Subgrid Energy Research Articles

Scalar energy fluctuations in Large-Eddy Simulation of turbulent flames: Statistical budgets and mesh quality criterion

Large-Eddy Simulation (LES) provides space-filtered quantities to compare with measurements, which usually have been obtained using a different filtering operation; hence, numerical and experimental results can be examined side-by-side in a statistical sense only. Instantaneous, space-filtered and statistically time-averaged signals feature different characteristic length-scales, which can be combined in dimensionless ratios. From two canonical manufactured turbulent solutions, a turbulent flame and a passive scalar turbulent mixing layer, the critical values of these ratios under which measured and computed variances (resolved plus sub-grid scale) can be compared without resorting to additional residual terms are first determined. It is shown that actual Direct Numerical Simulation can hardly accommodate a sufficiently large range of length-scales to perform statistical studies of LES filtered reactive scalar-fields energy budget based on sub-grid scale variances; an estimation of the minimum Reynolds number allowing for such DNS studies is given. From these developments, a reliability mesh criterion emerges for scalar LES and scaling for scalar sub-grid scale energy is discussed.

Multi-scale analysis of subgrid stress and energy dissipation in turbulent channel flow

In present study, the subgrid scale (SGS) stress and dissipation for multiscale formulation of large eddy simulation are analyzed using the data of turbulent channel flow at Re τ = 180 obtained by direct numerical simulation. It is found that the small scale SGS stress is much smaller than the large scale SGS stress for all the stress components. The dominant contributor to large scale SGS stress is the cross stress between small scale and subgrid scale motions, while the cross stress between large scale and subgrid scale motions make major contributions to small scale SGS stress. The energy transfer from resolved large scales to subgrid scales is mainly caused by SGS Reynolds stress, while that between resolved small scales and subgrid scales are mainly due to the cross stress. The multiscale formulation of SGS models are evaluated a priori, and it is found that the small–small model is superior to other variants in terms of SGS dissipation.

Large eddy simulation modelling of spray-induced turbulence effects

This study is focused on the development of spray models for large eddy simulations (LESs) of high injection pressure diesel sprays. Basic governing equations are numerically solved using the Lagrangian—Eulerian approach for liquid and gas phases respectively, in the KIVA-3V code. Non-reacting simulations of evaporating spray are performed using a dynamic structure LES model in which the subgrid stress tensor is modelled with a non-viscosity tensor coefficient. This is a one equation based model, in which an extra transport equation for the subgrid kinetic energy ( k) is solved. Subgrid scale energy exchange between droplets and the gas phase is identified as an important mechanism to capture the correct scaling of k, which eventually feeds back to both the spray droplets and the gas phase turbulent mixing. Hence, to account for spray-induced gas turbulence, a LES spray source model for k is developed. This model requires subfilter scale velocities, which are obtained by defiltering the filtered velocity field available from LES. Results are compared with corresponding RANS results and available experimental data for various physical spray variables such as spray penetration, gas phase penetration, droplet distribution, etc. The LES spray model was found to be important in predicting the correct liquid spray evolution and responds correctly to varying parameters such as nozzle size and ambient density.

Hybrid URANS/LES Turbulence Simulation of Vortex Shedding Behind a Triangular Flameholder

