Edward E. O'Brien

An approach to the autoignition of a turbulent mixture

Abstract This paper considers the turbulent homogeneous mixing of two reactants undergoing a one step, second order, irreversible, exothermic chemical reaction with a rate constant of the Arrhenius type. A statistically stationary turbulent velocity field is assumed given and unaffected by mass or heat production due to the chemical reaction. Relative density fluctuations are neglected. A Hopf-like functional formalism is presented, with application to both statistically inhomogeneous and statistically homogeneous flows. Single and double point probability density function differential equations are derived from those functional equations. The limit of very large activation energies is considered; a low degree of statistical correlation between temperature and concentration fields during the ignition period is hypothesized. After making use of the homogeneity assumption a closure problem is still present due to the nonlocalness of the molecular diffusion term. The problem is rendered closed by assuming a Gaussian conditional expected value for the temperature at a point given the temperature at a neighboring point. The closure is seen to preserve very important mathematical and physical properties. A linear first order hyperbolic differential equation with variable coefficients for the probability density function of the temperature field is obtained. A second Damkohler number based on Taylors microscale turns out to be an important controlling parameter. A numerical integration for different values of the second Damkohler number and the initial stochastic parameters is carried out. The mixture is seen to evolve towards an eventual thermal runaway, the detailed behavior however being different for different systems. Some peculiarities during the ignition period evolution are uncovered.

Functional formulation of nonisothermal turbulent reactive flows

The two familiar functional formalisms in turbulence are applied to the simultaneous turbulent mixing and chemical reaction of scalar fields. One‐step, second‐order, irreversible, exothermic chemical reactions with an Arrhenius‐type rate constant are considered. The problem is formally posed in terms of initial and boundary conditions. For the special case of equal mass‐diffusivities and a Lewis number of one the functional equations are decoupled into a turbulent binary mixing study and a reactive problem. The Lewis‐Kraichnan formalism is used in order to obtain exact functional solutions of the binary mixing case in final period turbulence. Driving forces are included in the thermal energy equation. These solutions are used to obtain detailed information about the binary mixing problem and the behavior of very rapidly reacting species in the final period.

Turbulent Mixing of Two Rapidly Reacting Chemical Species

The evolution of moments of the concentration of each of the species in a two‐species, very rapid, isothermal, irreversible, second‐order, chemical reaction in a homogeneous turbulence is described in terms of assumed initial distributions of the concentration fields. The fields decay in two stages. In the stage dominated by chemical kinetics, exact stochastic solutions are derived for a class of initial distributions. These solutions exhibit asymptotic concentration fields having an extremely high relative intensity and skewness associated with the spatial segregation of the species. In the second or diffusion controlled stage exact solutions are obtained in terms of the turbulent mixing of a nonreacting species when the molecular diffusivities of the species are equal. An approximate solution is proposed when they are unequal. In both cases the time scale of decay in the second stage is entirely characterized by turbulent mixing parameters. It is shown that in final period turbulence the reactants decay...

Turbulent shear flow mixing and rapid chemical reactions: an analogy

Two-species, irreversible, very rapid reactions, with mild heat release, in a turbulent shear flow are shown to be analogous to the transport of two non-reacting species by the same shear field. Expressions for the probability density functions of the reacting species, the product species and the reaction-generated thermal field are obtained in terms of the joint probability density functions of the two nonreacting species. As an example we have constructed, from recent measurements of temperature statistics at a cross-section in a heated jet, the meanand fluctuating concentration fields of the reacting species and the mean concentration of the product.

Isochoric turbulent mixing of two rapidly reacting chemical species with chemical heat release

The temperature and concentration fields produced by turbulent mixing of two reactants undergoing a one‐step, very rapid, irreversible, exothermic reaction are studied. Temperature and concentration fields decay in two stages: kinetically driven and diffusion controlled. Stochastic solutions are derived for the first stage, the asymptotic state of which displays all the characteristics of species segregration. In particlar, only a weak correlation between the concentration and temperature fields is generated in the first stage. For the special case of equal mass diffusivities and a Lewis number of one the solution to the second stage is obtained in terms of the solution of a nonreacting binary mixing problem. An approximate solution for Lewis numbers different from one is proposed. The temperature field seems to decay with an effective thermal diffusivity determined by the smaller of the molecular mass and thermal diffusivities.

