Filtered Density Function Research Articles

Velocity-scalar filtered mass density function for large eddy simulation of turbulent reacting flows

A methodology termed the “velocity-scalar filtered mass density function” (VSFMDF) is developed and implemented for large eddy simulation (LES) of variable-density turbulent reacting flows. This methodology is based on the extension of the previously developed “velocity-scalar filtered density function” method for constant-density flows. In the VSFMDF, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function of the velocity and scalar fields. An exact transport equation is derived for the VSFMDF in which the effects of SGS convection and chemical reaction are in closed forms. The unclosed terms in this equation are modeled in a fashion similar to that in Reynolds-averaged simulation procedures. A set of stochastic differential equations (SDEs) are considered which yield statistically equivalent results to the modeled VSFMDF transport equation. The SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô-Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed. In nonreacting flows, it is shown that the VSFMDF results agree well with those obtained by a “conventional” finite-difference LES procedure in which the transport equations corresponding to the filtered quantities are solved directly. The VSFMDF results are also compared with those obtained by the Smagorinsky closure, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that all of the first two moments including the scalar fluxes are predicted well by the VSFMDF. Moreover, the VSFMDF methodology is shown to be able to represent the variable density effects very well. The predictive capabilities of the VSFMDF in reacting flows are further demonstrated by LES of a reacting shear flow. The predictions show favorable agreement with laboratory data, and demonstrate several of the features as observed experimentally.

A particle formulation for treating differential diffusion in filtered density function methods

We present a new approach for treating molecular diffusion in filtered density function (FDF) methods for modeling turbulent reacting flows. The diffusion term accounts for molecular transport in physical space and molecular mixing in composition space. Conventionally, the FDF is represented by an ensemble of particles and transport is modeled by a random walk in physical space. There are two significant shortcomings in this transport model: (1) the resulting composition variance equation contains a spurious production term and (2) because the random walk is governed by a single diffusion coefficient, the formulation cannot account for differential diffusion, which can have a first-order effect in reacting flows. In our new approach transport is simply modeled by a mean drift in the particle composition equation. The resulting variance equation contains no spurious production term and differential diffusion is treated easily. Hence, the new formulation reduces to a direct numerical simulation (DNS) in the limit of vanishing filter width, a desirable property of any large-eddy simulation (LES) approach. We use the IEM model for mixing. It is shown that there is a lower bound on the specified mixing rate such that the boundedness of the compositions is ensured. We present a numerical method for solving the particle equations which is second-order accurate in space and time, obeys detailed conservation, enforces the realizability and boundedness constraints and is unconditionally stable.

On simulating scalar transport by mixing between Lagrangian particles

Lagrangian particles with mixing can be used as direct numerical simulations (DNS), large eddy simulations (LES), or filtered density function (FDF) methods depending on conditions of the simulations. We estimate major parameters associated with the DNS, LES, and FDF regimes and demonstrate that, under certain conditions specified in the paper, simulations using different mixing models approach the DNS limit.

Numerical study on the turbulent reacting flow in the vicinity of the injector of an LDPE tubular reactor

Three-dimensional (3-D), transient numerical simulations of the turbulent reacting flow in the vicinity of the initiator injection point of a low-density polyethylene (LDPE) tubular reactor using a large eddy simulation (LES) approach combined with a filtered density function (FDF) technique are presented. The numerical approach allows for detailed predictions of the turbulent flow field and the associated (passive and reactive) scalar mixing. The aim is to study the influence of the injector geometry and initiator injection temperature on the LDPE process in terms of product quality (average polymer chain length, and polydispersity) and process efficiency (such as initiator consumption).

Large eddy simulations of turbulent flames using the filtered density function model

Large eddy simulations (LES) of the Sandia/Sydney swirl burners (SM1 and SMA1) and the Sandia/Darmstadt piloted jet diffusion flame (Flame D) are performed. These flames are part of the database of turbulent reacting flows widely considered as benchmark test cases for validating turbulent-combustion models. In the simulations presented in this paper, the subgrid scale (SGS) closure model adopted for turbulence-chemistry interactions is based on the transport filtered density function (FDF) model. In the FDF model, the transport equation for the joint probability density function (PDF) of scalars is solved. The main advantage of this model is that the filtered reaction rates can be exactly computed. However, the density field, computed directly from the FDF solver and needed in the hydrodynamic equations, is noisy and causes numerical instability. Two numerical approaches that yield a smooth density field are examined. The two methods are based on transport equations for specific sensible enthalpy ( h s) and RT, where R is the gas constant and T is the temperature. Consistency of the two methods is assessed in a bluff-body configuration using Reynolds averaged Navier-Stokes (RANS) methodology in conjunction with the transported PDF method. It is observed that the h s method is superior to the RT method. Both methods are used in LES of the SM1 burner. In the near-field region, the h s method produces better predictions of temperature. However, in the far-field region, both methods show deviation from data. Simulations of the SMA1 burner and Flame D are also presented using the h s method. Some deficiencies are seen in the predictions of the SMA1 burner that may be related to the simple chemical kinetics model and mixing model used in the simulations. Simulations of Flame D show good agreement with data. These results indicate that, while further improvements to the methodology are needed, the LES/FDF method has the potential to accurately predict complex turbulent flames.

