3.6 Nozzle flow

Nozzle flow describes how a compressible fluid behaves as it passes through constricted or diverging passages. This topic is central to designing propulsion systems, wind tunnels, and any application involving high-speed gas flow. The core challenge is understanding how changes in nozzle geometry interact with pressure, temperature, and velocity across subsonic and supersonic regimes.

Nozzle flow fundamentals

A nozzle is a shaped duct that converts pressure energy into kinetic energy (or vice versa). The behavior of the flow depends heavily on whether it's subsonic or supersonic, and the analysis relies on compressible flow equations simplified by the isentropic flow assumption.

Subsonic vs supersonic flow

The dividing line between these two regimes is the local speed of sound, expressed through the Mach number ($M = V/a$, where $a$ is the local speed of sound).

• Subsonic flow ($M < 1$): Flow properties change smoothly and continuously. Density variations are relatively small, so the flow can sometimes be approximated as incompressible. Disturbances can propagate upstream, meaning downstream conditions influence the entire flow field.
• Supersonic flow ($M > 1$): Density changes are large, and compressibility effects dominate. Shock waves (thin regions of abrupt property changes) can form. Disturbances cannot travel upstream past the sonic point, which is why choked nozzles are insensitive to downstream pressure changes.

This upstream/downstream communication difference is what makes nozzle design so regime-dependent.

Compressible flow equations

When density variations matter, you need the full conservation equations for mass, momentum, and energy:

• Continuity: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$
• Momentum: $\rho \frac{D\vec{V}}{Dt} = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \vec{g}$
• Energy: $\rho \frac{De}{Dt} = -p(\nabla \cdot \vec{V}) + \overline{\overline{\tau}}:\nabla \vec{V} + \nabla \cdot (k\nabla T)$

These are closed by an equation of state such as the ideal gas law: $p = \rho R T$, where $R$ is the specific gas constant. For nozzle analysis, these equations are typically reduced to their steady, one-dimensional, inviscid forms.

Isentropic flow assumption

Isentropic means both adiabatic (no heat transfer) and reversible (no friction or entropy generation). Under this assumption, fluid properties are linked by simple algebraic relations:

• Pressure-density: $\frac{p}{p_0} = \left(\frac{\rho}{\rho_0}\right)^{\gamma}$
• Temperature-density: $\frac{T}{T_0} = \left(\frac{\rho}{\rho_0}\right)^{\gamma-1}$

Here, the subscript $0$ denotes stagnation (total) conditions, and $\gamma$ is the ratio of specific heats.

Real nozzle flows involve boundary layers and friction, so the isentropic assumption isn't perfectly accurate. But it provides an excellent first approximation for the core flow away from walls, and it's the foundation for nearly all nozzle design calculations.

Converging nozzles

A converging nozzle has a cross-sectional area that decreases in the flow direction. Its primary function is to accelerate subsonic flow.

Area-velocity relationship

For steady, one-dimensional flow, the continuity equation reduces to:

$\rho_1 A_1 V_1 = \rho_2 A_2 V_2$

As the area $A$ decreases, velocity $V$ must increase to conserve mass flow. This relationship holds straightforwardly in subsonic flow, where a smaller area always means a higher velocity.

The maximum velocity a converging nozzle can reach is $M = 1$ (sonic) at the narrowest cross-section, called the throat. You cannot accelerate a flow past Mach 1 with a converging-only geometry. To go supersonic, you need a diverging section downstream of the throat.

Pressure and temperature effects

As the flow accelerates through the nozzle, static pressure and temperature both drop. The isentropic relations expressed in terms of Mach number are:

• Pressure ratio: $\frac{p}{p_0} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{-\frac{\gamma}{\gamma-1}}$
• Temperature ratio: $\frac{T}{T_0} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{-1}$

The stagnation pressure $p_0$ and temperature $T_0$ remain constant throughout isentropic flow. Only the static values change as kinetic energy increases.

Mass flow rate calculation

The mass flow rate through a converging nozzle depends on the throat conditions. When the throat reaches sonic velocity, the mass flow rate is:

$\dot{m} = \frac{p_0}{\sqrt{T_0}} A^* \sqrt{\frac{\gamma}{R}} \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}$

The asterisk ($*$) denotes sonic ($M = 1$) conditions at the throat. Notice that $\dot{m}$ depends on $p_0$, $T_0$, and the throat area $A^*$, but not on the downstream pressure. Once the flow is choked, the mass flow rate is set entirely by upstream conditions and throat geometry.

