Lagrangian particles with mixing. I. Simulating scalar transport

The physical similarity and mathematical equivalence of continuous diffusion and particle random walk forms one of the cornerstones of modern physics and the theory of stochastic processes. The randomly walking particles do not need to posses any properties other than location in physical space. However, particles used in many models dealing with simulating turbulent transport and turbulent combustion do posses a set of scalar properties and mixing between particle properties is performed to reflect the dissipative nature of the diffusion processes. We show that the continuous scalar transport and diffusion can be accurately specified by means of localized mixing between randomly walking Lagrangian particles with scalar properties and assess errors associated with this scheme. Particles with scalar properties and localized mixing represent an alternative formulation for the process, which is selected to represent the continuous diffusion. Simulating diffusion by Lagrangian particles with mixing involves three main competing requirements: minimizing stochastic uncertainty, minimizing bias introduced by numerical diffusion, and preserving independence of particles. These requirements are analyzed for two limited cases of mixing between two particles and mixing between a large number of particles. The problem of possible dependences between particles is most complicated. This problem is analyzed using a coupled chain of equations that has similarities with Bogolubov–Born–Green–Kirkwood–Yvon chain in statistical physics. Dependences between particles can be significant in close proximity of the particles resulting in a reduced rate of mixing. This work develops further ideas introduced in the previously published letter Phys. Fluids 19, 031702 2007 . Paper I of this work is followed by Paper II Phys. Fluids 19, 065102 2009 where modeling of turbulent reacting flows by Lagrangian particles with localized mixing is specifically considered. © 2009 American Institute of Physics. DOI: 10.1063/1.3147925