No abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.

In assumed probability density function (pdf) methods of turbulent combustion, the shape of the scalar pdf is assumed a priori and the pdf is parametrized by its moments for which model equations are solved. In non-premixed flows the beta distribution has been a convenient choice to represent the mixture fraction in binary mixtures or a progress variable in combustion. Here the beta-pdf approach is extended to variable-density mixing: mixing between materials that have very large density differences and thus the scalar fields are active. As a consequence, new mixing phenomena arise due to 1) cubic non-linearities in the Navier-Stokes equation, 2) additional non-linearities in the molecular diffusion terms and 3) the appearance of the specific volume as a dynamical variable.The assumed beta-pdf approach is extended to transported pdf methods by giving the associated stochastic differential equation (SDE). This enables the direct computation of the scalar pdf in a Monte-Carlo fashion. Using the moment equations, derived from the governing SDE, we derive constraints on the model coefficients of the SDE that provide consistency conditions for binary material mixing. The beta distribution is shown to be a realizable, consistent and sufficiently general representation of the marginal pdf of the fluid density, an active scalar, in non-premixed variabledensity turbulent mixing. The moment equations derived from mass conservation are compared to the moment equations derived from the governing SDE. This yields a series of relations between the non-stationary coefficients of the SDE and the mixing physics. All rigorous mathematical consequences of assuming a beta-pdf for the fluid mass density.Our treatment of this problem is general: the mixing is mathematically represented by the divergence of the velocity field which can only be specified once the problem is defined. A simple example of the wide range of physical problems is isobaric, isothermal, large-density binary material mixing. A more complex one is mixing and combustion of non-premixed reactants in which the divergence is related to the source terms in the energy and species conservation equations. In this paper we seek to describe a theoretical framework to subsequent applications. We report and document several rigorous mathematical results, necessary for forthcoming work that deals with the applications of the current results to model specification, computation and validation of binary mixing of inert fluids.

Dispersion of a passive scalar from concentrated sources in fully developed turbulent channel flow is studied with the probability density function ͑PDF͒ method. The joint PDF of velocity, turbulent frequency and scalar concentration is represented by a large number of Lagrangian particles. A stochastic near-wall PDF model combines the generalized Langevin model of Haworth and Pope ͓Phys. Fluids 29, 387 ͑1986͔͒ with Durbin's ͓J. Fluid Mech. 249, 465 ͑1993͔͒ method of elliptic relaxation to provide a mathematically exact treatment of convective and viscous transport with a nonlocal representation of the near-wall Reynolds stress anisotropy. The presence of walls is incorporated through the imposition of no-slip and impermeability conditions on particles without the use of damping or wall-functions. Information on the turbulent time scale is supplied by the gamma-distribution model of van Slooten et al. ͓Phys. Fluids 10, 246 ͑1998͔͒. Two different micromixing models are compared that incorporate the effect of small scale mixing on the transported scalar: the widely used interaction by exchange with the mean and the interaction by exchange with the conditional mean model. Single-point velocity and concentration statistics are compared to direct numerical simulation and experimental data at Re = 1080 based on the friction velocity and the channel half width. The joint model accurately reproduces a wide variety of conditional and unconditional statistics in both physical and composition space.

The joint probability density function (PDF) of turbulent velocity and concentration of a passive scalar in an urban street canyon is computed using a newly developed particle-in-cell Monte Carlo method. Compared to moment closures, the PDF methodology provides the full one-point one-time PDF of the underlying fields containing all higher moments and correlations. The small-scale mixing of the scalar released from a concentrated source at the street level is modelled by the interaction by exchange with the conditional mean (IECM) model, with a micro-mixing time scale designed for geometrically complex settings. The boundary layer along no-slip walls (building sides and tops) is fully resolved using an elliptic relaxation technique, which captures the high anisotropy and inhomogeneity of the Reynolds stress tensor in these regions. A less computationally intensive technique based on wall functions to represent the boundary layers and its effect on the solution are also explored. The calculated statistics are compared to experimental data and large-eddy simulation. The present work can be considered as the first example of computation of the full joint PDF of velocity and a transported passive scalar in an urban setting. The methodology proves successful in providing high level statistical information on the turbulence and pollutant concentration fields in complex urban scenarios.