Solving the Momentum-Integral Equation

A powerful method for solving the boundary-layer equations involves representing the velocity profile (i. e., the parallel part of the velocity in terms of the coordinate normal to the undisturbed flow direction) by an approximate analytical expression and integrating the equations in the direction normal to the flow. Integrating the equations removes the normal independent variable, leaving an ordinary differential equation in the streamwise direction, as demonstrated in the previous subsection.

The expression for the velocity profile is represented by:

where f is any function of the dimensionless normal coordinate that satisfies the necessary boundary conditions. For example, it is likely that f must go to zero at the surface to satisfy the no-slip boundary condition.

We now rewrite Eq. 8.96 to put it in the form most convenient for calculations. The wall shear stress and boundary-layer thickness are related to the momentum thickness by:

where the primes denote derivatives relative to the dimensionless, normal boundary- layer variable (n = z/5). It also is useful to define a second shape factor based on the boundary-layer thickness, 5:

5

H ‘ = – (8.100)

0

Hence, the momentum-integral relationship, Eq. 8.96, can be written as:

H0 dU«2 + d („2 Vtf,

~ + dX (U0)=H0 f(0)-

The second term on the left side can be expanded and recombined with the first term to give:

where the equation was multiplied through by 20.

For simple problems such as flow over a flat plate and flows with favorable or only weakly unfavorable pressure gradients, the shape factor, H, can be approxi­mated as a constant. In such cases, Eq. 8.101 has an integrating factor:

(tfe2)2+H.

The momentum integral then becomes:

Integration yields:

This simple expression can be evaluated for the momentum thickness, and the shape factor, H, then can be used to determine the displacement thickness.