Filtered Mass Density Function Research Articles

Joint subgrid velocity-scalar filtered mass density function method for compressible turbulent flows

In large eddy simulations (LESs) and scalar filtered mass density functions, a gradient diffusion model is generally used for subgrid-scale (SGS) scalar flux modeling. However, this hypothesis is known to generate evident errors under certain conditions, particularly when the counter-gradient scalar transport effect dominates. Herein, a joint subgrid velocity-scalar filtered mass density function (VSGSSFMDF) method is developed to address this problem. The exact FMDF transport equation is derived in detail. Based on the order of the magnitude analysis and simply Langevin model, the FMDF transport equation is modeled, and a system of stochastic differential equations is, thus, proposed. Theoretical derivation and analysis suggest that both the first- and second-order moments are consistent. A compressible mixing layer and hydrogen/air-reactive mixing layer are simulated to verify the proposed method. Based on the diffusion model, a direct numerical simulation and an LES are performed for comparative verification. A generally reasonable SGS velocity distribution is obtained using the proposed VSGSSFMDF method. Consequently, the counter-gradient scalar transport effect is effectively simulated using this method. Overall, the VSGSSFMDF produces better results than the SFMDF and LES in both cases, particularly in the reactive case.

Large eddy simulation of supersonic mixing layers using a compressible filtered mass density function method

To verify the applicability and superiority of the compressible filtered mass density function method in real supersonic mixing flows, a classical experimental subsonic-supersonic spatial mixing layer is numerically investigated. In the present method, a transport equation for the scalar filtered mass density function is solved using the Lagrangian particle solver, while the filtered mass, momentum, and energy equations are fully coupled and solved by the large eddy simulation finite difference solver. First, a parametric study of the mixing model constant controlling the scalar mixing rate is conducted in the two-dimensional case. It is found that the mixing model constant has a significant effect on the root mean square profile of the mixture fraction. The appropriate mixing model constant for this supersonic mixing layer is given by comparing the root mean square results under three different mixing model constants. These results also show that the suitable mixing model constant used in compressible flows is much larger than that used in incompressible flows. Second, three-dimensional simulations are performed to verify the current method. For comparison, a traditional large eddy simulation with the gradient diffusion model is also conducted. The results show that the scalar mixing is better predicted by the filtered mass density function method than the gradient diffusion model. In particular, the mean and root mean square mixture fraction computed by the present method are in excellent agreement with the experimental data whereas the gradient diffusion model overestimates the peak root mean square mixture fraction by about 45%.

A Numerical Investigation of Mixing Models in LES-FMDF for Compressible Reactive Flows

The filtered mass density function (FMDF) model has been employed for large-eddy simulations (LES) of compressible high-speed turbulent mixing and reacting flows. However, the mixing model remains a pressing challenge for FMDF methods, especially for compressible reactive flows. In this work, a temporal development mixing layer with two different convective Mach numbers, Mc=0.4 and Mc=0.8, is used to investigate the mixing models. A simplified one-step reaction and a real hydrogen/air reaction are employed to study the mixing and turbulence-chemistry interaction. Two widely used mixing models, interaction by exchange with the mean (IEM) and Euclidean minimum spanning tree (EMST), are studied. Numerical results indicate that no difference is observed between the IEM and EMST models in simple reaction flows. However, for hydrogen/air reactions, the EMST model can predict the reaction more accurately in high-speed flow. For mixing models in compressible reactive flows, the requirement of localness preservation tends to be more essential as the convective Mach number increases. With the increase of compressibility, the sensitivity of the mixing model coefficient is reduced significantly. Therefore, the appropriate mixing model coefficient has a wider range. Results also indicate that a large error may result when using a fixed mixing model coefficient in compressible flows.

A conservative and consistent scalar filtered mass density function method for supersonic flows

