fluid_mechanics

708 Appendix A ■ Computational Fluid Dynamics and FlowLab
y
U
a
+
r
θ
j
Δy
(a)
+
Δ x
i
x
(b)
ψ
i, j + 1
Δy
ψ
i – 1, j
ψi, j
ψ
i + 1, j
Δ x
Δ x
Δy
ψ
i, j – 1
(c)
F I G U R E EA.2
storage may be required. Considerable work has gone into forming
appropriate grids 1Ref. 62. We consider a grid that is uniformly
spaced in the x and y directions, as shown in Fig. EA.2b.
As shown in Eq. 6.112, the exact solution to Eq. 1 1in terms
of polar coordinates r, u rather than Cartesian coordinates x, y2
is c Ur 11 a 2 r 2 2 sin u. The finite difference solution approximates
these stream function values at a discrete 1finite2
number of locations 1the grid points2 as c i, j , where the i and j indices
refer to the corresponding x i and y j locations.
The derivatives of c can be approximated as follows:
and
This particular approximation is called a forward-difference approximation.
Other approximations are possible. By similar reasoning,
it is possible to show that the second derivatives of c can
be written as follows:
and
0c
0x 1 ¢x 1c i1, j c i, j 2
0c
0y 1 ¢y 1c i, j1 c i, j 2
0 2 c
0x 2 1
1¢x2 2 1c i1, j 2c i, j c i1, j 2
0 2 c
0y 2 1
1¢y2 2 1c i, j1 2c i, j c i, j1 2
(2)
(3)
Thus, by combining Eqs. 1, 2, and 3 we obtain
0 2 c
0x 2 02 c
0y 2 1
1¢x2 2 1c i1, j c i1, j 2 1
1¢y2 2 1c i, j1
Equation 4 can be solved for the stream function at and to give
c i, j
c i, j1 2 2 a 1
1¢x2 2 1
1¢y2 2b c i, j 0
1
31¢y2 2 1c
231¢x2 2 1¢y2 2 i1, j c i1, j 2
4
1¢x2 2 1c i, j1 c i, j1 24
c i, j
Note that the value of depends on the values of the stream
function at neighboring grid points on either side and above and
below the point of interest 1see Eq. 5 and Fig. EA. 2c2.
To solve the problem 1either exactly or by the finite difference
technique2 it is necessary to specify boundary conditions for
points located on the boundary of the flow domain 1see Section
6.6.32. For example, we may specify that c 0 on the lower
boundary of the domain 1see Fig. EA.2b2 and c C, a constant,
on the upper boundary of the domain. Appropriate boundary conditions
on the two vertical ends of the flow domain can also be
specified. Thus, for points interior to the boundary Eq. 5 is valid;
similar equations or specified values of c i, j are valid for boundary
points. The result is an equal number of equations and unknowns,
c i, j , one for every grid point. For this problem, these equations
represent a set of linear algebraic equations for c i, j , the solution
x i
y j
(4)
(5)