A Supersonic Misconception

When studying injector design, an interesting question arises: why are most injectors designed for subsonic flow rather than supersonic? At first glance, supersonic gas injection seems advantageous—it could create choked flow, effectively acting as a one-way valve. This might allow us to reduce injector pressure drop while maintaining the same stability margin, thereby lowering upstream pressure requirements.

The hypothetical reasoning goes:

1. If the injector orifice is designed to choke (reach Mach 1 at the throat), flow becomes independent of downstream pressure fluctuations.
2. This choked condition acts as an acoustic barrier, isolating combustion chamber oscillations from the feed system.
3. Therefore, we could achieve the same combustion stability with lower injector ΔP, reducing upstream pressure demands.

In theory, this would be a win-win: better stability at lower cost.

However, this is fundamentally wrong. There’s no rocket injector uses supersonic injection, due to reasons below:

What Injectors Actually Do

According to Sutton & Biblarz (Ch. 8.1, p. 276):

“The functions of an injector are similar to those of a carburetor of an internal combustion engine. The injector has to introduce and meter the flow of liquid propellants to the combustion chamber, cause the liquids to be broken up into small droplets (a process called atomization), and distribute and mix the propellants in such a manner that a correctly proportioned mixture of fuel and oxidizer will result, with uniform propellant mass flow and composition over the chamber cross section.”

These major functions, metering, atomization, uniform distribution require a relatively stable, subsonic flow field. Supersonic flow would completely destroy these capabilities due to its extremely short residence time and complex shock structures.

The Core Injector Equations

Liquid Injection

For liquid propellants, injector orifice flow is governed by the single-phase, incompressible flow equation (Eq. 8-2, p. 281):

$$\dot{m} = C_d A \sqrt{2 \rho \Delta p}$$

where:

• $\dot{m}$ = mass flow rate
• $C_d$ = discharge coefficient (typically 0.6–0.8)
• $A$ = orifice area
• $\rho$ = propellant density
• $\Delta p$ = pressure drop across the injector

This equation assumes incompressible, subsonic flow. It works for liquids because density remains approximately constant.

Gas Injection

For gaseous propellants, the situation is different. At first glance, one might think the choked flow equation should be used:

$$\dot{m} = \frac{A p_t}{\sqrt{T_t}} \sqrt{\frac{\gamma}{R}} \left( \frac{\gamma+1}{2} \right)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

However, this is the choked flow formula, applicable only when the orifice has reached Mach 1. For subsonic gas injection, the correct approach uses the general compressible flow equation:

$$\dot{m} = A \cdot p_t \cdot \sqrt{\frac{\gamma}{RT_t}} \cdot M \left( 1 + \frac{\gamma-1}{2}M^2 \right)^{-\frac{\gamma+1}{2(\gamma-1)}}$$

where:

• $p_t$ = total (stagnation) pressure
• $T_t$ = total (stagnation) temperature
• $M$ = Mach number at the orifice exit
• $\gamma$ = specific heat ratio
• $R$ = specific gas constant

This equation is valid for any subsonic Mach number as well as sonic, when $M=1$. Actual injector designs keep $M < 1$, typically $M = 0.2$–$0.4$ for gas jets.

When gas is injected at supersonic speed, the supersonic jet entering the subsonic chamber forms a normal shock. Total pressure drops drastically, and the nonuniform flow field is incompatible with proper mixing. This led to the drop of combustion efficiency $\eta_c$, characteristic velocity $c^*$, which causing reduce of $I_{sp}$.

This supersonic flow creates extreme high heat flux, which is far exceeding regenerative, film, or ablative cooling capacity. In addition, there is the potential oscillation problem caused by shock waves.

Most importantly, it does not reduce the pressure drop requirement. To achieve choked flow:

$$\frac{p_{\text{upstream}}}{p_c} \geq \left(\frac{\gamma+1}{2}\right)^{\frac{\gamma}{\gamma-1}} \approx 1.89 \quad (\gamma=1.4)$$

Upstream pressure must be ≥1.89× $p_c$. The “ΔP savings” becomes an illusion.

This is why no engine uses supersonic injection.

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References:

• Sutton, G.P., & Biblarz, O. (2017). Rocket Propulsion Elements (8th ed.). Wiley. (Ch. 3, 8, 9)
• Harrje, D.T., & Reardon, F.H. (1972). Liquid Propellant Rocket Combustion Instability. NASA SP-194.