Lagrangian particle model

Hort, M., Devenish, B., and Thomson, D. (2002) Building affected dispersion development and initial performance of a new Lagrangian particle model. In Proceedings of 15th AMS Symposium on Boundary Layers and Turbulence. 

In stochastic Lagrangian particle models, the evolution of the concentration field is computed in a two-step process. First, the Eulerian velocity field in the region of interest must be calculated, either by solution of the Navier-Stokes equations or via an approximate method that satisfies mass consistency. The solution must also provide the local statistics of the velocity field. Individual particles are then released, and their position is updated over a time increment dt using an equation of the form (Wilson and Sawford 1996)... 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

Lagrangian mixing models Particle-field estimation... 

Perhaps the simplest Lagrangian micromixing model is the interaction by exchange with the mean (IEM) model for a CSTR. In addition to the residence time r, the IEM model introduces a second parameter tm to describe the micromixing time. Mathematically, the IEM model can be written in Lagrangian form by introducing the age a of a fluid particle, i.e., the amount of time the fluid particle has spent in the CSTR since it entered through a feed stream. For a non-premixed CSTR with two feed streams,100 the species concentrations in a fluid particle can be written as a function of its age as... 

We will also discuss Lagrangian PDF models for the composition PDF. In this case, the notional particles follow... 

For a Lagrangian PDF model, (6.162) should hold when /u, , x y is substituted for. /1j1,1,x11y- Thus, we will now look at what conditions are necessary to ensure that (6.162) holds for the notional-particle PDF. 

Note that (6.190) contains a number of conditional expected values that must be evaluated from the particle fields. The Lagrangian VCIEM model follows from (6.86), and has the same form as the LIEM model, but with the velocity, location-conditioned scalar mean (0 U, X )(U. X. t) in place of location-conditioned scalar mean ( X )(X. t) in the final term on the right-hand side of (6.190). 

In order to simulate the SDEs, we will introduce a large ensemble of notional particles that move through the simulation domain according to the Lagrangian PDF models. As an example, we will consider a single inert-scalar field in a one-dimensional domain. The position and composition of the th notional particle can be denoted by X n t) and 4i(n)(f), respectively. The SDEs for the Lagrangian composition PDF (with closures) become... 

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

Fig. 14. Computed motion of a collection of pseudo-Lagrangian particles due to the evolution and propagation of gas bubbles in a section of a hypothetical two-dimensional gas fluidized bed [From Pritchett, J.W., Blake, T.R., and Garg, S.K. A numerical model of gas fluidized beds. AIChE Symp. Sen 176, 74, 134 (1978). Reproduced with permission of the American Institute of Chemical Engineers. Copyright 1978 AIChE. All rights reserved.]...

At this point, the variables C and u are now used to connote the average (macroscopic) tracer concentration and velocity, respectively, and the overbar is dropped for convenience. Implicit in this formulation is the belief that fundamentally Lagrangian (particle) dispersion can be modeled as a continuum Eulerian phenomenon in a fashion analogous to the Fickian formulation of molecular transport by Brownian motion. This is a useful fiction for simple modeling exercises, but must be used with caution (see the next section on isopycnal diffusion). 

Multiphase flows Eulerian-Eulerian (EE) capabilities closure/drag laws/additional forces EE-granular flows model options/Eulerian-Lagrangian (EL) true/psuedo particle model s/UD ... 

Eulerian grid box or Lagrangian particle tracking methods using turbulence models (not suitable for real time emergency response computation), above canopy. Input from obstacle or continuum scale data (for local sources). Upwind data for distant sources. 

Seibert, P. (2001) Inverse modelling with a Lagrangian particle dispersion model application to point releases over limited time intervals. ImGryning and Schiermeier (eds.) Air pollution modeling and its application XIV, Plenum, NY, 381-389. 

The Department of Energy (DOE) has sponsored the development of two Lagrangian-particle-based dispersion models ... 

ADAPT-LODI, developed at Lawrence Livermore National Laboratory. The ADAPT model assimilates meteorological data provided by observations and models (in particular, by Coupled Ocean/Atmosphere Mesoscale Prediction System [COAMPS ]) to construct the wind and turbulence fields. Particle positions are updated using a Lagrangian particle approach that uses a skewed (non-Gaussian) probability density function (Nasstrom et al. 1999 Ermak and Nasstrom 2000). 

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