13.4 Oblique Shock Waves and Expansion Waves

Oblique Shock Waves and Expansion Waves

When supersonic flow hits a change in geometry, the flow can't adjust gradually the way subsonic flow does. Instead, it adjusts through discrete waves: oblique shock waves at concave corners and expansion waves at convex corners. These two wave types are the building blocks for analyzing almost any supersonic flow problem, from airfoil design to nozzle shaping.

Oblique Shock Waves

Oblique shocks vs expansion waves

Oblique shock waves form when supersonic flow encounters a concave corner (a compression ramp). The flow turns toward the surface, and properties change abruptly across the shock:

• Pressure, density, and temperature all increase
• Velocity decreases
• The flow deflects toward the surface by an angle $\theta$

Expansion waves form when supersonic flow encounters a convex corner. The flow turns away from the surface, and properties change gradually through a fan of Mach waves:

• Pressure, density, and temperature all decrease
• Velocity increases
• The flow deflects away from the surface by an angle $\theta$

A useful way to remember the distinction: shocks compress and slow the flow; expansion waves do the opposite. Shocks are abrupt (a single wave front), while expansion waves spread out over a fan-shaped region.

Oblique shock flow calculations

The key idea is that you can reuse the normal shock relations you already know by decomposing the flow into components normal and tangential to the shock front. The tangential component passes through unchanged; only the normal component "sees" the shock.

Step-by-step procedure:

1. Find the normal Mach number upstream. Given the upstream Mach number $M_1$ and the shock wave angle $\beta$ (measured from the upstream flow direction to the shock):

$M_{1n} = M_1 \sin\beta$

1. Apply normal shock relations. Use $M_{1n}$ in the standard normal shock tables or equations to find the downstream normal Mach number $M_{2n}$, as well as the pressure ratio $p_2/p_1$, temperature ratio $T_2/T_1$, and density ratio $\rho_2/\rho_1$.

2. Recover the full downstream Mach number. The downstream Mach number accounts for the fact that the flow has been deflected by $\theta$:

$M_2 = \frac{M_{2n}}{\sin(\beta - \theta)}$

Finding the wave angle $\beta$: If you know $M_1$ and the deflection angle $\theta$ (set by the geometry), use the $\theta$-$\beta$-$M$ relation:

$\tan\theta = 2\cot\beta\;\frac{M_1^2 \sin^2\beta - 1}{M_1^2(\gamma + \cos 2\beta) + 2}$

This equation is implicit in $\beta$, so you typically solve it iteratively or read $\beta$ from a $\theta$-$\beta$-$M$ chart. For a given $M_1$ and $\theta$, there are generally two solutions:

• The weak shock (smaller $\beta$), which is the one that usually occurs in practice. The downstream flow remains supersonic in most cases.
• The strong shock (larger $\beta$), where the downstream flow is subsonic. This solution requires special back-pressure conditions and is less common.

If $\theta$ exceeds the maximum deflection angle for a given $M_1$, no attached oblique shock solution exists and the shock detaches from the body.

Expansion Waves and Interactions

Expansion wave property determination

Expansion waves are isentropic, so there's no entropy increase and total pressure is conserved. The Prandtl-Meyer function $\nu(M)$ is the tool for these calculations.

Prandtl-Meyer function:

$\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\;\tan^{-1}\!\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} \;-\; \tan^{-1}\!\sqrt{M^2-1}$

This function gives the angle (in radians or degrees, depending on your table) through which a flow at Mach 1 must expand to reach Mach number $M$. Note that $\nu = 0$ at $M = 1$.

Step-by-step procedure:

1. Look up (or compute) the upstream Prandtl-Meyer angle $\nu_1$ corresponding to $M_1$.
2. Add the deflection angle to get the downstream value:

$\nu_2 = \nu_1 + \theta$

1. Invert the Prandtl-Meyer function to find $M_2$ from $\nu_2$. This is done with tables, charts, or a numerical solver since the function can't be inverted analytically.
2. Compute downstream properties using isentropic relations. Because total pressure $p_0$ and total temperature $T_0$ are constant across the expansion fan:

$\frac{p_2}{p_1} = \frac{(1 + \frac{\gamma-1}{2}M_1^2)^{\gamma/(\gamma-1)}}{(1 + \frac{\gamma-1}{2}M_2^2)^{\gamma/(\gamma-1)}}$

The same pattern applies for temperature and density ratios.

Wave interactions with boundaries

Waves don't just disappear when they hit a wall or a free boundary. How they reflect matters for real designs.

Oblique shock reflection from a solid wall:

• Regular reflection: The incident shock hits the wall and produces a reflected shock. The flow downstream of the reflected shock is turned back parallel to the wall. You solve for the reflected shock by treating the post-incident-shock flow as a new "upstream" condition and finding the $\beta$ that turns the flow back by the required angle.
• Mach reflection: If the required deflection for regular reflection exceeds the maximum deflection angle at the local Mach number, regular reflection is impossible. Instead, a normal shock segment (the Mach stem) forms perpendicular to the wall, connecting the incident and reflected shocks. The flow behind the Mach stem is subsonic.

Expansion wave reflection from a solid wall:

• An expansion fan reflects off a solid boundary as another expansion fan.
• The net effect is that the flow downstream of the reflected fan is again parallel to the wall, but at a higher Mach number and lower pressure than the incoming flow.

Applications in supersonic design

Supersonic airfoils use oblique shocks and expansion waves together to generate lift while managing drag:

• A diamond (double-wedge) airfoil is a classic example. The sharp leading edge creates attached oblique shocks on both surfaces. At the midpoint, the surface angle changes and expansion fans form. At the trailing edge, additional waves adjust the flow back to the freestream direction.
• The pressure difference between the upper and lower surfaces produces lift. Wave drag arises because the shocks are irreversible (entropy-producing), so minimizing shock strength is a central design goal.

Supersonic nozzles (converging-diverging, or De Laval nozzles) accelerate flow from subsonic to supersonic:

• The throat reaches $M = 1$, and the diverging section accelerates the flow further.
• If the nozzle is perfectly expanded, the exit pressure matches the back pressure and the flow exits cleanly. If not, oblique shocks or expansion fans form at the nozzle exit to adjust the pressure, producing either overexpanded or underexpanded jet patterns.

Understanding where shocks and expansion waves form in these systems, and how strong they are, is what lets you predict forces, heating, and performance in any supersonic design problem.