Flat plate boundary layer with uniform suction – integral method

The boundary layer on a flat plate with uniform suction (ue(x) = U = constant; vw = constant <0) displays a non–similar development. (see the figure below and section 4–5.2 in White).

Near the leading edge the boundary layer behaves like the Blasius solution, while it develops towards the uniform suction solution for large distance downstream. Properties of these two solutions according to the exact theory are:

1) Blasius solution: 2
2 0.66411w
x U U x

2) Suction profile: 1u( ) = e U
where: wyv  


Here we apply an integral analysis to compute the (approximate) development of the boundary layer, in particular of the momentum thickness. Under the considered conditions the integral momentum equation is given by:
2
2
w w w v vU d + S + dx U
where: =
wS U
and with: ( 0) 0x
The shear–parameter S is only a function of the shape of the velocity profile and not of its thickness.


a. The effect of the (constant) parameters U, vw and ν can be conveniently incorporated in a coordinate scaling:
2
wv= x U and * wv= (remember that 0wv!)
– Verify that with these definitions the integral momentum equation can be written as:
2* 2 * d = Sd
 with: *( 0) 0
– Determine from the exact solutions for the Blasius and suction boundary layer (given above) what the solution for * should be for small and large values of , respectively, and check if these two (asymptotic) results are in agreement with the character of the differential equation given here.

(1) VISCOUS FLOWS – AE 4120 5
b. The solution of the momentum integral equation requires the value of S, which is not a constant for the entire boundary layer, because the shape of the velocity profile develops with . We may attempt to simplify the problem by using a constant value of S. – Determine from the exact solution data given above the values of S for the Blasius solution
and for the suction profile. Integrate the momentum integral equation (numerically) over the domain0 5  , for each of these values of S. – Plot the results graphically as 2* . vs and compare it to the accurate results of a finite–difference solution of the problem, given in the table below. Also indicate in the graph the asymptotes of the exact solution as determined in part (a). Comment on the accuracy of the followed approach and the impact of the value of S?

c. As the shape variation has an impact on the computation of the boundary layer, it seems a logical step to try to improve the method by introducing a shape parameter λ, similar as in the method of Thwaites, such that:( ) S . Thwaites’method cannot be applied directly, however, as it does not take the effect of suction (both on the shape and in the momentum equation) into account.
– The original definition of the shape factor involves the curvature of the velocity profile near the wall:
2 2
2e w
u
u y

Evaluate the (differential) x–momentum equation at the wall to show how λ for the general case is related to the pressure gradient and the wall suction velocity. Verify that it reduces to the familiar expression of Thwaites when vw = 0. Derive that for the present problem (where ue(x) = U = constant) it can be directly related to S and *. Determine the values of λ for the Blasius and the suction profiles. – Assume that there is approximately a linear relation between S and λ: S A B , and determine the constants such that the Blasius and suction–profile values are reproduced. Show that for the current problem the result can be written as: *( ) S S . – Repeat the numerical integration, now with this modified (adaptive) relation for S. Compare the outcome with the previous results and comment if it matches your expectations.