Page 241 - Computational Fluid Dynamics for Engineers

230 7. Boundary-Layer Equations

7.5.1 Similar Laminar Flows
A certain class of external laminar flows admit similarity solutions, as discussed
in Section 7.3, for external velocity, u e , of the form given by Eq. (7.3.10) and with
boundary conditions in transformed variables independent of the downstream
distance x. The solutions of the continuity and momentum equations can be
obtained by solving the Falkner-Skan equation (7.3.11) subject to the boundary
conditions given by Eq. (7.3.7).
To demonstrate the application of BLP, we first consider the case of flow
with no mass transfer so that f w = 0, and solve Eq. (7.3.11) for values of m
of 1, 1/3, 0, —0.075 and —0.0904. The calculations can be carried out at one
^-station (NX = 1) for each value of m or performed for all the values of m.
In the former case, it is necessary to set the value of m at £ = 0, that is, P2(l)
in subroutine INPUT. In the latter, P2(NX) is set to require m values at five
^-stations, including P2(l), rather than calculating them from the input values
of £ and u e. With NXT = 5, X(I), UE(I) and RL can be assigned any value since
the Falkner-Skan equation is independent of these parameters. The parameter
(= CEL) [subroutine COEF] is set to zero at all ^-stations.
a n
The second choice is preferable since the results are obtained from a multi-
step calculation, and better convergence is achieved since initial estimates are
provided by the results from the previous ^-station.
1
The calculated velocity profiles f (= u/u e) as a function of the similarity
l 2
variable 77 [= (u e/vx) l y\ are shown in Fig. 7.5. Of particular note are the
m-values of unity and zero corresponding to stagnation and flat-plate flows,
m = 1/3 corresponding to the stagnation flow on an axisymmetric body, and
m = —0.0904 corresponding to the strongest adverse pressure gradient for which
ff
the flow remains attached with f (0) equal to zero. These calculations were
performed with a uniform grid (K = 1) and hi = 0.10.
To demonstrate the application of BLP to a similar laminar flow with mass
transfer, we consider a flat-plate flow with specified values of f w. It is seen
from Eq. (7.3.7a) that for similarity, the mass transfer velocity v w must vary as
1/y/x. To account for the nonzero value of f Wl one must make changes in the
IVPL subroutine since Eqs. (7.4.4) are for zero mass transfer. By rewriting Eq.
(7.4.4b) as

the mass transfer effects can be included in the solution of the Falkner-Skan
equation. As in the case of calculations for different values of m, the calculations
can also be performed with the two choices described above for flows with m = 0
for all ^-stations without mass transfer and, in this case, it is more convenient
to perform them with the first choice.
Figure 7.6 shows the calculated velocity profiles as a function of rj for sev-
eral values of mass transfer parameter f w with positive values corresponding to