Presumed Mapping Functions for Eulerian Modelling of Turbulent Mixing

The mapping closure of Chen et al. [Phys. Rev. Lett., 63, 1989] is a transported probability density function (PDF) method that has proven very efficient for modelling of turbulent mixing in homogeneous turbulence. By utilizing a Gaussian reference field, the solution to the mapping function (in homogeneous turbulence) can be found analytically for a range of initial conditions common for turbulent combustion applications, e.g. for binary or trinary mixing. The purpose of this paper is to investigate the possibility of making this solution a presumed mapping function (PMF) for inhomogeneous flows. The PMF in turn will imply a presumed mixture fraction PDF that can be used for a wide range of models in turbulent combustion, e.g. flamelet models, the conditional moment closure (CMC) or large eddy simulations. The true novelty of the paper, though, is in the derivation of highly efficient, closed algebraic expressions for several existing models of conditional statistics, e.g. for the conditional scalar dissipation/diffusion rate or the conditional mean velocity. The closed form expressions nearly eliminates the overhead computational cost that usually is associated with nonlinear models for conditional statistics. In this respect it is argued that the PMF is particularly well suited for CMC that relies heavily on manipulations of the PDF for consistency. The accuracy of the PMF approach is shown with comparison to DNS of a single scalar mixing layer to be better than for the β-PDF. Not only in the shape of the PDF itself, but also for all conditional statistics models computed from the PDF.     

A Generalised Multiple Mapping Conditioning Approach for Turbulent Combustion

This paper follows the evolution in understanding of the multiple mapping conditioning (MMC) approach for turbulent combustion and reviews different implementations of MMC models. As the MMC name suggests, the original version represents a consistent combination of CMC-type conditional equations (conditional moment closure) and generalised mapping closure. It seems that the strength of the MMC model, and especially that of its stochastic version, lies in a more general (and much more transparent) interpretation. In this new generalised interpretation, we can replace complicated derivations by physical reasoning and the model appears to be a natural extension of modelling approaches developed in recent decades. MMC can be seen as a methodology for enforcing certain known characteristics of turbulence on a conventional mixing model. This is achieved by localising the mixing operation in a reference space. The reference space variables are selected to emulate the properties of a turbulent flow which have a strong effect on reactive quantities. The best and simplest example is an MMC model which has a single reference variable emulating the mixture fraction. In diffusion flames turbulent fluctuations of reacting quantities are strongly correlated with fluctuations of the mixture fraction. By making mixing local in the reference mixture fraction space a CMC-type mixing closure is enforced. In the original interpretation of MMC the reference variables are modelled as Markov processes. Since the reference variables should emulate properties of turbulent flows as realistically as possible the next step, and the basis of generalised MMC, is to remove the Markovian restriction and set reference variables equal to traced Lagrangian quantities within DNS or LES flow fields. Indeed, no Markov value can emulate the mixture fraction better than the mixture fraction itself. (Using a Markov vector process of dimension higher than the number of conditioning variables represents a more economical alternative for producing reference variables in generalised MMC.) The generalised MMC approach effectively incorporates the mixture fraction-based models, the PDF methods and LES/DNS techniques into a single methodology with possibility of blending useful features developed previously for conventional models. The generalised approach to MMC stimulates a more flexible understanding of simulations using sparsely placed Lagrangian particles as tools that may provide accurate joint distributions of reactive scalars at relatively low computational cost. The physical reasoning behind the new interpretation of MMC is supported by example computations for a partially premixed methane/air diffusion flame (Sandia Flame D). The scheme utilises LES for the dynamic field and a sparse-Lagrangian filtered density function method with MMC mixing for the scalar field. Two different particle mixing schemes are tested. Simulations are performed using only 35,000 Lagrangian particles (of these only 10,000 are chemically active) on a single workstation. The relatively low computational cost allows the use of realistic chemical kinetics containing 34 reactive species and 219 reactions. Intended for publication in the special issue of Flow, Turbulence and Combustion arising from the 2nd ECCOMAS Thematic Conference on Computational Combustion held at Delft in mid-2007.     

