Stochastic models

Statistical descriptions introduced in Chapter 1 lead to well-defined averages (and higher-order moments) and their transport equations. Each transport equation contains moments of one higher order (e.g., the transport equation of the average contains second moments), characteristic of the closure problem in single-phase turbulence. Stochastic models are well suited to representing fluctuations about averages because they imply a probability distribution for the modeled variable, and thus imply a closure model for all moments. Stochastic models for describing the evolution of Lagrangian particle properties lead naturally to stochastic differential equations (SDEs). This chapter focuses on the application of SDEs in Euler–Lagrange frameworks. Motivated by the problem of inertial particle dispersion from a point source, the Langevin equation is introduced as the prototypical SDE to model inertial particle dispersion in turbulence. The transport equation for the probability density function is established as the basis for correspondence between the stochastic model and its physical counterpart. Subsequently, stochastic Lagrangian models for inertial particles interacting with intrinsic and pseudo-turbulence are summarized. Numerical schemes for integrating SDEs are described. Finally, opportunities for extending stochastic models to capture preferential concentration and clustering phenomena are briefly outlined.