Page 76 - Introduction to Computational Fluid Dynamics

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3 1D Conduction–Convection May 25, 2005 10:54

3.1 Introduction

Consider a 1D domain (0 ≤ x ≤ L) through which a fluid with a velocity u is
flowing. Then, the steady-state form of the first law of thermodynamics can be
stated as

∂q x
= S, (3.1)
∂x
where
∂T
q x = q conv + q cond = ρ C p uT − k . (3.2)
x
x
∂x
These equations are to be solved for two boundary conditions, T = T 0 at x = 0
and T = T L at x = L. It is further assumed that ρ u is a constant as are properties
C p and k.
Our interest in this chapter is to examine certain discretisational aspects as-
sociated with Equation 3.1. This is because in computational fluid dynamics
(momentum transfer) and in convective heat and mass transfer, we shall recur-
ringly encounter representation of the total flux in the manner of Equation 3.2.
Note that if u = 0, only conduction is present and the discretisations carried
out in Chapter 2 readily apply. However, difficulty is encountered when con-
vective flux is present. The objective here is to understand the difficulty and
to learn about commonly adopted measures to overcome it. In the last section
of this chapter, stability and convergence aspects of explicit and implicit proce-
dures for an unsteady equation in the presence of conduction and convection are
considered.

3.2 Exact Solution

Because our interest lies in examining the discretisational aspects associated with
convective–conductive flux, we take the special case of S = 0. For this case, an

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