In this study a detached eddy simulation (DES) model, which belongs to the group of hybrid URANS/LES turbulence models, is used for the simulation of vortex shedding behind a triangular obstacle. In the near wall region or in regions where the grid resolution is not sufficiently fine to resolve smaller structures, the two-equation RANS shear-stress transport (SST) model is used. In the other regions with higher grid resolution a LES model, which uses a transport equation for the turbulent subgrid energy, is applied. The DES model is first investigated for two standard test cases, namely decaying homogeneous isotropic turbulence and the backward facing step, respectively. For the decaying homogeneous isotropic turbulence test case the evolution of the energy spectra in wavenumber space for different times are studied for both the DES and a Smagorinsky type LES model. Different grid resolutions are analyzed with a special emphasis on the modeling constant connecting the filter length scale to the grid size. The results are compared to experimental data. The backward facing step test case is used to study the model behavior for a case with a transition region between a RANS modeling approach close to the wall and LES based modeling in the intense shear flow region. The final application is the simulation of the vortex shedding behind a triangular obstacle. First, the influence of the inlet condition formulation is studied in detail as they can have a significant influence especially for LES based models. Detailed comparisons between simulation and experiment for the flow structure past the obstacle and statistical quantities such as the shedding frequency are shown. Finally the additional temporal and spatial information provided by the DES model is used to show the predicted anisotropy of turbulence.

Preservation of statistical properties in large-eddy simulation of shear turbulence

We discuss how large-eddy simulation (LES) can be properly employed to predict the statistics of the resolved velocity fluctuations in shear turbulence. To this purpose ana posterioricomparison of LES data against filtered direct numerical simulation (DNS) is used to establish the necessary conditions that the filter scaleLFmust satisfy to achieve the preservation of the statistical properties of the resolved field. In this context, by exploiting the physical role of the shear scaleLS, the Kármán–Howarth equation allows for the assessment of LES data in terms of scale-by-scale energy production, energy transfer and subgrid energy fluxes. Even higher-order statistical properties of the resolved scales such as the probability density function of longitudinal velocity increments are well reproduced, provided the relative position of the filter scale with respect to the shear scale is properly selected. We consider here the homogeneous shear flow as the simplest non-trivial flow which fully retains the basic mechanism of turbulent kinetic energy production typical of any shear flow, with the advantage that spatial homogeneity implies a well-defined value of the shear scale while numerical difficulties related to resolution requirements in the near wall region are avoided.

A stochastic subgrid model with application to turbulent flow and scalar mixing

A new computationally cheap stochastic Smagorinsky model which allows for backscatter of subgrid scale energy is proposed. The new model is applied in the large eddy simulation of decaying isotropic turbulence, rotating homogeneous shear flow and turbulent channel flow at Reτ=360. The results of the simulations are compared to direct numerical simulation data. The inclusion of stochastic backscatter has no significant influence on the development of the kinetic energy in homogeneous flows, but it improves the prediction of the fluctuation magnitudes as well as the anisotropy of the fluctuations in turbulent channel flow compared to the standard Smagorinsky model with wall damping of CS. Moreover, the stochastic model improves the description of the energy transfer by reducing its length scale and increasing its variance. Some improvements were also found in isotropic turbulence where the stochastic contribution improved the shape of the enstrophy spectrum at the smallest resolved scales and reduced the time scale of the smallest resolved scales in better agreement with earlier observations.

Multifractal subgrid-scale modeling for large-eddy simulation. I. Model development and a priori testing

Results are presented from a new approach to modeling the subgrid-scale stresses in large-eddy simulation of turbulent flows, based on explicit evaluation of the subgrid velocity components from a multifractal representation of the subgrid vorticity field. The approach is motivated by prior studies showing that the enstrophy field exhibits multifractal scale-similarity on inertial-range scales in high Reynolds number turbulence. A scale-invariant multiplicative cascade thus gives the spatial distribution of subgrid vorticity magnitudes within each resolved-scale cell, and an additive cascade gives the progressively isotropic decorrelation of subgrid vorticity orientations from the resolved scale Δ to the viscous scale λν. The subgrid velocities are then obtained from Biot–Savart integrals over this subgrid vorticity field. The resulting subgrid velocity components become simple algebraic expressions in terms of resolved-scale quantities, which then allow explicit evaluation of the subgrid stresses τij*. This new multifractal subgrid-scale model is shown in a priori tests to give good agreement for the filtered subgrid velocities, the subgrid stress components, and the subgrid energy production at both low (ReΔ≈160) and high (ReΔ≈2550) resolved-scale Reynolds numbers. Implementing the model is no more computationally burdensome than traditional eddy-viscosity models. Moreover, evaluation of the subgrid stresses requires no explicit differentiation of the resolved velocity field and is therefore comparatively unaffected by discretization errors.

