Transported Probability Density Function Method Research Articles

Stochastic modeling of turbulent mixing based on a hierarchical swapping of fluid parcels

AbstractTurbulent mixing is an omnipresent phenomenon that constantly affects our everyday life and plays an important role in a variety of industrial applications. The simulation of turbulent mixing poses great challenges, since the full resolution of all relevant length and time scales is associated with an immense computational effort. This limitation can be overcome by only resolving the large‐scale effects and completely model the sub‐grid scales. The development of an accurate sub‐grid mixing model is therefore a key challenge to capture all interactions in the sub‐grid scales. At this place, the hierarchical parcel‐swapping (HiPS) model formulated by A.R. Kerstein [J. Stat. Phys. 153, 142–161 (2013)] represents a computationally efficient and scale‐resolving turbulent mixing model. HiPS mimics the effects of turbulence on time‐evolving, diffusive scalar fields. In HiPS, the diffusive scalar fields or a state space is interpreted as a binary tree structure, which is an alternative approach compared to the most common mixing models. Every level of the tree represents a specific length and time scale, which is based on turbulence inertial range scaling. The state variables are only located at the base of the tree and are treated as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by turbulent time scales associated with the sub‐trees. The mixing only takes places between adjacent fluid parcels and at rates consistent with the prevailing diffusion time scales. In this work, the HiPS model formulation for the simulation of passive scalar mixing is detailed first. Preliminary results for the mean square displacement, passive scalar probability density function (PDF) and scalar dissipation rate are given and reveal the strengths of the HiPS model considering the reduced order and computational efficiency. These model investigations are an important step of further HiPS advancements. The integrated auxiliary binary tree structure allows HiPS to satisfy a large number of criteria for a good mixing model. From this point of view, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

Simulation of turbulent premixed flames with machine learning - tabulated thermochemistry

The numerical integration of the differential equations describing chemical kinetics consumes the majority of computational time in combustion simulations that involve direct coupling of chemistry and flow, such as transported probability density function (PDF) methods, direct numerical simulation (DNS), conditional moment closure (CMC), unsteady flamelet, multiple mapping closure (MMC), thickened flame model, linear eddy model (LEM), partially stirred reactor (PaSR) as in OpenFOAM and laminar flame computation. This step can be accelerated by tabulation, and artificial neural networks (ANNs) have recently emerged as a powerful technique in this domain. To be applicable to a wide family of problems, an ANN tabulation approach must be based on data generated by an abstract process, rather than from the turbulent flame to be simulated. In the present work, the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) method (Ding et al., Combust. Flame 231, 111493, 2021) for non-premixed flames is taken as a basis to develop a thermochemistry tabulation method for premixed flames. In the spirit of maintaining an essentially random data set that still originates in meaningful composition states, a set of one-dimensional premixed flame simulations is employed to generate data that are used as starting points for a random data generation process and subsequently discarded. The approach is applied to large eddy simulations (LES) of the Cambridge/Sandia swirl burner in configurations five and six, with the transported PDF method employed to provide closure for the filtered reaction source terms and the stochastic fields method used for numerical solution. Very good agreement in both major and minor species is observed between the LES-PDF simulations using direct integration of the reaction source term and the ones with the ANNs. Furthermore, the average time taken for reaction source term computations is reduced by fourteen times, while memory requirements constitute only 1.4 MB.

Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model

AbstractTurbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

A feasibility study on the use of low-dimensional simulations for database generation in adaptive chemistry approaches

Large eddy simulation (LES)/ Probability Density Function (PDF) approaches are now well established and can be used for simulating challenging turbulent combustion configurations with strong turbulence chemistry interactions. Transported PDF methods are known to be computationally expensive compared to flamelet-like turbulent combustion models. The pre-partitioned adaptive chemistry (PPAC) methodology was developed to address this cost differential. PPAC entails an offline preprocessing stage, where a set of reduced models are generated starting from an initial database of representative compositions. At runtime, this set of reduced models are dynamically utilised during the reaction fractional step leading to computational savings. We have recently combined PPAC with in-situ adaptive tabulation (ISAT) to further reduce the computational cost. We have shown that the combined method reduced the average wall-clock time per time step of large-scale LES/particle PDF simulations of turbulent combustion by 39%. A key assumption in PPAC is that the initial database used in the offline stage is representative of the compositions encountered at runtime. In our previous study this assumption was trivially satisfied as the initial database consisted of compositions extracted from the turbulent combustion simulation itself. Consequently, a key open question remains as to whether such databases can be generated without having access to the turbulent combustion simulation. Towards answering this question, in the current work, we explore whether the compositions for forming such a database can be extracted from computationally-efficient low-dimensional simulations such as 1D counterflow flames and partially stirred reactors. We show that a database generated using compositions extracted from a partially stirred reactor configuration leads to performance comparable to the optimal case, wherein the database is comprised of compositions extracted directly from the LES/PDF simulation itself.

