Lagrangian grid

Figure 9.25. Lagrangian calculation of projectile impact—grid in Fig. 9.24 after rezoning.

With this approach, when an element becomes severly distorted, it is eliminated from the computational grid and becomes a free mass point. Clearly, care must be taken to avoid eliminating elements that could potentially influence the problem at some later time. An example of a three-dimensional Lagrangian calculation that uses the eroding element scheme is presented in the next section. 

M.J. Frits, Two-Dimensional Lagrangian Fluid Dynamics Using Triangular Grids, in Finite-Difference Techniques for Vectorized Fluid Dynamics Calculations (edited by D.L. Book), Springer-Verlag, New York, 1981. 

D.E. Burton, Conservation of Energy, Momentum, and Angular Momentum in Lagrangian Staggered-Grid Hydrodynamics, UCRL-JC-105926, Lawrence Livermore National Laboratory, Livermore, CA, 1990. 

Another Lagrangian-based description of micromixing is provided by multienvironment models. In these models, the well macromixed reactor is broken up into sub-grid-scale environments with uniform concentrations. A four-environment model is shown in Fig. 5.16. In this model, environment 1 contains unmixed fluid from feed stream 1 environments 2 and 3 contain partially mixed fluid and environment 4 contains unmixed fluid from feed stream 2. The user must specify the relative volume of each environment (possibly as a function of age), and the exchange rates between environments. While some qualitative arguments have been put forward to fit these parameters based on fluid dynamics and/or flow visualization, one has little confidence in the general applicability of these rules when applied to scale up or scale down, or to complex reactor geometries. 

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. 

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