Multifractal subgrid-scale modeling for large-eddy simulation. II. Backscatter limiting anda posteriorievaluation

Results are presented from a posteriori evaluations of momentum and energy transfer between the resolved and subgrid scales when the multifractal subgrid-scale model from Part I is implemented in a flow solver for large-eddy simulations of turbulent flows. The multifractal subgrid-stress model is used to evaluate the subgrid part τij* of the stress tensor, with the resolved part u¯iu¯j¯ evaluated by an explicit filter. It is shown that the corresponding subgrid and resolved contributions P* and PR to the resolved-scale energetics produce extremely accurate results for the combined subgrid energy production field P(x,t). A separate backscatter limiter is developed here that removes spurious energy introduced in the resolved scales by including physical backscatter, without sacrificing the high fidelity in the stress and energy production fields produced by the multifractal subgrid-scale model. This limiter makes small reductions only to those components of the stress that contribute to backscatter, and principally in locations where the gradients are large and thus the energy introduced by numerical errors is also largest. Control of the energy introduced by numerical error is thus accomplished in a manner that leaves the modeling of the subgrid-scale turbulence largely unchanged. The multifractal subgrid-scale model and the backscatter limiter are then implemented in a flow solver and shown to provide stable and accurate results in a posteriori tests based on large-eddy simulations of forced homogeneous isotropic turbulence at cell Reynolds numbers ranging from 160⩽ReΔ⩽106, as well as in simulations of decaying turbulence where the model and the limiter must adjust to the changing subgrid conditions.

Analysis of the energy budget in turbulent channel flow using orthogonal wavelets

The spatial and scale statistics of turbulence kinetic energy are examined in turbulent channel flow at Re τ = 300 using the orthonormal wavelet transform. The behaviour of the production, viscous and transfer terms is examined in terms of their variation with both space and scale. All terms are numerically large at wavenumbers at which a −5/3 slope is apparent in the velocity spectra, and they all exhibit significant spatial variability as evidenced by large flatnesses which increase with decreasing scale size. The flatness of terms involving transfer are particularly large. Attention focusses primarily on the sublayer and local-equilibrium regions: in the former, scale-to-scale flux is large and negative and consistent with conventional “backscatter” in Fourier space. In the linear sublayer, the flux is positive, consistent with Couette-like vortex stretching. The present work paves the way for close scrutiny of those components of the subgrid-scale stress that contribute most to the subgrid energy flux in large-eddy simulation of near-wall turbulent flows.

Large eddy simulation and ALE mesh motion in Rayleigh–Taylor instability simulation

Many large eddy simulation (LES) techniques have been developed for stationary computational meshes. This study applies a single equation LES to Arbitrary Lagrangian–Eulerian (ALE) simulations of Rayleigh–Taylor instability and investigates its effects. Behavior of LES is similar for Eulerian and ALE simulations for the test problem studied. However, the motion of the mesh can be tied to the subgrid scale model in the form of a relaxation weight based on subgrid scale energy. This increases mesh resolution in areas of high subgrid scale energy.

A vortex-based model for the subgrid flux of a passive scalar

A model for the flux of a passive scalar by the subgrid motions in the large-eddy simulation of turbulent flow is proposed within the framework of the stretched-vortex subgrid stress model. The model is based on an analytical solution for the winding of a scalar field by an elemental subgrid vortex. This gives a tensor gradient-diffusion expression for the local flux of the scalar with subgrid turbulent diffusivity which depends upon the subgrid energy, the local cell size, and the vortex orientation in space. The scalar-flux subgrid model is tested by comparison of the results of 323 large-eddy simulation of passive-scalar transport by forced isotropic turbulence in the presence of a mean scalar gradient, with the direct-numerical simulation results of Overholt and Pope [Phys. Fluids 8, 2128 (1996)]. The present large-eddy simulation results predict that at large Taylor–Reynolds numbers, the ratio of the scalar variance to the squared product of the scalar gradient with the dissipation length of the turbulence, is asymptotic to a nearly constant value c′2/(α1 Lε)2≈0.36.

