Hypersonic Flows

Abstract

It is generally well accepted that all those flows, having the Mach numbers more than 5, are termed as hypersonic flow. However, it is natural to ask, why this limit is set to Mach 5 only? why not at Mach 4 or even at Mach 6? One may also ask, how and in what respect the shock wave produced at hypersonic Mach number is distinct than the shock produced in a supersonic flow? To answer these questions, we must have a thorough understanding of the characteristics associated with a hypersonic flow regime. In this chapter, we have attempted to describe these attributes which, in turn, represent a “formal definition” of the hypersonic flow.

Appendices

Summary

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

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

Exercises

1.1 Descriptive Type Questions

• 1. Using Newtonian impact theory, demonstrate that, at hypersonic speeds, the stagnation pressure is approximately equal to two times the dynamic pressure.

• 2. Consider a flat plate in the airstream at Mach 20 at 35$^{\mathrm {\text {o}}}$ to the freestream direction, as sketched in Fig. 10.10. If the temperature of the freestream is 220 $\mathrm {K}$, then calculate the Mach number and static temperature downstream of the shock.

• 3. Consider a flat plate in hypersonic airstream at 35$^{\mathrm {\text {o}}}$ to the freestream direction. Utilizing Newtonian impact theory, calculate (a) the pressure coefficient, (b) the lift coefficient, (c) the drag coefficient.

• 4. Consider a sphere of diameter 0.5 $\mathrm {m}$ flying at a speed of 6 $\mathrm {kms^{-1}}$ at an altitude of 65 $\mathrm {km}$ above the sea level. Calculate (a) the freestream Mach number and the Reynolds number, (b) Assuming air to be in the thermodynamic equilibrium, determine the static pressure $\mathrm {\left( p_{2}\right) }$ and static temperature $\mathrm {\left( T_{2}\right) }$ downstream of the shock positioned ahead of the sphere.

• 5. Consider a flat plate in uniform hypersonic airstream at an angle of attack $\mathrm {\alpha }$. Using Newtonian flow theory, find the value of $\alpha$ for which the lift coefficient $\mathrm {\left( C_{L}\right) }$ is maximum. Also, find the expressions for drag coefficient $\mathrm {\left( C_{D}\right) }$ and lift-to-drag ratio $\mathrm {\left( \frac{C_{L}}{C_{D}}\right) }$ when $\mathrm {C_{L}=C_{L,\text {max}}}$.

• 6. Consider a blunt-nosed body flying in the air sufficiently above the sea level. Let the entropy increase along the stagnation streamline is 0.7 $\mathrm {KJkg^{-1}K^{-1}}$. Find (a) the freestream Mach number, (b) the strength of shock in the proximity of stagnation streamline.

• 7. Consider a hypersonic vehicle flying at Mach 22 at an altitude of 65 $\mathrm {km}$ above the sea level. What will the air temperature at the stagnation point on the forward end of the vehicle? Explain the accuracy of your answer. Can it be changed? Under what conditions?

• 8. Utilizing Newtonian impact theory, develop an expression for the drag on a sphere. If a sphere of diameter 40 $\mathrm {cm}$ is moving at Mach 8 through the air at an ambient pressure of 0.05 $\mathrm {kPa}$, calculate the drag on the sphere.

• 9. In a hypersonic wind tunnel, the test section is designed to operate at Mach 25. If the stagnation temperature in the settling chamber is 3500 $\mathrm {K}$, calculate the temperature in the test section. Also find the required minimum settling chamber temperature so that the condensation can be avoided in the test section. (Note: The liquefaction temperature of air is about 77 $\mathrm {K}$.)

• 10. If a sphere 2.0 $\mathrm {m}$ in diameter is flying at 6.5 $\mathrm {kms^{-1}}$ at an altitude of 65,000 $\mathrm {m}$ above the sea level. Calculate (a) the freestream Mach number $\mathrm {M_{a}}$, (b) the freestream Reynolds number $\mathrm {Re_{a}}$, (c) Assuming the air to be in thermodynamic equilibrium, find the values of static pressure $\mathrm {\left( p_{2}\right) }$, static temperature $\mathrm {\left( T_{2}\right) }$, stagnation pressure $\mathrm {\left( p_{02}\right) }$, and stagnation temperature $\mathrm {\left( T_{02}\right) }$ downstream of the bow-shock formed in front of the flying sphere.