A novel scalar filtered mass density function (SFMDF) method is developed for high-speed flows, especially for supersonic reactive flows. The total energy is proposed as the energy form for SFMDF, instead of the commonly used enthalpy or sensible enthalpy. Such an energy form is entirely consistent with the one typically used in large eddy simulation (LES) for fully compressible flows, so that the exact/modeled energy equations in both LES and SFMDF are readily identical. Moreover, the total energy can formulate the SFMDF energy transport equation in such a way that the high-speed source term is strictly conservative. Following the conservative formulation, numerically robust conservative schemes are readily available for flows with discontinuities. Tests in one-dimensional Euler equations show that the temperature redundantly obtained based on the total energy (with conservative high-speed source terms) shows better agreement with the analytical result than the one based on the enthalpy. The proposed LES-SFMDF method is further tested in a shock tube interacting with an isotropic turbulent flow, a compressible two-dimensional non-reactive temporally developing mixing layer, and a supersonic three-dimensional reactive temporally developing mixing layer. Results show that SFMDF with the total energy can considerably improve the temperature distribution in both non-reactive and reactive flows. The proposed LES-SFMDF method with the total energy predicts the turbulence–chemistry interaction better than LES-SFMDF with the enthalpy as well as LES with the well-stirred reactor model in supersonic combustion. This conservative and consistent SFMDF method can be readily extended to more sophisticated probability density function methods in high-speed flows.

LES/FMDF of turbulent reacting jet in cross-flow

Purpose This study aims to numerically investigate the flow features and mixing/combustion efficiencies in a turbulent reacting jet in cross-flow by a hybrid Eulerian-Lagrangian methodology. Design/methodology/approach A high-order hybrid solver is employed where, the velocity field is obtained by solving the Eulerian filtered compressible transport equations while the species are simulated by using the filtered mass density function (FMDF) method. Findings The main features of a reacting JICF flame are reproduced by the large-eddy simulation (LES)/FMDF method. The computed mean and root-mean-square values of velocity and mean temperature field are in good agreement with experimental data. Reacting JICF’s with different momentum ratios are considered. The jet penetrates deeper for higher momentum ratios. Mixing and combustion efficiency are improved by increasing the momentum ratio. Originality/value The authors investigate the flow and combustion characteristics in subsonic reacting JICFs for which very limited studies are reported in the literature.

Large eddy simulation of propane combustion in a planar trapped vortex combustor

Propane combustion in a trapped vortex combustor (TVC) is characterized via large eddy simulation coupled with filtered mass density function. A computational algorithm based on high order finite difference (FD) schemes, is employed to solve the Eulerian filtered compressible Navier-Stokes equations. In contrast, a Lagrangian Monte-Carlo solver based on the filtered mass density function is invoked to describe the scalar field. The impact of injection strategy on temperature distribution and flame structure in a planar single-cavity TVC is investigated. A fuel jet and an air jet are injected directly into the cavity from the forebody and the afterbody, respectively. Different injection schemes are contemplated by altering fuel and air jet locations representing the different flow and flame structures. The temperature distribution, along with cross-sectional averaged temperature and flame structure, are compared for fuel/air injection strategies. The temperature field reveals that configurations in which both air and fuel jets are located at the cavity-walls midpoint or adjacent to the cavity inferior wall, lead to a more uniform temperature distribution and lower maximum temperature with the latter configuration performing slightly better. While, the former configuration provides the closest cross-sectional averaged temperature to the adiabatic flame temperature. The reaction rate distributions show that the configurations mentioned above lead to a more contained flame, chiefly due to more efficient fuel-air mixing at lower regions of the cavity.

A hybrid discontinuous spectral element method and filtered mass density function solver for turbulent reacting flows

We present a novel hybrid scheme for the large eddy simulation (LES) of turbulent reacting flows. The scheme couples the discontinuous spectral element method (DSEM) solver for the unsteady compressible Navier-Stokes equations with a Monte Carlo particle filtered mass density function (FMDF) solver for the transport of reacting species. The method is capable of high-order simulations on unstructured grids. Mean particle estimate construction mimics the DSEM numerical procedure and utilizes variable basis functions. The scheme is tested on non-reacting and reacting Taylor-Green vortex flows. Studies of varying polynomial order, different basis functions for constructing particle estimates, and varying particle quantities are conducted. We demonstrate that a tent kernel, in conjunction with high polynomial order, produces the most accurate results. The chemically reacting simulations validate the hybrid scheme and demonstrate its applicability across a range of reaction regimes. The hybrid scheme's computational cost is 2.1 times the DSEM-LES solver.

