Hypersonic flow — Mach 5+, high temperature effects

WHAT is "hypersonic"?

WHY a number like 5? Because the temperature rise behind a shock scales with $M^2$. Around $M\approx 5$, post-shock temperatures climb past ~1000–1500 K, where oxygen vibration and then dissociation begin to matter. The label tracks physics, not a clean threshold.

HOW the temperature blows up: deriving stagnation temperature

Derivation from first principles (adiabatic, steady, no work):

Energy conservation along a streamline (steady adiabatic flow) says total enthalpy is constant: $h + \tfrac{1}{2}V^2 = h_0 = \text{const}$

WHY this step? With no heat added and no shaft work, the steady-flow energy equation reduces to "enthalpy + kinetic energy = constant." $h_0$ is the stagnation enthalpy.

For a calorically perfect gas $h = c_p T$, so: $c_p T + \tfrac{1}{2}V^2 = c_p T_0$

Divide by $c_p T$: $\frac{T_0}{T} = 1 + \frac{V^2}{2 c_p T}$

Now use $c_p = \dfrac{\gamma R}{\gamma-1}$ and the speed of sound $a^2 = \gamma R T$, so $\dfrac{V^2}{2c_pT} = \dfrac{V^2(\gamma-1)}{2\gamma R T} = \dfrac{(\gamma-1)}{2}\dfrac{V^2}{a^2} = \dfrac{\gamma-1}{2}M^2$.

WHY $\gamma$ changes: degrees of freedom and real-gas effects

By equipartition, internal energy per mole rises with the number of active degrees of freedom $f$: $e = \tfrac{f}{2}RT,\qquad c_v=\tfrac{f}{2}R,\qquad \gamma=\frac{c_p}{c_v}=1+\frac{2}{f}.$

WHY: more active modes $\Rightarrow$ larger $f \Rightarrow$ smaller $\gamma$.

HOW the shock layer gets thin: Mach number and density ratio

For a normal shock, the limiting density ratio as $M\to\infty$ (calorically perfect) is: $\frac{\rho_2}{\rho_1}\to \frac{\gamma+1}{\gamma-1}$

WHY: from the Rankine–Hugoniot relations, the velocity ratio $u_2/u_1 \to (\gamma-1)/(\gamma+1)$, and $\rho u$ is conserved, so density rises by the reciprocal. For $\gamma=1.4$ this caps at 6. With real-gas effects $\gamma$ falls, so $\rho_2/\rho_1$ can reach 15–20 — shock even closer.

Newtonian impact theory (the 80/20 hypersonic shortcut)

Derivation: A stream of density $\rho_\infty$, speed $V_\infty$ hits a surface inclined at angle $\theta$ to the flow. Mass flux hitting unit area $= \rho_\infty V_\infty \sin\theta$. Each parcel loses normal velocity $V_\infty\sin\theta$. Pressure = momentum flux destroyed: $p - p_\infty = (\rho_\infty V_\infty \sin\theta)(V_\infty\sin\theta)=\rho_\infty V_\infty^2\sin^2\theta.$

Define $C_p = \dfrac{p-p_\infty}{\tfrac12\rho_\infty V_\infty^2}$:

WHY blunt is better for re-entry

Connections

• Normal and Oblique Shock Waves — source of the post-shock $T$, $\rho$ jumps used here.
• Stagnation Properties & Isentropic Relations — origin of $T_0/T = 1+\tfrac{\gamma-1}{2}M^2$.
• Rankine–Hugoniot Relations — gives the $(\gamma+1)/(\gamma-1)$ density limit.
• Real Gas Thermodynamics & Dissociation — why $\gamma$ falls and energy hides.
• Boundary Layers & Aerodynamic Heating — stagnation heat flux and TPS design.
• Supersonic Flow & Area-Mach Relations — the regime just below hypersonic.

Concept Map

Hinglish (regional understanding)