3.3 Isentropic flow

Isentropic flow assumptions

Isentropic flow is one of the most useful idealizations in compressible aerodynamics. By assuming the flow is both adiabatic (no heat transfer) and reversible (no friction or dissipation), you get a flow where entropy stays constant. That single constraint lets you link pressure, temperature, and density changes to Mach number through clean, closed-form equations.

These assumptions obviously don't hold perfectly in real flows, but they're surprisingly accurate in regions away from boundary layers and shock waves. Most nozzle and diffuser design starts with isentropic analysis, then accounts for real-world losses afterward.

Adiabatic process

No heat is transferred between the fluid and its surroundings. This lets you apply the first law of thermodynamics directly to relate changes in temperature, pressure, and density without worrying about external heat sources or sinks.

Reversible process

The flow has no irreversible losses from friction, turbulence, or mixing. Because no entropy is generated, the entropy of the fluid stays constant throughout the flow. That's where the name "isentropic" comes from: iso (constant) + entropic (entropy).

Inviscid flow

The fluid is treated as having zero viscosity. This eliminates boundary layers, flow separation, and all viscous shear effects. The governing equations reduce from the full Navier-Stokes equations to the simpler Euler equations.

Steady flow

Flow properties at any point don't change with time. Mathematically, all partial derivatives with respect to time are zero. This lets you use steady-state forms of the continuity, momentum, and energy equations, which is what makes nozzle and diffuser analysis tractable.

Isentropic flow properties

Under isentropic conditions, every flow property can be expressed as a function of a single variable: the local Mach number. The key relations connect static properties (what you'd actually measure in the moving flow) to stagnation properties (what you'd measure if you brought the flow to rest isentropically).

Stagnation vs. static properties

Stagnation (total) properties are the temperature, pressure, and density the fluid would have if it were brought to rest without any losses. Static properties are the actual local values in the moving flow.

The isentropic relations connecting them are:

$\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2} M^2$

$\frac{p_0}{p} = \left(1 + \frac{\gamma - 1}{2} M^2\right)^{\frac{\gamma}{\gamma - 1}}$

$\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma - 1}{2} M^2\right)^{\frac{1}{\gamma - 1}}$

Here, $\gamma$ is the ratio of specific heats (1.4 for air), and $M$ is the local Mach number. Notice that the pressure and density ratios are just the temperature ratio raised to different powers. That comes directly from the isentropic relation $p / \rho^\gamma = \text{const}$.

Mach number effects

The Mach number $M = V/a$ (flow velocity divided by local speed of sound) is the single most important parameter in compressible flow. The isentropic relations above show that as $M$ increases, static temperature, pressure, and density all drop relative to their stagnation values.

• Subsonic ($M < 1$): Compressibility effects are mild. Below about $M = 0.3$, density changes are less than ~5%, and the flow is effectively incompressible.
• Sonic ($M = 1$): The flow velocity equals the local speed of sound. This is the dividing line where the flow behavior changes fundamentally.
• Supersonic ($M > 1$): Compressibility dominates. Density, temperature, and pressure drop significantly, and the area-velocity relationship reverses (more on that below).

Critical conditions

Critical conditions (denoted with a superscript $*$) refer to the state where $M = 1$. For air with $\gamma = 1.4$:

• Critical temperature ratio: $T^*/T_0 = 2/(\gamma + 1) = 0.8333$
• Critical pressure ratio: $p^*/p_0 = (2/(\gamma + 1))^{\gamma/(\gamma - 1)} = 0.5283$
• Critical density ratio: $\rho^*/\rho_0 = (2/(\gamma + 1))^{1/(\gamma - 1)} = 0.6340$

These values are worth memorizing. They tell you the exact conditions at the throat of a choked nozzle and set the benchmark for whether a nozzle will reach sonic flow.

Isentropic flow in nozzles

Nozzles accelerate or decelerate flow by changing the cross-sectional area. The key governing relationship is the area-velocity relation, which for isentropic flow states:

$\frac{dA}{A} = (M^2 - 1)\frac{dV}{V}$

This equation has a critical implication: the effect of an area change on velocity reverses depending on whether the flow is subsonic or supersonic.

• Subsonic flow ($M < 1$): $M^2 - 1 < 0$, so decreasing area increases velocity (just like a garden hose nozzle).
• Supersonic flow ($M > 1$): $M^2 - 1 > 0$, so increasing area increases velocity. This is counterintuitive but fundamental.

