Pdf methods for turbulent reactive flows

The type of analysis used in Section 5. Analysis of the stochastic reorientation model are equal, we obtain The effect of the stochastic reorientation model 1 Eqs 5. The final term is Eq. It is expedient to assume that B is a linear function of v since this leads to a model that is For the homogeneous case, the gradients of the compatible with Reynolds stress models.

A Poisson support for the linearity assumption is provided by equation for the fluctuating pressure is obtained by the exact result Eq. Before considering the determination of the tensor These contributions to p' are denoted by pm and p : C, let us examine the form of the model, Combining respectively. The effect of the rapid pressure on the joint pdf of velocity Eq. Cqmlj Oq-- Uq. Povw: remains normal and, conversely, any other distribu- tion does not relax to a normal distribution.

For the other processes considered, the lack of relaxation was considered to be a defect in the model. But rapid distortion theory 89 shows that the rapid pressure has a deterministic effect on the turbulence, and hence, does not cause the pdf to become Gaussian. A completely satisfactory model for the non-dimen- sional tensor C has yet to be developed. The best available model is obtained by relating C to a fourth- order tensor A that appears in Reynolds-stress Fla.

But their identity see, for example, Ref. In this process, both surface integrals on C--except that it be finite. But, if the turbulence vanish. In view of this, it would be preferable to model C directly.

The corre- 5. R with bounding surface S, Fig. In the limit as R tends to infinity, this equation defines 5. The convergence of the integral In the previous two sections a model for the evolu- is not in doubt since, as r tends to infinity, tion of the velocity joint pdf fu is described.

In this section an required solution alternative model for the evolution offu is described. As in the previous 5. This solution is In terms of the velocity increment of a stochastic useful because it shows explicitly the role of the mean particle, velocity gradient and because several properties of Bmq can be deduced. It is evident from Eq. Both G and B can depend upon time Thus, in Eq.

The first term on the right-hand side of Eq. PoPE U x,t is not uniform. It is useful, therefore, to modelled equation yield the correct energy decay rate consider g v; 0 - - t h e Eulerian joint pdf of the velocity Eq. And, as noted in case of homogeneous anisotropic turbulence with uni- Section 5.

The The coefficient B is determined by requiring that in most general model for G contains far more scalar a limited sense the Langevin equation be a direct coefficients than could possibly be determined with model of fluid particle behavior. Consider a time confidence. The anisotropy tensor b is defined by Eq. In the alternative model presented in Sections From the Langevin model Eq. Then, By comparing Eqs 5. As a consequence, H is not subject to many s2.

In the determination of the eleven coefficients in For homogeneous isotropic turbulence without Eqs 5. The tensor must be isotropic Go oc 6o. Thus, in this sub- With the five conditions Eqs 5. But when the modelled Reynolds stress equation is deduced from Eq. Thus two For the homogeneous flows considered, the joint degrees of indeterminacy remain. The This last equation simplifies Eq. The models presented above are modified in such a 5. The four remaining revealed in the behavior of the scalar fluxes.

For this case experimental data. It is more informative to study equation, Eq. If mixing takes place, then to Eqs 5. Values of C, in the range 0. Thus the experimental data do show that r decay constant is decays, but the value of the decay constant C, is uncertain--if, indeed, it is a constant.

Molecular mixing is a microscale process. We adopt this fluid particle's composition changes, it does so by assumption without reservation or further discussion. However, we tentatively of particles separated by the Kolmogorov scale--are adopt the assumption in order to explore its conse- strongly correlated. But in the stochastic mixing quence. The resulting joint selected at random as before.

N, the change in then performed as before. Eqs 5. But we wish to apply the joint When correlated mixing occurs, two of the N pdf equation to more complicated flows--inhomo- stochastic particles denoted by p and q are selected geneous, variable-density reactive flows.

Indeed, the at random without replacement and their properties virtue of the pdf equation is that it facilitates the are replaced by: treatment of these complications.

In this section the extension to general flows is discussed. At the jet exit convective transport in physical space. In models are needed. Nevertheless, the term appears in delta function of magnitude 1 -? Integration over the delta function and over the con- The next three terms represent the effects of tinuous distribution yields dissipation and pressure fluctuations that have been modelled in Sections 5.