Scalar transport in plane mixing layers

This paper describes the application of the Eulerian, single-point, single-time joint-scalar probability density function (PDF) equation for predicting the scalar transport in mixing layer with a high-speed and a low-speed stream. A finite-volume procedure is applied to obtain the velocity field with the k-ε closure being used to describe turbulent transport. The scalar field is represented through the modelled evolution equation for the scalar PDF and is solved using a Monte Carlo simulation. The PDF equation employs gradient transport modelling to represent the turbulent diffusion, and the molecular mixing term is modelled by the LMSE closure. There is no source term for chemical reaction as only an inert mixing layer is considered here. The experimental shear layer data published by Batt is used to validate the computational results despite the fact that comparisons between experiments and computational results are difficult because of the high sensitivity of the shear layer to initial conditions and free stream turbulence phenomena. However, the bimodal shape of the RMS scalar fluctuation as was measured by Batt can be reproduced with this model, whereas standard gradient diffusion calculations do not predict the dip in this profile. In this work for the first time an explanation is given for this phenomenon and the importance of a micromixing model is stressed. Also it is shown that the prediction of the PDF shape by the LMSE model is very satisfactory. Received on 27 October 1998     

Flow and Mixing Fields for Transported Scalar PDF Simulations of a Piloted Jet Diffusion Flame (‘Delft Flame III’)

Numerical simulation results are presented for ‘Delft Flame III’, a piloted jet diffusion flame with strong turbulence–chemistry interaction. While pilot flames emerge from 12 separate holes in the experiments, the simulations are performed on a rectangular grid, under the assumption of axisymmetry. In the first part of the paper, flow and mixing field results are presented with a non-linear first order k–ε model, with the transport equation for ε based on a modeled enstrophy transport equation, for cold and reactive flows. For the latter, the turbulence model is applied in combination with pre-assumed β-PDF modeling for the turbulence–chemistry interaction. The mixture fraction serves as conserved scalar. Two chemistry models are considered: chemical equilibrium and a steady laminar flamelet model. The importance of the turbulence model is highlighted. The influence of the chemistry model is noticeable too. A procedure is described to construct appropriate inlet boundary conditions. Still, the generation of accurate inlet boundary conditions is shown to be far less important, their effect being local, close to the nozzle exit. In the second part of the paper, results are presented with the transported scalar PDF approach as turbulence–chemistry interaction model. A C₁ skeletal scheme serves as chemistry model, while the EMST method is applied as micro-mixing model. For the transported PDF simulations, the model for the pilot flames, as an energy source term in the mean enthalpy transport equation, is important with respect to the accuracy of the flow field predictions. It is explained that the strong influence on the flow and mixing field is through the turbulent shear stress force in the region, close to the nozzle exit.     

Joint PDF Closure of Turbulent Premixed Flames

In this paper, a novel model for turbulent premixed combustion in the corrugated flamelet regime is presented, which is based on transporting a joint probability density function (PDF) of velocity, turbulence frequency and a scalar vector. Due to the high dimensionality of the corresponding sample space, the PDF equation is solved with a Monte-Carlo method, where individual fluid elements are represented by computational particles. Unlike in most other PDF methods, the source term not only describes reaction rates, but accounts for “ignition” of reactive unburnt fluid elements due to propagating embedded quasi laminar flames within a turbulent flame brush. Unperturbed embedded flame structures and a constant laminar flame speed (as expected in the corrugated flamelet regime) are assumed. The probability for an individual particle to “ignite” during a time step is calculated based on an estimate of the mean flame surface density (FSD), latter gets transported by the PDF method. Whereas this model concept has recently been published [21], here, a new model to account for local production and dissipation of the FSD is proposed. The following particle properties are introduced: a flag indicating whether a particle represents the unburnt mixture; a flame residence time, which allows to resolve the embedded quasi laminar flame structure; and a flag indicating whether the flame residence time lies within a specified range. Latter is used to transport the FSD, but to account for flame stretching, curvature effects, collapse and cusp formation, a mixing model for the residence time is employed. The same mixing model also accounts for molecular mixing of the products with a co-flow. To validate the proposed PDF model, simulation results of three piloted methane-air Bunsen flames are compared with experimental data and very good agreement is observed.     

Second-Order Conditional Moment Closure Simulations of Autoignition of an n-heptane Plume in a Turbulent Coflow of Heated Air

Autoignition of an n-heptane plume in a turbulent coflow of heated air has been studied using the conditional moment closure (CMC) method with a second-order closure for the conditional chemical source term. Two different methodologies have been considered: (i) the Taylor expansion method, in which the second order correction was based on the solution of the full covariance matrix for the 31 reactive species in the chemical mechanism and hence was not limited to a few selected reactions, and (ii) the conditional PDF method, in which only the temperature conditional variance equation has been solved and its PDF assumed to be a β-function. The results compare favorably with experiment in terms of autoignition location. The structure of the reaction zone in mixture fraction space has been explored. The relative performance of the two methodologies is discussed.     