Developments in Formulation and Application of the Filtered Density Function

An overview is presented of the state of progress in large eddy simulation of turbulent combustion via the filtered density function (FDF). This includes the formulations based on both the joint velocity-scalar FDF, and the marginal scalar FDF. In the former, the most up-to-date and comprehensive form of the model is presented along with its applications for LES of some relatively simple flows. In the latter, results are presented of the most recent LES of a complex turbulent flame. Both of the models are described in the context of a variable density flow via consideration of the filtered mass density function (FMDF).

Large-Eddy Simulations as a Design Tool for Gas Turbine Combustion Systems

In recent years, large-eddy simulation (LES) has emerged as a promising next-generation computational tool for design applications. Compared to Reynolds-averaged Navier-Stokes simulation, the advantages that LES offers, in terms of improved turbulent mixing and ability to predict unsteady phenomena, makes it attractive for gas turbine combustor applications. An overview of recent activities related to LES being pursued at Rolls-Royce is provided. The LES methodology is reviewed both from numerical and modeling perspectives. The state of the art in subgrid-scale (SGS) modeling and spray modeling is discussed. Recently developed SGS reaction-rate closures such as the transport filtered density function method are reviewed and discussed. Validation experiments performed on several benchmark flows (nonreacting and reacting) and practical gas turbine combustor configurations are presented. Finally, ongoing developments for improving the LES methodology are highlighted.

Filtered Density Function for Subgrid Scale Modeling of Turbulent Combustion

Abstract : This research was concentrated primarily on developments and applications of the filtered density function (FDF) for subgrid scale (SGS) modeling of turbulent reacting flows. During the past three years, this work addressed: (1) development of the joint velocity-scalar filtered mass density function (VSFMDF), (2) development of the joint frequency-velocity-scalar filtered mass density function (FVS-FMDF), and (3) implementation of the scalar filtered mass density function (SFMDF) and VSFMDF for large eddy simulation of complex turbulent flames. This is a final report of our activities sponsored by AFOSR under Grant FA9550-06-1-0015.

Formulation Of A Two-phase Filtered DensityFunction Approach For Large Eddy Simulation

Two-phase velocity-scalar filtered density function (FDF) methods have recently been extended for large eddy simulation (LES) of separated two-phase flows [1]. Several terms in the two-phase FDF transport equation are unclosed and must be modeled for practical application of the two-phase FDF theory. Some terms, such as conditional dissipation in momentum, scalar, and momentum-scalar phase spaces are analogous to those in the single-phase velocity-scalar FDF formulation for which closures are readily available. Phase-coupling and interphase conversion terms, however, which are unique to the two-phase FDF formulation are unclosed and must b e modeled. There are, then, th ree d istin ct types o f unclo sed terms in th e two-phase FDF formulation: conditional dissipation terms, phase-coupling terms and interphase conversion terms. In this study, the mathematical formulation of this approach is presented and conditional dissipation, phase-coupling and interphase conversion terms are identified and their connection to standard LES closures identified . 1 Two-phase FDF transport equation Two-phase probability density function (PDF) methods have recently been formulated and successfully applied for dispersed/continuum phase flows [2, 3, 4]. Probability density function methods are based on defining a suitable set of flow realizations over which the PDF may be defined in the ensemble statistics context and therefore in practice are devoid of any spatial information about the two-phase flow. That is to say, turbulent flow field information in the PDF approach must be given in the time-averaged or RANS context, and consequently coupling to a RANS calculation for the carrier gas field is common [4, 2]. Filtered density func© 2005 WIT Press WIT Transactions on Engineering Sciences, Vol 50, www.witpress.com, ISSN 1743-3533 (on-line) Computational Methods in Multiphase Flow III 275

