stochastic subgrid scale parameterizations. Starting from a high-resolution finite-difference discretization of some model equations, the new approach is based on splitting the model variables into fast, small-scale and slow, large-scale modes by averaging the model discretization over neighboring grid cells. After that, the fast modes are eliminated by applying a stochastic mode reduction procedure. This procedure is a generalization of the mode reduction strategy proposed by Majda, Timofeyev & Vanden-Eijnden, in that it allows for oscillations in the closure assumption. The new parameterization is applied to the forced Burgers equation and is compared with a Smagorinsky-type subgrid scale closure.

Consistent subgrid scale modelling for lattice Boltzmann methods

AbstractThe lattice Boltzmann method has become a widely used tool for the numerical simulation of fluid flows and in particular of turbulent flows. In this frame the inclusion of subgrid scale closures is of crucial importance and is not completely understood from the theoretical point of view. Here, we propose a consistent way of introducing subgrid closures in the BGK Boltzmann equation for large eddy simulations of turbulent flows. Based on the Hermite expansion of the velocity distribution function, we construct a hierarchy of subgrid scale terms, which are similar to those obtained for the Navier–Stokes equations, and discuss their inclusion in the lattice Boltzmann method scheme. A link between our approach and the standard way on including eddy viscosity models in the lattice Boltzmann method is established. It is shown that the use of a single modified scalar relaxation time to account for subgrid viscosity effects is not consistent in the compressible case. Finally, we validate the approach in the weakly compressible case by simulating the time developing mixing layer and comparing with experimental results and direct numerical simulations.

Computational fluid dynamics modelling of fire

An overview of a methodology for simulating fires and other thermally-driven, low-speed flows is presented. The model employs a number of simplifications of the governing equations that allow for relatively fast simulations of practical fire scenarios. The hydrodynamic model consists of the low Mach number large-eddy simulation subgrid closure with either a constant or dynamic coefficient eddy diffusivity. Combustion is typically treated as a mixing-controlled, single-step reaction of fuel and oxygen. The radiation transport equation is written in terms of a spectrally-averaged grey gas. Applications of the model include the design of fire protection systems in buildings and the reconstruction of actual fires.

A sub‐grid scale closure for nonlinear hillslope sediment transport models

Hillslope sediment transport models express the sediment flux at a point as a function of some topographic attributes of the system, such as slope, curvature, soil thickness, etc., at that point only (referred here as “local” transport models) or at an appropriately defined vicinity of that point (referred here as “nonlocal” transport models). Typically, topographic attributes are computed from digital elevation data (DEMs) and thus their estimates depend on the DEM resolution (1 m, 10 m, 90 m, etc.) rendering any sediment flux computation scale‐dependent. Often calibration compensates for this scale‐dependence resulting in effective parameterizations with limited physical meaning. In this paper, we demonstrate the scale‐dependence of local nonlinear hillslope sediment flux models and derive a subgrid scale closure via upscaling. We parameterize the subgrid scale closure in terms of the low resolution, resolved topographic attributes of the landscape, thus allowing the reliable computation of a scale‐independent sediment flux from low resolution digital elevation data. We also show that the accuracy of the derived subgrid scale closure model depends on the dimensionless erosion rate and the dimensionless relief of any given basin. Finally, we present theoretical arguments and demonstrate that the recently proposed nonlocal sediment flux models are scale‐independent. These concepts are demonstrated via an application on a small basin (MR1) of the central Oregon Coast Range using high‐resolution lidar topographic data.

Partially premixed reacting acetone spray using LES and FGM tabulated chemistry

Large Eddy Simulations of dilute spray-air reacting two phase flow are performed following an Eulerian–Lagrangian approach. The method includes a full two-way coupling in which phase properties and spray source terms are interchanging between the two phases within each coupling time step. To achieve sub-grid scale closures in the filtered equations used for LES of the carrier phase, the Smagorinsky model with dynamic procedure is applied for the flow field while an eddy diffusivity approach is used for the scalar fluxes. The phase transition of droplets is captured using a non-equilibrium evaporation model. The evaporating droplets are tracked using a Langragian procedure. They are injected in a polydisperse manner and generated in time dependent boundary conditions. The combustion is described by a reaction progress variable and a mixture fraction as well as their variances in the line of the Flamelet Generated Manifold method. The ultimate objective of this work is to appraise the ability of this LES-based spray module developed here to retrieve the spray flow and combustion properties of the configuration under study. The configuration consists of a spray jet issued into a pilot flame and a co-flowing atmospheric air in which droplets traverse a pre-evaporation distance and release part of their mass before reaching the combustion zone. Thereby the spray pre-evaporation turns the combustion regime from diffusion to partially premixed. The fuel used is liquid acetone, which is modeled by a detailed reaction mechanism including 84 species and 409 elementary reactions. A series of spray established at different operating conditions are investigated, analyzed and compared. Analysis allows to gain a deep insight into the process ongoing. Comparisons include exhaust gas temperature, droplet velocities and corresponding fluctuations, droplet mean diameters and spray volume flux at different distances from the exit planes. An overall good agreement is reported.

Large Eddy Simulation of a thermal mixing tee in order to assess the thermal fatigue

The present paper deals with thermal fatigue phenomenon, and more particularly with the numerical simulation using Large Eddy Simulation technique of a mixing tee, for which experimental thermal statistics are available. The sensitivity to the sub-grid scale closure is first evaluated by comparing the experimental statistics with the numerical results obtained via both the Smagorinsky and the structure-function models. Because of a difference of temporal resolution between the experiment and the simulation, the direct comparison of the fluctuations is not possible. Therefore, a methodology based on filtering the numerical results is proposed in order to achieve a proper comparison. The comparison of the numerical results with the experiment suggests that slight better predictions are obtained with the structure-function model even if the dependency of the results to the sub-grid scale model is low. Then, the possibility to reduce the fluid computational domain by prescribing synthetic turbulence at the inlet is tested. First results are encouraging and underline the advantage of considering this technique instead of a standard noise at the entrance of the domain. All the simulations are conducted with the commercial CFD code STAR-CD.

From Large-Eddy Simulation to Direct Numerical Simulation of a lean premixed swirl flame: Filtered laminar flame-PDF modeling

Large-Eddy Simulations (LES) and Direct Numerical Simulation (DNS) are applied to the analysis of a swirl burner operated with a lean methane–air mixture and experimentally studied by Meier et al. [ 19]. LES is performed for various mesh refinements, to study unsteady and coherent large-scale behavior and to validate the simulation tool from measurements, while DNS enables to gain insight into the flame structure and dynamics. The DNS features a 2.6 billion cells unstructured-mesh and a resolution of less than 100 microns, which is sufficient to capture all the turbulent scales and the major species of the flame brush; the unresolved species are taken into account thanks to a tabulated chemistry approach. In a second part of the paper, the DNS is filtered at several filter widths to estimate the prediction capabilities of modeling based on premixed flamelet and presumed probability density functions. The similarities and differences between spatially-filtered laminar and turbulent flames are discussed and a new sub-grid scale closure for premixed turbulent combustion is proposed, which preserves spectral properties of sub-filter flame length scales. All these simulations are performed with a solver specifically tailored for large-scale computations on massively parallel machines.