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Modeling of The Reactant Conversion Rate In a Turbulent Shear Flow
Abstract
Results are presented of direct numerical simulations (DNS) of spatially developing shear flows under the influence of infinitely fast chemical reactions of the type A + B yields Products. The simulation results are used to construct the compositional structure of the scalar field in a statistical manner. The results of this statistical analysis indicate that the use of a Beta density for the probability density function (PDF) of an appropriate Shvab-Zeldovich mixture fraction provides a very good estimate of the limiting bounds of the reactant conversion rate within the shear layer. This provides a strong justification for the implementation of this density in practical modeling of non-homogeneous turbulent reacting flows. However, the validity of the model cannot be generalized for predictions of higher order statistical quantities. A closed form analytical expression is presented for predicting the maximum rate of reactant conversion in non-homogeneous reacting turbulence.
... The PDF of the passive scalar with β kernel density function is in good agreement with the results obtained with DNS (Frankel et al., 1992; Madnia et al., 1991). For (ψ ∈ [0, 1]) ...
One of the important attractions of employing the transported probability density function (PDF) methods for turbulent reacting flows is the fact that the chemical source terms are treated exactly. The composition probability density function (PDF) evolves with convective transport in physical space due to the mean velocity (macromixing), the turbulent diffusivity which transports the composition PDF in physical space (mesomixing), the transport in composition space due to molecular mixing (micromixing) and chemical reactions. The key model term of our interest in this study in the transport PDF equation is the molecular mixing term, which describes how molecular diffusion affects both the shape of the PDF and the rate of scalar variance decay. Molecular mixing is expressed using the Fokker-Planck model (Fox, 2003, 1994), which is an extension of the interaction-by-exchange-with-the-mean (IEM) model. The IEM model, widely used in chemical-reaction engineering and computational combustion due to its simple form, assumes the linear relaxation of the scalar towards its mean, while the Fokker-Plank model considers, in addition, the effect of the differential diffusion process to mimic the mixing. Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. An extended quadrature-based moment method (EQMOM) (Yuan et al., 2012; Chalons C. Fox R.-O. Massot M., 2010) is used to close the transport equation for the composition PDF. The β kernel density function is used for EQMOM since the scalar composition is represented by a scalar bounded between -1 and 1. The PDF predicted by the EQMOM model in the same conditions studied in the direct numerical simulation of Eswaran and Pope (1988) are reported and compared to the DNS predictions.
References:
Chalons, C., Fox, R. O., Massot, M., 2010. A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework. Studying Turbulence Using Numerical Simulation Databases, Center for Turbulence Research, Summer Program 2010, Stanford University 347–358.
Eswaran, V., Pope, S.B., 1988. Direct numerical simulations of the turbulent mixing of a passive scalar. Physics of Fluids 31, 506–520.
Fox, R.O., 2003. Computational Models for Turbulent Reacting Flows. Cambridge University Press.
Fox, R.O., 1994. Improved Fokker–Planck model for the joint scalar, scalar gradient PDF. Physics of Fluids 6, 334–348.
Yuan, C., Laurent, F., Fox, R.O., 2012. An extended quadrature method of moments for population balance equations. Journal of Aerosol Science 51, 1–23.
The second edition of Computational Fluid Dynamics represents a significant improvement from the first edition. However, the original idea of including all computational fluid dynamics methods (FDM, FEM, FVM); all mesh generation schemes; and physical applications to turbulence, combustion, acoustics, radiative heat transfer, multiphase flow, electromagnetic flow, and general relativity is still maintained. The second edition includes a new section on preconditioning for EBE-GMRES and a complete revision of the section on flowfield-dependent variation methods, which demonstrates more detailed computational processes and includes additional example problems. For those instructors desiring a textbook that contains homework assignments, a variety of problems for FDM, FEM and FVM are included in an appendix. To facilitate students and practitioners intending to develop a large-scale computer code, an example of FORTRAN code capable of solving compressible, incompressible, viscous, inviscid, 1D, 2D and 3D for all speed regimes using the flowfield-dependent variation method is made available.
