# Interdisciplinary Applied Mathematics

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The third step is simulation of collisions via a probabilistic process. Since only a small portion of the molecules is simulated, and the motion and collision processes are decoupled, the probabilistic treatment becomes necessary. A common collision model is the no-time-counter (NTC) technique of Bird (1994), which is used in conjunction with the subcell technique, where the collision rates are calculated on the basis of the DSMC cell, and the collision pairs are selected within the subcells. This improves the accuracy of the method by maintaining the collisions of molecules near to each other (Oran et al., 1998).

The last step is the sampling of macroscopic flow properties, which is done within a cell, and the appropriate macroscopic values are presented at cell centers. The macroscopic properties for unsteady flow conditions are obtained by ensemble averaging of many independent calculations. For steady flows, time-averaging can also be used.

###### 15.1.1 Limitations and Errors in DSMC

While DSMC is a very powerful approach, and in fact the only possible numerical approach in simulating high Knudsen number rarefied flows in complex geometries, there are several possible limitations and error sources that the reader should be aware of:

1.    Finite    Cell    Size:    The    typical    DSMC    cell    should be    about    one-

third of the local mean free path. Values of cell sizes larger than this may result in erroneously enhanced diffusion coefficients. In DSMC one cannot directly specify the dynamic viscosity of the fluid. The dynamic viscosity is calculated via diffusion of linear momentum. (Breuer et al., 1995) have performed one-dimensional Rayleigh flow problems in the continuum flow regime. They discovered that for cell sizes larger than one mean free path the apparent viscosity of the fluid increases. Also, the dependence of viscosity and thermal conductivity on the cell size has been studied using the Green-Kubo theory (Alexander et al., 1998). Specifically, it was shown that the transport coefficients deviate from the Enskog values for dilute gas as the square of the cell size Ay. For example, for the viscosity coefficient it was found that

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