Mechanics of Fluids

322 Boundary layers, wakes and other shear layers
(see Section 7.6.3) and is the cause of the flow separation which occurs
there unless the angle of divergence is very small.
Separation of the boundary layer greatly affects the flow as a whole. In particular
the formation of a wake of disturbed fluid downstream, in which the
pressure is approximately constant, radically alters the pattern of flow. The
effective boundary of the flow is then not the solid surface but an unknown
shape which includes the zone of separation. Because of the change in the
pattern of flow the position of minimum pressure may be altered, and the
separation point may move upstream from where the pressure was originally
a minimum (e.g. point C in Fig. 8.8).
Once a laminar layer has separated from the boundary it may become
turbulent. The mixing of fluid particles which then occurs may, in some
circumstances, cause the layer to re-attach itself to the solid boundary so
that the separation zone is an isolated bubble on the surface. Although not
a common occurrence this does sometimes happen at the leading edge of a
surface where excessive roughness causes separation of the laminar layer,
which is followed by a turbulent layer downstream.
8.8.2 Predicting separation in a laminar boundary layer
Predicting the position at which separation may be expected is clearly important
yet there is at present no exact theory by which this may readily be done.
However, the momentum eqn 8.10 allows some valuable approximate results
to be obtained, especially for laminar boundary layers. One method,
due to the English mathematician Sir Bryan Thwaites (1923–), is simple to
use yet remarkably accurate.
Expanding the first term on the right of eqn 8.10 and then isolating the
term containing dθ/dx we get
ϱu 2 dθ
m
dx = τ0 − 2ϱum
dum
dx
From this, multiplication by 2θ/µum gives
um
ν
dum ∗
θ − ϱ umδ
dx
d
dx (θ 2 ) = 2θτ0
�
− 2 2 +
µum
δ∗
�
θ 2 dum
θ ν dx
(8.31)
To integrate eqn 8.31 we need to be able to correlate τ0/µum and δ ∗ with
the momentum thickness θ. For a laminar layer τ0/µ = (∂u/∂y)y=0 and
so is obtainable from the velocity distribution. Thwaites examined many
exact and approximate solutions for velocity distribution in laminar layers
under various values of pressure gradient. His analysis showed that the righthand
side of eqn 8.31 is to a close approximation simply a function of the
dimensionless quantity (θ 2 /ν)dum/dx = λ say. Moreover the function is
given very nearly by the linear relation 0.45 − 6λ, the range of choice for
each numerical coefficient being only about 2% of its value. Substituting