Exploring the beta distribution in variable-density turbulent mixing

Abstract

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 nonlinearities in the Navier–Stokes equation, (2) additional nonlinearities 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 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. 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 and 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.

Keywords:

• probability density function method
• variable-density turbulence
• transition to turbulence
• active scalar mixing
• beta distribution

Acknowledgements

J. Waltz is gratefully acknowledged for a series of informative discussions. This work was performed under the auspices of the US Department of Energy under the Advanced Simulation and Computing Program.