Subgrid-scale Variance Research Articles

Conditionally filtered diffusion of mixture fraction and temperature in turbulent partially premixed flames

Recent studies have shown that the subgrid-scale (SGS) mixture fraction has different structures and probability distributions for different SGS scalar variances. We study the effects of these structures on the scalar diffusion and temperature diffusion in the context of large-eddy simulation (LES) of turbulent combustion. Line images obtained in turbulent partially premixed (Sandia) flames are used to analyze the scalar and temperature diffusion filtered conditionally on the scalar and temperature, which must be correctly modeled in LES. The results show that for small SGS variance the scalar and temperature diffusion have a relatively simple structure. For large SGS variance the diffusion is much more complex, with the flamelet structure and local extinction playing important roles. The results show that it is important that mixing models for filtered density function methods be able to account for the different SGS mixture fraction and temperature structures for small and large SGS variance.

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.

Implementation Issues of the Conditional Moment Closure Model in Large Eddy Simulations

Different ways of transferring information regarding the mixture fraction, its sub-grid scale variance and the scalar dissipation rate are examined in terms of a Large Eddy Simulation (LES)/Conditional Moment Closure (CMC) calculation. In such a simulation, information must be transferred from a fine LES grid to a usually coarser CMC grid. Different options of calculating conditional and unconditional quantities in the CMC resolution are assessed by filtering experimental mixture fraction and scalar dissipation rate data at various resolutions. It was found that when a presumed shape for the Filtered Density Function at the CMC resolution is used, special care must be given to the mixture fraction variance. It was also found that the Amplitude Mapping Closure model can be used for the conditional scalar dissipation rate. LES/CMC with detailed chemistry of a bluff-body stabilised burner was performed using two different ways of calculating the turbulent diffusivity. The structure of the flame is realistic, with little difference noticed when using the two diffusivities.

Spatial Weighting and Iterative Projection Methods for EOFs

Abstract Often there is a need to consider spatial weighting in methods for finding spatial patterns in climate data. The focus of this paper is on techniques that maximize variance, such as empirical orthogonal functions (EOFs). A weighting matrix is introduced into a generalized framework for dealing with spatial weighting. One basic principal in the design of the weighting matrix is that the resulting spatial patterns are independent of the grid used to represent the data. A weighting matrix can also be used for other purposes, such as to compensate for the neglect of unrepresented subgrid-scale variance or, in the form of a prewhitening filter, to maximize the signal-to-noise ratio of EOFs. The new methodology is applicable to other types of climate pattern analysis, such as extended EOF analysis and maximum covariance analysis. The increasing availability of large datasets of three-dimensional gridded variables (e.g., reanalysis products and model output) raises special issues for data-reduction methods such as EOFs. Fast, memory-efficient methods are required in order to extract leading EOFs from such large datasets. This study proposes one such approach based on a simple iteration of successive projections of the data onto time series and spatial maps. It is also demonstrated that spatial weighting can be combined with the iterative methods. Throughout the paper, multivariate statistics notation is used, simplifying implementation as matrix commands in high-level computing languages.

A PDF-based hybrid prognostic cloud scheme for general circulation models

A new cloud parameterization based on prognostic equations for the subgrid-scale fluctuations in temperature and total water content is introduced for global climate models. The proposed scheme, called hybrid prognostic cloud (HPC) parameterization, employs simple probability density functions (PDFs) to the horizontal subgrid-scale inhomogeneity, allowing them to vary in shape in response to small-scale processes such as cumulus detrainment and turbulent mixing. Simple tests indicate that the HPC scheme is highly favorable as compared to a diagnostic scheme in terms of the cloud fraction and cloud water content under either uniform or non-uniform forcing. The relevance of the HPC scheme is investigated by implementing it in an atmospheric component model of the climate model MIROC with a coarse resolution of T42. A comparison of the short-term integrations between the T42 model and a global cloud resolving model (GCRM) reveals that the HPC scheme can reproduce, to a certain degree, the subgrid-scale variance and skewness of temperature and total water content simulated in the GCRM. It is also found that the HPC scheme significantly alters the climatological distributions in cloud cover, precipitation, and moisture, which are all improved from the model using a conventional diagnostic cloud scheme.

