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

• Hans Wengle
Part of the NATO · Challenges of Modern Society book series (NATS, volume 3)

## 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
$$\frac{\partial c}{\partial t}+u\frac{\partial c}{\partial x}+v\frac{\partial c}{\partial y}+w\frac{\partial c}{\partial z}=\frac{\partial }{\partial x}\left( {{K}_{x}}\frac{\partial c}{\partial x} \right)+\frac{\partial }{\partial y}\left( {{K}_{y}}\frac{\partial c}{\partial y} \right)+\frac{\partial }{\partial z}\left( {{K}_{z}}\frac{\partial c}{\partial z} \right)$$
(1)
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 Kx, Ky and Kz 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)
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).

## 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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