6.4 Normal and oblique shock waves

Normal and oblique shock waves are fundamental phenomena in supersonic flow. They form when supersonic flow hits an obstruction or changes direction, producing abrupt jumps in pressure, density, and temperature. Understanding these shocks is essential for designing supersonic aircraft, engine inlets, and wind tunnels.

Both shock types obey the same conservation laws and are described by the Rankine-Hugoniot relations. The critical difference: normal shocks always slow the flow to subsonic speeds, while oblique shocks can leave the downstream flow supersonic. This distinction drives much of supersonic aerodynamic design, from inlet compression systems to thin airfoil analysis.

Characteristics of normal shock waves

Formation in supersonic flow

Normal shock waves form when supersonic flow (upstream Mach number $M_1 > 1$) encounters an obstruction or a sudden change in flow conditions. You'll see them in supersonic wind tunnels, rocket nozzle exits, and aircraft inlet systems. The shock itself is extremely thin, typically only a few molecular mean free paths across, yet it produces dramatic changes in every flow property.

Discontinuous changes across the shock

Across a normal shock, flow properties change nearly instantaneously:

• Pressure, density, and temperature all increase sharply
• Velocity drops from supersonic to subsonic
• These jumps happen over such a small distance that we treat them as mathematical discontinuities in the flow field

The gas molecules simply cannot adjust gradually to the imposed conditions, so the flow "snaps" to a new equilibrium state.

Increase in pressure, density, and temperature

The magnitude of these increases depends on two things: the upstream Mach number $M_1$ and the specific heat ratio $\gamma$ of the gas. Higher upstream Mach numbers produce stronger shocks with larger property jumps. Physically, the kinetic energy of the supersonic flow is partially converted into thermal energy (internal energy of the gas), which is why temperature rises so significantly.

Decrease in velocity to subsonic

The downstream Mach number after a normal shock is always less than 1. This is not optional; it's a direct consequence of satisfying conservation of mass and momentum simultaneously. As $M_1$ increases, $M_2$ decreases, but it never reaches zero. For very strong shocks ($M_1 \to \infty$), the downstream Mach number approaches a limiting value of $\sqrt{\frac{\gamma - 1}{2\gamma}}$, which is approximately 0.378 for air ($\gamma = 1.4$).

Irreversible process and entropy rise

Normal shocks are inherently irreversible. Entropy increases across every normal shock, and this entropy rise quantifies the loss of usable energy in the flow. You cannot "undo" a shock without adding energy from an external source. This irreversibility is why stagnation pressure drops across a shock, and it's the reason engineers work hard to minimize shock strength in supersonic devices.

Governing equations for normal shocks

Conservation of mass, momentum, and energy

The three governing equations for a normal shock, applied between stations 1 (upstream) and 2 (downstream), are:

• Mass: $\rho_1 u_1 = \rho_2 u_2$
• Momentum: $p_1 + \rho_1 u_1^2 = p_2 + \rho_2 u_2^2$
• Energy: $h_1 + \frac{1}{2}u_1^2 = h_2 + \frac{1}{2}u_2^2$

Here $\rho$ is density, $u$ is velocity, $p$ is static pressure, and $h$ is specific enthalpy. These are the one-dimensional, steady, adiabatic forms with no body forces. Everything about normal shock behavior follows from solving these three equations simultaneously.

Rankine-Hugoniot relations

The Rankine-Hugoniot relations combine the conservation equations to express downstream properties purely in terms of $M_1$ and $\gamma$:

• Pressure ratio: $\frac{p_2}{p_1} = \frac{2\gamma M_1^2 - (\gamma - 1)}{\gamma + 1}$
• Density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2}{(\gamma - 1)M_1^2 + 2}$

These are the key working relations. Notice that the density ratio has an upper bound: as $M_1 \to \infty$, $\rho_2/\rho_1 \to (\gamma+1)/(\gamma-1)$, which equals 6 for air. Pressure, by contrast, grows without bound.

Mach number relations

The downstream Mach number is related to the upstream Mach number by:

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

You can verify that plugging in $M_1 = 1$ gives $M_2 = 1$ (infinitely weak shock, no change). For any $M_1 > 1$, you get $M_2 < 1$. As $M_1 \to \infty$, $M_2$ approaches $\sqrt{(\gamma - 1)/(2\gamma)}$.