Application Of The Two-phase Filtered DensityFunction Approach For LES Of A 2D DropletLaden Turbulent Mixing Layer

In this study, the two-phase velocity-scalar filtered density function transport equation for large eddy simulation is considered in the limit of a continuum-dispersed phase two-phase flow. For a sufficiently small dispersed particulate phase, all quantities conditionally filtered within the dispersed phase may be disregarded leaving only terms conditionally filtered on the phase interface. These conditionally surface-filtered terms account for phase-coupling between the dispersed and continuum phases of the flow. Closure models are presented and implemented for a two-phase system consisting of a water droplet laden 2d temporally developing mixing layer. Marginal FDF transport equations are presented for each phase and a statistically equivalent set of Ito stochastic differential equations (SDE) are derived from each marginal FDF equation. Simulations are conducted via a full stand-alone Lagrangian particle Monte-Carlo method with closure models to account for subgrid scale (SGS) mixing and inter-phase conversion processes. The effect of variable Stokes number on turbulent dispersion of evaporating and non-evaporating droplets is discussed. 1 Marginal FDF transport equations Two-phase velocity-scalar filtered density function (TVSFDF) methods have recently been developed for large eddy simulation (LES) of separated two-phase flows [1]. In this approach, unclosed terms arise in the marginal FDF transport equations in three classes: quantities conditionally filtered within the phase, quantities conditionally filtered on the phase interface, and inter-phase conversion terms © 2005 WIT Press WIT Transactions on Engineering Sciences, Vol 50, www.witpress.com, ISSN 1743-3533 (on-line) Computational Methods in Multiphase Flow III 283

Large-eddy simulation of a bluff-body-stabilized non-premixed flame using a recursive filter-refinement procedure

A large-eddy simulation (LES) of a bluff-body-stabilized flame has been carried out using a new strategy for LES grid generation. The recursive filter-refinement procedure (RFRP) has been used to generate optimized clustering for variable density combustion simulations. A methane–hydrogen fuel-based bluff-body-stabilized experimental configuration has been simulated using state-of-the-art LES algorithms and subfilter models. The combustion chemistry is described using a precomputed, laminar flamelet model-based look-up table. The GRI-2.11 mechanism is used to build the look-up table parameterized by mixture fraction and scalar dissipation rate. A beta function is used for the subfilter mixture fraction filtered density function (FDF). The simulations show good agreement with experimental data for the velocity field. Time-averaged profiles of major species and temperature are very well reproduced by the simulation. The mixture fraction profiles show excellent agreement at all locations, which helps in understanding the validity of flamelet assumption for this flame. The results indicate that LES computations are able to quantitatively predict the flame structure quite accurately using the laminar flamelet model. Simulations tend to corroborate experimental evidence that local extinction is not significant for this flame.

Turbulent mixing in a tubular reactor: Assessment of an FDF/LES approach

AbstractThe aim of the present research was to investigate the feasibility of a probability density function (PDF) methodology combined with a large eddy simulation (LES) for turbulent reactive mixing in industrial geometries. In order to incorporate the PDF transport equation into a LES context, a filtered version was employed, denoted as filtered density function (FDF) equation. Using FDF/LES, the yield of mixing‐sensitive, parallel competitive reactants was numerically studied in a tubular reactor with perpendicular inlet operated at Re = 4,000 for varying Damköhler numbers and feed‐stream concentrations. The “interaction by exchange with the mean” (IEM) model was used to close the filtered conditional scalar energy dissipation rate (SED) appearing in the FDF equation. The modeling assumptions were assessed by means of laser induced fluorescence (LIF) experimental data, providing the mean conserved concentration field in a horizontal and vertical center plane downstream the injector, and concentration PDF's at several downstream positions. In the vicinity of the injector, the FDF/LES model slightly overpredicts both the spreading of jet and the dissipation of small scale fluctuations, which was attributed to the overprediction of the turbulent diffusivity in the transition region of the laminar jet to the the turbulent wake of the injector. Further downstream, however, the global macro transport of the reactants was qualitatively well predicted, and the sensitivity of the yield to the Damköhler number and feedstream concentration showed consistent behavior. © 2005 American Institute of Chemical Engineers AIChE J, 51: 725–739, 2005

Large eddy simulation (2D) of spatially developing mixing layer using vortex-in-cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion-velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two-dimensional vortex-in-cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte-Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root-mean-square velocity and vorticity fluctuations profiles and negative cross-stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root-mean-square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Prognostic Equations for Cloud Fraction and Liquid Water, and Their Relation to Filtered Density Functions

