Filtered Density Function Model Research Articles

Lagrangian filtered density function modeling of a turbulent stratified flame combined with flamelet approach

To simulate turbulent flames with high accuracy at low computational cost, Rieth et al. [“A hybrid flamelet finite-rate chemistry approach for efficient LES with a transported FDF,” Combust. Flame 199, 183–193 (2019)] have developed a hybrid method combining a combustion sub-grid model with assumed filtered density function (FDF) with a transported FDF approach. The present paper extends the hybrid approach to a stratified flame from the Cambridge stratified flame series. In contrast to the conventional Lagrangian FDF transport approach, the hybrid model applies Lagrangian particles to solve FDF transport only in selected regions, while an assumed FDF is applied in the remaining domain. With the hybrid model, the overall number of particles is strongly reduced compared to the conventional Lagrangian FDF transport model, promising great savings in computational cost. To provide a basis for the comparisons, simulations with assumed FDF or transported FDF only have also been performed. The present work aims to show the advantage of the Lagrangian transported FDF and the hybrid approach for a highly stratified flame, one of the most challenging members of the well-known Cambridge stratified flame series. Different criteria are tested for triggering the switch-over between the methods to maximize the efficiency of the hybrid approach, where basic flame quantities such as mixture fraction were predicted well with the assumed FDF model, and the temperature and mass fraction of carbon monoxide were predicted better by the hybrid method, featuring the transported FDF technique.

Exploring subgrid-scale variance models in LES of lab-scale methane fire plumes

Large Eddy Simulation (LES) of the lab-scale methane fire plumes investigated experimentally by McCaffrey is performed using the steady laminar flamelet/presumed beta filtered density function model on grids of different resolution ranging from the Taylor length scale to about six times the Kolmogorov length scale. This work focuses on investigating existing subgrid (SGS) mixing models for mixture fraction variance prediction. Three different models based on the local equilibrium assumption, the variance transport equation (VTE) and the second moment transport equation (STE) are assessed. In the non-equilibrium modelling (VTE and STE), the scalar dissipation rate is modelled with an algebraic expression involving an SGS mixing time-scale. The comparison of the solutions is based on the convergence properties of LES statistics for mixture fraction, temperature and axial velocity with respect to the filter width. The simulations show that the equilibrium algebraic model is not suitable for purely buoyant flows. On the other hand, simulations performed with the transport models show that grids coarser than 1 cm cannot resolve adequately the natural laminar instability near the edge of the plume that governs the formation of large-scale vortex and, therefore, underestimate the mixing process, especially in the lower part of the continuous flame. For grid resolutions finer than 1 cm, the STE model is less sensitive to grid refinement than the VTE formulation and differences between the two models are reduced with grid refinement. The STE model predicts also a stronger mixing, resulting in a slightly larger lateral expansion of the fire plume. Predicted solutions by the two models are in quantitative agreement with the experimental data in terms of axial temperature, velocity and temperature fluctuations.

Survey of Turbulent Combustion Models for Large-Eddy Simulations of Propulsive Flowfields

A general review of turbulent combustion modeling closures applicable to large eddy simulations (LES) is provided. The focus is on “regime-independent” models able to provide turbulent combustion closures ranging from purely premixed to purely non-premixed and all regimes between these two limits. Special emphasis is placed on primary propulsion applications, including liquid rocket engines, diesel engines, gas turbines, and scramjets. These applications span a large range of physical phenomena including both idealand real-gas behavior, single-phase and multi-phase combustion, relatively low Mach number to supersonic and hypersonic combustion, and relatively simple geometries to highly complex geometries. Three classes of models are identified as possibly providing such broad based modeling closures: flamelet-library/presumed probability density function (PDF) models, linear eddy based models (LEM), and transported PDF or filtered density function (FDF) based models. This review focuses both on fundamental physical assumptions that apply across all of the models and assumptions that apply to each of the models individually. Namely, assumptions regarding the presumed size of the large dimensional turbulent scalar manifold apply to all of the models; however, flamelet models almost always presume only a few dimensions are necessary to yield adequate representation of the larger, turbulent manifold. In contrast, LEM and FDF models are not, in theory, bound by any manifold size assumptions (i.e. direct calculation of the turbulent scalar manifold is possible); however, due to current computational limitations, these models often employ manifold reduction techniques which are usually not as restrictive as those used by flamelet models. Individual assumptions associated with the specific formulation of each model are also analyzed. From these discussions, additional novel results testing some of the fundamental physical assumptions associated with each model are provided from a unique database of DNS of high pressure turbulent reacting temporally developing shear flames. The DNS database includes simulations of H2/O2, H2/Air, and Kerosene/Air flames with both detailed and reduced chemistry. The DNS include real property models, a real-gas equation of state, and generalized heat and mass diffusion derived from non-equilibrium thermodynamics. The simulations span a large range of Reynolds numbers and pressures (up to 125 atm), with resolutions approaching 1 billion grid points. Finally, some general comments towards the future challenges related to LES combustion modeling are offered.

