Theoretical Developments

For the case of a flat plate, approximate analytical results are available for both weak and strong interaction (excluding the neighborhood of the leading edge).

Following Hayes and Probstein [1] and Cox and Crabtree [3], the displacement thickness for weak interaction on a flat plate is given by (as discussed before)

Thus, the equivalent body slope due to the displacement effect is

where Rex = poUx/^o.

On the other hand, the pressure ratio for oblique shocks is

where K = M0 d6*/dx.

For (M0 d6*/dx)2 ^ 1, the pressure ratio over a wedge is obtained from

p d6* Ye ( Tw

— = 1 + YM0 = 1 + 0.664 + 1.73—W x (12.232)

P0 dx 2 Te

where x = M03VC/V Rex.

For an insulated surface (Tw/T0 ^ 1) and a very cold wall (Tw/T0 ^ 0), the above results are reduced to

In the case of strong interaction, the pressure formula is based on strong oblique

shock, i. e.

(12.234)

It can be shown that for a consistent solution, the displacement thickness and induced pressure distribution must comply to the following behaviors, S* ~ x-3/4, p ~ x-1/2.

Let p/p0 ~ kx-1/2, then

Hence

Therefore

The role of parameter x is clear for both interaction regions.

Comparison with experimental results are reasonable (see Hayes and Probstein [1]).

Later, a more comprehensive theory has been developed based on a triple deck structure, see reviews by Stewartson [145], Rothmayer and Smith [146], Sychev [147] and Brown et al. [148]. In this theory, upstream influence (suggested earlier by Lighthill [149]) as well as flow reversal and separation are studied in details.