A hybrid stochastic Lagrangian – cellular automata framework for modelling fire propagation in inhomogeneous terrains

A stochastic model motivated by the Lagrangian transported probability density function method for turbulent reacting flows and the cellular automata approach for forest fires was put together to simulate propagation of fires in terrains with inhomogeneous composition. In contrast to the usual cellular automata models for fires where the probability of ignition is prescribed, here the ignition of cells is determined by a random walk that mimics turbulent convection and diffusion of the hot gases and firebrands from upwind and neighbouring fire fronts. Radiation is also included. The model is aimed at speed of computation while approximating the key physics through only a few terrain-related inputs and tunable parameters representing fire intensity, hot gas and ember decay timescales, cell ignition delay and local turbulence. These parameters were calibrated against controlled fire experiments and the model was then used to give reasonable predictions for fires of increasing complexity. The presented framework allows improvements for more accurate representation of the flammable material characteristics, fire-induced flow modifications, and most other phenomena present in fires, hence providing an extendable and simple yet physically-realistic novel modelling approach.

A mixing timescale model for differential mixing in transported probability density function simulations of turbulent non-premixed flames

The modeling of scalar mixing timescale remains a primary challenge in the transported probability density function (TPDF) method. The variation of scalar mixing timescale among species, i.e., differential mixing, results from the difference in molecular diffusivity and reaction-induced scalar gradient. Nevertheless, the vast majority of TPDF studies on turbulent non-premixed flames simply apply a single mixing timescale determined by the mixture fraction. In this work, a reaction-induced differential mixing timescale (RIDM) model for the mixing timescale of individual species in turbulent non-premixed flames is proposed. The key idea of the RIDM model is to approximate the relative magnitude of the species dissipation rates by using their values in laminar flamelets. A direct numerical simulation dataset of a temporally evolving non-premixed ethylene flame is employed to thoroughly evaluate the model performance via a priori and a posteriori tests. Results show that specifying a single mixing timescale for all species results in a poor prediction of the species dissipation rate and thus the failure to predict the overall combustion process. By accounting for the difference in molecular diffusivity, a slightly better prediction can be obtained, but the improvement is very limited, illustrating that simply modeling the difference due to molecular diffusivities for differential mixing is not sufficient. In comparison, the RIDM model exhibits superior performance in both a priori and a posteriori tests. Moreover, all the components of the RIDM model are readily available in the TPDF method, making the RIDM model a promising candidate employed in practice.

A Lagrangian-based flame index for the transported probability density function method

We propose a new flame index for the transported probability density function (PDF) method. The flame index uses mixing flux projections of Lagrangian particles on mixture fraction and progress variable directions as the metrics to identify the combustion mode, with the Burke-Schumann solution as a reference. A priori validation of the flame index is conducted with a series of constructed turbulent partially premixed reactors. It indicates that the proposed flame index is able to identify the combustion mode based on the subgrid mixing information. The flame index is then applied the large eddy simulation/PDF datasets of turbulent partially premixed jet flames. Results show that the flame index separate different combustion modes and extinction correctly. The proposed flame index provides a promising tool to analyze and model the partially premixed flames adaptively.