Crude closure for flow with topography through large-scale statistical theory

Crude closure algorithms based on equilibrium statistical theories are developed here for prototypical geophysical flows involving barotropic flow over topography. These algorithms are developed utilizing the simplest energy-enstrophy statistical theory for flow with topography. Only a single parameter, the energy, is tracked by the algorithm and the entire flow structure is predicted through the equilibrium statistical state. In particular, no explicit parametrization of a sub-grid scale energy spectrum is utilized in the algorithm. The predictions of the crude closure algorithm are compared with direct pseudo-spectral numerical simulations of the barotropic flow equations with random small-scale forcing and dissipation for a variety of random topographies in basin, channel and periodic geometries. In most situations studied here, the energy is tracked within small errors by the crude closure, while the velocity errors rarely exceed 10% provided that the enstrophy/energy ratio is not large or growing significantly in time. Examples are also introduced where the crude closure algorithm based on the energy-enstrophy theory fails; in these circumstances, a crude closure algorithm based on more sophisticated equilibrium statistical theories is introduced as a possible remedy.

On Subgrid Models and Filter Operations in Large Eddy Simulations

Abstract Large eddy simulations use a subgrid model, which is characterized by a length scale that is often related to the scale of the computational mesh by a numerical constant, Cs. Mason and Callen argued that this subgrid model and its length scale define and impose the filter operation of the simulation. They saw Cs as a measure of numerical accuracy. Others have sought to link the filter operation to the computational mesh and have viewed Cs as needing determination for correct implementation. Here tests with a high resolution of 224 × 224 × 200 grid points are found to confirm Mason and Callen’s view. These simulations are also used together with lower-resolution simulations to illustrate the degree of convergence achieved. Some erroneous features of the simulations are identified through this test. For the case of buoyant convection, the buoyancy dependence of the subgrid model is further examined. Most available subgrid models allow for buoyancy fluxes changing the level of the subgrid energy but...

An analysis of subgrid-resolved scale interactions with use of results from direct numerical simulations

Subgrid nonlinear interaction and energy transfer are analyzed using direct numerical simulations of isotropic turbulence. Influences of cutoff wave number at different ranges of scale on the energetics and dynamics have been investigated. It is observed that subgrid-subgrid interaction dominates the turbulent dynamics when cut-off wave number locates in the energy-containing range while resolved-subgrid interaction dominates if it is in the dissipation range. By decomposing the subgrid energy transfer and nonlinear interaction into ‘forward’ and ‘backward’ groups according to the sign of triadic interaction, we find that individually each group has very large contribution, but the net of them is much smaller, implying that tremendous cancellation happens between these two groups.

A vortex-based subgrid stress model for large-eddy simulation

A class of subgrid stress (SGS) models for large-eddy simulation (LES) is presented based on the idea of structure-based Reynolds-stress closure. The subgrid structure of the turbulence is assumed to consist of stretched vortices whose orientations are determined by the resolved velocity field. An equation which relates the subgrid stress to the structure orientation and the subgrid kinetic energy, together with an assumed Kolmogorov energy spectrum for the subgrid vortices, gives a closed coupling of the SGS model dynamics to the filtered Navier–Stokes equations for the resolved flow quantities. The subgrid energy is calculated directly by use of a local balance between the total dissipation and the sum of the resolved-scale dissipation and production by the resolved scales. Simple one- and two-vortex models are proposed and tested in which the subgrid vortex orientations are either fixed by the local resolved velocity gradients, or rotate in response to the evolution of the gradient field. These models are not of the eddy viscosity type. LES calculations with the present models are described for 323 decaying turbulence and also for forced 323 box turbulence at Taylor Reynolds numbers Rλ in the range Rλ≃30 (fully resolved) to Rλ=∞. The models give good agreement with experiment for decaying turbulence and produce negligible SGS dissipation for forced turbulence in the limit of fully resolved flow.

