Tutorial EG501V Computational Fluid Dynamics (AY 2023/24)
Tutorial 1. Convection-diffusion equation
Convection and diffusion are two mechanisms for transporting a scalar (which can be thought of as a concentration of a chemical agent, or temperature). In this Tutorial we will consider the scalar to be a concentration c (unit kg/m3). Convective transport means that the scalar goes with the flow; diffusive transport is the process of evening out concentration gradients (transport
from areas with high to areas with low concentration). In mathematical terms:
Convective transport :
Diffusive transport:
The symbol stands for the scalar flux. It is the amount of scalar passing through a unit surface per unit time: kg/(m2·s); it is a vector since it has direction. The vector u is the velocity of the flow, the coefficient Γ is the diffusion coefficient (unit m2/s). In this example we assume that the scalar does not react, i.e. c does not change as a result of a chemical reaction, it only changes as a result of transport.
Your assignment
Based on a mass balance of the scalar over a small two-dimensional volume dx×dy×W
(similar to what was shown in LN01, Figure 1.2), derive that the concentration c satisfies the following equation
This is known as the convection-diffusion equation.
Take inspiration from the section Mass Balance that starts on page 2 of LN01.