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

Abstract

We extend the generalized Langevin model (D.C. Haworth and S.B. Pope, A generalized Langevin model for turbulent flows, Phys. Fluids 29 (1986) p. 387), 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 nonuniform 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. Our aim is to develop a joint model for variable-density pressure-gradient-driven turbulence and mixing, such as occurs due to the Rayleigh–Taylor instability. A previous paper (J. Bakosi and J.R. Ristorcelli, Exploring the beta distribution in variable-density turbulent mixing, J. Turbul. 11 (2010), p. 1) discussed VD mixing and developed a stochastic Lagrangian model equation for the mass-density. Second in the series, this paper 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 joint model for the coupled problem and validate it by numerically computing joint statistics of a Rayleigh–Taylor flow at several Atwood numbers.

Keywords:

• probability density function method
• Langevin equation
• variable-density turbulence
• pressure-gradient-driven flows
• small-scale anisotropy

Acknowledgements

J. Waltz and J. D. Schwarzkopf are gratefully acknowledged for a series of informative discussions. This work was performed under the auspices of the U.S. Department of Energy under the Advanced Simulation and Computing Program.