Additional examples of the application of the momentum integral equation

V* U’

For the general solution of the momentum integral equation it is necessary to resort to computational methods, as described in Section 7.11. It is possible, however, in certain cases with external pressure gradients to find engineering solutions using the momentum integral equation without resorting to a computer. Two examples are given here. One involves the use of suction to control the boundary layer. The other concerns determining the boundary-layer properties at the leading-edge stagnation point of an aerofoil. For such applications Eqn (7.59) can be written in the alter­native form with H = 6*/в:

When, in addition, there is no pressure gradient and no suction, this further reduces to the simple momentum integral equation previously obtained (Section 7.7.1, Eqn (7.66)), i. e. Cr = 2(d0/dx).

Example 7.7 A two-dimensional divergent duct has a total included angle, between the plane diverging walls, of 20°. In order to prevent separation from these walls and also to maintain a laminar boundary-layer flow, it is proposed to construct them of porous material so that suction may be applied to them. At entry to the diffuser duct, where the flow velocity is 48 ms"1 the section is square with a side length of 0.3 m and the laminar boundary layers have a general thickness (<5) of 3 mm. If the boundary-layer thickness is to be maintained constant at this value, obtain an expression in terms of x for the value of the suction vel­ocity required, along the diverging walls. It may be assumed that for the diverging walls the laminar velocity profile remains constant and is given approximately by ii= 1.65 j7 _ 4.30/ + 3.65)’.

For bodies with sharp leading edges such as flat plates the boundary layer grows from zero thickness. But in most engineering applications, e. g. conventional aero­foils, the leading edge is rounded. Under these circumstances the boundary layer has a finite thickness at the leading edge, as shown in Fig. 7.27a. In order to estimate the

i. e.

Equation (7.16) gives

Then

dU,

Finally

Boundary-layer edge	Boundary-layer edge
Fig. 7.27 Boundary-layer flow in the vicinity of the fore stagnation point

initial boundary-layer thickness it can be assumed that the flow in the vicinity of the stagnation point is similar to that approaching a flat plate oriented perpendicularly to the free-stream, as shown in Fig. 7.27b. For this flow Ue = cx (where c is a constant) and the boundary-layer thickness does not change with x. In the example given below the momentum integral equation will be used to estimate the initial boundary-layer thickness for the flow depicted in Fig. 7.27b. An exact solution to the Navier-Stokes equations can be found for this stagnation-point flow (see Section 2.10.3). Here the momentum integral equation is used to obtain an approximate solution.

Example 7.8 Use the momentum integral equation (7.59) and the results (7.64a’, b’, d) to obtain expressions for <5, <5*, 9 and Q. It may be assumed that the boundary-layer thickness does not vary with x and that Ue = cx.

= —c = const, v

Hence 9 = const, also and Eqn (7.59) becomes

After rearrangement this equation simplifies to

0.00022Л3 + 0.01045Л2 – 0.3683Л + 2 = 0

or

It is known that A lies somewhere between 0 and 12 so it is relatively easy to solve this equation by trial and error to obtain

Using Eqns (7.64a’, b’, c’) then gives