We provide a brief overview of some Monte Carlo methods which have been used to simulate systems with a turbulent fluid component. We discuss two main classes of simulation approaches: an "Eulerian" class which is based on a random velocity field model defined on a fixed coordinate system, and a "Lagrangian" class in which the motion of fluid particles and immersed particles are instead stochastically modeled. The main aim of this article is to expose these novel simulation approaches to turbulent and complex fluid systems to a broader readership familiar with stochastic processes and to provide some pointers to the literature.
IntroductionA continuing challenge is the numerical simulation of turbulent systems, in which the fluid is forced sufficiently vigorously so as to generate a disordered and unpredictable structure over a wide range of scales [47,74]. The key difficulty is the inability for even large computers to resolve all of the active scales in systems with strong turbulence in a direct numerical simulation based on the fundamental Navier-Stokes equations [43,54]. A numerical representation of turbulence, however, is crucial in computer simulation studies in many applied fields, ranging from atmosphere-ocean dynamics (including weather and climate prediction) [11,43,47], combustion [7,33,45,62,80,85], turbulent diffusion [48], mixing processes and numerous other engineering situations [36,54]. These applications do not require a turbulence simulation with full fidelity -only certain key features of the turbulence need to be represented. On the other hand, models which account for turbulence only through an enhanced diffusivity coefficient are often insufficient [43,54,77]. This is particularly the case in reaction and mixing processes where the fine-scale fluctuations in the immersed chemical species play an important role, and these are completely ignored in a crude eddy diffusion model and not well represented in large eddy simulations [33,45].Monte Carlo approaches are an attractive option for turbulence simulations due both to their capacity for investigating systems with many degrees of freedom and to their natural generation of a disordered velocity field structure and irregular particle trajectories. Indeed, it is difficult to conceive of a deterministic mechanism for generating a velocity field with disordered fluctuations over a wide range of scales, other than by an expensive direct numerical simulation which resolves all those scales! Most recent methods which have been used to simulate turbulent systems can be classified into two categories. In the first, which we call the "Eulerian" fluid simulation approach, a velocity field is generated over a prescribed spatial domain, but by a direct stochastic construction 1