Investigation of subgrid-scale mixing of mixture fraction and temperature in turbulent partially premixed flames

The filtered mass density function (FMDF) of mixture fraction and temperature used in large eddy simulation (LES) of turbulent combustion is studied experimentally using line images obtained in turbulent partially premixed methane flames (Sandia flames D and E). Cross-stream filtering is employed to obtain the FMDF and other filtered variables. The mean of the FMDF conditional on the subgrid-scale (SGS) scalar variance at a given location are found to vary from unimodal to bimodal, corresponding to quasi-equilibrium distributed reaction zones and laminar flamelets (including extinguished flamelets), respectively. The conditionally filtered mixture fraction dissipation for small SGS variances has a relatively weak dependence on the mixture fraction, and is not sensitive to temperature for extinguished samples. For large SGS variance the large dissipation is concentrated in the cliffs and increases with decreasing temperature. The conditionally filtered temperature dissipation for small SGS variances is the highest for intermediate temperature. For large SGS variance the dependence is more complex and the pilot gas appears to be playing an important role. The different SGS mixture fraction structures for small and large SGS variances, as reflected by the unimodal and bimodal FMDF, have a strong impact on the small-scale mixing and turbulence–chemistry interaction, as reflected by the results for the conditionally filtered temperature dissipation. The results have implications for understanding and modeling multiple reactive scalar SGS mixing.

Comparison between LES results and experimental data in reacting flows

Resolved flow statistics extracted from large eddy simulations (LES) do not directly compare to unfiltered flow statistics determined, for example, from experiments. Scalar statistics in reacting flows are investigated here from a priori tests based on experimental data. Mean scalar fields are found to be quite well estimated from LES, taking into account mass-weighted statistics, but resolved variances are strongly affected by the filtering operation and decrease roughly linearly when the filter size Δ increases. Accordingly, subgrid scale scalar variances should be modelled to compare LES and experimental results. A simple gradient model is found to provide very good estimates of the mean subgrid scale variance of the progress variable.

DNS of a premixed turbulent V flame and LES of a ducted flame using a FSD-PDF subgrid scale closure with FPI-tabulated chemistry

Two complementary simulations of premixed turbulent flames are discussed. Low Reynolds number two-dimensional direct numerical simulation of a premixed turbulent V flame is first performed, to further analyze the behavior of various flame quantities and to study key ingredients of premixed turbulent combustion modeling. Flame surface density, subgrid-scale variance of progress variables, and unresolved turbulent fluxes are analyzed. These simulations include fully detailed chemistry from a flame-generated tabulation (FPI) and the analysis focuses on the dynamics of the thin flame front. Then, a novel subgrid scale closure for large eddy simulation of premixed turbulent combustion (FSD-PDF) is proposed. It combines the flame surface density (FSD) approach with a presumed probability density function (PDF) of the progress variable that is used in FPI chemistry tabulation. The FSD is useful for introducing in the presumed PDF the influence of the spatially filtered thin reaction zone evolving within the subgrid. This is achieved via the exact relation between the PDF and the FSD. This relation involves the conditional filtered average of the magnitude of the gradient of the progress variable. In the modeling, this conditional filtered mean is approximated from the filtered gradient of the progress variable of the FPI laminar flame. Balance equations providing mean and variance of the progress variable together with the measure of the filtered gradient are used to presume the PDF. A three-dimensional larger Reynolds number flow configuration (ORACLES experiment) is then computed with FSD-PDF and the results are compared with measurements.

Experimental study of velocity-scalar filtered joint density function for LES of turbulent combustion

The velocity-scalar filtered joint density function (FJDF) used in large eddy simulation (LES) of turbulent combustion is experimentally studied. Measurements are made in the fully developed region of an axisymmetric turbulent jet using an array consisting of three X-wires and resistance-wire temperature sensors. Filtering in the cross-stream and streamwise directions is realized by using the array and by invoking Taylor’s hypothesis, respectively. The means of the FJDF conditional on the subgrid-scale (SGS) turbulent kinetic energy and the SGS scalar variance at a given location range from close to joint normal to bimodal with the peaks separated in both velocity and scalar spaces, which correspond to qualitatively different mixing regimes. For close to joint normal FJDFs, the SGS fields are well mixed. For bimodal FJDFs, the conditionally filtered scalar diffusion and dissipation strongly depend on the SGS velocity and scalar, consistent with a combination of diffusion layers and plane strain in the SGS fields, which is similar to the counter-flow model for laminar flamelets. The results suggest that in LES, both mixing regimes could potentially be modeled accurately. The velocity field affects the SGS variance and the filtered scalar dissipation rate primarily by changing the degree of nonequilibrium of the SGS scalar and the SGS time scale, respectively. This study further demonstrates the importance of including velocity in mixing models.