Transported PDF modeling of compressible turbulent reactive flows by using the Eulerian Monte Carlo fields method

Although the transported probability density function (PDF) method has been developed for decades, its application has been mainly focused on the low-Mach number flow problems. This work extends the transported PDF method to compressible flow problems. The Eulerian Monte Carlo fields (EMCF) solution method is employed to solve the transported PDF equation for compressible flow problems. A pseudo stagnation enthalpy is introduced and its stochastic partial differential equation is derived to ensure total energy conservation numerically. A new mixing model called interaction by partial exchange with mean (IPEM) is introduced to expand the available choices of mixing models for the EMCF method. The consistency of the EMCF method is examined for solving the transported PDF equation. Numerical implementation details are discussed, such as the density coupling between the compressible flow solver and the EMCF solver, discretization schemes for the mixing terms and the stochastic terms. The implemented compressible flow solver coupled with the EMCF solver is verified and validated in a series of test cases with increasing level of complexity, ranging from a statistically one-dimensional turbulent mixing layer to a self-excited resonance model rocket combustor. It is observed that in general with the increase of compressibility, there is an increase in the sensitivity of the modeling results to the different models and algorithms. This makes it necessary to develop a thorough understanding of the model sensitivity in order to develop a robust and accurate simulation solver for highly compressible turbulent reactive flows. The thermo-acoustic instability inside the model rocket combustor case is captured reasonably, which demonstrates the overall capability of the developed compressible turbulent combustion solver based on the transported PDF method.

An investigation on the impact of small-scale models in gasoline direct injection sprays (ECN Spray G)

The work presents a numerical investigation of gasoline direct injection and the resulting early development of spray plumes from an eight-hole injector (Engine Combustion Network Spray G). The objective is to evaluate the impact on the droplet size distribution (DSD) statistics from the assumed model physics, particularly for the small scales. Two modelling approaches are compared: Eulerian–Lagrangian spray atomisation with adaptive mesh refinement and a stochastic fields transported probability density function method. The two models simulate the small scales and sub-grid droplet physics with different approaches, but based on the same concept of transport of liquid surface density. Both approaches predict similar liquid distributions in the near-field comparable to experimental measurements. The spray break-up patterns are very similar and both models reproduce quasi-log-normal droplet distributions, with same overall Sauter mean diameters. The Eulerian–Lagrangian spray atomisation with probability density function approach shows different break-up behaviour between droplets originating from the dilute region and those originating from the dense core region. The transition from Eulerian to Lagrangian can be observed in the Eulerian–Lagrangian spray atomisation with adaptive mesh refinement predicted distribution with an abrupt change in the DSD. Both methods are able to produce similar DSD below filter width/grid size resolution.

Transported PDF simulation of auto-ignition of a turbulent methane jet in a hot, vitiated coflow

In the current work, the auto-ignition of a turbulent round methane jet is studied numerically by means of a transported probability density function (PDF) method. The methane jet is issued into a hot, vitiated coflow, where it ignites to form a steady lifted flame. For this flame, experimental data of hydroxyl, temperature and mixture fraction are provided in the area where the fuel auto-ignites. To model this experiment, the transport equation for the thermochemical PDF is solved using a hybrid finite volume / Lagrangian Monte-Carlo method. Turbulence is modelled using the k-ϵ turbulence model including a jet-correction. Computational results are compared to experimental data in terms of mean quantities, variances and lift-off height. Moreover, the structure of the one-point, one-time marginal PDF of temperature is analysed and compared to experimental data which are provided in this work. It is found that the transported PDF method in conjunction with the k-ϵ model is capable of reproducing these statistical data very well. In particular the effect of ignition on the marginal PDF of temperature can be well reproduced with this approach. To further analyse the relevant processes in the evolution of the temperature PDF, a statistically homogeneous system is studied both numerically and analytically.

Turbulent electric current in the marine convective atmospheric boundary layer

The atmospheric boundary layer considered a part of the planetary global electric circuit is known to exhibit a most dramatically changing in space and time atmospheric electrical variables. Extremely low specific electrical conductivity in the lower atmosphere is accompanied by the work of the electric generator associated with turbulent dynamo realizing itself through the space charge transport by boundary layer flows. The electromotive force of this generator acting in fair weather atmospheric regions provides a contribution to the global potential drop between the lower ionosphere and the earth's surface. In this study, the vertical electric current density produced by the turbulent transfer of space charge is investigated in the cases of the buoyancy-dominated and shear-dominated convective boundary layer (CBL) over the oceans based on second-order dissipation-conditioned Lagrangian stochastic model (LSM) applied near air-sea interface and conjugated with first-order LSM in the rest of the CBL. The total electric charge flux is treated in terms of mechanical transfer of atmospheric ions and charged aerosol particles by Lagrangian turbulence together with the drift of small ions in the self-consistent electric field. This approach based on a transported probability density function method offers persuasive advantages for quantitative estimates of local perturbations in the fair-weather electric field created by the atmospheric space charge carried by turbulent flows. It is found that the space charge density vertical profile only slightly depends on turbulence everywhere except for the wavy boundary layer about 10 m thick, a new parameterization of space charge vertical profile is proposed. Based on the simulation results, it can be predicted that in the buoyancy-dominated CBLs the electric field must be greater than in the sheared CBLs with other things being equal. Turbulent electromotive force and changes in total downward atmospheric electric current density are quantified. Simulated vertical profiles of electrical conductivity, space charge, electric field, and turbulent current density reasonably well reproduce previously reported observation results.

