the velocity-dissipation joint-pdf [5,6]. Solutions of these equations have been successfully obtained using Monte A particle method which applies the probability density function (PDF) method to compressible turbulent flows is presented. Solution Carlo algorithms in which the fluid is represented by a large of the PDF equation is achieved using a Lagrangian/Monte Carlo set of stochastic particles having properties that evolve in approach. A unique feature of the method is its ability to calculate time. The majority of applications, however, are limited the mean pressure gradient directly from the particles using a gridto incompressible flows, and only recently has work been free approach. This is accomplished by applying techniques bordone in extending and applying the method to compressrowed from the field of smoothed particle hydrodynamics. Furtherible flows having pressure induced density variations more, these techniques have been implemented using a recently discovered algorithm which greatly reduces the computational work [7, 8]. Related to this is the need to develop a general in 1D. The particle method also incorporates a variance-reduction method for calculating the mean pressure field within the technique which can significantly reduce statistical error in first and Monte Carlo algorithm. A few approaches exist, but these second moments of selected mean flow quantities. When combined are limited in scope: for thin shear flows the mean pressure with a second-order accurate predictor/corrector scheme, the rehas been calculated by invoking boundary-layer approxisulting particle method provides a feasible way to obtain accurate mations [9], and for statistically stationary incompressible PDF solutions to compressible turbulent flow problems. Results have been obtained for a variety of quasi-1D flows to demonstrate flows the pressure has been obtained from a Poisson equathe method's robustness. These include solutions to both statistition [10]. Recently, a pressure-correction algorithm has cally stationary and nonstationary problems, and use both periodic been devised and implemented which is applicable to either and characteristic-based inflow/outflow boundary conditions.
constant density flows or statistically stationary variable
Convergence of the method with respect to four different kinds of density flows [11]. A different approach is to couple the numerical errors has also been studied. Detailed results are pre-Monte Carlo code to a finite-volume code which calculates sented which confirm the expected convergence behavior of each error. ᮊ 1997 Academic Press the mean pressure [8,[12][13][14]. While this approach can combine advantages of both methods, the coupling becomes complex for compressible reacting flows and this