For flows remote from walls, it is assumed that the rapid-pressure models still apply. Further, the analysis of non-turbulent: Section 5. Consequently, the models Eqs 5. But there are two problems. First, the turbulent and non-turbulent fluids is that their representation of turbulence by a single scale most behavior is quite different. Consequently, a gain in likely limits the validity of the approach to reasonably accuracy can be expected if different models are used simple flows?

This can be done in scarce and difficult to perform, a modelled transport two ways. The first way that followed by Kollmann equation for e cannot be adequately tested. The modelling that has been Reynolds-stress models may suffer from the discussed so far is appropriate to the turbulent part fr, inaccurate or at least uncertain determination of e. The rate Scale information can be incorporated by con- at which non-turbulent fluid becomes turbulent is sidering multi-point pdf's, or the joint pdf of state determined by the transport equation for 7.

The rate of present work. The calculated intermittency factor and conditional where statistics agree well with the experimental data. The measurements of Lin and Lin 1os and those of Warhaft and Lumley 1o9 show 5. Dissipation rates that C, is not a universal constant. Wartiaft 11o performed an experiment in which the Since the kinetic energy k can be determined from the scalar fluctuations were induced differently than in joint pdf, a knowledge of either z or e is sufficient.

In Refs and In another experiment performed by Warhaft is implicitly modelled by the stochastic mixing model grid turbulence with scalar fluctuations was passed to be through an axisymmetric contraction. After the con- traction C, was found to be in the range 1. Is Eq. The purpose of this subsection is It is abundantly clear that, with measured values in to outline the problems raised by these questions. Since k is proposed--by Newman et al. POPE sistent with Warhaft's data. The major effect is of allowing C, to remain constant.

The Poisson The discussion above centers on scalar fluctuations equation for the pressure becomes in decaying grid turbulence. B6guier et al. For a constant-density Newtonian fluid, only the first Thus, at present, Eq. Not only are there additional Finally, we draw attention to the fact that none of influences on p', but p' also affects the pdf in different the models presented depends upon the molecular ways. For example, the term transport properties.

However, many flows of interest are at moderate Reynolds equation. In constant-density flows the divergence of numbers. It is possible to include a Reynolds number velocity is zero and so pressure fluctuations do not affect the kinetic energy. It is emphasised, however, At moderate Reynolds number, the different that even in variable-density flows the terms per- molecular diffusivities of different species are known to have a significant effect in some reactive flows.

Again, this is a 5. Laminar flamelets topic that has yet to be addressed in pdf methods. Variable density tional to the turbulent time scale z. This assumption In the low Mach number flows considered, the modified by the comments of Section 5. But rapid reactions can also variations of a factor of five or more are not produce steep composition gradients. This effect is uncommon. It is clear, therefore, that the density studied in more detail for an idealised premixed variations have a significant effect on the flow, and turbulent flame.

The conditional Lagrangian equation cf. Libby and Bray t t6 have suggested that this effect is the cause of counter- FIG. None of the scale. Pdf methods for turbulent flows fuel-air mixture and unity in the fully burnt products. Assuming simple gradient diffusion, the transport equation for 4 is Eq. Quantity h O against O, Eq. With these boundary conditions there its orientation or of a superimposed uniform fluid is a steady one-dimensional solution to Eq.

It is clear from the last expression in Eq. Thus, from the above three equations we obtain Then, mixing proceeds at a rate inversely proportional to the turbulence time scale z in accord with the stochastic mixing model. But the left-hand side of be simply connected. Turbulence strains this sheet Eq. Thus, locally, the flame sheet behaves like a plane premixed laminar flame. In the transport equation for the joint pdf. With the assumption of laminar flamelet combustion this becomes 0.

We thus reach the remarkable conclusion that for laminar flamelet combustion the reaction and mixing terms appear in closed form. Using this result Eq. PoPE have made joint pdf calculations to compare the 6.

Discrete Representation properties of premixed flames in flamelet and distri- buted combustion. We consider the turbulent flow within a three- dimensional volume of physical space--the solution domain. At time t, the mass of fluid within the solution 6. Introduction in state space, each particle representing a fixed mass Throughout the development, the ability to solve Am. The joint pdf can be represented com- putationally by the discrete representation, and its 6.