Direct numerical simulation of turbulent mixing of a passive scalar in pipe flow

Turbulent mixing of a passive scalar in fully developed turbulent pipe flow has been investigated by means of a Direct Numerical Simulation (DNS). The scalar is released from a point source located on the centreline of the pipe. The domain size of the concentration field has been chosen large enough to capture the different stages of turbulent mixing. Results are presented for mean concentration profiles, turbulent fluxes, concentration fluctuations, probability density functions and higher-order moments. To validate the numerical simulations the results are compared with experimental data on mixing in grid-turbulence that have been reported in the literature. The agreement between the experimental measurements and the computations is satisfactory. We have also considered the Probability Density Function (PDF). For small diffusion times and positions not on the plume centreline, our results lead to a PDF of an exponential form with a large peak at zero concentration. When the diffusion time increases, the PDF shifts from a exponential to a more Gaussian form.     

Modeling of the Joint PDF of a Scalar and Its Gradient in Turbulent Mixing

A joint probability distribution function of a conservative scalar (mixture fraction) and its gradient is predicted numerically. Statistical moments of this function are compared to their approximations, direct numerical simulation data, and also to the results obtained by simplified models for a conditional rate of scalar dissipation, the surface density function, and the one-point PDF of scalar fluctuation under homogeneous isotropic turbulence. The results allow to evaluate the performance of existing statistical micromixing models.     

DNS of Mixing and Reaction of Two Species in a Turbulent Channel Flow: A Validation of the Conditional Moment Closure

We consider the chemical reaction in a turbulent flow for the case that the time scale of turbulence and the time scale of the reaction are comparable. This process is complicated by the fact that the reaction takes place intermittently at those locations where the species are adequately mixed. This is known as spatial segregation. Several turbulence models have been proposed to take the effect of spatial segregation into account. Examples are the probability density function (PDF) and the conditional moment closure (CMC) models. The main advantage of these models is that they are able to parameterize the effects of turbulent mixing on the chemical reaction rate. As a price several new unknown terms appear in these models for which closure hypothesis must be supplied. Examples are the conditional dissipation 〈 χ ∣ φ 〉, the conditional diffusion 〈 κ ∇² φ ∣ u, φ 〉 and the conditional velocity 〈 u ∣ φ 〉. In the present study we investigate these unknown terms that appear in the PDF and CMC model by means of a direct numerical simulation (DNS) of a fully developed turbulent flow in a channel geometry. We present the results of two simulations in which a scalar is released from a continuous line source. In the first we consider turbulent mixing without chemical reaction and in the second we add a binary reaction. The results of our simulations agree very well with experimental data for the quantities on which information is available. Several closure hypotheses that have been proposed in the literature, are considered and validated with help of our simulation results. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of Mixing Models for use in Simulations of Turbulent Spray Combustion

The present study concerns the investigation of different mixing models for use in the transported probability density function (PDF) modeling of turbulent (reacting) spray flows. The modeling of the turbulent mixing and other characteristic scalar variables such as gas enthalpy using transported (joint) PDFs has become an important method to describe turbulent (reacting) spray flows since the evaporation process causes the PDF of the mixture fraction to deviate from the widely used β function, which is typically used in models for turbulent gas flows. In the PDF transport equation, the molecular mixing does not appear in closed form so that modeling strategies are required. For gas combustion, the interaction-by-exchange-with-the-mean (IEM) model, the modified Curl (MC) model, and the Euclidean minimum spanning tree (EMST) models are used. More recently, a new mixing model, the PSP model, which is based on parameterized scalar profiles has been developed. The present study focuses on the use and analysis of the IEM, MC and PSP models for turbulent spray flames. For this purpose, the models are reconsidered with respect to the evaporation process that must be included and evaluated if spray combustion is considered. For model evaluation, turbulent ethanol/air spray flames are simulated, and the results are compared to experimental data by A. Masri, University of Sydney, Australia.     