Numerical investigation of flow field effects on fuel–air mixing in a non-reacting trapped vortex combustor with different injection arrangements

Large eddy simulation (LES) of turbulent mixing in a trapped vortex combustor (TVC) is carried out using hybrid Eulerian–Lagrangian methodology. The Eulerian filtered compressible transport equations are solved for the resolved velocity and the composition field is provided by the filtered mass density function (FMDF). The reliability of the solver is assessed by investigating a TVC like configuration and the results compare well with the available experimental data. The consistency between Eulerian and Lagrangian parts is statistically quantified proving the performance of the LES/FMDF model. A planar single-cavity TVC is studied in which the fuel and air jets are issued into the cavity from the forebody and the afterbody, respectively. Various injection schemes are considered by changing fuel and air jet locations introducing different vortical structures and mixing qualities. The vortical structure analysis along with various quantitative measures such as mean cavity and near stoichiometric equivalence ratios, global fuel distribution and mixing efficiency curves are invoked to compare fuel/air injection strategies. The primary objective of this study is to identify the arrangement that provides most efficient mixing among the considered arrangements in non-reacting conditions. The predicted results show that configurations with both air and fuel jets adjacent to the mainstream or adjacent to the cavity inferior wall leads to more homogeneous mixture and a better global fuel distribution, with the former configuration performing slightly better. While the latter configuration holds larger local near stoichiometric regions inside the cavity and its interface with mainstream. These results are commensurate with expectations based on the vortical structures analysis and the injected fuel trajectory inside the cavity.

LES/FMDF of premixed methane/air flow in a backward-facing step combustor

In the present study, a hybrid Eulerian-Lagrangian methodology is utilized for large eddy simulation (LES) of premixed fuel/air flow over a three-dimensional backward facing step (BFS). The fluid dynamic features are obtained by solving the Eulerian filtered compressible transport equations while the species are predicted by using the filtered mass density function method (FMDF). Some scalar fields are duplicated in FD and MC solvers to examine the numerical consistency between them. A good agreement is achieved by comparing the essential characteristics of the BFS flow (such as the mean and RMS values of the velocity and temperature fields and also the reattachment length) obtained from numerical results with the measurements. This ensures that the proposed hybrid method is reliable for studying the reacting flow in relatively complex combustion systems. Additionally, the performance of several SGS models are assessed, and the results indicate that the dynamic Smagorinsky and WALE models are superior to standard Smagorinsky and MKEV models.

Assessment of subgrid scale mixing models used in LES at high pressures

ABSTRACTMolecular mixing models for conditional scalar diffusion (CSD) in the filtered mass density function (FMDF) transport equation used in Large Eddy Simulations are evaluated against direct numerical simulation (DNS) of spatially evolving jets. A database of DNS of turbulent mixing slot jets of and is developed. The formulation includes compressible form of the governing equations, a generalized diffusion model, a real gas equation of state and realistic property models. Simulations are conducted at initial pressures between 1-100 atm and Reynolds numbers of 850, 1300. The CSD term in the FMDF transport equation is calculated from the DNS. An a priori analysis of Interaction by Exchange of Mean, Modified Curl and Mapping Closure models for CSD is conducted. At high pressures, significant deviations are observed between the CSD measured from the DNS and those predicted by the models. The source of the deviations is determined to be the mixing frequency used in the models. The significance of this term and its variation with change in pressure and diffusion models is studied. The primary objective of this work is to assess the applicability of the aforementioned models to flows at large pressures and determine the causes for deviation in prediction.

Self-contained filtered density function

The filtered density function (FDF) closure is extended to a format to include the subgrid scale (SGS) statistics of all of the hydro-thermo-chemical variables in turbulent flows. These are the thermodynamic pressure, the specific internal energy, the velocity vector, and the composition field. In this format, the model is comprehensive and facilitates large eddy simulation (LES) of flows at both low and high compressibility levels. A transport equation is developed for the joint pressure-energy-velocity-composition filtered mass density function (PEVC-FMDF). In this equation, the effect of convection appears in closed form. The coupling of the hydrodynamics and thermochemistry is modeled via a set of stochastic differential equation (SDE) for each of the transport variables. This yields a self-contained SGS closure. For demonstration, LES is conducted of a turbulent shear flow with transport of a passive scalar. The consistency of the PEVC-FMDF formulation is established, and its overall predictive capability is appraised via comparison with direct numerical simulation (DNS) data.

An energy-consistency-preserving large eddy simulation-scalar filtered mass density function (LES-SFMDF) method for high-speed flows

An energy-consistency-preserving large eddy simulation-scalar filtered mass density function (LES-SFMDF) method is developed to improve the existing LES-SFMDF method in high-speed flows, especially supersonic flows. The high-speed source term in the SFMDF transport equation is analysed theoretically from a new point of view, and then several primary principles are proposed for the LES-SFMDF to achieve a good consistency even along discontinuities. Based on these principles and further theoretical analysis, the high-speed source term of the enthalpy in the SFMDF is modelled and computed from both MC and LES variables rather than the usually used solely LES variables. This new LES-SFMDF method is used for simulating the flows in a shock tube and in a subsonic temporally developing mixing layer. This method shows a better particle energy consistency than the existing method when applied across discontinuities in supersonic laminar flows. Unlike the existing method, with this energy-consistency-preserving LES-SFMDF method, particle energy consistency is consistent with particle mass consistency so that particle energy consistency can benefit from particle velocity correction. This method also demonstrates robustness for various numbers of particles.