Stagnation pressure ratio vs Mach number

Stagnation (total) pressure always decreases across a normal shock, reflecting the entropy rise. The ratio is:

$\frac{p_{02}}{p_{01}} = \left[\frac{(\gamma + 1)M_1^2}{2 + (\gamma - 1)M_1^2}\right]^{\frac{\gamma}{\gamma - 1}} \left[\frac{\gamma + 1}{2\gamma M_1^2 - (\gamma - 1)}\right]^{\frac{1}{\gamma - 1}}$

At $M_1 = 1$, this ratio equals 1 (no loss). It drops rapidly as $M_1$ increases. For example, at $M_1 = 2$ in air, $p_{02}/p_{01} \approx 0.72$, meaning 28% of the stagnation pressure is lost. This is exactly why supersonic inlets use oblique shocks first: they produce smaller losses per unit of flow turning.

Oblique shock waves

Formation by flow deflection

Oblique shock waves form when supersonic flow is deflected by a sharp corner, wedge, or compression ramp. The flow turns through a deflection angle $\theta$, and the shock sits at a wave angle $\beta$ measured from the upstream flow direction. Unlike normal shocks, oblique shocks are the more common type encountered in practice because most real surfaces deflect the flow gradually rather than stopping it head-on.

Oblique vs normal shock waves

The key differences are:

• An oblique shock sits at an angle $\beta$ to the flow; a normal shock is perpendicular ($\beta = 90°$)
• Downstream flow after an oblique shock can remain supersonic; after a normal shock it's always subsonic
• Property changes across an oblique shock are less severe for the same upstream Mach number

A normal shock is actually a special case of an oblique shock where $\beta = 90°$ and $\theta = 0$. Both satisfy the same conservation equations; the only difference is that for oblique shocks, you use the normal component of velocity ($u_1 \sin\beta$) in those equations.

Shock angle and deflection angle

The relationship between $\theta$, $\beta$, and $M_1$ is given by the $\theta$-$\beta$-$M$ relation:

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

For a given $M_1$ and $\theta$, this equation yields two solutions for $\beta$, corresponding to the weak and strong shock cases. The shock angle increases with both increasing deflection angle and increasing upstream Mach number. There is also a maximum deflection angle $\theta_{\max}$ for each $M_1$; beyond it, no attached oblique shock solution exists and a detached bow shock forms instead.

Weak vs strong shock solutions

For each combination of $M_1$ and $\theta$ (below $\theta_{\max}$), two oblique shock solutions exist:

• Weak shock: smaller $\beta$, lower pressure ratio, downstream flow usually remains supersonic ($M_2 > 1$)
• Strong shock: larger $\beta$, higher pressure ratio, downstream flow is subsonic ($M_2 < 1$)

In nearly all practical situations, the weak shock solution is the one that occurs. Strong shocks require special downstream boundary conditions (like high back pressure) to be realized. When solving problems, assume the weak solution unless told otherwise.

Oblique shock properties

Downstream Mach number and pressure

The trick to computing oblique shock properties is recognizing that only the velocity component normal to the shock undergoes the jump. You apply the normal shock relations using $M_{n1} = M_1 \sin\beta$ as the "effective" upstream Mach number.

The downstream Mach number satisfies:

$M_2^2 \sin^2(\beta - \theta) = \frac{1 + \frac{\gamma - 1}{2}M_1^2 \sin^2 \beta}{\gamma M_1^2 \sin^2 \beta - \frac{\gamma - 1}{2}}$

The pressure ratio is:

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

Pressure, density, and temperature ratios

All property ratios across an oblique shock depend on $M_1 \sin\beta$, not $M_1$ alone:

• Pressure ratio: $\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1}(M_1^2 \sin^2 \beta - 1)$
• Density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2 \sin^2 \beta}{2 + (\gamma - 1)M_1^2 \sin^2 \beta}$
• Temperature ratio: $\frac{T_2}{T_1} = \frac{p_2}{p_1} \cdot \frac{\rho_1}{\rho_2}$

All three ratios increase with increasing shock angle and upstream Mach number. At $\beta = 90°$, these reduce exactly to the normal shock relations.

Supersonic flow downstream of the shock

For weak oblique shocks, the downstream Mach number $M_2 > 1$. This is the crucial advantage over normal shocks: the flow stays supersonic, so you can place additional oblique shocks in series. Supersonic inlet designs exploit this by using a sequence of oblique shocks to compress the flow incrementally, with much less total pressure loss than a single normal shock would produce. The downstream flow direction is deflected by angle $\theta$ relative to the upstream direction.

