Modified Curl Model Research Articles

A pairwise mixing model with kernel constraint and its appraisal in transported PDF simulations of ethylene flames

Modeling micro-mixing remains a primary challenge for the transported probability density function (TPDF) method. In this study, the pairwise mixing with kernel constraint (KerM) model is proposed and evaluated in TPDF simulations of a turbulent non-premixed ethylene flame. The new model features controllable localness by selecting mixing particle pairs with a probability that depends on their distance in composition space, normalized by a characteristic kernel size. To evaluate the model in a controlled setting, TPDF simulations were carried out with the mean velocity, turbulent diffusivity and scalar mixing frequencies extracted from a direct numerical simulation (DNS) dataset. The predictions of the mean statistics and the conditional PDFs illustrate that the KerM model exhibits asymptotic behavior to the modified Curl (MC) and the Euclidean minimum spanning trees (EMST) model, respectively, depending on the kernel size. For the flame considered, results for KerM approach those for EMST when the kernel size is less than 0.01, while they approach those for MC when the kernel size is larger than 1.0. Given a kernel size of 0.1, KerM yields a notably better prediction of the overall combustion process than the conventional MC model. The predictions of the mean temperature and species mass fractions by KerM are similar to those by EMST, while KerM better predicts the conditional PDF of temperature than EMST that overdamps the conditional fluctuations. In addition, its complexity with the number of particles per cell is instead of O(Np2) for EMST. By incorporating different mixing timescales among species, the KerM-DD model better predicts the mean temperature field at the time instant close to local extinction than the KerM and EMST models. This work demonstrates that the KerM model, apart from being simple and robust in terms of implementation, features the advantages of being flexible in localness and being capable of incorporating differential diffusion.

Predicting jet ignitability using a PDF transport model

For the first time the probability of ignition is calculated successfully using a PDF transport model. At present the model is only accurate where jet intermittency effects are small, away from the jet boundary. A conclusion of this investigation is the probability of ignition field is sensitive to the micro-mixing model used. In this article, four micro-mixing models are implemented and the modified Curl model demonstrated to be the most accurate for the jet ignitability experiments simulated.

Numerical analysis of nitrogen oxides in turbulent lifted H2/N2 cabra jet flame issuing into a vitiated coflow

Abstract This paper gives an in-depth insight into NOX (NO, NO2, and N2O) formation of H2/N2 turbulent Cabra jet flame issuing into a hot vitiated coflow. The joint composition probability density function (PDF) was employed to model the combustion and to specify the characteristics of the flame (i.e., scalar variables, concentration of species etc.). The turbulent transport term was modelled by Reynold-Average-Naiver-Stokes (RANS) SSG and molecular mixing was modelled by modified curl model. A combustion mechanism including 13 species and 34 reactions was employed to define the thermochemical state of the flame. The chemical reaction terms were resolved and accelerated by In Situ Adaptive Tabulation (ISAT). The simulation was performed at different equivalence ratios (ER), fuel jet nitrogen content ( Y N 2 , C ), coflow ( T C ) and jet temperatures ( T J ), coflow oxygen ( Y O 2 , C ) and water contents ( Y H 2 O , C ). Results reveal NOX is composed of 30% NO2 and 70% NO in the burner. Reaction rate analysis at different operating points in the ignition kernel demonstrates that N + O H ⇌ N O + H and N O 2 + H ⇌ N O + O H are dominant reactions in NO formation, while N O + H O 2 ⇌ N O 2 + O H is the main reaction in NO2 formation.

Micromixing model performance for nonreacting flows using a consistent Monte Carlo method

ABSTRACTIn this paper, several micromixing models are applied to the prediction of turbulent nonreacting flows. All micromixing models predict the mean mixture fraction and root mean squared (RMS) mixture fraction well. For higher mixture fraction moments, there is a tendency to overpredict the skewness and kurtosis field. The exception to this is a modification to the Langevin model which produces predictions more consistent with the measured fields. Where intermittency effects do not dominate the flow, the modified Curl model and the limited Langevin model can accurately predict the measured probability density function. Of the micromixing models tested for nonhomogeneous flows, the modified Curl model represents an appropriate balance between predictive capabilities, ease of implementation, and short run times for simulating nonreacting flows.

