Stephen B. Pope

PDF methods for turbulent reactive flows

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 velocity-composition joint pdf. For a variable-density flow with arbitrarily complex and nonlinear reactions, it is remarkable that in this equation the effects of convection, reaction, body forces and the mean pressure gradient appear exactly and so do not have to be modelled. Even though the joint pdf is a function of many independent variables, its transport equation can be solved by a Monte Carlo method for the inhomogeneous flows of practical interest. A second method that is described briefly is to solve a modelled transport equation for the composition joint pdf. The objective of the paper is to provide a comprehensive and understandable of the theoretical foundations of the pdf approach.

Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space

A general procedure for simplifying chemical kinetics is developed, based on the dynamical systems approach. In contrast to conventional reduced mechanisms no information is required concerning which reactions are to be assumed to be in partial equilibrium nor which species are assumed to be in steady state. The only “inputs” to the procedure are the detailed kinetics mechanism and the number of degrees of freedom required in the simplified scheme. (Four degrees of freedom corresponds to a four-step mechanism, etc.) The state properties given by the simplified scheme are automatically determined as functions of the coordinates associated with the degrees of freedom. Results are presented for the CO/H2/air system. These show that the method provides accurate results even in regimes (e.g., at low temperatures) where conventional mechanisms fail.

Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation

A computational technique is described and demonstrated that can decrease by three orders of magnitude the computer time required to treat detailed chemistry in reactive flow calculations. The method is based on the in situ adaptive tabulation (ISAT) of the accessed region of the composition space - the adaptation being to control the tabulation errors. Test calculations are performed for non-premixed methane - air combustion in a statistically-homogeneous turbulent reactor, using a kinetic mechanism with 16 species and 41 reactions. The results show excellent control of the tabulation errors with respect to a specified error tolerance; and a speed-up factor of about 1000 is obtained compared to the direct approach of numerically integrating the reaction equations. In the context of PDF methods, the ISAT technique makes feasible the use of detailed kinetic mechanisms in calculations of turbulent combustion. The technique can also be used with reduced mechanisms, and in other approaches for calculating rea...

Ten questions concerning the large-eddy simulation of turbulent flows

In the past 30 years, there has been considerable progress in the development of large-eddy simulation (LES) for turbulent flows, which has been greatly facilitated by the substantial increase in computer power. In this paper, we raise some fundamental questions concerning the conceptual foundations of LES and about the methodologies and protocols used in its application. The 10 questions addressed are stated at the end of the introduction. Several of these questions highlight the importance of recognizing the dependence of LES calculations on the artificial parameter Δ (i.e. the filter width or, more generally, the turbulence resolution length scale). The principle that LES predictions of turbulence statistics should depend minimally on Δ provides an alternative justification for the dynamic procedure.

A more general effective-viscosity hypothesis

A discussion of the applicability of an effective-viscosity approach to turbulent flow suggests that there are flow situations where the approach is valid and yet present hypotheses fail. The general form of an effective-viscosity formulation is shown to be a finite tensor polynomial. For two-dimensional flows, the coefficients of this polynomial are evaluated from the modelled Reynolds-stress equations of Launder, Reece & Rodi (1975). The advantage of the proposed effective-viscosity formulation, equation (4.3), over isotropie-viscosity hypotheses is that the whole velocity-gradient tensor affects the predicted Reynolds stresses. Two notable consequences of this are that (i) the complete Reynolds-stress tensor is realistically modelled and (ii) the influence of streamline curvature on the Reynolds stresses is incorporated.

Direct numerical simulations of the turbulent mixing of a passive scalar

The evolution of scalar fields, of different initial integral length scales, in statistically stationary, homogeneous, isotropic turbulence is studied. The initial scalar fields conform, approximately, to ‘‘double‐delta function’’ probability density functions (pdf ’s). The initial scalar‐to‐velocity integral length‐scale ratio is found to influence the rate of the subsequent evolution of the scalar fields, in accord with experimental observations of Warhaft and Lumley [J. Fluid Mech. 88, 659 (1978)]. On the other hand, the pdf of the scalar is found to evolve in a similar fashion for all the scalar fields studied; and, as expected, it tends to a Gaussian. The pdf of the logarithm of the scalar‐dissipation rate reaches an approximately Gaussian self‐similar state. The scalar‐dissipation spectrum function also becomes self‐similar. The evolution of the conditional scalar‐dissipation rate is also studied. The consequences of these results for closure models for the scalar pdf equation are discussed.

