Page 233 - Computational Fluid Dynamics for Engineers

222 7. Boundary-Layer Equations

7.4 Computer Program BLP

This section describes a boundary layer program (BLP) for two-dimensional
external flows in which the solutions of the continuity and momentum equa-
tions are obtained in terms of Falkner-Skan variables. The program, given in
Appendix B, is appropriate to two-dimensional flows without separation and
mass transfer but, with minor modifications, can be used for flows with mass
transfer, axisymmetric flows, free shear flows, and flows with heat transfer as
discussed in [2].
In BLP, the calculations start at the leading edge, £ = 0, where the flow is
laminar and becomes turbulent at any ^-location by specifying the transition
location. The solution procedure requires the specification of the dimensionless
pressure gradient ra(£) which can be obtained from the specified dimensionless
external velocity distribution u e(£).
BLP consists of a MAIN routine, which contains the logic of the computa-
tions, and subroutines: INPUT, IVPL, GROWTH, COEF3, SOLV3, OUTPUT
and EDDY. The following subsections describe the function of each subroutine.
A listing for each routine is given in Appendix B.

7.4.1 M A I N

BLP solves the linearized form of the equations. Thus an iteration procedure
in which the solution of Eqs. (7.3.19) and (7.3.21) is obtained for successive
estimates of the velocity profiles is needed with a subsequent need to check the
convergence of the solutions. A convergence criterion based on vo which corre-
sponds to f!^ is usually used and the iterations, which are generally quadratic
for laminar flows, are stopped when
|£vo(=DELV(l))| <ei (7.4.1)

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with e\ taken as 10 . For turbulent flows, due to the approximate linearization
procedure used for the turbulent diffusion term, the rate of convergence is not
quadratic and solutions are usually acceptable when the ratio of |<5^o/^o| is less
than 0.02. With proper linearization, quadratic convergence of the solutions can
be obtained as described in [2].
After the convergence of the solutions, the OUTPUT subroutine is called
and the profiles F, U, V and B, which represent the variables / j , UJ, Vj and bj
are shifted.

7.4.2 Subroutine I N P U T

The solution procedure requires the generation of a grid normal to the surface,
77-grid, and along the surface, £-grid. The latter requirement is usually satisfied