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212 7. Boundary-Layer Equations

In this way the solution procedure for the momentum equation and continuity
equation, Eq. (2.4.33), which is renumbered for convenience

+ (7 L3)
d- X 8-y=° -
is the same for laminar and turbulent flows with an algebraic eddy-viscosity
formulation. The solution procedure is general and can be used to solve the
turbulent boundary-layer equations with turbulence models other than those
based on an algebraic eddy viscosity formulation as discussed in [1,2].

7.2 Standard, Inverse and Interaction Problems

Equations (7.1.1) and (7.1.3) apply to internal and external flows. In the latter
case they are often solved for prescribed pressure p(x) or external velocity u e(x)
(see Section 2.7) that are related to each other by Bernoulli's equation,

_ I^E = u— (7 2 1)
Q dx e dx
so that a specification of one variable implies the other. This procedure, some-
times referred to as the standard problem, can be used to solve Eqs. (7.1.1) and
(7.1.3) subject to the boundary conditions discussed in Section 2.7 provided that
boundary-layer separation, which corresponds to vanishing wall shear, does not
occur. If the wall shear vanishes at some x-location, the solutions break down
and convergence cannot be obtained. This is referred to as the singular behavior
of the boundary-layer equations at separation. For laminar flows, the behavior
of the wall shear r w close to the separation point x S) has been shown to be of
the form
' | ) - ( — ) V 2 (7-2.2)

by Goldstein [3], who considered a uniformly retarded flow past a semi-infinite
plate and showed that, with the relation given above, there is no real solution
downstream of separation; the normal velocity component v becomes infinite at
x s. Goldstein also pointed out that the pressure distribution around the sepa-
ration point cannot be taken arbitrarily and must satisfy conditions associated
with the existence of reverse flow downstream of separation. This is consistent
with the fact that the standard boundary-layer equations are parabolic in space,
whereas flow separation introduces the elliptic nature of the flowfield.
The inviscid-pressure distribution for flow past a circular cylinder leads to
a velocity maximum at 90° and separation at about 105° from the front stag-
nation point, whereas viscous flow in the subcritical range of Reynolds number
leads to a velocity maximum at 70° and separation at 83°. As can be seen from
Fig. 7.1, viscosity modifies the inviscid flow, which cannot serve even as a first