Effect of subgrid models on the computed interscale energy transfer in isotropic turbulence

Direct and large eddy simulations of forced and decaying isotropic turbulence have been performed to investigate the behavior of subgrid models. Various subgrid models have been analyzed (i.e. Smagorinsky's eddy viscosity model, dynamic eddy viscosity model, dynamic one-equation model for the subgrid kinetic energy and scale-similarity model). A priori analysis showed that the subgrid stress and the subgrid energy flux predicted by the scale similarity model, and subgrid kinetic energy model (with fixed coefficients) correlate reasonably well with exact data, while the Smagorinsky's eddy viscosity model showed relatively poor agreement. However, the correlation for the scale similarity model decreased much more rapidly with decrease in grid resolution when compared to the subgrid kinetic energy model. The subgrid models were then used to carry out large-eddy simulations for a range of Reynolds number. When dynamic evaluation was incorporated, the correlation improved significantly. The dynamic subgrid kinetic energy model showed, consistently, a higher correlation for a range of Reynolds number when compared to the dynamic eddy viscosity model. These results demonstrate the capabilities of the dynamic one-equation model.

Small-scale properties of nonlinear interactions and subgrid-scale energy transfer in isotropic turbulence

Using results of direct numerical simulations of isotropic turbulence the subgrid-scale energy transfer in the physical space is calculated exactly employing a spectral decomposition of the velocity field into large (resolved) and small (unresolved) scales. Comparisons with large-scale quantities reveal large qualitative correlations between regions of subgrid transfer and the boundaries of regions of large vorticity production. This suggests a novel analysis of the nonlinear term, where it is decomposed into four components determined by four combinations of the resolved and unresolved velocity and vorticity fields. It is found that there is a 90% vector-correlation between the subgrid transfer and the component of the full transfer associated with the resolved velocity and unresolved vorticity, but that 90% of the total subgrid energy production is determined by the component associated with the unresolved velocity and resolved vorticity. These results suggest subgrid-scale models that have higher correlation values with the exact subgrid scale terms than a number of other physical quantities that are traditionally considered to govern the dynamics of the large scales of turbulence. The distinguishing feature of the new models is that they emphasize the nonlinear dynamics of the smallest resolved scales in the modeling procedures.

On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number ofRλ≈ 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Usinga prioritesting, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgrid-scale model. It is based on the ‘resolved stress’ tensorLij, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During thea prioritest, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during large-eddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardinaet al.(1980). A relationship betweenLijand a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, anad hocweighting function that depends on the alignment betweenLijand the strain-rate tensor, is introduced. A ‘dynamic’ version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

Tests of subgrid models in the near-wall region using represented velocity fields

The 2.5-dimensional model of the turbulent field near a wall, proposed by Hatziavramidis & Hanratty (1979) and modified by Chapman & Kuhn (1981), has been used to test the subgrid models of Schumann (1973, 1975) and Moin & Kim (1982). The results are disquieting, both trends and orders of magnitude sometimes being seriously in error. It also appears that the contribution of the subgrid energy to the pseudopressure calculated in large-eddy simulations can be large, although this contribution is usually neglected. On the positive side, Leonard's model for the Leonard stress is extremely good, and Schumann's synthetic boundary condition is also found to be reliable.These results must be taken with a grain of salt, since the tests reported in §5 show that the 2.5-dimensional model cannot reproduce important characteristics of the turbulence in the neighbourhood of y+ = 40.