Experimental investigation of scalar-scalar-dissipation filtered joint density function and its transport equation

The filtered joint density function (FJDF) of a conserved scalar and its dissipation rate and the FJDF transport equation are studied experimentally in the fully developed region of a turbulent jet with a jet Reynolds number Rej=UjDj/ν of 40 000. Filtered variables are obtained by a two-dimensional box filter having a size of 90 scalar dissipation scales, realized by using three discrete sensors aligned in the cross-stream direction and by invoking Taylor’s hypothesis. One-dimensional filters are also used to examine the effects of filter scales. The results show that for small and large subgrid-scale (SGS) scalar variance the FJDF and its dynamics have qualitatively different characteristics. The conditional mean FJDF for small SGS variance is generally unimodal and the dependence of the dissipation on the SGS scalar is moderate. The dissipation rate is diffused toward its conditional mean and its production rate is generally smaller than its destruction rate. These characteristics are similar to those of scalar-scalar-dissipation joint probability density functions in a fully developed turbulent flow. On the other hand, the conditional FJDF for large SGS variance is bimodal and the dissipation shows strong dependence on the SGS scalar. Diffusion of the dissipation toward values much larger than its conditional mean is observed. The production of the dissipation generally exceeds its destruction, indicating nonequilibrium scalar dissipation. These characteristics are found to be associated with diffusion-layer-like structure in the SGS scalar for large SGS variance. In addition to the SGS scalar variance, the time scale of the SGS scalar also affects the FJDF. A current assumed FJDF model is also examined. The results provide new fundamental understanding of the structure of the small-scale scalar, which is important for improving large eddy simulation of nonpremixed turbulent combustion as well as for studying local extinction and reignition phenomena.

Conditionally filtered scalar dissipation, scalar diffusion, and velocity in a turbulent jet

Conditionally filtered conserved scalar (temperature) dissipation (CFD), diffusion, and conditionally filtered velocity are studied experimentally in the fully developed region of a turbulent jet with a jet Reynolds number Rej=UjDj/ν of 40 000. These variables are the unclosed terms in the transport equation of the conserved scalar filtered density function. One-dimensional box filters of widths Δ ranging from 30 to 496 scalar dissipation scales (ηφ) as well as a two-dimensional box filter (Δ/ηφ=90) which consists of three discrete sensors are used to obtain filtered variables. Taylor’s hypothesis is used to perform the streamwise filtering. The means of these conditionally filtered variables conditional on the resolvable-scale scalar fluctuations 〈φ〉L and the subgrid-scale (SGS) variance 〈φ″2〉L indicate two regimes of the SGS scalar mixing. For large SGS variance (〈φ″2〉L/〈φ″2〉>1) the SGS scalar exhibits similarities to scalars in initially binary mixing at early times, suggesting existence of diffusion-layer-like structure and nonequilibrium SGS scalar. The nonequilibrium is further evidenced by the observed large SGS mechanical-to-scalar time scale ratio and large SGS variance production-to-dissipation ratio. These characteristics have not been previously observed in fully developed flows. For small SGS variance the conditionally filtered variables indicate well-mixed SGS scalar. The results also show that the SGS advection is only significant at large SGS variance. Our measurements also suggest that the dependence of the scalar dissipation (and the filtered dissipation) on large-scale scalar fluctuations is caused primarily by the fluctuations in the spectral transfer and that the conditional CFD is directly affected mostly by scales near the filter scale. The present study suggests that the nonequilibrium effects related to sheet-like structures are important for modeling the conditionally filtered variables.