In As time proceeds, the statesof the stochasticparticles this section an algorithm is outlined to solve the change--some particles may leave the solution modelled joint pdf equation for an unsteady, variable- domain and some may enter as prescribed by the density, three-dimensional turbulent reactive flow.

The discrete mass density Variants Of the algorithm for simpler flows are function, and the discrete pdf of the stochastic particle outlined in Section 6. It is assumed that the 6. The discrete representation, on which the solution In the numerical solution, the states of the N algorithm is based is reviewed briefly in Section 6. Nonetheless, the connection between marches in time with small time steps At. In the time the discrete representation and the joint pdf is best interval At, several different processes simultaneously viewed in terms of expectations: affect the evolution of the joint pdf.

Stochastic mixing takes dition place during the second fractional step which is described in Section 6. The algorithm described provides the solution in 6. Eqs 3. Then an ensemble average over the particles 6. But this method--though con- satisfied. As described in Section 6. Fractional Steps means from the discrete representation of the joint pdf. It is a generalization of the right-hand side of Eq. We define The three fractional steps 6.

We examine the third step since P3 is the simplest of the operators. For finite At, Eq. For each time sections. While we have defined each step as an explicit step, the method is defined by the three fractional steps: Euler step Eqs 6. Using Eqs 6. POPE implement. Clearly At should be chosen to be significantly smaller This is simply achieved since Eq.

Integrating these equafi ons for a time then the nth particle is selected for mixing. A more At using the explicit Euler method yields to first-order efficient, general, and complicated algorithm is given in in At : Ref.

Thus in the first fraction step, the states of each of the The pairs of particles mix, as in the homogeneous N t stochastic particles change independently case, according to Eq. This completes the second according to Eq. The computational work needed to evaluate Eqs 6. For each particle in an inert constant-density flow, about 6. Third Fractional Step: Convection and forty arithmetic operations ale needed to perform the Mean Pressure Gradient integration.

In this subsection we position variables tr can be reduced to three or less then describe both the numerical solution of Eq. Equa- pair of particles mixing is tions 6. Then, Eq. For simplicity and without loss of generality we consider a single particle i. Then, with a mass 6. POPE iii the function h x Eq.

The number of samples N is chosen to The algorithm described previously provides a be Figure 6. In order to compare 6, Some scatter is evident, but the overall agreement theory with experiment, it is desirable to determine all is satisfactory: the r.

Joint pdf's of more than two variables r. Not only is the inaccuracy cannot readily be presented graphically. In the first fractional step, O, stability. The x-axis algorithm. Thus there are 4L polynomial coef- It is shown that the straight-forward method of ficients. At the L - 1 points between the L intervals, the approximating means by ensemble averages over cells is function and its first two derivatives are continuous.

But the method of least-squares cubic splines provides a more accurate alternative. O Q-- 0. A more sophisticated technique that overcomes these diffi- 0 culties is to use smoothing cubic splines, x2o. Figures 6. It is immedi- ately apparent that this approximation is inadequate: FIG. To estimate the pdffrom a finite number of samples is a standard problem for which many algorithms have This condition determines 3 L - 1 of the coefficients, been proposed, see for example Refs An efficient algorithm using All the methods mentioned must compromise B-splmes is described by de Boor.

But the 0. The approxi- - - estimated pdf is an average over the cell, so that if the mation to the second derivative shown on Fig. The r. Clearly then, least-squares cubic splines provide an efficient method of determining mean quantities. But the optimum Figure 6. POPE 3. Gaussian; from least-squares cubic splines with samples. Gaussian; from least-squares cubic splines with 10, samples.

It may be seen that The accuracy with which means can be determined the agreement with the Gaussian is better than for the from the particle properties depends upon the particle histograms, but it is still not very good.

In addition, the number density. The corresponding discrete representa- increased by a factor of 10 i. It may be tion of the joint pdfis seen that the agreement is now satisfactory.

From this it follows that for an axisymmetric flow, the If, in a given coordinate system, the joint pdfdoes not particle number density in the r-z plane is vary in one coordinate direction, then the flow is statistically two-dimensional.

But since for be modified to account for particles of different axisymmetric flows no mean property depends on 0, weights. Thus, the location of the particle is adequately becomes still simpler.