Choked flow conditions

Choked flow occurs when $M = 1$ at the throat. Once this happens:

1. The mass flow rate reaches its maximum for the given $p_0$, $T_0$, and $A^*$.
2. Lowering the back pressure further does not increase $\dot{m}$ or change any upstream flow properties.
3. Information from downstream cannot propagate upstream past the sonic throat.

For air ($\gamma = 1.4$), choking occurs when the back-pressure-to-stagnation-pressure ratio drops to about 0.528. This critical ratio is $\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}$.

Diverging nozzles

A diverging nozzle has an increasing cross-sectional area. On its own, a diverging duct decelerates subsonic flow (acting as a diffuser). But when placed downstream of a sonic throat, it accelerates the flow to supersonic speeds.

De Laval nozzle design

The De Laval nozzle (named after Carl Gustaf Patrick de Laval) is a converging-diverging (CD) nozzle. It works in three stages:

1. Converging section: Subsonic flow accelerates toward the throat.
2. Throat: Flow reaches exactly $M = 1$.
3. Diverging section: Supersonic flow continues to accelerate as the area increases.

The key insight is that the area-velocity relationship reverses at $M = 1$. In subsonic flow, decreasing area speeds the flow up. In supersonic flow, increasing area speeds the flow up. The De Laval nozzle exploits this reversal. Its geometry is carefully contoured to minimize losses and achieve the desired exit Mach number.

Expansion and compression waves

In the diverging section, the flow can undergo expansion or compression depending on the back pressure:

• Expansion waves form when the flow needs to accelerate further. Pressure drops and Mach number increases. These are gradual, isentropic processes.
• Compression waves form when the back pressure is too high for fully supersonic flow. These can coalesce into shock waves if the pressure mismatch is large enough.

Shock waves in nozzles

When the back pressure exceeds the design value, a normal shock can form inside the diverging section. Across a normal shock:

• Pressure and density jump up sharply.
• Velocity and Mach number drop (the flow transitions from supersonic to subsonic).
• Entropy increases, so the process is irreversible and total pressure is lost.

Oblique shocks occur at the nozzle exit when the exit pressure doesn't match the ambient pressure and the flow encounters a geometric deflection. Oblique shocks are inclined at an angle to the flow and produce smaller property changes than normal shocks.

The location of a normal shock inside the nozzle depends on the back pressure. As back pressure decreases from the no-flow condition, the shock moves downstream toward the exit, eventually being pushed out of the nozzle entirely at the design condition.

Over-expanded vs under-expanded flow

These terms describe what happens when the nozzle exit pressure doesn't match the ambient pressure:

• Over-expanded ($p_e < p_{amb}$): The flow has expanded too much. Outside the nozzle, oblique shocks form to compress the flow back up to ambient pressure. This reduces efficiency and can cause flow separation inside the nozzle if the over-expansion is severe.
• Under-expanded ($p_e > p_{amb}$): The flow hasn't expanded enough. Expansion fans form outside the nozzle exit to further reduce the pressure. The remaining pressure energy that could have been converted to kinetic energy inside the nozzle is wasted, resulting in lost thrust.

Neither condition is ideal. Maximum performance occurs when $p_e = p_{amb}$.

Nozzle performance parameters

Several parameters quantify how well a nozzle converts pressure energy into directed kinetic energy.

Thrust and momentum considerations

The thrust equation for steady, one-dimensional flow is:

$F = \dot{m}(V_e - V_0) + (p_e - p_0)A_e$

• $\dot{m}$: mass flow rate
• $V_e$: exit velocity, $V_0$: inlet (or freestream) velocity
• $p_e$: exit pressure, $p_0$: ambient pressure
• $A_e$: exit area

The first term is momentum thrust (from accelerating the fluid). The second term is pressure thrust (from any pressure imbalance at the exit). At the design condition ($p_e = p_0$), pressure thrust is zero and all thrust comes from momentum change.