An extended quadrature method of moments using the β kernel density function (β-EQMOM) is used to approximate solutions to the evolution equation for univariate and bivariate composition probability distribution functions (PDFs) of a passive scalar for binary and ternary mixing. The key element of interest is the molecular mixing term, which is described using the Fokker–Planck (FP) molecular mixing model. The direct numerical simulations (DNSs) of Eswaran and Pope [“Direct numerical simulations of the turbulent mixing of a passive scalar,” Phys. Fluids 31, 506 (1988)] and the amplitude mapping closure (AMC) of Pope [“Mapping closures for turbulent mixing and reaction,” Theor. Comput. Fluid Dyn. 2, 255 (1991)] are taken as reference solutions to establish the accuracy of the FP model in the case of binary mixing. The DNSs of Juneja and Pope [“A DNS study of turbulent mixing of two passive scalars,” Phys. Fluids 8, 2161 (1996)] are used to validate the results obtained for ternary mixing. Simulations are performed with both the conditional scalar dissipation rate (CSDR) proposed by Fox [Computational Methods for Turbulent Reacting Flows (Cambridge University Press, 2003)] and the CSDR from AMC, with the scalar dissipation rate provided as input and obtained from the DNS. Using scalar moments up to fourth order, the ability of the FP model to capture the evolution of the shape of the PDF, important in turbulent mixing problems, is demonstrated. Compared to the widely used assumed β-PDF model [S. S. Girimaji, “Assumed β-pdf model for turbulent mixing: Validation and extension to multiple scalar mixing,” Combust. Sci. Technol. 78, 177 (1991)], the β-EQMOM solution to the FP model more accurately describes the initial mixing process with a relatively small increase in computational cost.
This paper provides an overview of progress in turbulent reacting flow modeling for combustors. Some available models and methods which are compatible with the computational resources available in Taiwan are suggested in the following. Moment closure models provide the most feasible approach to solving for the fields of velocity and passive scalars. The laminar flamlet method incorporated with the presumed pdf models is a promising choice for dealing with the interaction between turbulence and combustion. The stochastic Eulerian-Lagrangian models are suggested for use in finding solutions of two-phase turbulent flows. Although there continues to be much uncertainty about the modeling of key physical and mathematical processes, the better numerical resolution which has recently become achievable due to advances in numerical methodology and hardware has led to generally improved simulations of industrial combustors.
The second edition of Computational Fluid Dynamics represents a significant improvement from the first edition. However, the original idea of including all computational fluid dynamics methods (FDM, FEM, FVM); all mesh generation schemes; and physical applications to turbulence, combustion, acoustics, radiative heat transfer, multiphase flow, electromagnetic flow, and general relativity is still maintained. This unique approach sets this book apart from its competitors and allows the instructor to adopt this book as a text and choose only those subject areas of his or her interest. The second edition includes a new section on preconditioning for EBE-GMRES and a complete revision of the section on flowfield-dependent variation methods, which demonstrates more detailed computational processes and includes additional example problems. For those instructors desiring a textbook that contains homework assignments, a variety of problems for FDM, FEM, and FVM are included in an appendix. To facilitate students and practitioners intending to develop a large-scale computer code, an example of FORTRAN code capable of solving compressible, incompressible, viscous, inviscid, 1D, 2D, and 3D for all speed regimes using the flowfield-dependent variation method is made available.
A family of Probability Density Functions (PDF's) generated by Johnson-Edgeworth Translation (JET) is used for statistical modeling of the mixing of an initially binary scalar in isotropic turbulence. The frequencies obtained by this translation are shown to satisfy some of the characteristics of the PDF's generated by the Amplitude Mapping Closure (AMC) (Kraichnan, 1989; Chen et al., 1989). In fact, the solution obtained by one of the members of this family is shown to be identical to that developed by the AMC (Pope, 1991). Due to this similarity and due to the demonstrated capabilities of the AMC, a justification is provided for the use of other members of JET frequencies for the modeling of the binary mixing problem. This similarity also furnishes the reasoning for the applicability of the Pearson Family (PF) of frequencies for modeling of the same phenomena. The mathematical requirements associated with the applications of JET in the modeling of the binary mixing problem are provided, and all the results are compared with data generated by Direct Numerical Simulations (DNS). These comparisons indicate that the Logit-Normal frequency portrays some subtle features of the mixing problem better than the other closures. However, none of the models considered (JET, AMC, and PF) are capable of predicting the evolution of the conditional expected dissipation and/or the conditional expected diffusion of the scalar field in accordance with DNS. It is demonstrated that this is due to the incapability of the models to account for the variations of the scalar bounds as the mixing proceeds. A remedy is suggested for overcoming this problem which can be useful in probability modeling of turbulent mixing, especially when accompanied by chemical reactions. While in the context of a single-point description the evolution of the scalar bounds cannot be predicted, the qualitative analytical-computational results portray a physically plausible behavior.
