Air Pollution Modeling and Its Application II pp 575-589 | Cite as

# Numerical Solution of Three-Dimensional and Time-Dependent Advection-Diffusion Equations by “Orthogonal Collocation on Multiple Elements”

Chapter

## Abstract

To predict the dynamic behavior of trace contaminants in given three-dimensional and time-dependent flow fields, a transport equation must be solved of the general form where c is the mean concentration of a chemically inert contaminant, u,v, and w are the given components of the mean wind field, and K The x-direction is a so-called “marching”-direction, and significant savings of computing time are possible if (2) is solved instead of (1). Therefore, in this paper, we will consider the numerical solution of equation (2).

$$ \frac{\partial c}{\partial t}+u\frac{\partial c}{\partial x}+v\frac{\partial c}{\partial y}+w\frac{\partial c}{\partial z}=\frac{\partial }{\partial x}\left( {{K}_{x}}\frac{\partial c}{\partial x} \right)+\frac{\partial }{\partial y}\left( {{K}_{y}}\frac{\partial c}{\partial y} \right)+\frac{\partial }{\partial z}\left( {{K}_{z}}\frac{\partial c}{\partial z} \right) $$

(1)

_{x}, K_{y}and K_{z}are given eddy diffusivities. The concentration field may be three-demensional and time-varying due to the three-demensional and time-dependent flow field and, in general, the eddy-diffusivities are also functions of position and time. In a wind field not varying in time, the concentration field may still be time-dependent due to unsteady source emission. In many cases, the turbulent transport in the main wind direction, say x-direction, may be neglected in comparison to the advective transport in the same direction, and it is sufficient to solve the less general form$$ \frac{\partial c}{\partial t}+u\frac{\partial c}{\partial x}+v\frac{\partial c}{\partial y}+w\frac{\partial c}{\partial z}=\frac{\partial }{\partial y}\left( K\frac{\partial c}{\partial y} \right)+\frac{\partial }{\partial z}\left( {{K}_{z}}\frac{\partial c}{\partial z} \right) $$

(2)

## Keywords

Concentration Field Collocation Point Multiple Element Orthogonal Collocation Weighted Residual Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

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## Copyright information

© Plenum Press, New York 1983