Assessment of Presumed/Transported Probability Density Function Methods for Rocket Combustion Simulations

Combustion models of different fidelity are applied to a seven-element gaseous methane/oxygen subscale rocket combustion chamber. The covered region of thermo-chemical states is analyzed for two non-adiabatic flamelet-based methods, from which one is well established and the other one has just recently been proposed for hydrogen/oxygen combustion. Of particular interest is their applicability in situations with strong heat losses. Consequently, the tabulated combustion models are linked with a presumed probability density function approach, and the results are assessed using a transported probability density function method. An analytically reduced chemical reaction mechanism with 13 species and 73 reactions is used for all models in order to allow for direct comparison of turbulence–chemistry interaction. The results are evaluated with respect to temperature, gas composition, and wall heat transfer and they are brought in context with their respective computational costs. Finally, the results are compared to available experimental data in terms of wall heat flux and pressure measurements. Locally and temporally resolved results provide additional insights to the experiment.

Consistency and convergence of Eulerian Monte Carlo field method for solving transported probability density function equation in turbulence modeling

The transported probability density function (PDF) method is an attractive model for the closure of turbulent mixing and turbulent reactive flows. The transported PDF method suffers from the curse of dimensionality and an efficient numerical implementation of the method has always been a research topic of great importance. The Eulerian Monte Carlo field (EMCF) method (also termed as the stochastic field method) has been proposed as an efficient solution approach for solving the PDF transport equation for two decades. In this work, we revisit the EMCF method and examine its mathematical consistency analytically and numerically. It is found that the EMCF method is not mathematically consistent with the PDF transport equation that the method intends to solve. This creates a serious inconsistency issue and causes uncertainties in the yielded numerical solutions by EMCF. It is imperative to examine the effect of the inconsistency. We evaluate the effect of the inconsistency in a simplified turbulent mixing layer test case. Corrections to remedy the mathematical inconsistency are proposed and examined. The effectiveness of the corrections is demonstrated numerically through convergence testing. The effect of the Reynolds number on the inconsistency is explored. The impact of the inconsistency is also investigated in a thermal wake behind a line source in grid turbulence to assess the importance of the issue in real turbulence problems.

Variance consistent mean shift particle model for treating differential molecular diffusion in transported PDF methods for turbulent reactive flows

A mean shift (MS) particle model has been developed before to account for molecular diffusion in the transported probability density function (PDF) methods for turbulent reactive flows (McDermott and Pope, Journal of Computational Physics, 2007, 226, 947–993). The MS model is superior over the traditional random walk model for its capabilities to incorporate differential molecular diffusion and to yield a consistent DNS limit of scalar variance production/dissipation. The MS model, however, yields inconsistent transport of the scalar variance because of the missing molecular diffusion in the resultant variance transport equation. To yield fully consistent transport of scalar mean and variance, a variance consistent mean shift (VCMS) model is developed in this work. The VCMS model formulations are developed for both a single-scalar system and a multi-scalar system. Boundedness violation of particle scalars exists in both the MS and VCMS models and causes a realizability issue of the models. A correction algorithm (called MP correction by McDermott and Pope) has been developed before to correct the boundedness violation in the MS model. The MP correction is found to produce overcorrection to the particles, resulting in an underprediction of the scalar variance. In addition, it is only applicable to the IEM mixing model. In this work, three new boundedness correction algorithms are developed based on the strategies of a least-decay factor (LDF), clipping at scalar bounds (CSB), and clipping with paired mixing (CPM). The accuracy and consistency of the VCMS models are examined and validated thoroughly in a turbulent mixing layer problem. The effectiveness and accuracy of the different boundedness correction algorithms are also examined and assessed in the mixing layer problem. The applicability of the CPM correction to other mixing models (MCurl and EMST) is also demonstrated.

