A flow is called hypersonic if the flow Mach number is greater than 5, i.e., the flow speed is five times or more than the acoustic speed.

For a fixed flow turning angle, the increase of Mach number eventually decreases the wave angle. That is, when the flow Mach number is increased the shock comes closer to the body surface. The flow field between the shock wave and the body surface is defined as the shock layer. At very high Mach numbers, shock layer will be very thin and may be even close to the body giving rise to a possibility of an interaction with the boundary layer on the surface. This phenomenon is referred to as shock–boundary layer interaction, which is more pronounced at low Reynolds number where boundary layer is comparatively thicker.

Whenever the flow past a solid surface, a thin viscous layer develops over the surface. At high Mach numbers, the hypersonic stream possesses a large amount of kinetic energy, which gets retarded by viscous actions within the boundary layer. A portion of the lost kinetic energy is utilized in increasing the internal energy of the gas which, in turn, increases the temperature of the boundary layer. This phenomenon is known as viscous dissipation.

The thickness of the boundary layer

\(\mathrm {\left( \delta \right) }\) at a distance

\(\mathrm {x}\) from the leading edge is defined by

$$ \mathrm {\delta \propto \frac{x}{\sqrt{Re_{x}}}} $$

For a hypersonic boundary layer,

$$ \mathrm {\delta \propto \frac{x}{\sqrt{Re_{a}}}M_{a}^{2}} $$

That is, the boundary layer thickness

\(\mathrm {\left( \mathrm {\delta }\right) }\) varies as the square of

\(\mathrm {M_{a}}\) and thus,

\(\mathrm {\mathrm {\delta }}\) will be excessively large at hypersonic Mach numbers.

We know that whenever the supersonic stream turns into itself, a shock wave is produced. Shock is an extremely thin region which has the thickness of the order of \(\mathrm {10^{-5}}\) cm, where the viscosity and thermal conductivity are the important mechanism making the shock process irreversible. Because of this irreversibility associated with the shock wave, stagnation pressure across the shock decreases with increase in the Mach number, while the static pressure, static density, and static temperature rise. The hypersonic shock wave remains stationary if the static pressure downstream of the shock is sufficiently high.

In the limit of high Mach number, i.e.,

\(\mathrm {M\gg \text {1}}\), the density ratio across the shock produced at hypersonic Mach number is

$$\begin{aligned} \mathrm {\frac{\rho _{2}}{\rho _{1}}}\,&\mathrm {=\frac{\left( \gamma +1\right) }{\left( \gamma -1\right) }} \end{aligned}$$

and the pressure ratio is

$$ \mathrm {\mathrm {\frac{p_{2}}{p_{1}}}=\left( \frac{2\gamma }{\gamma +1}\right) M_{1}^{2}\sin ^{2}\beta } $$

In addition, the temperature ratio across the shock is

$$\begin{aligned} \mathrm {\frac{T_{2}}{T_{1}}}\,&\mathrm {=\frac{2\gamma \left( \gamma -1\right) }{\left( \gamma +1\right) ^{2}}M_{1}^{2}\sin ^{2}\beta } \end{aligned}$$

Together with high Mach number

\(\mathrm {\left( M_{1}\gg \text {1}\right) }\) and small angle approximations, the relation between shock wave angle and flow turning angle is

$$\begin{aligned} \mathrm {\beta }\,&\mathrm {=\left[ \frac{\gamma +1}{2}\right] \theta } \end{aligned}$$

For air

\(\mathrm {\left( \gamma =\text {1.4}\right) }\),

$$\begin{aligned} \mathrm {\beta }\,&\mathrm {=\text {1.2}\theta } \end{aligned}$$

Thus, at hypersonic Mach numbers for small flow turning angles, the wave angle is just 20% larger than the deflection angle.

For hypersonic flows, the pressure coefficient

\(\mathrm {\left( C_{p}\right) }\) is

$$\begin{aligned} \mathrm {C_{p}}\,&\mathrm {=\left[ \frac{4}{\gamma +1}\right] \sin ^{2}\beta } \end{aligned}$$

For expansion waves at high but finite Mach numbers, we have

$$\begin{aligned} \mathrm {\theta }\,&\mathrm {=\frac{2}{\left( \gamma -1\right) }\left( \frac{1}{M_{1}}-\frac{1}{M_{2}}\right) } \end{aligned}$$

where

\(\theta \) is the flow turning angle and

\(\mathrm {M_{1}}\) and

\(\mathrm {M_{2}}\) are the Mach numbers upstream and downstream of the expansion fan. Also, for the same assumption, the pressure ratio across the expansion fan will be

$$\begin{aligned} \mathrm {\frac{p_{2}}{p_{1}}}\,&\mathrm {=\left[ 1-\left( \frac{\gamma -1}{2}\right) M_{1}\theta \right] ^{\frac{\gamma }{\gamma -1}}} \end{aligned}$$

In hypersonic flows, the similarity parameter

\(\mathrm {\left( K\right) }\) is defined as

$$\begin{aligned} \mathrm {K}\,&\mathrm {=M\theta } \end{aligned}$$

Thus, if two different flow problems have same values of

\(\mathrm {K}\) then they are similar flows and will have like solutions.

In this chapter, we have discussed two hypersonic local surface inclination methods: Newtonian and modified Newtonian theories. They are used to predict the local surface pressure as a function of the surface inclination angle with respect to the incoming freestream direction. For high Mach numbers, the pressure coefficient predicted by direct Newtonian method is

$$ \mathrm {\mathrm {C_{p}}=2\sin ^{2}\theta } $$

and by the modified Newtonian approach is

$$ \mathrm {C_{p}}=\mathrm {C_{p,\text {max}}}\sin ^{2}\theta $$

where

$$\begin{aligned} \mathrm {\mathrm {C_{p,\text {max}}}=}\,&\mathrm {\left\{ \left[ \frac{\left( \gamma +1\right) ^{2}}{4\gamma }\right] ^{\frac{\gamma }{\left( \gamma -1\right) }}\frac{4}{\left( \gamma +1\right) }\right\} } \end{aligned}$$