Large-Eddy Simulation of Turbulent Mixing of a Jet in Cross-Flow

An Eulerian–Lagrangian mathematical/computational methodology is employed for large-eddy simulation (LES) and detailed study of turbulent mixing in jet in cross-flow (JICF) configuration. Accurate prediction of mixing in JICF is crucially important to the development of advanced combustion systems. A high-order multiblock finite difference (FD) computational algorithm is used to solve the Eulerian velocity and pressure equations in a generalized coordinate system. The composition field, describing the mixing, is obtained from the filtered mass density function (FMDF) and its stochastic Lagrangian Monte-Carlo (MC) solver. Our simulations are shown to accurately predict the important flow features present in JICF such as the counter-rotating vortex pair (CVP), horseshoe, shear layer, and wake vortices. The consistency of the FD and MC parts of the hybrid LES/FMDF model is established for the simulated JICF in various conditions, indicating the numerical accuracy of the model. The effects of parameters influencing the jet penetration, entrainment, and turbulent mixing such as the jet velocity profile, and jet pulsation are investigated. The results show that the jet exit velocity profile significantly changes the trajectory and mixing of injected fluid. The jet pulsation is also shown to enhance the mixing depending on the flow Strouhal number. The LES/FMDF results are shown to be in good agreement with the available experimental data, confirming the reliability of LES/FMDF method for numerical simulation of turbulent mixing in complex flow configurations.

Turbulent mixing in non-isothermal jet in crossflow

Large eddy simulation (LES) of turbulent mixing in a non-isothermal jet in crossflow (JICF) configuration is conducted with a hybrid Eulerian–Lagrangian mathematical/computational methodology. In this methodology, a high-order finite difference (FD) multi-block method is used to solve the Eulerian filtered Navier–Stokes equations in a generalized coordinate system. The composition field is described by the filtered mass density function (FMDF) and its equivalent stochastic Lagrangian equations, which are solved by a Lagrangian Monte Carlo (MC) method. The consistency of the Eulerian and Lagrangian parts of LES/FMDF is established for isothermal and non-isothermal JICFs. It is demonstrated that the instantaneous and averaged scalar concentration and temperature fields as obtained from LES-FD and FMDF-MC data are very similar. The LES/FMDF results are also in good agreement with the available experimental data, indicating the accuracy and reliability of LES/FMDF model. The numerical results show that the variation in temperature has a significant effect on the flow and mixing in JICF. By decreasing the jet to crossflow temperature ratio for fixed jet and crossflow velocities, the jet penetration, the spreading rate, the entrainment and the mixing are considerably improved.

Two-phase filtered mass density function for LES of turbulent reacting flows

AbstractThis paper describes a new computational model developed based on the filtered mass density function (FMDF) for large-eddy simulation (LES) of two-phase turbulent reacting flows. The model is implemented with a unique Lagrangian–Eulerian–Lagrangian computational methodology. In this methodology, the resolved carrier gas velocity field is obtained by solving the filtered form of the compressible Navier–Stokes equations with high-order finite difference (FD) schemes. The gas scalar (temperature and species mass fractions) field and the liquid (droplet) phase are both obtained by Lagrangian methods. The two-way interactions between the phases and all the Eulerian and Lagrangian fields are included in the new two-phase LES/FMDF methodology. The results generated by LES/FMDF are compared with direct numerical simulation (DNS) data for a spatially developing non-reacting and reacting evaporating mixing layer. Results for two more complex and practical flows (a dump combustor and a double-swirl burner) are also considered. For all flows, it is shown that the two-phase LES/FMDF results are consistent and accurate.

Filtered density function simulation of a realistic swirled combustor

The scalar filtered mass density function (SFMDF) methodology is employed for large eddy simulation (LES) of the PRECCINSTA burner from DLR. This burner involves a swirling flow configuration and provides a good model of the combustor in gas turbines. In SFMDF, the effects of unresolved scalar fluctuations are taken into account by considering the probability density function of subgrid scale scalar quantities. The modeled SFMDF transport equation is solved numerically via a Lagrangian Monte Carlo scheme coupled with a finite volume flow solver on unstructured grids. The simulated results are assessed via comparison with experimental data and show reasonable agreements. This demonstrates the capability of SFMDF for LES of complex flows, and warrants future applications of the methodology for LES of practical combustor configurations. This work represents one of the first attempts in implementing FDF for LES of a practical gas turbine combustor.