Experimental data of Bilger et al. (1991) pertaining to the compositional structure of a turbulent reactive scalar mixing layer are reproduced by a mathematical computational procedure utilizing the Pearson family (PF) of univariate and multivariate scalar probability density functions (pdfs). Aided by comparisons against these data and extensive additional data generated here by direct numerical simulations (DNS) of a spatially developing reacting mixing layer, an appraisal is made of the applicability and the extent of validity of the PF for statistical description of the reactant fluctuations. In accord with the experiment, a chemical reaction of the type A + B →Products is simulated in isothermal, incompressible flows. A wide range of the Damkohler number is considered including both frozen and equilibrium chemistry limits. The comparison of predicted results with laboratory data indicates that the PF generated pdfs are convenient for modeling the influence of turbulence on the mean reactant conversion rate. In particular, the Dirichlet frequency parameterized with the “scalar-energy” provides a reasonable means of portraying the multivariate scalar pdf. A more detailed comparative assessment of the model predictions against DNS data confirms the applicability of the Dirichlet pdf. However, the model portrays some drawbacks, the consequences of which are appraised by laboratory and DNS data.
Turbulent scalar mixing in the near field of a tee mixer (transverse jet in a pipe) was studied using particle image velocimetry and planar laser induced fluorescence. Two jet-to-pipe velocity ratios r = 3.06 and r = 5.04 were investigated for a turbulent pipe flow at a Reynolds number of 20,850. Laser measurement techniques are described, typical flow structures in the near field of tee mixer are examined, and turbulence statistics and scalar statistics are calculated. The general behavior of mixing is analyzed from the jet expansion and the concentration decay along the jet centerline, and the flow condition for jet impingement on the pipe wall is estimated. The mixing mechanism in the near field is revealed from the variation of the scalar PDF across the jet. The mixing on the upstream side of the jet is dominated by large-scale structures and on the downstream side by small eddies and turbulent diffusion, where the β-PDF is an excellent approximation.
An investigation of the statistical description of binary mixing and/or reaction between a carrier gas and an evaporated vapor species in two-phase gas-liquid turbulent flows is performed through both theoretical analysis and comparisons with results from direct numerical simulations (DNS) of a two-phase mixing layer. In particular, the validity and added complications of extending single-point assumed probabolity density function (PDF) methods to two-phase flows involving evaporating droplets as sources of vapor are addressed. Noting that Favre density-weighted averaging is the most convenient form for moment transport equations for these flows, algebraic relationships are derived for the ratios of the Favre and nonweighted scalar means and variances. Comparisons with the DNS results indicate that the mixing layer centerline where the root mean square (rms) density fluctuation is >12% of the mean density. It is therefore considered appropriate to use Favre moments for the nonweighted PDF closure. A transport equation for effects. The DNS results indicate that one of these terms due to scalar-source correlations is predominantly responsible for scalar variance production, whereas the remaining three terms are of negligible magnitude. Finally, the β PDF, which is known to represent well the DNS generated mixture fraction statistics for single-phase mixing, is shown to be a poor representation for mixing of vapor resulting from droplet evaporation.
