Page 110 - Computational Fluid Dynamics for Engineers

96 4. Numerical Methods for Model Parabolic and Elliptic Equations

The numerical solution of the parabolic and elliptic equations with finite-
difference methods are discussed in Sects. 4.4 and 4.5, respectively, following
a brief description of the discretization of derivatives with finite-differences in
Section 4.3. The methods in Section 4.4 and Chapter 5 can be either implicit and
explicit in contrast to the numerical methods for elliptic equations in Section
4.5 which are always implicit.

4.2 Model Equations

A general idea about the model equations used in computational fluid dynamics
can be obtained by considering the simplified form of the conservation equations
for one- and two-dimensional flows. For example, for one-dimensional flows
neglecting the body force in Eq. (2.2.18) and heat transfer term q^ in Eq.
(2.2.15b), Eqs. (2.2.12b), (2.2.18) and (2.2.15b) can be written as

t + ! ( « 0 = 0 (1.2.1)

9 , N 5 / 9 x da xx 4 d ( du\ , A rt .

de de\ du d (,dT\ du /AnnS
k
p
oi + u o- )= ~ o~ x + o~ { ^) + a **o- x (4 2 3)
- -
x
x
If the terms containing the velocity u in Eq. (4.2.3) are neglected by assuming
that either the fluid is at rest or the flow velocity is small, then the energy
equation for a constant thermal conductivity k and specific heat C v can be
written as
a
- m = o ^ (42 4)
-
This equation is known as the one-dimensional unsteady heat conduction equa-
tion and is a typical parabolic equation in time in the (x, t) space.
If the velocity u in the continuity equation (4.2.1) is assumed constant, then
this equation takes the typical form of a linear convection equation

describing the transport of mass by a flow velocity of u. It is a first-order hy-
perbolic equation.
If the velocity is not negligible but the flow is incompressible, it follows from
the continuity equation that u is constant. The energy equation (4.2.3) becomes

2
dT dT d T /AC^ n,
a
+u
-m o^- o^ (4 2 6)
- '