A mixing timescale model for TPDF simulations of turbulent premixed flames

Transported probability density function (TPDF) methods are an attractive modeling approach for turbulent flames as chemical reactions appear in closed form. However, molecular micro-mixing needs to be modeled and this modeling is considered a primary challenge for TPDF methods. In the present study, a new algebraic mixing rate model for TPDF simulations of turbulent premixed flames is proposed, which is a key ingredient in commonly used molecular mixing models. The new model aims to properly account for the transition in reactive scalar mixing rate behavior from the limit of turbulence-dominated mixing to molecular mixing behavior in flamelets. An a priori assessment of the new model is performed using direct numerical simulation (DNS) data of a lean premixed hydrogen–air jet flame. The new model accurately captures the mixing timescale behavior in the DNS and is found to be a significant improvement over the commonly used constant mechanical-to-scalar mixing timescale ratio model. An a posteriori TPDF study is then performed using the same DNS data as a numerical test bed. The DNS provides the initial conditions and time-varying input quantities, including the mean velocity, turbulent diffusion coefficient, and modeled scalar mixing rate for the TPDF simulations, thus allowing an exclusive focus on the mixing model. The new mixing timescale model is compared with the constant mechanical-to-scalar mixing timescale ratio coupled with the Euclidean Minimum Spanning Tree (EMST) mixing model, as well as a laminar flamelet closure by Pope and Anand (1984). It is found that the laminar flamelet closure is unable to properly capture the mixing behavior in the thin reaction zones regime while the constant mechanical-to-scalar mixing timescale model under-predicts the flame speed. The EMST model coupled with the new mixing timescale model provides the best prediction of the flame structure and flame propagation among the models tested, as the dynamics of reactive scalar mixing across different flame regimes are appropriately accounted for.

Modeling combustion under engine combustion network Spray A conditions with multiple injections using the transported probability density function method

In this study, numerical simulations of an n-dodecane spray flame—known as Spray A—with multiple injections (0.5 ms injection/0.5 ms dwell/0.5 ms injection) have been carried out using the transpor...

Modelling n-dodecane spray and combustion with the transported probability density function method

An n-dodecane spray in temperature and pressure conditions typical of diesel engines, known as Spray A, is modelled by the transported probability density function (TPDF) method coupled with a time-dependent Reynolds-averaged k–∊ turbulence model and a Lagrangian discrete phase model of the liquid spray. To establish a baseline for comparisons, non-reacting cases are first studied. Good results are obtained for the vapour penetration, the mean and variance of fuel mixture fraction, and velocity profiles, with variations in ambient density and injection pressure. These comparisons are more extensive than previous studies due to new experimental data being available. Reacting cases are then investigated for a number of ambient conditions and injection parameters, employing a reduced chemical kinetic model. The chemical mechanism incorporates an OH∗ sub-mechanism (Hall and Petersen, 2006) which enables a direct comparison with experimental measurements of the lift-off length that are based on OH∗ chemiluminescence. To assess the importance of interactions between turbulence and chemistry, the results from the PDF model are compared to the measurements and to those from a well-mixed model that ignores turbulent fluctuations. Variations of ambient temperature, ambient oxygen concentration, ambient density, and injection pressure are considered. In all cases the PDF model with the EMST mixing model and Cϕ=1.5 shows an excellent agreement with the experimental lift-off length and presents improved results compared with the well-mixed model. Ignition delay is however over-predicted by both the PDF method and well-mixed models. Available shock tube data suggests that this may be due to the chemical kinetic model over-predicting ignition delay at higher pressures.

Stabilization and liftoff length of a non-premixed methane/air jet flame discharging into a high-temperature environment: An accelerated transported PDF method

A particle-based transported probability density function (PDF) method with a novel chemistry acceleration technique is developed in this work. The technique is based on the chemistry coordinate mapping (CCM) approach that was proposed in our previous works for accelerating direct numerical simulations (DNS) of partially premixed combustion. The method is first validated using Sandia flames D and F. It is shown that PDF-CCM results converge toward those obtained without CCM as phase-space resolution increases. PDF-CCM is then applied to simulate methane/air lifted jet flames in vitiated coflow reported in experiments by Cabra et al. (2005). It is shown that combustion is initiated in the form of auto-ignition in very fuel-lean gases where the gas velocity is low (the residence time is long) and the gas temperature is high (the ignition delay time is short). The ignition delay of the mixture below the liftoff position scales well with the liftoff height at different coflow temperature conditions. The combustion process above the liftoff height can develop into different modes depending on the coflow temperature. For high-temperature coflow, a premixed-burned combustion is formed above the liftoff height, which involves fuel-lean to fuel-rich burning modes; a triple-flame structure eventually is formed a few nozzle diameters above the liftoff position. For low-temperature coflow, the ignition delay and the liftoff height are sufficiently large to allow premixing between fuel and oxidizer before the onset of high-temperature combustion; in this case, a lean-to-stoichiometric premixed burn combustion is established downstream of the liftoff height, and no obvious triple-flame structure is formed at this condition.