Results are presented of Direct Lagrangian Simulations (DLS) of an unsteady, two-dimensional planar mixing layer by means of the “transport-element method.” The simulated results are used to understand the mixing behavior in a non-reacting flow, and to assess the limiting rate of the reactant conversion in a layer under the influence of a non-heat releasing reaction of the type A + B → Products. Both unforced and harmonically forced layers are considered. It is shown that harmonic perturbation results in a significant enhancement of mixing at initial stages of the flow development. However, it does not have a substantial influence at later stages. In all the cases considered the statistical results generated by DLS are shown to compare well with experimental data. Therefore, based on the DLS results the extent of validity of some of the turbulence closures of reacting flows is assessed. With this assessment it is demonstrated that the limiting rate of reactant conversion can be accurately predicted by the use of a Beta density model for the probability distribution of an appropriately defined Shvab-Zeldovich variable. Also, it is shown that the gradient diffusion model works reasonably well in accounting for the effects of turbulent convective fluxes.
Two assumed probability density functions (pdfs) are employed for computing the effect of temperature fluctuations on chemical reaction. The pdfs assumed for this purpose are the Gaussian and the beta densities of the first kind. The pdfs are first used in a parametric study to determine the influence of temperature fluctuations on the mean reaction-rate coefficients. Results indicate that temperature fluctuations significantly affect the magnitude of the mean reaction-rate coefficients of some reactions depending on the mean temperature and the intensity of the fluctuations. The pdfs are then tested on a high-speed turbulent reacting mixing layer. Results clearly show a decrease in the ignition delay time due to increases in the magnitude of most of the mean reaction rate coefficients.
The presently obtained closed-form analytical expressions, which predict the limiting rate of mean reactant conversion in homogeneous turbulent flows under the influence of a binary reaction, are derived via the single-point pdf method based on amplitude mapping closure. With this model, the maximum rate of the mean reactant's decay can be conveniently expressed in terms of definite integrals of the parabolic cylinder functions. The results obtained are shown to be in good agreement with data generated by direct numerical simulations.
An overview is presented of modern computational techniques for the numerical treatments of chemically reacting unsteady shear layers. Emphasis is placed upon the discussions on the physical phenomena described by the simulations of these flows, together with an explanation of the capabilities and limitations of various CFD techniques in capturing these phenomena. A brief review of the contributions within the past decade in this area of research is furnished in conjunction with some of the issues raised in the position paper (Swithenbank et aL, 1989).
Direct numerical simulations have been employed to study the effects of large coherent structures in a perturbed, time-dependent, spatially developing, reacting mixing layer. It is shown that the vortex dynamics has a significant influence on the enhancement of the chemical reactions, convolution of the reaction surface, and increase of product formation in the reaction zone of the layer. Furthermore, examination of some of the statistical quantities obtained from the results of direct numerical simulations indicates the asymmetric nature of the mechanism of mixing processes in the mixing zone of the layer.
Direct Numerical Simulations (DNS) are performed of a homogeneous turbulent flow under the influence of a chemical reaction of the type A + B→ Products. The generated results are statistically analyzed to describe the process of mixing in an analogous plug flow reactor configuration. The results of the statistical analysis indicate that the Probability Density Functions (PDF's)of a conserved Shvab-Zeldovich variable, characterizing the mixing process, can be well approximated by a Beta distribution. This provides a justification for implementing this density for predicting the limiting bounds of the unmixedness in homogeneous reacting turbulence.
A simple mathematical model is presented for the prediction of the limiting bounds of the unmixedness parameter in non-premixed homogeneous reacting turbulence. This model is based on a Beta distribution for the PDF of a conserved Shvab-Zeldovich variable and generates statistical results which show similar trends to those previously obtained by computational experiments based on direct numerical simulations.
Experimental investigations of shear layer instability have shown that some obviously essential features of the instability properties cannot be described by the inviscid linearized stability theory of temporally growing disturbances. Therefore an attempt is made to obtain better agreement with experimental results by means of the inviscid linearized stability theory of spatially growing disturbances. Thus using the hyperbolic-tangent velocity profile the eigenvalues and eigenfunctions were computed numerically for complex wave-numbers and real frequencies. The results so obtained showed the tendency expected from the experiments. The physical properties of the disturbed flow are discussed by means of the computed vorticity distribution and the computed streaklines. It is found that the disturbed shear layer rolls up in a complicated manner. Furthermore, the validity of the linearized theory is estimated. The result is that the error due to the linearization of the disturbance equation should be larger for the vorticity distribution than for the velocity distribution, and larger for higher disturbance frequencies than for lower ones. Finally, it can be concluded from the comparison between the results of experiments and of both the spatial and temporal theory by Freymuth that the theory of spatially-growing disturbances describes the instability properties of a disturbed shear layer more precisely, at least for small frequencies.
