fluid_mechanics

706 Appendix A ■ Computational Fluid Dynamics and FlowLab
H
h
D T
h 2
h
h 3
D
V
h i –1
h i
h i – h i –1
Δt
0 Δt 2Δt
i = 1 2 3 i – 1 i
t
(a)
(b)
h
H
0.8H
0.6H
Δt = 0.2
Exact: h = He -t
Δt = 0.1
0.4H
0.2H
0
0.0 0.2 0.4 0.6 0.8 1.0
(c)
F I G U R E EA.1
t
By combining Eqs. 2 and 3 we obtain
or
D 2 gh
32m/ aD T dh
D b2 dt
dh
dt Ch
where C gD 4 32m/D 2 T is a constant. For simplicity we assume
the conditions are such that C 1. Thus, we must solve
dh
h with h H at t 0
dt
The exact solution to Eq. 4 is obtained by separating the variables
and integrating to obtain
h He t
However, assume this solution was not known. The following finite
difference technique can be used to obtain an approximate
solution.
(4)
(5)
As shown in Fig. EA.1b, we select discrete points 1nodes or
grid points2 in time and approximate the time derivative of h by
the expression
dh
dt ` h i h i1
tt i
¢t
where ¢t is the time step between the different node points on the
time axis and h i and h i1 are the approximate values of h at nodes i
and i 1. Equation 6 is called the backward-difference approximation
to dhdt. We are free to select whatever value of ¢t that we wish.
1Although we do not need to space the nodes at equal distances, it is
often convenient to do so.2 Since the governing equation 1Eq. 42 is an
ordinary differential equation, the “grid” for the finite difference
method is a one-dimensional grid as shown in Fig. EA.1b rather than
a two-dimensional grid 1which occurs for partial differential equations2
as shown in Fig. EA.2b, or a three-dimensional grid.
Thus, for each value of i 2, 3, 4, . . . we can approximate the
governing equation, Eq. 4, as
h i h i1
¢t
h i
(6)