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Direct Numerical Simulation of the Unmixedness in a Homogeneous Reacting Turbulent Flow
Abstract
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.
... From the first two moments of the concentration it is then possible to get a presumed, continuous or discrete, PDF. The beta-function, as presumed continuous PDF, was verified and recommended by Rhodes 3 , Frankel, Madnia et al. 4 and Madnia, Frankel et al. 5 and is commonly used in chemical reaction rate closures (Li and Toor 6 , Ba³dyga 1 , Bilger 7 ). The purpose of the present study is to compare the multi-scale models to experiments performed with Planar laser induced fluorescence (PLIF) in a pipe. ...
Planar laser induced fluorescence measurements were performed on a passive tracer coaxially injected in the center of a larger pipe carrying intermediate to fully turbulent flows. Axial distributions of mean concentration and concentration variance for the tracer are presented. Results were compared with theoretical predictions of two multiple-timescale models for concentration variance. Calculations were performed with computational fluid dynamics. Finally concentration distributions were compared to beta-PDFs. INTRODUCTION Turbulent mixing is central to the understanding of reactive flows, and to the calculations of selectivity, product quality and yield of fast chemical reactions in turbulent flows. Most currently employed models for scalar mixing, ranging from simple moment closures to full probability density function (PDF) methods, require information concerning the coupling between the turbulence time scale and the scalar dissipation time scale for closure. This information is provided by the turbulent mixer model of Ba³dyga 1 and the spectral relaxation model of Fox 2 , which are both multiple-timescale mixing models for the concentration variance of a passive scalar. From the first two moments of the concentration it is then possible to get a presumed, continuous or discrete, PDF. The beta-function, as presumed continuous PDF, was verified and recommended by Rhodes 3 , Frankel, Madnia et al. 4 and Madnia, Frankel et al. 5 and is commonly used in chemical reaction rate closures (
... From the first two moments of the concentration it is then possible to get a presumed, continuous or discrete, PDF. The beta-function, as presumed continuous PDF, was verified and recommended by Rhodes (1975), Frankel et al. (1991) and Madnia et al. (1991), and is commonly used in chemical reaction rate closures (Li and Toor, 1986;Bałdyga, 1989;Bilger, 1989). ...
Planar laser induced fluorescence measurements were performed on a passive tracer coaxially injected in the center of a larger pipe carrying intermediate to fully turbulent flows. Axial distributions of mean concentration and concentration variance for the tracer are presented. Results were compared with theoretical predictions of two multiple-time-scale models for concentration variance. Calculations were performed with computational fluid dynamics. Finally concentration distributions were compared with beta-PDFs.
In the context of modeling turbulent scalar mixing using probability density function (PDF) methods, the treatment of molecular mixing is of paramount importance. The conditional moment closure (CMC) offers a high-fidelity description for molecular mixing in nonpremixed flows. Recent work has demonstrated that first-order CMC can be implemented numerically using the moments of the conditioning variable and first-order joint moments of the scalar of interest. When solving the CMC using, for example, quadrature-based moment methods (QBMM), a functional form must be chosen for the conditional scalar dissipation rate (CSDR) of the conditioning variable. In prior work, the CSDR was chosen to produce a β-PDF for the conditioning variable (mixture fraction) at steady state. This choice has the advantage that the system of moment equations used in QBMM-CMC can be written in closed form. In this work, an alternative choice for the CSDR is investigated, namely, the amplitude mapping closure (AMC). With the AMC, the moment equations can be closed using the quadrature method of moments incorporated into a realizable ordinary differential equation solver. Results are compared with the β-CSDR closure for binary, passive scalar mixing in homogeneous single- and disperse-phase turbulent flows. It is also demonstrated that the moment formulation of CMC provides a straightforward method for modeling the effect of differential diffusion in the context of CMC.
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.
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 effect of micromixing on parallel reactions was studied experimentally and interpreted theoretically using closure models. A solution of soduim hydroxide was contacted with a pre-mixture of hydrochloric acid and ethyl chloroacetate solutions in a tubular reactor equipped with an impingement mix-head. Two different impingement mix-heads were applied: the first was T. shaped with two feeding nozzles; the second was equipped with six feeding nozzles. The selectivity of the parallel reactions was then dependent upon the mix-head used, the Reynolds number, the concentretion of reactants and the volume ratio of reagent solutions. The effect of the volume ratio of the reagant solutions on the selectivity was used as the test effect for closure models. It is shown that simple closures based on moments are not able to describe properly the behaviour of parallel reactions. A closure method based on the probability density functions of the concentrations of chemically inert species and on the interpolation of the local concentrations of reactants between the asymptotes on infinitely slow and infinitely fast chemical reactions is verified experimentally.
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.
A recent article, made a comparative study of several turbulence closures for predicting the mean rate of reactant conversion in a chemical reaction of the type A+B right arrow products in homogeneous turbulent flows. In this work, we use recent developments in the approach based on the probability density function (PDF) method for treating the problem considered. We propose a simple algebraic relation for predicting the limiting rate of mean reactant conversion. (from Authors)
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.
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 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.