Measurements of conserved scalar filtered density function in a turbulent jet

Conserved scalar (temperature) filtered density function (FDF) is studied experimentally in the fully developed region of a turbulent jet with Taylor-microscale Reynolds numbers of 293 and 190. We obtain the FDFs using one-dimensional box filters of widths Δ ranging from 30 to 248 scalar dissipation scales (ηφ) as well as a two-dimensional box filter (Δ/ηφ=90) which consists of three discrete sensors. Taylor’s hypothesis is used to perform streamwise filtering operations. The mean conserved scalar FDF conditioned on the resolvable-scale scalar fluctuations 〈φ〉L and the subgrid scale (SGS) variance 〈φ″2〉L (log-normally distributed) is found to be bimodal when 〈φ″2〉L/〈φ″2〉 is large, indicating that the conditional SGS mixing is nearly binary. For small 〈φ″2〉L/〈φ″2〉 (<1) the conditional FDF is approximately Gaussian. The kurtosis of the conditional FDF decreases with increasing SGS variance and is independent of the filter widths for large SGS variance. The bimodal distribution can be symmetric or asymmetric depending on the curvature of the resolvable-scale scalar. As the SGS variance increases, the conditional scalar differences for separations comparable to the filter widths also change from Gaussian to bimodal distributions. At the same time the conditional SGS scalar changes from approximately isotropic to strongly anisotropic. The results show that the contributions to the bimodal distributions come primarily from the SGS scales comparable to the filter width. It is remarkable that similarities exist between the bimodal conditional FDFs obtained here in a fully developed jet and bimodal probability density functions observed in the early stages of binary scalar mixing. The present study provides a physical basis for the assumed FDF method for conserved scalars used in large-eddy simulation of turbulent combustion.

Subgrid scale variance and dissipation of a scalar field in large eddy simulations

Subgrid scale (SGS) variance of a scalar field in large eddy simulations is only properly defined in relation to a probability density function. This solves a reported problem in the variance definition [Cook and Riley, Phys. Fluids 6, 2868 (1994); Cook, Riley, and Kosály, Combust. Flame 109, 332 (1997)] and allows to write a simple evolution equation for the scalar variance. This equation shows that a recently proposed model for scalar dissipation in terms of the large-scale gradients [Pierce and Moin, Phys. Fluids 10, 3041 (1998)] implies dissipation and production canceling out, preventing variance decay and complete mixing at SGS level. An alternative simple model for dissipation in terms of a SGS mixing characteristic time is proposed and tested here.

An evaluation of the assumed beta probability density function subgrid-scale model for large eddy simulation of nonpremixed, turbulent combustion with heat release

The assumed beta distribution model for the subgrid-scale probability density function (PDF) of the mixture fraction in large eddy simulation of nonpremixed, turbulent combustion is tested, a priori, for a reacting jet having significant heat release (density ratio of 5). The assumed beta distribution is tested as a model for both the subgrid-scale PDF and the subgrid-scale Favre PDF of the mixture fraction. The beta model is successful in approximating both types of PDF but is slightly more accurate in approximating the normal (non-Favre) PDF. To estimate the subgrid-scale variance of mixture fraction, which is required by the beta model, both a scale similarity model and a dynamic model are used. Predictions using the dynamic model are found to be more accurate. The beta model is used to predict the filtered value of a function chosen to resemble the reaction rate. When no model is used, errors in the predicted value are of the same order as the actual value. The beta model is found to reduce this error by about a factor of two, providing a significant improvement.

Mixing Models for Large-Eddy Simulation of Nonpremixed Turbulent Combustion

The application of mixture fraction based models to large-eddy simulations (LES) of nonpremixed turbulent combustion requires information about mixing at length scales not resolved on the LES grid. For instance, the large-eddy laminar flamelet model (LELFM) takes the subgrid-scale variance and the filtered dissipation rate of the mixture fraction as inputs. Since chemical reaction rates in nonpremixed turbulence are largely governed by the mixing rate, accurate mixing models are required if mixture fraction methods are to be successfully used to predict species concentrations in large-eddy simulations. In this paper, several models for the SGS scalar variance and the filtered scalar dissipation rate are systematically evaluated a priori using benchmark data from a DNS in homogeneous, isotropic, isothermal turbulence. The mixing models are also evaluated a posteriori by applying them to actual LES data of the same flow. Predictions from the models that depend on an assumed form for the scalar energy spectrum are very good for the flow considered, and are better than those from models that rely on other assumptions.

A dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar

The dynamic procedure is applied to the problem of modeling the subgrid-scale variance and dissipation rate of a conserved scalar in large eddy simulations of turbulent reacting flows. A simple scaling relation for the subgrid-scale variance is proposed, and the coefficient of the scaling law is obtained using the dynamic procedure. The variance dissipation rate is modeled by assuming equilibrium with the local variance production rate, which is obtained using a dynamic model. Example model predictions are obtained using actual large eddy simulation data, and the subgrid variance predicted by the dynamic model is compared to results obtained using a scale similarity model. Generalization of the approach to multiple scalars and nonconserved scalars is briefly discussed.

Subgrid-scale modeling for turbulent reacting flows

The Large Eddy Simulation of non-premixed, turbulent, reacting flows is addressed. A new subgrid-scale chemistry model, previously proposed for incompressible, isothermal flows, is extended to the case of compressible combustion with multi-step, Arrhenius-rate reactions. The chemistry model predicts filtered chemical species concentrations and filtered reaction rates in a turbulent flow. It accounts for finite-rate chemistry by invoking the laminar flamelet approximation and employs an assumed form for the subgrid or “Large Eddy” Probability Density Function (LEPDF) of a mixture-fraction. It also uses an assumed counterflow form for the local scalar dissipation rate. Inputs to the chemistry model are the Favre-filtered mixture-fraction, its subgrid-scale variance, and filtered dissipation rate. The model is evaluated using (256) 3 point Direct Numerical Simulations of incompressible, nonisothermal decaying turbulence with a single-step reaction. Results indicate that, as the activation temperature is increased, the accuracy of the model degrades in an absolute sense but improves relative to an equilibrium chemistry assumption. Finally, it is also demonstrated that the assumed Beta distribution for the LEPDF yields reasonably accurate results for low (realistic) stoichiometric values of the mixture-fraction.

Investigation of Modeling for Non-Premixed Turbulent Combustion

A method for predicting filtered chemical species concentrations and filtered reaction rates in Large-Eddy Simulations of non-premixed, non-isothermal, turbulent reacting flows has been demonstrated to be quite accurate for higher Damkohler numbers. This subgrid-scale model is based on flamelet theory and uses presumed forms for both the dissipation rate and subgrid-scale probability density function of a conserved scalar. Inputs to the model are the chemistry rates, the Favre-filtered scalar, and its subgrid-scale variance and filtered dissipation rate. In this paper, models for the filtered dissipation rate and subgrid-scale variance are evaluated by filtering data from 5123-point Direct Numerical Simulations of a single-step, isothermal reaction developing in the isotropic, incompressible, decaying turbulence field of Comte-Bellot and Corrsin. Both the subgrid-scale variance and the filtered dissipation rate models (the ”sub-models”) are found to be reasonably accurate. The effect of the errors introduced by the sub-models on the overall model is found to be small, and the overall model is shown to accurately predict the spatial average of the filtered species concentrations over a wide range of times.

A laminar flamelet approach to subgrid-scale chemistry in turbulent flows

A method is presented whereby filtered chemical species may be modeled in large eddy simulations (LES) of nonpremixed, turbulent combustion. The model is based on the concept of laminar flamelets, and assumes that a filtered mixture-fraction, as well as its subgrid-scale variance and filtered dissipation rate, are known at each grid cell of an LES. The model makes use of the subgrid-scale or “large-eddy” probability density function of the mixture-fraction, which is assumed to follow a beta-distribution. Also, an assumed functional form for the scalar dissipation rate is employed. The model is evaluated by filtering data from direct numerical simulations (DNS) of homogeneous, isotropic, decaying turbulence. Results show that the model, termed the large-eddy, laminar flamelet model (LELFM), is reasonably accurate and that the accuracy improves with increasing Damköhler number.

Determination of the constant coefficient in scale similarity models of turbulence

Scale similarity models are discussed for the subgrid-scale stress tensor and subgrid-scale variance of a passive scalar in a turbulent flow. The modeling involves sampling of the resolved fields by applying a spatial filter at a scale greater than the grid spacing of the computational mesh. A method is described for determining the constant of proportionality present in each model. The model constant for the subgrid-scale stress tensor is determined such that the subgrid-scale turbulent kinetic energy is exact on average. The constant is shown to depend on the size of the sampling filter, the turbulence Reynolds number and the degree of resolution of the large eddies.