Stochastic discrete particle method

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

In recent years, new discrete-particle methods have been developed for modeling physical and chemical phenomena occurring in the mesoscale. The most popular are grid-type techniques such as cellular automata (CA), LG, LBG, diffusion- and reaction-limited aggregation [37] and stochastic gridless methods, e.g., DSMC used for modeling systems characterized by a large Knudsen number [41], and SRD [39]. Unlike DSMC, in SRD collisions are modeled by simultaneous stochastic rotation of the relative velocities of every particle in each cell. 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

Brownian Dynamics (BD) methods treat the short-term behavior of particles influenced by Brownian motion stochastically. The requirement must be met that time scales in these simulations are sufficiently long so that the random walk approximation is valid. Simultaneously, time steps must be sufficiently small such that external force fields can be considered constant (e.g., hydrodynamic forces and interfacial forces). Due to the inclusion of random elements, BD methods are not reversible as are the MD methods (i.e., a reverse trajectory will not, in general, be the same as the forward using BD methods). BD methods typically proceed by discretization and integration of the equation for motion in the Langevin form... 

Discrete models treat individual atoms, molecules, or particles and can be deterministic or stochastic. Examples of the former include MD simulations. Examples of the latter are various MC methods, BD, DPD, DSMC, and LB simulations. There are different ensembles in which these simulations can be performed, depending on the quantities that one is interested in computing. 

However, since the stochastic method deals with a discrete number of particles, large numbers of particles must be used to accurately represent low, but finite concentrations of intermediates. This is especially important for models such as the Oregonator model, where the behavior of the model depends heavily on the intermediate concentrations. For example, in the Oregonator model, a typical value for the critical Y concentration is 3 X 10 M, five orders of magnitude lower than the reactant and product concentrations. In order for this critical concentration to be modeled accurately, 10 particles must be followed. This means the critical concentration is represented by about 10-20 particles. This can cause the stochastic method to begin slowing down, but it is still possible to do calculations. 

Particle-based simulation techniques include atomistic MD and coarse-grained molecular dynamics (CG-MD). Accelerated dynamics methods, such as hyperdynamics and replica exchange molecular dynamics (REMD), are very promising for circumventing the timescale problem characteristic of atomistic simulations. Structure and dynamics at the mesoscale level can be described within the framework of coarse-grained particle-based models using such methods as stochastic dynamics (SD), dissipative particle dynamics (DPD), smoothed-particle hydrodynamics (SPH), lattice molecular dynamics (LMD), lattice Boltzmann method (IBM), multiparticle collision dynamics (MPCD), and event-driven molecular dynamics (EDMD), also referred to as collision-driven molecular dynamics or discrete molecular dynamics (DMD). 

In the DSMC method, collisions between the discrete phase particles are not detected by deterministic contact point calculations, but rather stochastically from the local number density, relative velocities, and sizes of neighboring particles. This method is applicable when the duration of a coUision is much shorter than the mean free time between two consecutive coUisions. We have shown how this method can be apphed to predict the spatial and temporal evolution of a spray of liquid droplets in a gas, using correlations for binary droplet interactions obtained from experiment and DNSs. 

In RANS-based simulations, the focus is on the average fluid flow as the complete spectrum of turbulent eddies is modeled and remains unresolved. When nevertheless the turbulent motion of the particles is of interest, this can only be estimated by invoking a stochastic tracking method mimicking the instantaneous turbulent velocity fluctuations. Various particle dispersion models are available, such as discrete random walk models (among which the eddy lifetime or eddy interaction model) and continuous random walk models usually based on the Langevin equation (see, e.g.. Decker and... 

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