Compressible flows with r.m.s. velocities of the order of the speed of sound are studied with direct numerical simulations using a pseudospectral method. We concentrate on turbulent homogeneous flows in the two-dimensional case. The fluid obeys the Navier-Stokes equations for a perfect gas, and viscous terms are included explicitly. No modelling of small scales is used. We show that the behaviour of the flow differs sharply at low compared with high r.m.s. Mach number Ma, with a transition at Ma = 0.3. In the large scales, temporal exchanges between longitudinal and solenoidal modes of energy retain an acoustical character; they lead to a slowing down of the decrease of the Mach number with time, which occurs with interspersed plateaux corresponding to quiescent periods. When the flow is initially supersonic, the small scales are dominated by shocks behind which vortices form. This vortex production is particularly prominent when two strong shocks collide, with the onset of shear turbulence in the region downstream of the collision. However, at the resolutions reached by our code on a 256 × 256 uniform grid, this mechanism proves insufficient to bring vortices into equipartition with shocks in the small-scale tail of the energy spectrum.
New subgrid-scale models for the large-eddy simulation of compressible turbulent flows are developed based on the Favre-filtered equations of motion for an ideal gas. A compressible generalization of the linear combination of the Smagorinsky model and scale-similarity model (in terms of Favre-filtered fields) is obtained for the subgrid-scale stress tensor. An analogous thermal linear combination model is also developed for the subgrid-scale heat flux vector. The three dimensionless constants associated with these subgrid-scale models are obtained by correlating with the results of direct numerical simulations of compressible isotropic turbulence performed on a 96 to the third power grid using Fourier collocation methods. Extensive comparisons between the direct and modeled subgrid-scale fields are provided in order to validate the models. Future applications of these compressible subgrid-scale models to the large-eddy simulation of supersonic aerodynamic flows are discussed briefly.
Turbulent mixing in this type mixer is experimentally investigated. Probability density distribution function of ''c is measured by using digital counter and proves to be normally distributed. This fact affirms the basis of Toor''s theory. Concentration spectrum has a narrow -5/3 power region, Toor''s theory concerning the relation between concentration fluctuation and conversion of very rapid reaction is experimentally proved. Initial mixing rate is studied on the basis of it. Dependence of the mixing time constant on U, a, dN and Dm are measured. Their effects are generally explained by modified Corrsin''s equation.
The principles of mixing on a macroscopic level have been known and used for a number of years for the scale-up of mixing processes. However, these scale-up methods have been empirical in nature due to the lack of information about mixing theory on a microscopic level. The complexity of problems involving turbulence, such as mixing, has made a quantitative description difficult. However, the modern turbulence approach has led to a much deeper understanding of the field of mixing and of related turbulent scalar transport processes. The analysis allows us to define measurable mixing criteria. A knowledge of the parameters of the turbulence provides the information necessary to estimate the degree of mixing. The parameters can often be estimated from the geometry of the flow system and from simple empirical relationships. With a knowledge of the mixing, the problem of chemical kinetics, when mixing cannot be ignored, can be studied. A theoretical approach is possible that with limited assumptions allows one to predict the course of second-order chemical reactions. Recent advances in this area arc reviewed, as well as the results and predictions for a specific flow reactor.
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.
In turbulent flow the equations governing the development of conversion vs. distance are not closed because of the contribution of the concentration fluctuations to the average reaction rate. Toor's hypothesis permits closure from data obtained for mixing without reaction. This paper presents a critical review of the derivation of the hypothesis and discusses its validity.
Previously published conversion data for turbulent diffusion-controlled acid-base reactions in two different multijet reactors are used to extract the mixing parameters, tracer probability density function, and variance. The PDF is obtained with this instantaneous reaction method by differentiating twice the reactive mixing plot of mean reactant concentrations. The limiting behavior for the cases of complete segregation and complete mixing is derived and turns out to be piecewise linear on the reactive mixing plot. The variance is shown to be twice the area between the actual reactive mixing curve and the limiting curve for complete mixing; it can be computed by numerical integration without any foreknowledge of the PDF shape. The method, in principle, allows detection of Dirac delta functions (which represent unmixed virgin fluid) and indeed shows significant amounts of virgin fluid close to the inlet of one of the reactors.
The authors provide a generalization of G. Kosaly's result and, by use of direct numerical simulation (DNS), further investigate the validity of Toor's hypothesis and the revisions proposed by Kosaly. In this application DNS refers to the numerical solution of the aerothermodynamic transport equations of turbulent reacting flows for the detailed time development of the flow field. This technique uses no closure modeling, and no assumptions are made pertaining to the turbulent behavior of the fluid. DNS has proven very useful in computational studies of turbulence and is used here as a means of performing experiments on the computer. This approach offers advantages over laboratory experiments in that the instantaneous values of the hydrochemical variables are known at all locations at each time step, allowing better statistical analysis of the data.
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 Non-Premixed Reaction
- H L Toor
Toor, H.L., The Non-Premixed Reaction; A + B-Producrs, in Turbulence in Miring Operations, R.S. Brodkey. ed., Academic Press, New York, NY (1975).