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2.7 Boundary Conditions 71

stream extending to y = — oo (Fig. 2.5b), in which case the viscous region is
called a "free shear layer." The variation of the static pressure p with x within
the shear layer depends on the shape of the solid body, and, additionally on the
displacement effect of the shear layer, which may either be small enough to be
neglected or can be accounted for, for example, by the interactive procedures to
be described in Chapter 7. In this case there are three boundary conditions for
the velocity field that must be specified, two at the wall, y = 0, and the other
at the boundary-layer edge, y = 6. The conditions at the wall are usually the
specification of normal (v) and tangential (u) components of velocity, and that
at the edge specifies u as the external velocity u e(x). They can be summarized
as
y = 0, u = 0, v = v w(x) (2.7.3a)

2/-><5, u = u e{x) (2.7.3b)
where 6 is sufficiently large so that du/dy at the boundary-layer edge is small,
- 4
say around 10 . The transpiration velocity, v w(x), may be either suction or
injection. On a nonporous surface it is equal to zero.
For boundary-layer flows, the energy equation, being second order, re-
quires two boundary conditions, one at the wall, y = 0, and one at the thermal
boundary-layer edge, ^ , which may be smaller or larger than the hydrodynamic
boundary-layer thickness, 6. For a Prandtl number, Pr, less than unity, 6t > <5,
and conversely for Pr greater than unity, 6t < 6. The boundary conditions at
the wall may correspond to either specified wall temperature, T w(x), or wall
heat flux, q w(x), and with the requirement at the boundary-layer edge, they
can be summarized as

= 0, T = T w(x), or (?p\ =-q w(x)/k (2.7.4a)
y
(2.7.4b)
y = S u T = T e
where T e is constant for an incompressible flow but is a function of x for a
compressible flow with pressure gradient.
In free shear layers (Figs. 2.5b and c), the external velocity must be specified
on both edges. The difficulties associated with the v boundary condition in free
shear layers are less obvious. If the flow is symmetrical (Fig. 2.5a), no problem
arises: the initial symmetrical velocity profile is specified and v is required to
be zero on the centerline and u = u e at one edge. If the flow is not symmetrical
(Fig. 2.5b), a boundary condition for v cannot be found from consideration of
the shear layer and the boundary-layer equations alone. In the real flow the
behavior of v outside the shear layer depends on the ^-component equation of
motion and the continuity equation, applied throughout the flow and not merely
in the shear layer. Of course, a similar problem occurs in determining u e either
in the boundary layer or wake of Fig. 2.5a; the latter problem is both more