Towards an Extension of TFC Model of Premixed Turbulent Combustion

In order to determine the mean rate of product creation within the framework of the Turbulent Flame Closure (TFC) model of premixed combustion, the model is combined with a simple closure of turbulent scalar flux developed recently by the present authors based on the flamelet concept of turbulent burning. The model combination is assessed by numerically simulating statistically planar, one-dimensional, developing premixed flames that propagate in frozen turbulence. The mean rate of product creation yielded by the combined model decreases too slowly at the trailing edges of the studied flames, with the effect being more pronounced at longer flame-development times and larger ratios of rms turbulent velocity u′ to laminar flame speed S _L . To resolve the problem, the above closure of turbulent scalar flux is modified and the combination of the modified closure and TFC model yields reasonable behaviour of the studied rate. In particular, simulations indicate an increase in the mean combustion progress variable associated with the maximum rate by u′/S _L , in line with available DNS data. Finally, the modified closure of turbulent scalar flux is validated by computing conditioned velocities and turbulent scalar fluxes in six impinging-jet flames. The use of the TFC model for simulating such flames is advocated.     

Conditional Moment Closure for Large Eddy Simulations

A conditional moment closure (CMC) based combustion model for large-eddy simulations (LES) of turbulent reacting flow is proposed and evaluated. Transport equations for the conditionally filtered species are derived that are consistent with the LES formulation and closures are suggested for the modelling of the conditional velocity, conditional scalar dissipation and the fluctuations around the conditional mean. A conventional β-probability density distribution of the scalar is used together with dynamic modelling for the sub-grid fluxes. The model is validated by comparison of simulations with measurements of a piloted, turbulent methane-air jet diffusion flame.     

Spatial resolution effects on the measurement of scalar variance and scalar gradient in turbulent nonpremixed jet flames

The effect of spatial averaging is important for scalar gradient measurements in turbulent nonpremixed flames, especially when the local dissipation length scale is small. Line imaging of Raman, Rayleigh and CO-LIF is used to investigate the effects of experimental resolution on the scalar variance and radial gradient in the near field of turbulent nonpremixed CH₄/H₂/N₂ jet flames at Reynolds numbers of 15,200 and 22,800 (DLR-A and B) and in piloted CH₄/air jet flames at Reynolds numbers of 13,400, 22,400 and 33,600 (Sandia flames C/D/E). The finite spatial resolution effects are studied by applying the Box filter with varying filter widths. The resulting resolution curves for both scalar variance and mean squared-gradient follow nearly the same trends as theoretical curves calculated from the model turbulence kinetic energy spectrum of Pope. The observed collapse of resolution curves of mean squared-gradient for nearly all studied cases implies the shape of the dissipation spectrum is approximately universal. Fluid transport properties are shown to have no effect on the dissipation resolution curve, which implies that the dissipation length scale inferred from the square gradient is equivalent to the length scale for the scalar dissipation rate, which includes the diffusion coefficient. With the Box filter, the required spatial resolution to resolve 98% of the mean dissipation rate is about one−two times of the dissipation cutoff length scale (analogous to the Batchelor scale in turbulent isothermal flows). The effects of resolution on the variances of mixture fraction, temperature, and the inverted Rayleigh signal are also compared. The ratio of the filtered variance to the true variance is shown to depend nearly linearly on the probe resolution. The inverted Rayleigh scattering signal can be used to study the resolution effect on temperature variance even when the Rayleigh scattering cross section is not constant. The experimental results also indicate that these laboratory scale turbulent jet flames have small effective Reynolds numbers, such that there is some direct interaction of the large (energy containing) and small (dissipative) scalar length scales, especially for the near field case at x/d = 7.5 of the piloted Sandia flames C/D/E.     

Measurements of Scalar Variance, Scalar Dissipation, and Length Scales in Turbulent Piloted Methane/Air Jet Flames

One-dimensional (line) measurements of mixture fraction, temperature, and scalar dissipation in piloted turbulent partially premixed methane/air jet flames (Sandia flames C, D, and E) are presented. The experimental facility combines line imaging of Raman scattering, Rayleigh scattering, and laser-induced CO fluorescence. Simultaneous single-shot measurements of temperature and the mass fractions of all the major species (N₂, O₂, CH₄, CO₂, H₂O, CO, and H₂) are obtained along 7 mm segments with a nominal spatial resolution of 0.2 mm. Mixture fraction, ξ, is then calculated from the measured mass fractions. Ensembles of instantaneous mixture fraction profiles at several streamwise locations are analyzed to quantify the effect of spatial averaging on the Favre average scalar variance, which is an important term in the modeling of turbulent nonpremixed flames. Results suggest that the fully resolved scalar variance may be estimated by simple extrapolation of spatially filtered measurements. Differentiation of the instantaneous mixture fraction profiles yields the radial contribution to the scalar dissipation, χ _r = 2D _ξ(?ξ/?r)², and radial profiles of the Favre mean and rms scalar dissipation are reported. Scalar length scales, based on autocorrelation of the spatial profiles of ξ and χ _r , are also reported. These new data on this already well-documented series of flames should be useful in the context of validating models for sub-grid scalar variance and for scalar dissipation in turbulent flames.     