Large eddy simulation of turbulent spray combustion

The two-phase filtered mass density function (FMDF) method is employed for large eddy simulation (LES) of high speed evaporating and combusting n-heptane sprays using simple (global) and complex (skeletal) chemical kinetic mechanisms. The resolved fluid velocity and pressure fields are obtained by solving the filtered compressible Navier–Stokes equations with high-order Eulerian finite difference methods. The liquid spray and gas scalar (temperature and species mass fractions) fields are both obtained by Lagrangian stochastic models. The chemistry calculation is accelerated by incorporating the parallel in situ adaptive tabulation (ISAT) method. There are two-way interactions among Eulerian and Lagrangian fields. Simulations of evaporating sprays with and without combustion indicate that the two-phase LES/FMDF results are consistent and compare well with the available experimental data at different gas temperatures and oxygen concentrations. The spray controlled flame tends to move away from a diffusion flame structure toward a premixed one as the oxygen concentration decreases and/or the ambient gas temperature increases because of changes in spray-induced turbulence and mixing. The LES/FMDF results for ignition delay show more sensitivity to the chemical kinetic model at lower gas temperatures due to slower reaction and stronger turbulence–chemistry interactions. The liftoff length is less sensitive to the kinetics.

Large-eddy simulations of turbulent flows in internal combustion engines

Large-eddy simulations (LES) of turbulent spray combustion in internal combustion engines are conducted with a new hybrid Eulerian–Lagrangian methodology. In this methodology, the Eulerian filtered compressible gas equations are solved in a generalized curvilinear coordinate system with high-order finite difference schemes for the turbulent velocity and pressure. However, turbulent mixing and combustion are simulated with the two-phase compressible scalar filtered mass density function (FMDF) and Lagrangian spray models. The LES/FMDF model is used for simulation of in-cylinder turbulent flow and spray mixing and combustion in three flow configurations: (1) a poppet valve in a sudden expansion, (2) a simple piston–cylinder assembly with a stationary valve, and (3) a realistic single-cylinder, direct-injection spark–ignition (DISI) engine. The flow statistics predicted by the LES model are shown to compare well with the available experimental data for (1) and (2). The computed in-cylinder flow for the third configuration shows significant cycle-to-cycle variations. The flow in the DISI engine is also highly inhomogeneous and turbulent during the intake stroke, but becomes much more homogeneous and less turbulent in the compression stroke. Simulations of spray, mixing and combustion in the DISI engine indicate the applicability of the LES/FMDF model to internal combustion engines and other complex combustion systems.

Subgrid-scale mixing of mixture fraction, temperature, and species mass fractions in turbulent partially premixed flames

The subgrid-scale (SGS) mixing of mixture fraction, temperature, and species mass fraction in turbulent partially premixed (Sandia) flames is studied. We focus on the effects of the SGS mixing regimes and the spatial relationships among the scalars on the SGS mixing of the species mass fractions. High-resolution lines images are used to obtain the scalar filtered mass density function and the conditionally filtered diffusion. The results show that for small SGS variance the scalar fields have a relatively simple structure. For large SGS variance the structure is more complex. The spatial relationships among the scalars result in different SGS scalar structures and SGS mixing characteristics, with the mass fraction of CO2 being the simplest and that of CO being the most complex. The results in this study present a challenging test for SGS mixing models.

The FDF or LES/PDF method for turbulent two-phase flows

In this paper, a new formalism for the filtered density function (FDF) approach is developed for the treatment of turbulent polydispersed two-phase flows in LES simulations. Contrary to the FDF used for turbulent reactive single-phase flows, the present formalislm is based on Lagrangian quantities and, in particular, on the Lagrangian filtered mass density function (LFMDF) as the central concept. This framework allows modeling and simulation of particle flows for LES to be set in a rigorous context and various links with other approaches to be made. In particular, the relation between LES for particle simulations of single-phase flows and Smoothed Particle Hydrodynamics (SPH) is put forward. Then, the discussion and derivation of possible subgrid stochastic models used for Lagrangian models in two-phase flows can set in a clear probabilistic equivalence with the corresponding LFMDF.