Mach angle and Mach cone

The Mach angle $\mu$ is the angle of the weakest possible disturbance (a Mach wave) in supersonic flow:

$\sin \mu = \frac{1}{M}$

A Mach wave is the limiting case of an oblique shock as $\theta \to 0$ and the shock strength approaches zero. In three dimensions, Mach waves from a point source form a Mach cone with half-angle $\mu$. Any oblique shock must have a wave angle $\beta$ greater than the Mach angle $\mu$ of the upstream flow, meaning the shock always lies inside the Mach cone.

Oblique shock applications

Supersonic thin airfoil theory

Supersonic thin airfoil theory treats the disturbances caused by a thin, sharp-edged airfoil as small perturbations. The upper and lower surfaces each produce weak oblique shocks or expansion fans depending on the local surface angle. Using linearized oblique shock and expansion relations, you can calculate the pressure distribution and, from that, the lift and drag coefficients. This theory predicts that supersonic airfoils produce wave drag even in inviscid flow, unlike subsonic airfoils.

Supersonic inlet design

A well-designed supersonic inlet decelerates the incoming airflow to subsonic speeds before it reaches the engine compressor, while recovering as much stagnation pressure as possible. The typical approach:

1. Use one or more oblique shocks (via ramps or cones) to gradually slow and compress the flow
2. Terminate with a weaker normal shock to bring the flow subsonic
3. Use a subsonic diffuser to further decelerate the flow to the engine face

Each oblique shock produces a smaller stagnation pressure loss than a single normal shock at the same freestream Mach number. The more oblique shocks you use, the better the pressure recovery, but the inlet geometry becomes more complex.

Shock-expansion theory

Shock-expansion theory handles supersonic flow over airfoils with finite thickness and sharp corners more accurately than linearized thin airfoil theory. The method works as follows:

1. At the leading edge, an oblique shock forms (compression)
2. At convex corners, the flow undergoes a Prandtl-Meyer expansion (isentropic turning)
3. You apply the oblique shock relations and Prandtl-Meyer function sequentially around the body to find pressure on each surface segment
4. Integrate the pressure distribution to get lift and drag

Because the expansion fans are isentropic (no entropy increase), only the shocks contribute to wave drag.

Shock-boundary layer interaction

When an oblique shock impinges on a viscous boundary layer, the sudden adverse pressure gradient can cause significant problems:

• Boundary layer thickening as the flow decelerates
• Separation bubbles where the flow detaches from the surface and reattaches downstream
• Increased heat transfer in the reattachment region
• Unsteady oscillations of the shock and separation point

These interactions are particularly important in supersonic inlet design, where shock-induced separation can cause engine unstart, and on control surfaces, where separation degrades effectiveness. Managing these interactions often involves boundary layer bleed, vortex generators, or careful geometric shaping.

Detached and bow shocks

Blunt body shock formation

When a supersonic flow encounters a blunt body (like a spherical nose cone or rounded leading edge), the required deflection angle exceeds $\theta_{\max}$ for the given Mach number. No attached oblique shock can satisfy the $\theta$-$\beta$-$M$ relation, so instead a detached bow shock forms ahead of the body. This curved shock is nearly normal to the flow along the centerline and becomes progressively more oblique toward the periphery.

Subsonic region behind the bow shock

Because the bow shock is nearly normal at the centerline, the flow directly behind it is subsonic. This subsonic region, called the shock layer, has high pressure, density, and temperature. Moving away from the centerline, the shock becomes more oblique, and the post-shock flow transitions back to supersonic. The thickness of the subsonic shock layer depends on the body geometry and freestream Mach number: higher Mach numbers produce a thinner shock layer that wraps more tightly around the body.

Shock standoff distance

The shock standoff distance $\delta$ is the gap between the bow shock and the body surface along the centerline. It depends on:

• Body geometry: blunter bodies produce larger standoff distances
• Freestream Mach number: higher $M_1$ reduces the standoff distance
• Specific heat ratio $\gamma$: lower $\gamma$ (as in high-temperature dissociating flows) increases standoff distance

For simple geometries like spheres, empirical correlations exist, but complex shapes typically require numerical (CFD) solutions. A larger standoff distance means a thicker shock layer and generally higher drag.

Entropy layer and flow separation

The bow shock has varying strength across its span: strongest (normal) at the centerline, weakest (oblique) at the edges. This means each streamline passing through the shock experiences a different entropy rise. The result is an entropy layer near the body surface where streamlines that crossed the strongest part of the shock carry high entropy (low stagnation pressure, high temperature).

This entropy gradient, combined with the adverse pressure gradient on the body's aft portion, can trigger flow separation and recirculation. Controlling this separation is critical in the design of reentry vehicles, blunt atmospheric probes, and supersonic parachutes, where both aerodynamic heating and drag characteristics depend strongly on the shock layer structure.