Performance of transported PDF mixing models in a turbulent premixed flame

Modeling of premixed turbulent flames is challenging due to the effects of strong turbulence-chemistry interaction. In the transported probability density function (TPDF) methods, chemical reactions are treated exactly, while molecular mixing needs to be modeled. In the present study, the performance of three widely used mixing models, namely the Interaction by Exchange with the Mean (IEM), Modified Curl (MC), and Euclidean Minimum Spanning Tree (EMST) models, are assessed using direct numerical simulation (DNS) data of a lean premixed hydrogen–air slot jet flame simulated at Sandia. The DNS provides initial conditions and time varying input quantities, including the mean velocity, turbulent diffusion coefficient, and scalar mixing rate for the TPDF simulations. A number of progress variable definitions are explored, as well as the commonly used constant mechanical-to-scalar mixing timescale model. It is found that the EMST model provides the best prediction of the flame structure and flame propagation speed out of the models tested. The IEM model implies a qualitatively incorrect conditional mean and RMS diffusion rate, while the MC model can qualitatively capture the conditional mean diffusion rate. Only the EMST model can accurately predict the conditional mean diffusion rate for this flame, which can be attributed to its enforcement of mixing that is local in composition space. Finally, a parametric study on the mechanical-to-scalar timescale ratio is performed. It is found that the optimal choice for the timescale ratio varies by a factor of 2 for the two DNS cases study, despite the cases having the same configuration. Therefore, this commonly used approach does not appear to be viable for turbulent premixed flames and further attention to mixing timescale models for reactive scalars is merited.

Effect of molecular transport on PDF modeling of turbulent non-premixed flames

Comprehensive studies are performed to evaluate the effect of molecular transport on the probability density function (PDF) modeling of turbulent non-premixed flames. The widely used Sandia piloted jet flame E is chosen as the validation test case. Three configuration cases are considered: neglecting the molecular transport (Case-A), considering the molecular transport (Case-B), and considering both the molecular transport and its effect on mixing (Case-C). Extensive testing is done to assess the effect of molecular transport on the performance of the different mixing models and on the prediction of the flame extinction limit. It is found that the molecular transport has a significant effect on the prediction of the upstream mixing layer between the pilot stream and the coflow, where the turbulence level is weak or it is laminar and hence the molecular transport dominates the turbulent transport. Accounting for the molecular transport increases the models’ resistance (for the IEM model and the modified Curl model) to flame extinction and yields a burning flame with a lower mixing rate that can cause global extinction without molecular transport. A modification to the mixing frequency is introduced to account for the effect of the molecular diffusion on mixing. With such a modification, the modeled flame sustainability is further enhanced and a lower mixing rate can be used, which enhances the ability of the IEM model and the modified Curl model which were previously found to yield a burning flame solution only at a highly increased rate of mixing.

J056041 確率密度関数法による液相格子乱流中の反応性スカラー拡散の数値計算 : 連続競争反応の場合([J05604]乱流における運動量,熱,物質の輸送現象(4))

The aim of this study is to elucidate numerically characteristics of a diffusion field with a competitive-consecutive reaction in a liquid grid-turbulence. Numerical simulation for a nearly homogeneous diffusion field of reactive scalars has been conducted by probability density function (PDF) method. PDF method is a one of promising approach for turbulent reacting flows because the chemical source term appears in a closed form. For the molecular mixing, two models (Curl's model and the modified Curl model) are applied. At first, the results of nonreactive field and reactive field of one-step reaction are compared. It is found that the mean concentrations of reactive species become smaller than those of nonreactive species because of consumption by chemical reaction. However, the simulation results overestimate the effect of the chemical reaction as a whole.

An efficient PDF calculation of flame temperature and major species in turbulent non-premixed flames

In spite of recent developments in the PDF calculations of turbulent flames, the high computational time required to implement PDF simulations makes it intractable in practical applications. Therefore, it is important to design and select different parameters for PDF calculation of most important quantities, i.e. temperature and major species means, in an efficient manner. The ingredients of the present model are a standard k – ε turbulence closure for modeling flow field and a joint composition PDF closure for the scalar fields. A modified Curl model is applied to consider molecular mixing in PDF transport equation and a simplified two-step mechanism which lowers the computational cost is incorporated to describe the chemistry. The flow field is solved numerically using an upwind discretization for the convective terms and a central discretization for the diffusion terms by coupling it with an Eulerian Monte Carlo algorithm to solve PDF transport equation. To show the superiority of the current PDF calculations over traditional moment-closure methods commonly used in practical applications, simulation is also performed by RANS method which shows large discrepancies, especially in prediction of maximum flame temperature (on the basis of present results, predicted flame temperature has 26 % error via RANS method and 8 % error via PDF method). Stoichiometric flame length predicted by RANS has 10 % error while, by PDF method, this error is negligible and about 0.6 % . The effect of coefficient C Φ on the modified Curl model is also investigated and it is concluded that the commonly used value C Φ = 2.0 is the best choice for the case of study. The numerical results obtained reveal that Westbrook–Drier mechanism is working very well in fuel-lean ( F < F st ) non-premixed combustion and also it predicts the total heat released in methane combustion in a very good agreement with the experiment.

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

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

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.