But a transformation of simple. U , V p are determined as functions the transport equations for statistical quantities of only two variables, and that in the stochastic mixing dependent upon a single spatial variable. Thus, for a step three-dimensional cells are replaced by two- self-similar plane jet, Pope 42'43 solved the joint pdf dimensional cells. The simplification-- metric flows, the cells are two-dimensional areas on the which is analogous to the boundary-layer approxi- r-z plane, the kth centered at rk,Zk having an area Ak.

Consider, for example, the statistically two-dimensional turbulent mixing layer Now with NkIt particles in the kth cell, each one formed between two parallel streams, Fig.

The joint representing a mass Am, the mean density is pdf equation is to be solved in the solution domain f l r k. A fluid particle Hence, the particle number density on the r-z plane is path in x-space is sketched on Fig. Sketch of a two-dimensional mixing layer showing a fluid particle path. First, the stream surface d. Consequently each stochastic particle the surfaces a, b and c, but not on the surface d.

An represents a fixed amount of axial momentum rather algorithm that exploits these observations is now out- than a fixed mass. Second, in each fractional step, the nth lined. Section 6. This alternative approach is described in this section. The that for. The first term on the. The stochastic mixing model Section 5. In This is the simplest possible consistent gradient- the velocity-composition joint pdf equation this diffusion model for momentum transport.

The analo- process appears in closed form. First, because of the use of mined from the solution to the modelled composition the isotropic viscosity hypothesis Eq. The 7. Janicka et al. Consequently it can 7. The main con- clusions are now summarized, and the attributes of the velocity-composition joint pdfapproach are discussed in Section 8.

Summary where the operator P2 corresponds to the stochastic At low Mach number, a turbulent flow of a single- mixing model. Any thermo- described in Sections 6. At achieved since the left-hand side of Eq. Any random 7.

Hence from Eqs 4. Consequently, the mass density function can be represented by N samples of such a random vector 7. N, Eq. Each In order for the diffusion process Eq. The expected particle number density with the right-hand side set to zero. Pdf methods for turbulent flows The joint pdf equation can also be derived by a stochastic mixing model and the stochastic reorienta- Lagrangian approach.

The Lagrangian conditional tion model. This is the stochastic reorientation model, pairs of particles are transition density for turbulent reactive flows: the randomly selected and are randomly reorientated in mass density function at time t is equal to the mass- velocity space Eq. Both energy and momentum probability-weighted integral of JL over all initial are conserved in this process. The effect of these two states, Eq.

But there is only one model. Mean-velocity gradients give rise to rapid- such deterministic system. In this system, the N pressure fluctuations. The interaction models is presented in Eq. These conditional diffusion process, Eq. For consistency with Kolmogorov's scaling paths. A model for the system of stochastic particles whose evolution is tensor Go has been proposed Eqs 5. For simply computed and in which the pdf evolves in the homogeneous turbulence, the resulting pdf equation same way as the pdf of fluid particles.

Poisson yields a joint normal distribution with the Reynolds- processes and diffusion processes described in stress evolution closely matching experimental data. Section 4. The aim of the modelling is there- equation for unsteady, three-dimensional, variable- fore to produce a joint normal distribution whose first density flows.

The velocities are modelled by the and second moments evolve correctly. Langevin equation, and the scalar dissipation by the For homogeneous turbulence, the joint pdf stochastic mixing model.

The stochastic mixing model is used to according to the stochastic mixing model. And in the produce this effect. The indirect Alternative models are presented for the effects of algorithm used is based on the observation that the the fluctuating pressure gradient and viscous dissi- stochastic particle number density in physical space pation. The first is based on stochastic particle-inter- should be proportional to the mean fluid density.

The action models, and the second on the Langevin mean pressure is determined--as the solution of an equation. Subtle connections geneous turbulence with no mean-velocity gradients, between the mean pressure, the mean continuity the decay of turbulence energy and the return to equation, and consistency conditions are discussed in isotropy of the Reynolds stresses is simulated by the Section 4.

POPE The algorithm provides a solution to the pdf equa- 8. Advantages of the pdf approach tion in terms of stochastic particle properties.

A con- Since the joint pdf provides a complete one-point ceptually simple way to obtain mean quantities is to statistical description of the flow field, the pdf divide physical space into cells, and then to form approach has advantages over other one-point ensemble averages over the particles in each cell.