Closure for Stochastically Distributed Second‐Order Reactants

A closure at the third‐order moment is presented for the problem of the decay of reactants which obey a second‐order equation, and whose initial description is given stochastically. The closure satisfies prescribed realizability conditions for all possible initial assignments of the mean, the mean square fluctuations, and third‐order moments of the concentration field. The closure is applied in two typical examples, and in each case the results agree satisfactorily with exact stochastic solutions.

On the flux of heat through laminar wavy liquid layers

Dynamically passive transfer of heat across a layer of liquid supporting a progressive, periodic surface wave is approached from a Lagrangian viewpoint. The model layer considered is the region between two constant pressure surfaces of a Gerstner wave and the thermal boundary conditions are that the average temperature of any surface particle remains constant and that there is horizontal homogeneity of the average temperature field. It is shown that fluctuations in the temperature of any particle are negligibly small for ordinary liquids and a uniformly valid approximation to the average temperature of each particle is presented. The extent to which the flux of heat through the layer is augmented is computed for typical cases and it is shown to be at most doubled. Indication is given of extensions of the method to other kinds of progressive waves and to situations in which the boundary conditions are unsteady and spatially inhomogeneous.

Postulate of Statistical Independence for Decaying Reactants in Homogeneous Turbulence

An independence hypothesis is proposed which predicts the evolution of the amplitude of Fourier modes of the concentration fluctuation field for a substance undergoing a second‐order single‐species isothermal reaction in homogeneous turbulence. The hypothesis is that the time history of such an amplitude is a product of its time history due to reaction and its time history due to turbulent mixing and diffusion. A sufficient condition for the hypothesis to be applicable is that the spectral decay due to reaction alone exhibits wavenumber similarity with a constant length scale. When the initial fluctuation intensity is high enough, the initial decay of the mean concentration (mean‐square concentration) as predicted by the hypothesis gives a lower (upper) bound on the actual initial rate of decay. It is also demonstrated by numerical calculation of a particular initial isotropic concentration spectrum in the absence of turbulence that the independence hypothesis and the predictions of a valid closure for th...

A parallelized Eno procedure for direct numerical simulation of compressible turbulence

In this paper, a finite-difference based ENO (essentially nonoscillatory) procedure has been chosen for the direct numerical simulation (DNS) of compressible turbulence. The implementation of the ENO scheme follows the relatively efficient procedure in Shuet al. (1992), but the latter has been modified in the present paper to admit scalar conservation equations and to run on the iPSC/860 Paragon parallel supercomputer. DNS results with our procedure are in excellent agreement with pseudo-spectral and Padé approximation calculations in two and three dimensions. This is the case for a variety of initial conditions for compressible turbulence. The parallel algorithms presented are simple but quite efficient for DNS, with a speedup that approaches the theoretical value. Some of the attractive features include 1) minimum communication whereby a processor only communicates with two neighbors, 2) almost one hundred percent load balancing, 3) a checker-board approach to solve the Poisson equation reduces communication by a factor of approximately 2, and, 4) obtaining turbulence statistics is based on a ‘global collect’ approach, which is implemented to ensure that a single number, rather than a large matrix of numbers, is communicated between processors. The ENO code presented in this paper should be quite useful in its own right, while the parallel implementation should allow the simulation of fairly realistic problems.

Turbulent Diffusion of Rapidly Reacting Chemical Species

Publisher Summary This chapter explains statistical mechanics of reactants in turbulence, which can be seen as a prospect for a useful application of the results to atmospheric transport. There are several special features of concentration as a random variable that must be taken into account while describing the turbulent diffusion of reacting species. A procedure developed by Orszag for turbulence dynamics, and subsequently shown to be inadequate for representing phase mixing, appears to be better suited for describing reactive decay. An alternative approach to the advection of reacting scalar fields in turbulence is to avoid moment formulations altogether and to study functional differential equations of the type already developed for the velocity field. For very rapidly reacting species, the reaction takes place in thin reaction zones separating species A from species B, and understanding the nature of these zones in turbulence is vital to understanding the role of turbulence on very fast reactions.