Closure of the Transport Equation for the Probability Density Funcfion of Turbulent Scalar Fields

The transport equation for the probability density function (pdf)P of a scalar variable in a turbulent field is derived and various closure approximations for the turbulent convective and the molecular transport term are discussed. For the special case of a turbulent diffusion flame, for which the density, temperature and composition can be considered as a function of a scalar variable f, the transport equation for Pf (z) is closed using the conditional mean of the velocity for the turbulent convective term and an integral model for the molecular transport term. Preliminary results are presented and compared with a two-parameter form of the pdf Pf (z). Introduction Probability density functions (pdf) play an increasingly significant role in the theoretical treatment of turbulent flows. Lundgren [ 1 ] devised a method to derive a hierarchy of transport equations for N-point probability density fuctions (pdfs) for fluctuating variables in a turbulent incompressible flow. Fox [12] and Dopazo and O'Brien [5], [7] extended this method to compressible and reacting flows. Dopazo and O'Brien suggested closure assumptions for the onepoint pdf equation based on a quasi-Gaussian property of conditional moments, which was successful for nearly Gaussian problems [6]. The Lundgren approach has been applied to turbulent flames by Pope [4]. This author constructed the transport equation for the pdf of a scalar quantity f describing the instantaneous composition of a turbulent diffusion flame under certain simplifying assumptions. Beside, he suggested an interesting closure-procedure for the unknown term arising from molecular transport that will be discussed below. Paper presented at the Colloquium on "Probability Distribution Function-Methods for Turbulent Flows" in Aachen, Germany, August, 29-30, 1977. 0340-0204/79/0004-0047S02.00 © Copyright by Walter de Gruyter & Co. · Berlin · New York . 48 J. Janicka, W. Kolbe, W. Kollmann In this paper the transport equation for the pdf of a scalar quantity f in a turbulent field is considered. Closure assumptions for two different unknown terms occurring in this equation are discussed in the special case of a turbulent diffusion flame, the composition, density and temperature of which can be related to the scalar quantity f, [9]. For each term a new closure approximation is suggested. The physical significance of all approximations involved is investigated. Results of test calculations are presented for both the homogeneous case and the turbulent diffusion flame. 1 . Transport equation for the probability density of a scalar For turbulent flows, a hierarchy of transport equations for a probability density function (pdf) f(x, t) at a finite number of points in x-space can be derived using a method devised by Lundgren [1 ]. The structure of this hierarchy is similar to the well-known BBGKY-hierarchy in statistical mechanics. The closure approximations for the BBGKY-hierarchy, such as the Kirkwood closure for simple liquids [2], do not work satisfactorily for the turbulent case because there essential conditions are not fulfilled (for details cp. [3]). Therefore, a different approach towards constructing closure approximations is suggested. It is based mainly on physical assumptions about the turbulent transport mechanism. Consider a normalized scalar function f(x, t) of space coordinates ÷ = (÷á, α = 1 , 2, 3) and time t, that satifies the following equation Here ñ is the density which is a function of f(x, t), Ã is the molecular transport coefficient and S0 is the source term. The velocity í = (íá , α. = 1,2,3) satisfies the Navier-Stokes equations 9va dva 8p 8f<y/3 o? -·"'̂ — d£ + -3^· «=1.2,3, (2) where 9Vo, 3V 2 OVe + ' (3) For a given point (x, t) within the domain of the turbulent flow we define an instantaneous pdf as 3 P(za,z)^6(f(x,t)-z) Ð ä(íá(÷,ß)-æá), (4) á=1 2 In case of a turbulent diffusion flame, the function f due to Spalding [13], can be interpreted as mixture fraction of different components. J. Non-Equilib. Thermodyp. , Vol. 4, 1979, No. 1 Closure of stochastic transport equations 49 where ä(·) is the Dirac pseudofunction. Ñ can be interpreted as pdf of the realization (f, va) of the flow field at (x, t). Therefore it depends on f(x, t) and va (x, t). Averaging over all realizations of the turbulent flow yields the pdf P*(za, z, x, t) = <P(za, z, f, íá)> , (5) which is the simultaneous one-point pdf of the conserved scalar quantity f and the velocity va at (x, t). To derive a transport equation for P* we only need elementary properties of the averaging process (and avoid delicate mathematical questions connected with the probability measure of a turbulent flow field). Differentiation of eq. (5) yields (6) · at at az at aza Inserting (1) and (2) into (6) leads to