A critical review of the modern computational methods for solving the transport equations of turbulent reacting single-phase flows is presented. Primary consideration is given to those methods which lead to ‘model-free’ simulations while some attention is devoted to ‘turbulence modeling’. Emphasis is placed upon the role of supercomputers and how their increased computational capacities may be exploited to allow better simulations of the physics of turbulent reactive flows. Comparisons between the commonly employed computational schemes for simulating these flows are given, with the advantages and the limitations associated with each scheme being highlighted. Examples are presented of recent applications of model-free simulations to a variety of unsteady reacting flows, with detailed discussions on the physical phenomena captured by such simulations. Due to the nature of this review, experimental contributions are mentioned only in the context of providing empirical evidence. References are made to other contributions which are not directly related to the computational efforts in order to provide a reasonably comprehensive bibliography for those interested in pursuing various topics in greater detail. Predictions of future accomplishments, as well as some suggestions for future work, are also given.
The spectral-element method, a numerical scheme that combines the accuracy of spectral methods with the versatility of finite element techniques, has been employed to study the mechanisms of mixing and chemical reactions in, a diffusion flame stabilized on a two-dimensional planar mixing layer. The results of simulations of the harmonically forced, spatially developing flow are statistically analyzed to examine the compositional structure of the flame near quenching. The results indicate that as the flame approaches extinction, the mean and the rms values of the reactant concentrations decrease while those of the product concentration and temperature increase. This behavior is enhanced by increasing the hydrodynamics characteristic time (reducing the local Damkohler number) and is consistent with that observed experimentally.
The processes of entrainmentt and mixing are investigated in reacting and non-reacting, uniform density, liquid mixing layers over a wide range of Reynolds numbers. In non-reacting cases, a passive scalar technique is used to measure the probability density function (pdf) of the composition field. Chemically reacting experiments employ a diffusion-limited acid-base reaction to directly measure the extent of mixing. The diagnostics are based entirely on the laser induced fluorescence technique. The fluorescence signal is measured by self-scanning linear photodiode arrays using high speed, real-time computer data acquisition. The system is capable of yielding species concentration data with a spatial resolution of 100 µm and a temporal resolution of 0.8 msec. Results show that the vortical structures in the mixing layer initially roll up with a large excess of high speed fluid in the cores. During the mixing transition, not only does the amount of mixed fluid increase, but the composition also changes. It is found that the pdf of the mixed fluid, above the mixing transition, is quite uniform across the entire transverse extent of the layer. Furthermore, it is asymmetric and biased toward the high speed fluid. Experimental evidence indicates that the turbulent transport, in the cases studied, is dominated by large scale structures and is not adequately described by standard gradient-diffusion models. The fluid composition in the mixing layer, suggested by the present results, is in qualitative agreement with many aspects of the recent theoretical model of Broadwell and Breidenthal. The amount of product formed in the layer is compared to Mungal's measurements in gas, and, it is observed that the liquid layer has about 50% less product. The mean concentration of the mixed fluid, for a mixing layer at a velocity ratio of 0.38, becomes constant at 0.57 above the mixing transition. This corresponds to an entrainment ratio of 1.32, in agreement with the gaseous result of Konrad at the same velocity ratio.
An account is given of the implementation of the spectral-element technique for simulating a chemically reacting, spatially developing turbulent mixing layer. Attention is given to experimental and numerical studies that have investigated the development, evolution, and mixing characteristics of shear flows. A mathematical formulation is presented of the physical configuration of the spatially developing reacting mixing layer, in conjunction with a detailed representation of the spectral-element method's application to the numerical simulation of mixing layers. Results from 2D and 3D calculations of chemically reacting mixing layers are given.
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Numerical Simulation of a Spatially Developing, Forced Plane Mixing Layer
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Experiments on Turbulent Mixing and Chemical Reactions in a Liquid Mixing Layer
- M M Koochcsfahani