Momentum-Integral Equations For The Boundary Layer
To employ boundary layer concepts in real engineering designs, we need approximate methods that would quickly lead to an answer even if the accuracy is somewhat less.
Karman and Pohlhausen devised a simplified method by satisfying only the boundary conditions of the boundary layer flow rather than satisfying Prandtl's differential equations for each and every particle within the boundary layer. We shall discuss this method herein.
Consider the case of steady, two-dimensional and incompressible flow, i.e. we shall refer to Eqs (28.10) to (28.14). Upon integrating the dimensional form of Eq. (28.10) with respect to y = 0 (wall) to y = δ (where δ signifies the interface of the free stream and the boundary layer), we obtain
or,
(29.10)
The second term of the left hand side can be expanded as
or, by continuity equation
or,
(29.11)
Substituting Eq. (29.11) in Eq. (29.10) we obtain
(29.12)
Substituting the relation between
and the free stream velocity
for the inviscid zone in Eq. (29.12) we get
which is reduced to
Since the integrals vanish outside the boundary layer, we are allowed to increase the integration limit to infinity (i.e . )
or,
(29.13)
Substituting Eq. (29.6) and (29.7) in Eq. (29.13) we obtain
(29.14)
where is the displacement thickness
is momentum thickness
Equation (29.14) is known as momentum integral equation for two dimensional incompressible laminar boundary layer. The same remains valid for turbulent boundary layers as well.
Needless to say, the wall shear stress
will be different for laminar and turbulent flows.
The term
signifies space-wise acceleration of the free stream. Existence of this term means that free stream pressure gradient is present in the flow direction.
For example, we get finite value of
outside the boundary layer in the entrance region of a pipe or a channel.For external flows, the existence of
depends on the shape of the body.
During the flow over a flat plate,
and the momentum integral equation is reduced to
by continuity equation 
and the free stream velocity
for the inviscid zone in Eq. (29.12) we get
. )
is the displacement thickness 
is momentum thickness 
will be different for laminar and turbulent flows.
signifies space-wise acceleration of the free stream. Existence of this term means that free stream pressure gradient is present in the flow direction. 
outside the boundary layer in the entrance region of a pipe or a channel.
depends on the shape of the body. 
and the momentum integral equation is reduced to 