Simulation of finite-rate chemical kinetics in a turbulent bluff body flame

A turbulent combustion model using a Monte Carlo scalar PDF transport method with an adaptive number of PDF particles in each computational cell is coupled to an experimental version of a commercial CFD solver. While the molecular mixing is always modelled using a modified Curl model, the chemical source terms are calculated using a global 4-step reaction model and two different implementations of the method of intrinsic low-dimensional manifolds (ILDM), respectively. The quality of the chemistry representation by the three reduced chemical mechanisms is tested against detailed methane mechanisms using plug-flow reactor calculations. It is confirmed that the two ILDM implementations reproduce the chemical trajectories of the detailed reaction mechanism in scalar space quite well as expected, while the characteristic time scales of the global mechanism differ considerably. The full PDF transport CFD code is applied to a methane bluff-body diffusion flame stabilised by a recirculation zone. The calculations show a consistent improvement of predictions, particularly of temperature and CO concentrations of the PDF transport method compared to an presumed PDF calculation assuming equilibrium chemistry. Nevertheless, the CO concentrations are still considerably over predicted by all methods in such a highly strained flame, indicating that the mixing and turbulence models need to be improved.

On the Development of a Distributed Reaction Zones Model for Turbulent Premixed Flame of Lean Mixtures in a Closed Vessel

Turbulent premixed flame propagation in a closed spherical vessel with a homogeneous and isotropic turbulent flow field is modelled using the one-point transport equation for the probability density function (pdf) in the thermodynamic space. The pdf equation is discretized using the control volume formulation on a one-dimensional spherical grid. Then the discretized equation is solved using the Monte Carlo method to follow particles with variable volumes on a grid moving with the expansion radial velocity. The mixing term is modelled using the Curl model, and the modified Curl model. The source term is simulated delerministically using a one-step chemical reaction mechanism. Transient solutions are obtained for spherical benzene-air flames using frozen turbulent flow. The pressure rise, the mass fraction burned, the flame speed, the entrainment speed, the burning speed, and the burned gas speed are calculated. The calculated entrainment speeds are compared with the ones reported in Bradley et al (1992) for KLc>5.3 where distributed reaction zones are assumed to govern the flame propagation. The effects of different turbulent flow field (uvS1) and the effect of the product (KLe) are discussed.

Coupling of detailed and ILDM-reduced chemistry with turbulent mixing

The simulation of turbulent reactive flows is still a challenging problem. The direct numerical solution of the governing conservation equations is computationally expensive and for technical systems still not feasible. Therefore, simplifications both for the kinetics and the turbulence are necessary. A separate treatment of the turbulence and the chemistry, however, neglects the coupling of these processes. Thus, the present work is focused on the coupling of turbulence and chemistry. To investigate the influences of different submodels on the full system, several submodels were used. For the turbulent mixing, the modified Curl model and the interaction by exchange with the mean (IEM) model were chosen, and for the chemistry, a detailed mechanism was compared with a two-dimensional intrinsic low-dimensional manifolds (ILDM) reduced kinetic scheme. As a test case, a turbulent partially stirred homogeneous reactor with a syngas air system was chosen, with the intention to have a system governed by turbulence and chemistry only. The coupling of the kinetics with the turbulent mixing was analyzed both for detailed and reduced kinetics. It is shown for reduced kinetics, that oversimplified models can lead to considerable errors, and methods to overcome this problem are presented.

Numerical Simulation of Reactive-Scalar Mixing Layer in Liquid Phase by the PDF Method. Evaluation of Stochastic Process Models for Molecular Diffusion.

The reactive-scalar mixing layer in grid-generated turbulence is studied numerically by the probability density function (PDF) method. The chemical reaction in this study is a second-order and irreversible reaction. The simulation was made for finite Damkohler number, then the frozen and equilibrium limits were determined by using the conserved scalar theory. The simplified Langevin model is used for the velocity of stochastic particles. For the molecular mixing, three models (Curl's model, the modified Curl model and the binomial Langevin model) are applied. The results are compared with the experimental data in the liquid phase. It was found that as a whole, the simulation results of the mean concentration, the concentration fluctuation and the concentration covariance show good agreements with the experimental data. Comparing three models, Curl's model tends to overestimate the effect of the chemical reaction, but the modified version can overcome this defect. The binomial Langevin model can calculate the PDF shape most successfully.

Theoretical Remarks on a Phenomenological Model of Turbulent Mixing

Abstract In the pdf approach to turbulent combustion the molecular diffusion is to be modeled. In recent years, several authors have employed the Curl model (Curl, 1963) for closure. A modified version of the Curl model has been introduced by Pope (1982b). The modified model contains an unspecified function A∥α) [cf. Eq. (la)] which has been defined in the context of Monte-Carlo simulation. The paper Clarifies the physical meaning of this function by relating it to the physics of the initial phase of mixing of an inert scalar in homogeneous turbulence. A relationship is established between the modified Curl model and the linear mean square estimation closure introduced by Dopazo and O'Brien (1976). The asymptotic behavior of the pdf is discussed.