In closures. The principal advantages are that reaction Section 6. For constant-density homogeneous flows, the For simpler flows--one and two-dimensional flows, models for the effects of the fluctuating pressure self-similar flows, and boundary-layer-type flows-- gradient and for molecular mixing are compatible variants of the solution algorithm are described with Reynolds-stress models and hence are equally in Section 6.

For inhomogeneous flows, the joint pdf The composition joint pdf approach is discussed in equation can be expected to be more accurate since Section 7. In this approach, the mean velocity and additional models are required in the Reynolds-stress turbulence fields are determined using a standard equations. Thus even for inert, constant-density flows. Model improvements for simple flows. A solution algorithm for the modelled composition transport equation is de- Several aspects of the modelling need to be refined, scribed in Section 7.

The treatment of scalar dissipation by the improved stochastic mixing is only partially satisfactory. Discussion Prandtl or Schmidt number effects are not accounted In the following four subsections we discuss: the for nor can the model be justified when reaction times inherent limitations of one-point closures; the are of the order of the Kolmogorov time scale or advantages of the pdf approach compared to other smaller.

But little is known about the nature and mag- 8. Limitations of one-point closures nitude of variable-density effects. In formation see Section 2. In modelling the joint pdf the past, modelling has been guided by theory and equation, the time scale z has to be specified either experimental data.

But it is now possible to solve the directly, or indirectly through a modelled transport governing flow equations for homogeneous turbu- equation for e. The direct specification of scale lence at Reynolds numbers comparable to those of information is only possible in simple flows, and the wind-tunnel experiments.

From the computed flow validity of the modelled dissipation equation has often fields, multi-point statistical information can be been called into question e. In addition, as extracted, and this provides an additional source of discussed in Section 5. In the joint pdf equation, the incapable of cor. While Lagrangian statistics are extremely diffi- These inherent problems do not invalidate the cult to obtain experimentally, it is relatively easy to approach.

For simple shear flows, for example, the use obtain them from direct numerical simulations. Computationalconsiderations simply proportional to r. Many Monte Carlo calculations from the modelling inadequacies have yet to be have been performed, both for the velocity-com- quantified, tSeveral multi-point pdf methods have position joint pdf 42 46 and for the composition joint been suggested.

Flame 27, Heat Transf Fluid Mech. Non- solution algorithm has been described Section 6 : the equilib. McNuTT, D. Thesis, Massachusetts Institute of prove practicable for more difficult cases. Technology Acknowledgments--For comments and suggestions on the GM, P.

Kollmann, P. A, Libby and W. Reynolds as well as to the PopE, S. I am grateful to D. Haworth for Doexzo, C. Fluids 18, Conference paper. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Pronchick and S. Google Scholar. Libby and F. Springer-Verlag, New York, Launder and D.

Spalding, Mathematical Models of Turbulence. Academic Press, CrossRef Google Scholar. Mellor and T. Launder, E. Reece, and H. Vol 18 ed C. Yih , Academic Press, New York, Kollman , The Combustion Institute, Pittsburg, Pennsylvania, , pp — Mongia and K. Mongia, R. DOI: Pope Published Mathematics Progress in Energy and Combustion Science Abstract The aim of the methods described is to calculate the properties of turbulent reactive flow fields.

At each point in the flow field, a complete statistical description of the state of the fluid is provided by the velocity-composition joint pdf. This is the joint probability density function pdf of the three components of velocity and of the composition variables species mass fractions and enthalpy.

The principal method described is to solve a modelled transport equation for the… Expand. View via Publisher. Save to Library Save.

Create Alert Alert. Share This Paper. Background Citations. Methods Citations. Results Citations. Citation Type. Has PDF. Publication Type. More Filters. The pdf approach to turbulent flow. Probability density function pdf methods provide a complete statistical description of turbulent flow fields at a single point or a finite number of points.

Turbulent convection and finite-rate … Expand. The scalar probability density function PDF transport method is being increasingly used for simulations of turbulent reacting flows. The key feature of this method is that turbulence-chemistry … Expand. SummaryA mathematical model for two-phase turbulent reactive flows is presented which is based on considering both phases in Lagrangian manner.

The mechanical and thermodynamical properties of the … Expand.