This paper derives prognostic equations for cloud fraction and specific liquid water content. Such equations are carried by host models in order to provide more accurate forcing of microphysical and radiative parameterizations. The starting point of the derivations is a prognostic equation for the filtered density function (FDF) of a moisture variable. The FDF is the probability density function (PDF) of subgrid (i.e., filtered) fluctuations. The resulting equations for cloud fraction and liquid water contain unclosed terms. Many of these terms can be closed if the relevant joint FDF is known. In addition, there emerge from the derivation two dissipation terms that are not closed even if the FDF of moisture is known. These dissipation terms do not appear in equations for conserved variables such as total water content. The paper then compares various closures in the literature for turbulent flux, source, and dissipative mixing terms. Intuition suggests that dissipative mixing cannot usually increase liquid water content but that dissipative mixing can sometimes increase cloud fraction. The paper discusses two closures for dissipative mixing: one is based on the linear mean-square estimation (LMSE) model, and the other is due to Tiedtke. The LSME model permits cloud fraction to increase, as expected, but the Tiedtke model does not.

Velocity-scalar filtered density function for large eddy simulation of turbulent flows

A methodology termed the “velocity-scalar filtered density function” (VSFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function (PDF) of the velocity and scalar fields. An exact transport equation is derived for the VSFDF in which the effects of the SGS convection and chemical reaction are closed. The unclosed terms in this equation are modeled in a fashion similar to that typically used in Reynolds-averaged simulation procedures. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by a finite difference LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VSFDF results are compared with those obtained by the Smagorinsky model, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that the values of both the SGS and the resolved components of all second order moments including the scalar fluxes are predicted well by VSFDF. The sensitivity of the calculations to the model’s (empirical) constants are assessed and it is shown that the magnitudes of these constants are in the same range as those employed in PDF methods.

A subgrid-scale mixing model for large-eddy simulations of turbulent reacting flows using the filtered density function

The filtered density function approach [Colucci et al., Phys. Fluids 10, 499 (1999); Jaberi et al., J. Fluid Mech. 401, 85 (1999)] for large-eddy simulations of initially nonpremixed turbulent reacting flows is extended to encompass the flamelet regime. This is done by developing a model which describes the effect of realistic, Arrhenius chemical kinetics on the subgrid mixing time scale for reactive scalars. The model is based on mapping closure and flamelet modeling and can readily be applied to many existing micro-mixing models where the time scales of the reacting scalars appear explicitly in the formulation. Testing of the model using spatially filtered direct numerical simulation data of a turbulent reacting jet [Boersma, Center for Turbulence Research Annual Research Briefs (Stanford University/NASA Ames, 1999), pp. 59–72] show a significant improvement over the generally used estimate of substituting the time scale of the reacting scalars by that of a conserved scalar. The impact of internal intermittency on the performance of the new model is discussed.

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.

Velocity filtered density function for large eddy simulation of turbulent flows

A methodology termed the “velocity filtered density function” (VFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function of all of the components of the velocity vector. An exact transport equation is derived for the VFDF in which the effects of the SGS convection appear in closed form. The unclosed terms in this transport equation are modeled. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by an Eulerian LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VFDF results are compared with those obtained via several existing SGS closures. These results are also analyzed via a priori and a posteriori comparisons with results obtained by direct numerical simulation of an incompressible, three-dimensional, temporally developing mixing layer.

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.

Large Eddy Simulation (2D) of a Reacting Plane Mixing Layer Using Filtered Density Function Closure

The filtered density function (FDF) is implemented for a two-dimensional, large eddy simulation (LES) of a gas phase, spatially developing, reacting and non-reacting, constant-density, plane mixing layer in a flow regime prior to the mixing transition where the flow is mainly two-dimensional. The unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities following the FDF approach. In the derived FDF transport equation, the effect of chemical reactions appears in a closed form. The Lagrangian Monte Carlo scheme is used to solve the FDF transport equation. The applicability and performance of the FDF for LES of a reacting plane mixing layer are assessed by comparisons with experimental measurements. In non-reacting flow, the calculated mean streamwise velocity profiles and mean mixture fraction profiles relax to self-similarity, which is in satisfactory agreement with the measurements. In reacting flow, the FDF calculation provided a satisfactory accuracy in comparison with measurements of mean reactant and product concentration. The increase in the total amount of product formation in the flip case demonstrates the asymmetric characteristics of the entrainment and mixing characteristics in the mixing layer.