Mechanics of Fluids

320 Boundary layers, wakes and other shear layers
whence, for a flat plate with zero pressure gradient,
kc ≏ ν
�
� �
N
=
τ0/ϱ
�
N 1/2 ν
0.332u 2 m Re−1/2
x
� 1/2 = 1.74N1/2 ν
um
Re 1/4
x
(8.30)
Thus kc increases as x 1/4 . That is, as the distance from the leading edge
increases and the boundary layer grows thicker, a greater roughness is needed
to upset its stability. When the transition Reynolds number is reached, however,
the laminar layer becomes turbulent whatever the roughness of the
surface. Since the frictional force is less with a laminar boundary layer than
with a turbulent one, the transition should be delayed as long as possible
if minimum friction drag is to be achieved. Therefore the surface should be
as smooth as possible near the leading edge, where the boundary layer is
thinnest, although greater roughness may be tolerated further downstream.
8.8 EFFECT OF PRESSURE GRADIENT
8.8.1 Separation and flow over curved surfaces
We have so far considered flow in which the pressure outside the boundary
layer is constant. If, however, the pressure varies in the direction of flow, the
behaviour of the fluid may be greatly affected.
Favourable pressure Let us consider flows over a curved surface as illustrated in Fig. 8.8. (The
gradient radius of curvature is everywhere large compared with the boundary-layer
thickness.) As the fluid is deflected round the surface it is accelerated over the
left-hand section until at position C the velocity just outside the boundary
layer is a maximum. Here the pressure is a minimum, as shown by the graph
below the surface. Thus from A to C the pressure gradient ∂p/∂x is negative
and the net pressure force on an element in the boundary layer is in the forward
direction. (For the curved surface, x is a curvilinear coordinate along
the surface and y is perpendicular to it.) Such a pressure gradient is said to
be favourable: it counteracts to some extent the ‘slowing down’ effect of the
boundary on the fluid, and so the rate at which the boundary layer thickens
is less than for a flat plate with zero pressure gradient ( at a corresponding
value of Rex).
Separation
Beyond C, however, the pressure increases, and so the net pressure force
on an element in the boundary layer opposes the forward flow. Although
the pressure gradient ∂p/∂x has practically the same value throughout the
cross-section of the boundary layer, its most significant effect is on the fluid
closest to the surface. This is because the fluid there has less momentum than
fluid further out, and so when its momentum is reduced still more by the
net pressure force the fluid near the surface is soon brought to a standstill.
The value of ∂u/∂y at the surface is then zero as at D. Further downstream,
for example, at E, the flow close to the surface has actually been reversed.
The fluid, no longer able to follow the contour of the surface, breaks away