Modeling of heat transfer and differential diffusion in transported PDF methods

In order to model the conditional diffusive heat and mass fluxes in the joint probability density function (PDF) transport equation of the thermochemical variables, the diffusive fluxes are decomposed into their Favre mean and fluctuation. While the mean flux appears to be closed, the contributions of fluctuating fluxes are modeled with the interaction by exchange with the mean (IEM) model. Usually, the contribution of the Favre averaged diffusive fluxes is neglected at high Reynolds numbers. Here, however, this term is included to account for molecular mixing in regions, where turbulent mixing is negligible. This model approach is applied in steady state Reynolds Averaged Navier–Stokes (RANS)/transported PDF calculations to simulate the heat transfer of wall bounded flows as well as the stabilization of a hydrogen–air flame at the burner tip. For both flow problems it is demonstrated that molecular transport is recovered in regions where turbulent mixing vanishes. In wall bounded flows this is the case in the viscous sublayer. Heat transfer studies show, that “mixing models” based on the high Reynolds number assumption fail to compute correctly the temperature field and the heat flux close to the wall. A similar situation occurs at the flame root of the investigated turbulent hydrogen-air jet flame, where turbulent mixing is still too weak to achieve a fast mixing of reactants. In this area differential diffusion effects are observed in the experiment, i.e. superequilibrium temperatures and nonlinear relations between the elemental mixture fractions of hydrogen and oxygen. It will be shown, that the presented model can successfully reproduce these effects, which underlines the necessity to include Favre averaged molecular diffusive fluxes in transported PDF methods.

A DNS evaluation of mixing models for transported PDF modelling of turbulent nonpremixed flames

Transported probability density function (TPDF) methods are well suited to modelling turbulent, reacting, variable density flows. One of the main challenges to the successful deployment of TPDF methods is accurately modelling the unclosed molecular mixing term. This study examines three of the most widely used mixing models: the Interaction by Exchange with the Mean (IEM), Modified Curl (MC) and Euclidean Minimum Spanning Tree (EMST) models. Direct numerical simulation (DNS) data-sets were used to provide both initial conditions and inputs needed over the course of the runs, including the mean flow velocities, mixing frequency, and the turbulent diffusion coefficient. The same chemical mechanism and thermodynamic properties were used, allowing the study to focus on the mixing model. The simulation scenario was a one-dimensional, nonpremixed, turbulent jet flame burning either a syngas or ethylene fuel stream that featured extinction and reignition. This test scenario was selected because extinction and reignition phenomena are sensitive to the mixing model. Three DNS cases were considered for both the syngas and ethylene cases with a parametric variation of Reynolds and Damköhler numbers, respectively. Extinction events became more prevalent with increasing Reynolds number in the syngas cases and with decreasing Damköhler number in the ethylene cases. The model was first tested with the mixing frequency defined from the dissipation rate and variance of mixture fraction. With this definition, for the syngas cases this study finds that the TPDF method is successful at predicting flame extinction and reignition using all three mixing models for the relatively lower and intermediate Reynolds number cases, but that all models under-predict reignition in the relatively higher Reynolds number case. In the ethylene fuelled cases, only the EMST mixing model correctly predicts the reignition event for the two higher Damköhler number cases, however, in the lowest Damköhler number case the EMST model over-predicts reignition and the IEM and MC models under-predict it. Mixing frequency was then modelled based on the turbulence frequency and a model constant Cϕ, the ratio of scalar to mechanical mixing rates. The DNS cases were reexamined with this definition and the results suggested that the optimal value for Cϕ is mixing model and case dependent. In particular, it was found in the ethylene case considered that reignition could be achieved with the IEM and MC models by adjusting the value of Cϕ.