Experimental assessment of presumed filtered density function models

Measured filtered density functions (FDFs) as well as assumed beta distribution model of mixture fraction and “subgrid” scale (SGS) scalar variance z′′2¯, used typically in large eddy simulations, were studied by analysing experimental data, obtained from two-dimensional planar, laser induced fluorescence measurements in isothermal swirling turbulent flows at a constant Reynolds number of 29 000 for different swirl numbers (0.3, 0.58, and 1.07). Two-dimensional spatial filtering, by using a box filter, was performed in order to obtain the filtered variables, namely, resolved mean and “subgrid” scale scalar variance. These were used as inputs for assumed beta distribution of mixture fraction and top-hat FDF shape estimates. The presumed beta distribution model, top-hat FDF, and the measured filtered density functions were used to integrate a laminar flamelet solution in order to calculate the corresponding resolved temperature. The experimentally measured FDFs varied with the flow swirl number and both axial and radial positions in the flow. The FDFs were unimodal at flow regions with low SGS scalar variance, z′′2¯< 0.01, and bimodal at regions with high SGS variance, z′′2¯> 0.02. Bimodal FDF could be observed for a filter size of approximately 1.5-2 times the Batchelor scale. Unimodal FDF could be observed for a filter size as large as four times the Batchelor scale under well-mixed conditions. In addition, two common computational models (a gradient assumption and a scale similarity model) for the SGS scalar variance were used with the aim to evaluate their validity through comparison with the experimental data. It was found that the gradient assumption model performed generally better than the scale similarity one.

Large Eddy Simulation of piloted pulverized coal combustion using the velocity-scalar joint filtered density function model

Abstract A piloted pulverized coal jet flame was simulated by Large Eddy Simulation (LES). For the sub-grid scale (SGS) effects of the turbulence velocity and the scalar fluctuation of the gas flow on the coal particles, the velocity-scalar joint filtered density function (VSJFDF) model was used. Other models such as the governing equations for particles and coal combustion sub-models were also presented. Simulation results with the VSJFDF model, non-VSJFDF model and those by Franchetti et al. [7] were compared with the experimental data, including the mean and RMS (root mean square) velocity, temperature and species concentrations. The results showed that VSJFDF can distinctly improve the predictions, both in quantity and quality. The location where coal particles start burning could be roughly predicted along the centerline of the flame by VSJFDF model.

Numerical investigation of evaporating isopropyl alcohol spray with LES–FDF model

A model based on the framework of possibility density function (PDF) was used to simulate the sub-grid scale eddies “seen” by the droplets in an evaporating isopropyl alcohol spray. LES was used to simulate the fluid flow. The motion and evaporation of the droplets were solved in Lagrangian method and the filtered density function (FDF) transport equation was adopted to solve the sub-grid scale velocity of the gas phase surrounding the droplets. The comparison between the numerical and experimental results shows that the LES–FDF model can distinctly improve the precision of the simulation by considering the influence of sub-grid turbulence on the motion and evaporation of the droplets. Further studies on the effects of pressure and temperature on the droplet evaporation in turbulence were also carried out.

Large Eddy Simulation of Premixed Turbulent Flames Using the Probability Density Function Approach

In the present study a Large Eddy Simulation and Filtered Density Function model is applied to three premixed piloted turbulent methane flames at different Reynolds Numbers using the Eulerian stochastic fields approach. The model is able to reproduce the flame structure and flow characteristics with a low number of fields (between 4 and 16 fields). The results show a good agreement with experimental data with the same closures employed in non-premixed combustion without any adjustment for combustion regime. The effect of heat release on the flow field is captured correctly. A wide range of sensitivity studies is carried out, including the number of fields, the chemical mechanism, differential diffusion effects and micro-mixing closures. The present work shows that premixed combustion (at least in the conditions under study) can be modelled using LES-PDF methods.. Finally, the ability of the model to predict flame quenching is studied. The model can accurate capture the conditions at which combustion is not sustainable and large pockets of extinction appear.

Numerical simulation of gas–particle dispersion in a backward-facing step flow with LES–FDF model

The particle dispersion in a backward-facing step flow was numerically studied. Large eddy simulation (LES) was used to simulate the gas flow and a filtered density function (FDF) transport equation was derived to solve the sub-grid scale eddies seen by the Lagrangian tracked particles. The LES/FDF model was validated by experiment and the evolution process of the vortex structures was clearly demonstrated. The numerical results of particle dispersion obtained with and without the FDF model were compared; it showed that the FDF model could apparently reflect the influence of sub-grid scale eddies on the particle dispersion in the gas–particle two-phase flow.

Large eddy simulation of bluff-body stabilized flames using a multi-environment filtered density function model

A multi-environment turbulent combustion model, originally developed for Reynolds averaged Navier–Stokes equations (RANS), was extended to large eddy simulation (LES) and was used to model a bluff-body stabilized flame series (HM1 and HM3) that ranges from near equilibrium to near global extinction. The multi-environment filtered density function model (MEFDF) depicts the filtered density function (FDF) as a weighted summation of a small number of multi-dimensional Dirac delta functions in composition space. It is derived from the transport FDF equation using the direct quadrature method of moments (DQMOM). The MEFDF method with multiple reactive scalars retains the unique property of the joint FDF model of treating the chemical source term exactly. It was coupled with a 19-species reduced mechanism based on quasi-steady-state assumption and an In Situ adaptive tabulation (ISAT) algorithm for chemical source term integration. The model successfully captures the level of local extinction of the two flames exhibited in experiment. The predicted radial profiles of mixture fraction, temperature and CO mass fraction are in good agreement with measurements. The current work highlights the potential of LES/MEFDF approach as a tool for modeling of turbulent flame stability and combustion dynamics as well as pollutant emissions.

LES/FDF simulation of particle dispersion in a gas-particle two phase plane wake flow

A filtered density function (FDF) transport equation was derived for the fluid velocity seen by the particles in gas-particle two-phase flow. An LES/FDF simulation of a two-phase plane wake flow was carried out. The simulation results were compared with both the experimental photograph and the simulation results without using the FDF model, and proved that the LES/FDF model can clearly improve the spatial dispersion of the particle phase.

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.