1.2 Multiple Choice Questions

• 1. The hypersonic similarity parameter $\mathrm {\left( K\right) }$ is defined as

  • (a) $\mathrm {M+\theta }$

  • (b) $\mathrm {M-\theta }$

  • (c) $\mathrm {M\theta }$

  • (d) $\mathrm {\mathrm {\frac{M}{\theta }}}$

• 2. Consider a flat plate placed in hypersonic airstream $\left( \gamma =\text {1.2}\right)$ at an inclination of 20$\mathrm {^{o}}$. According to modified Newtonian theory, the maximum pressure coefficient will be

  • (a) 0.8

  • (b) 1

  • (c) 1.2

  • (d) 1.9

• 3. An airstream $\left( \gamma =\text {1.4}\right)$ turns around an expansion corner such that the upstream and downstream Mach numbers are $\mathrm {6}$ and $\mathrm {9}$, respectively. The flow turning angle is

  • (a) 0.22$\mathrm {^{o}}$

  • (b) 0.28$\mathrm {^{o}}$

  • (c) 0.32$\mathrm {^{o}}$

  • (d) 0.38$\mathrm {^{o}}$

• 4. In the limit of high Mach number $\mathrm {\left( M\gg \text {1}\right) }$, the density ratio of a gas $\mathrm {\mathrm {\left( \gamma =1.2\right) }}$ flow across a shock wave is

  • (a) 3

  • (b) 5

  • (c) 7

  • (d) 11

• 5. For high Mach number and small angle approximations, the relation between shock angle and the flow turning angle is

  • (a) $\mathrm {\beta =\left[ \frac{\gamma +1}{2}\right] \theta }$

  • (b) $\mathrm {\beta =\left[ \frac{\gamma -1}{2}\right] \theta }$

  • (c) $\mathrm {\theta =\left[ \frac{\gamma +1}{2}\right] \beta }$

  • (d) $\mathrm {\theta =\left[ \frac{\gamma -1}{2}\right] \beta }$

• 6. Airstream at Mach $\mathrm {7}$, encounters a shock with the wave angle of 15$\mathrm {^{o}}$. If the static pressure upstream of the shock wave is 50 $\mathrm {kPa}$, the static pressure downstream of the shock will be

  • (a) 155 $\mathrm {kPa}$

  • (b) 167 $\mathrm {kPa}$

  • (c) 186 $\mathrm {kPa}$

  • (d) 198 $\mathrm {kPa}$

• 7. For the hypersonic flow over a flat plate at 10$\mathrm {^{o}}$, the pressure coefficient predicted by direct Newtonian method is

  • (a) 0.028

  • (b) 0.048

  • (c) 0.068

  • (d) 0.088

• 8. A gas $\mathrm {\left( \gamma =\text {1.3}\right) }$ flow at Mach 6 turns around the expansion corner of turning angle 0.3$\mathrm {^{o}}$. The pressure ratio across the expansion fan is

  • (a) 0.10

  • (b) 0.15

  • (c) 0.25

  • (d) 0.30

• 9. The temperature upstream of a shock wave in Mach 6.5 airstream is 500 $\mathrm {K}$. If the wave angle is 20$\mathrm {^{o}}$, the air temperature downstream of the shock is

  • (a) 330 $\mathrm {K}$

  • (b) 368 $\mathrm {K}$

  • (c) 381 $\mathrm {K}$

  • (d) 392 $\mathrm {K}$

• 10. Consider a hypersonic flow over a flat plate of length 2 $\mathrm {m}$ at zero angle of attack. If the Reynolds number is increased, the boundary layer thickness at a distance 1 $\mathrm {m}$ from the leading edge will

  • (a) increase

  • (b) decrease

  • (c) remain same

  • (d) become infinitely large

1.3 Keys

• 1. (c)

• 2. (d)

• 3. (b)

• 4. (d)

• 5. (a)

• 6. (a)

• 7. (b)

• 8. (c)

• 9. (d)

• 10. (b)