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7.2 Standard, Inverse and Interaction Problems 213

C P

Fig. 7.1. Pressure distribution around a cir-
cular cylinder in the subcritical and super-
0 60 120 180 240 300 360
critical range of Reynolds numbers qoo =
8° iA?v£.

approximation to the actual flow. In the case of streamlined bodies, where sepa-
ration takes place close to the rear stagnation point, the solution of the inviscid
flow equations can serve as a good approximation to real flow as discussed in
detail in [4]. For completeness, it is briefly discussed below.
The boundary-layer equations are not singular at separation when the ex-
ternal velocity or pressure is computed as part of the solution. Catherall and
Mangier [5] were the first to show that in two-dimensional steady laminar flows,
the modification of the external velocity distribution near the region of flow
separation leads to solutions free of numerical difficulties. Prescribing the dis-
placement thickness as a boundary condition at the boundary-layer edge, that is,

y = (5, 8*(x) — given (7.2.3)

in addition to those given by Eq. (2.7.3) with no mass transfer, they were able
to integrate the boundary-layer equations through the separation location and
into a region of reverse flow without any evidence of singularity at the sep-
aration point. This procedure for solving the boundary-layer equations for a
prescribed displacement thickness distribution, with the external velocity or
pressure computed as part of the solution is known as the inverse problem. This
observation of Catherall and Mangier has led to other studies by various inves-
tigators of inverse solutions of the boundary-layer equations; these are obtained
by prescribing distributions of displacement thickness or wall shear. Further-
more, it has been demonstrated in [6] that for flows with separation bubbles,
these solutions are in good agreement with the solutions of the Navier-Stokes
equations.
A problem associated with the use of these inverse techniques for external
flows is the lack of a priori knowledge of the required displacement thickness
or wall shear. The appropriate value must be obtained as part of the overall
problem from interaction between the boundary layer and the inviscid flow. In