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).

Подпись: 6* ~ Theoretical Developments Подпись: w Подпись: (12.229)
Theoretical Developments

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)

Подпись: d 6* dx Theoretical Developments Подпись: (12.230)

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

where Rex = poUx/^o.

Подпись: p = 1 . Y(Y_ P0 4 Подпись: — K 2 + Y K2 Theoretical Developments Подпись: (12.231)

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.

Подпись: P0 Подпись: 1 + 0.28x, for insulated wall 1 + 0.078x, for cold wall Подпись: (12.233)

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.

Подпись:Theoretical DevelopmentsПодпись: P_ P0 (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.

Подпись: S* ~ e ^0.664 + 1.73 Theoretical Developments Подпись: 3/4 Подпись: (12.235)
Theoretical Developments

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

Theoretical Developments Theoretical Developments Theoretical Developments

Hence

Подпись: and Theoretical Developments Theoretical Developments

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.

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