Specific impulse and efficiency

Specific impulse ($I_{sp}$) measures how efficiently a propulsion system uses propellant:

$I_{sp} = \frac{F}{\dot{m}g}$

where $g$ is gravitational acceleration (9.81 m/s²). Units are seconds. A higher $I_{sp}$ means more thrust per unit of propellant consumed. For example, liquid hydrogen/oxygen engines achieve $I_{sp}$ values around 450 s, while solid rocket motors are typically 250-280 s.

Nozzle efficiency ($\eta_n$) compares the actual kinetic energy at the exit to the kinetic energy that would result from ideal isentropic expansion to the same pressure ratio. Losses from boundary layers, flow non-uniformities, and incomplete expansion all reduce $\eta_n$ below 1.0. Well-designed rocket nozzles typically achieve $\eta_n > 0.95$.

Nozzle pressure ratio effects

The nozzle pressure ratio (NPR) is $p_0 / p_{amb}$. It determines the flow regime:

• NPR below the critical value (~1.89 for air): Flow remains subsonic throughout. No choking occurs.
• NPR at the critical value: Flow just reaches $M = 1$ at the throat (onset of choking).
• NPR above critical but below design value: Flow is choked at the throat and supersonic in part of the diverging section, but a normal shock exists inside the nozzle. The exit flow is subsonic.
• NPR at the design value: Fully isentropic supersonic flow throughout the diverging section, with $p_e = p_{amb}$.
• NPR above the design value: Under-expanded flow at the exit.

Optimum nozzle expansion

Optimum expansion means $p_e = p_{amb}$, which eliminates the pressure thrust term and maximizes overall efficiency. The nozzle geometry that achieves this is set by the area ratio $A_e / A^*$, which is a function of the design Mach number and $\gamma$:

$\frac{A_e}{A^*} = \frac{1}{M_e}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M_e^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$

In practice, a nozzle can only be perfectly expanded at one altitude (one ambient pressure). Rocket nozzles are typically designed for optimum expansion at some representative altitude along the trajectory, accepting over- or under-expansion at other points.

Nozzle flow applications

Rocket propulsion systems

Rocket engines are the most direct application of CD nozzle theory. Hot combustion gases (temperatures of 2500-3500 K) expand through the nozzle to generate thrust.

• Liquid-propellant rockets (e.g., SpaceX Merlin, RS-25) use bell-shaped nozzles with carefully contoured walls to maximize performance while keeping the nozzle length manageable.
• Solid-propellant rockets (e.g., Space Shuttle SRBs) often use simpler conical or mildly contoured nozzles, since the internal ballistics of the solid grain already constrain the design.

Nozzle area ratios for rocket engines range from about 10:1 (sea-level engines) to over 200:1 (upper-stage engines operating in near-vacuum).

Jet engine exhaust nozzles

Jet engines use nozzles matched to their operating speed range:

• Convergent nozzles are standard on subsonic transport aircraft (e.g., commercial airliners). They're simple, lightweight, and sufficient because the exhaust doesn't need to reach high supersonic speeds.
• Convergent-divergent nozzles with variable geometry are used on supersonic fighters and afterburning engines. The variable throat and exit areas allow the nozzle to operate efficiently across a wide range of throttle settings and flight Mach numbers.

Supersonic wind tunnels

Supersonic wind tunnels use CD nozzles to accelerate the test gas to a specific Mach number in the test section.

• Contoured nozzles designed using the method of characteristics produce uniform, parallel flow with minimal disturbances.
• Adjustable nozzles with flexible walls or movable blocks allow the facility to change the test section Mach number without swapping hardware.

The quality of the test data depends directly on how uniform the nozzle flow is, making nozzle design one of the most critical aspects of wind tunnel engineering.

Gas dynamic lasers

Gas dynamic lasers (GDLs) use CD nozzles for a very different purpose: rapid cooling. The gas mixture expands through the nozzle, dropping in temperature fast enough to create a population inversion (more molecules in an excited state than the ground state), which is the condition needed for lasing.

• Supersonic diffusers downstream of the laser cavity decelerate the flow and recover static pressure.
• Aerodynamic windows (thin gas films or porous walls) separate the laser cavity from the surroundings while minimizing optical distortion of the beam.