Page 111 - Computational Fluid Dynamics for Engineers

4.2 Model Equations 97

This equation is still parabolic in time in the (x, t) space and is called the
convection-diffusion equation due to the simultaneous presence of convection
by the velocity u and diffusion through the diffusivity coefficient a.
If the effect of the pressure gradient on the momentum equation is neglected,
Eq. (4.2.2) becomes
2
du du d u , * „ ^
+u
- -
m e- =^ (4 2 7)
x
which has the structure of a convection-diffusion equation but is nonlinear. It
is known as Burger's equation, and it contains the full nonlinearity of the one-
dimensional flow equations. For an inviscid flow, it becomes
du du ,, ^ ^
+u
m d- =° (4 2 8)
- -
x
and is known as the "inviscid" Burger's equation. Unlike Eq. (4.2.7), which is
parabolic, this equation is hyperbolic. It is also nonlinear, as is Eq. (4.2.7).
The discussion on constructing model equations for simplified one-dimensional
flows can also be extended to simplified two-dimensional flows. For this purpose,
for two-dimensional inviscid flows, the Laplace equation given by Eq. (2.4.20)
2
2
d 6 + =0 (429)
d 6
n£ 4 -
which is a typical elliptic equation, describes an isotropic diffusion in the (x,y)
space.
A similar equation results from Eq. (2.2.11), which is a parabolic in time in
the (x,y, £) space. Assuming that the temperature field is in a medium at rest,
then in the presence of the source term q^, Eq. (2.2.11) reduces to the Poisson
equation
2
2
d T d T 1 . /. o ^ x
' -
7^ + W = -k qh (4 2 10a)
An equation similar to this is obtained from Eq. (2.2.9) for a fully developed
laminar channel flow in a rectangular duct. Noting that Du/Dt is zero, and
neglecting the body force,
2
2
d u d u I dp
dx 2 dy 2 ji dx
If the temperature in Eq. (4.2.10a) is time dependent, and the medium is
again at rest, then the unsteady heat conduction equation in two dimensions
and with source term becomes

This equation is parabolic in time in the (x,?/,£) space. Its solutions at large
time values should approach the solutions of Eq. (4.2.10a).