Vortex-In-Cell and Probability Density Function Approach for a Passive Scalar Field in a Mixing Layer

A Reynolds-averaged simulation based on the vortex-in-cell (VIC) and the transport equation for the probability density function (PDF) of a scalar has been developed to predict the passive scalar field in a two-dimensional spatially growing mixing layer. The VIC computes the instantaneous velocity and vorticity fields. Then the mean-flow properties, i.e. the mean velocity, the root-mean-square (rms) longitudinal and lateral velocity fluctuations, the Reynolds shear stress, and the rms vorticity fluctuations are computed and used as input to the PDF equation. The PDF transport equation is solved using the Monte Carlo technique. The convection term uses the mean velocities from the VIC. The turbulent diffusion term is modeled using the gradient transport model, in which the eddy diffusivity, computed via the Boussinesq's postulate, uses the Reynolds shear stress and gradients of mean velocities from the VIC. The molecular mixing term is closed by the modified Curl model.

The computational results were compared with two-dimensional experimental results for passive scalar. The predicted turbulent flow characteristics, i.e. mean velocity and rms longitudinal fluctuations in the self-preserving region, show good agreement with the experimental measurements. The profiles of the mean scalar and the rms scalar fluctuations are also in reasonable agreement with the experimental measurements. Comparison between the mean scalar and the mean velocity profiles shows that the scalar mixing region extends further into the free stream than does the momentum mixing region, indicating enhanced transport of scalar over momentum. The rms scalar profiles exhibit an asymmetry relative to the concentration centerline, and indicate that the fluid on the high-speed side mixes at a faster rate than the fluid on the low-speed side. The asymmetry is due to the asymmetry in the mixing frequency cross-stream profiles. Also, the PDFs have peaks biased toward the high-speed side.     

湍流被动标量研究的最新进展

被动标量场的统计性质, 在湍流理论以及湍流燃烧、污染物防治等工程领域都有非常重要的意义. 最近十几年, 大量的实验测量、数值模拟和理论分析结果表明, 标量场具有很多自身独特的性质, 甚至有些性质并不依赖于速度场本身. 这就促使人们对传统经典理论进行重新认识、修正或者提出新的理论来取代. 本文对标量场的各向异性、标量和标量耗散率的概率密度函数(PDF)、标量场的空间结构及演化过程、还有标量小尺度混合模型等几个方面进行综述.      

Simulation of Dilute Acetone Spray Flames with LES-CMC Using Two Conditional Moments

Large-eddy simulations (LES) have been coupled with a conditional moment closure (CMC) method for the computation of a series of turbulent spray flames. An earlier study by Ukai et al. (Proc. Combust. Inst. 34(1),1643–1650, 2013) gave reasonable results for the prediction of temperature and velocity profiles, but some limitations of the method became apparent. These limitations are primarily related to the upper limit in mixture fraction space. In order to enhance the applicability of the LES-CMC model, this paper proposes a two-conditional moment approach to account for the existence of pre-evaporated fuel by introducing two sets of conditional moments based on different mixture fractions. The two-conditional moment approach is first tested for a non-reacting test case. The results indicate that the spray evaporation induces relatively large conditional fluctuations within a CMC cell, and one set of conditional moments might not be sufficient. The upper limit of the mixture fraction space is dynamically selected for the solution of the second set of conditional moments, and the corresponding CMC solution in a CFD cell is estimated by interpolation between the two conditional moments weighted by the amount of vapour emitted within the domain. The cell-filtered value is given by integration of the conditional moment across mixture fraction space using a bounded β-FDF for the distribution of the scalar. As a result, the fuel concentration profiles given by LES and the two-conditional moment approach are shown to agree well. Then, the two-conditional moment approach is applied to four different flame configurations. The comparison of LES cell quantities and conditionally averaged moments indicates that the two sets of conditional moments are necessary for accurate predictions in zones where gas phase mixture fraction is significantly increased by droplet evaporation within the computational domain. The unconditional temperature profiles clearly show that the new approach improves the predictions of mean temperature especially along the centerline. Also, the better predictions of the temperature field improve the accuracy of the predicted mean axial droplet velocities. Overall, good agreement with the experimental results is found for all four cases, and the methodology is shown to be applicable to flames with a relatively wide range of fuel vapour concentrations.