An examination of forcing in direct numerical simulations of turbulence

Abstract A spectral forcing scheme is developed to provide the means to obtain statistically stationary velocity fields in direct numerical simulations of homogeneous, isotropic turbulence. Tests of the forcing scheme show that the details of the forcing do not have a significant effect on the small-scale structure of the velocity fields. Forced turbulent simulations are used to determine the effects of the time-step, and the spatial resolution of the grid, on the computations.

Filtered density function for large eddy simulation of turbulent reacting flows

A methodology termed the “filtered density function” (FDF) is developed and implemented for large eddy simulation (LES) of chemically reacting turbulent flows. In this methodology, the effects of the unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities. A transport equation is derived for the FDF in which the effect of chemical reactions appears in a closed form. The influences of scalar mixing and convection within the subgrid are modeled. The FDF transport equation is solved numerically via a Lagrangian Monte Carlo scheme in which the solutions of the equivalent stochastic differential equations (SDEs) are obtained. These solutions preserve the Ito-Gikhman nature of the SDEs. The consistency of the FDF approach, the convergence of its Monte Carlo solution and the performance of the closures employed in the FDF transport equation are assessed by comparisons with results obtained by direct numerical simulation ...

Computations of turbulent combustion: Progress and challenges

We review the significant progress that has been made in the development and use of turbulent combustion models applicable to practical combustion devices. Recent work has focused on the development of methods that can treat finite-rate kinetics in a realistic yet tractable way, so that local extinction and related phenomena can be studied. Direct numerical simulation cannot be used for this purpose, because it is computationally intractable; and the potential of large-eddy simulation is far from clear because combustion reactions give rise to a severe closure problem. PDF methods, on the other hand, overcome the major closure problems, and they have been shown to be tractable for complex flows and with realistic finite-rate kinetics. A simple explanation of pdf methods is presented. It is shown that the single modelled equation for the joint pdf of velocity, dissipation and composition provides a closure for turbulent combustion. Reaction and convection are treated exactly, while the modelling is performed in a Lagrangian setting, by constructing deterministic or stochastic models for the evolution of fluid-particle properties. Examples of recent pdf calculations are described, including those based on four-step mechanisms for methane. Extension of pdf methods to include composition gradients is discussed, with a view to improving the modelling of molecular diffusion.

Lagrangian statistics from direct numerical simulations of isotropic turbulence

A comprehensive study is reported of the Lagrangian statistics of velocity, acceleration, dissipation and related quantities, in isotropic turbulence. High-resolution direct numerical simulations are performed on 64 3 and 128 3 grids, resulting in Taylor-scale Reynolds numbers R λ in the range 38-93. The low-wavenumber modes of the velocity field are forced so that the turbulence is statistically stationary. Using an accurate numerical scheme, of order 4000 fluid particles are tracked through the computed flow field, and hence time series of Lagrangian velocity and velocity gradients are obtained. The results reported include: velocity and acceleration autocorrelations and spectra; probability density functions (p.d.f.s) and moments of Lagrangian velocity increments; and p.d.f.s, correlation functions and spectra of dissipation and other velocity-gradient invariants. It is found that the acceleration variance (normalized by the Kolmogorov scales) increases as R ½ λ - a much stronger dependence than predicted by the refined Kolmogorov hypotheses. At small time lags, the Lagrangian velocity increments are distinctly non-Gaussian with, for example, flatness factors in excess of 10. The enstrophy (vorticity squared) is found to be more intermittent than dissipation, having a standard-deviation-to-mean ratio of about 1.5 (compared to 1.0 for dissipation). The acceleration vector rotates on a timescale about twice the Kolmogorov scale, while the timescales of acceleration magnitude, dissipation and enstrophy appear to scale with the Lagrangian velocity timescale.