Exploring the beta distribution in variable-density turbulent mixing

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). 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 variable-density 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. 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. 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.Comment: Added two paragraphs to Introduction + minor changes, Accepted in Journal of Turbulence, July 19, 201

Extending the Langevin model to variable-density pressure-gradient-driven turbulence

We extend the generalized Langevin model, originally developed for the Lagrangian fluid particle velocity in constant-density shear-driven turbulence, to variable-density (VD) pressure-gradient-driven flows. VD effects due to non-uniform mass concentrations (e.g. mixing of different species) are considered. In the extended model large density fluctuations leading to large differential fluid accelerations are accounted for. This is an essential ingredient to represent the strong coupling between the density and velocity fields in VD hydrodynamics driven by active scalar mixing. The small scale anisotropy, a fundamentally "non-Kolmogorovian" feature of pressure-gradient-driven flows, is captured by a tensorial stochastic diffusion term. The extension is so constructed that it reduces to the original Langevin model in the limit of constant density. We show that coupling a Lagrangian mass-density particle model to the proposed extended velocity equation results in a statistical representation of VD turbulence that has important benefits. Namely, the effects of the mass flux and the specific volume, both essential in the prediction of VD flows, are retained in closed form and require no explicit closure assumptions. The paper seeks to describe a theoretical framework necessary for subsequent applications. We derive the rigorous mathematical consequences of assuming a particular functional form of the stochastic momentum equation coupled to the stochastic density field in VD flows. A previous article discussed VD mixing and developed a stochastic Lagrangian model equation for the mass-density. Second in the series, this article develops the momentum equation for VD hydrodynamics. A third, forthcoming paper will combine these ideas on mixing and hydrodynamics into a comprehensive framework: it will specify a model for the coupled problem and validate it by computing a Rayleigh-Taylor flow.Comment: Accepted in Journal of Turbulence, Jan 7, 201

Joint PDF modelling of turbulent flow and dispersion in an urban street canyon

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 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.Comment: Accepted in Boundary-Layer Meteorology, Feb. 19, 200

A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation, modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (e.g. the mean pressure) and to track particles. All three aspects regarding the grid make use of the finite element method (FEM) employing the simplest linear FEM shape functions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean model is adopted. An adaptive algorithm that computes the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.Comment: Accepted in Journal of Computational Physics, Feb. 20, 200