Page 228 - Computational Fluid Dynamics for Engineers

7.3 Numerical Method for the Standard Problem 217

rj = yju e/ux y, ij)(x, y) = y/u evx /(x, 7?) (7.3.4)

Noting the chain rule relations (subsection 2.2.4), it can be shown that, with a
prime denoting differentiation with respect to 77, and with

£ = £ u e = ^ (7.3.5)
Li UQQ
Eq. (7.3.3) and its boundary conditions, Eq. (7.3.1), can be written as
2
(bf'Y + ^ / / " + m[l - (/') ] = £ [f^ - " | 0 (7.3.6)
/

/
77 = 0, / ' = 0, /(£,0) = „(£) = - ^ / ? — d? (7.3.7a)
»? = %, / ' = 1 (7.3.7b)
Here r? e corresponds to a transformed boundary-layer thickness, yju e/vx8, or
y/u eRL/t;8/L; RL is a Reynolds number based on reference velocity UOQ and
length L\ m is a dimensionless pressure-gradient parameter defined by

ra= — — - (7.3.8)

The velocity components u and v are related to the dimensionless stream func-
tion /(£,/7) by
u = u e / ' (7.3.9a)

?J = —yJU eVX /u ex + —- + / — (7.3.9b)
/w ex dx dx dx \
For laminar flows with the external velocity of the form

= Cx m (7.3.10)
u e
with C and m constants, and with boundary conditions on / and ' independent
/
of x, the left-hand side of Eq. (7.3.6) reduces to
r+rn + \ ffH + m[1_ {fl)2]=Q ( 7 3 n )

which is called the Falkner-Skan equation. Its solutions, which are limited to
—0.0904 < m < 00, are independent of £ and can be used to generate the initial
conditions needed for the solution of the boundary-layer equations, as shall be
shown later.
There are several numerical methods that can be used to solve the boundary-
layer equations. Finite-difference methods offer the greatest flexibility, and those
of Crank-Nicolson and Keller, discussed in Section 4.4, have been widely used.
The latter method provides significant advantages over the former method and
will be used to solve the boundary-layer equations in this chapter, as well as
the stability equation to be discussed in Chapter 8.
To solve Eqs. (7.3.6) and (7.3.7) with Keller's box method, we follow the
four steps discussed in subsection 4.4.3.