Converging nozzles

A converging nozzle has a decreasing cross-sectional area in the flow direction. For subsonic inlet flow, the fluid accelerates as the area shrinks. The maximum Mach number occurs at the exit.

If the pressure ratio $p_{\text{exit}}/p_0$ is above the critical value (0.5283 for air), the exit flow is subsonic and the exit pressure matches the back pressure. If the back pressure drops to or below the critical ratio, the exit reaches $M = 1$ and the nozzle is choked. Lowering the back pressure further doesn't increase the mass flow rate or change conditions inside the nozzle.

Diverging nozzles

A diverging section alone can only do one of two things, depending on the inlet conditions:

• If the inlet flow is subsonic, the diverging section acts as a diffuser (decelerates the flow, increases pressure).
• If the inlet flow is supersonic, the diverging section accelerates the flow to higher Mach numbers.

You cannot accelerate a subsonic flow to supersonic speeds with a diverging section alone.

Converging-diverging (CD) nozzles

A CD nozzle is the only geometry that can accelerate flow from subsonic to supersonic. Here's how it works:

1. Subsonic flow enters the converging section and accelerates.
2. The flow reaches $M = 1$ at the throat (minimum area).
3. The flow continues to accelerate in the diverging section, now at supersonic speeds.

The exit Mach number is determined by the area ratio $A_{\text{exit}}/A^*$, where $A^*$ is the throat area. The area-Mach relation is:

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

For any area ratio greater than 1, this equation has two solutions: one subsonic and one supersonic. Which solution the flow actually follows depends on the back pressure.

Choked flow

Choked flow occurs when $M = 1$ at the throat. Once choked, the mass flow rate is at its maximum and is given by:

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

Two things to notice here:

• The mass flow rate depends only on stagnation conditions ($p_0$, $T_0$) and the throat area $A^*$. Downstream pressure doesn't matter.
• To increase mass flow through a choked nozzle, you must either raise $p_0$, lower $T_0$, or physically enlarge the throat.

This is why rocket engines and jet engines operate with choked nozzles: it gives predictable, maximum mass flow regardless of ambient conditions.

Isentropic flow in diffusers

Diffusers do the opposite of nozzles: they decelerate the flow and convert kinetic energy into a static pressure rise. The same area-velocity relation governs their behavior, just applied in reverse.

Subsonic diffusers

A subsonic diffuser is simply an increasing-area duct. As the area grows, subsonic flow decelerates and static pressure rises. The performance depends on:

• Area ratio: Larger ratios give more deceleration but risk flow separation.
• Wall divergence angle: Too steep an angle causes the boundary layer to separate, destroying the pressure recovery. Typical half-angles are kept below about 7°.
• Inlet Mach number: Higher inlet Mach numbers mean more kinetic energy to recover.

Subsonic diffusers are found in wind tunnel return sections and the intake ducts of subsonic jet engines.

Supersonic diffusers

Decelerating a supersonic flow to subsonic speeds is much harder than the subsonic case. In practice, the deceleration involves shock waves, which are inherently non-isentropic. A typical supersonic diffuser uses:

1. One or more oblique shocks to gradually reduce the Mach number.
2. A normal shock to bring the flow from (weakened) supersonic to subsonic.
3. A subsonic diffuser section to further decelerate and recover pressure.

Each shock causes a total pressure loss, so the design goal is to minimize shock strength. A series of weak oblique shocks loses far less total pressure than a single strong normal shock.

Normal shock waves

A normal shock is a very thin region (on the order of a few mean free paths) where the flow abruptly transitions from supersonic to subsonic. Across a normal shock:

• Mach number drops ($M_1 > 1 \rightarrow M_2 < 1$)
• Static pressure, temperature, and density all increase
• Total (stagnation) pressure decreases (entropy increases, so the process is not isentropic)

The stronger the incoming supersonic flow (higher $M_1$), the larger the jumps in properties and the greater the total pressure loss. Normal shock relations are tabulated in normal shock tables and are not isentropic, but they frequently appear alongside isentropic analysis because real supersonic systems almost always involve shocks somewhere.

Isentropic flow with area change

The connection between area change and flow properties is central to compressible flow design. Every variable (velocity, pressure, temperature, density) can be tracked through a variable-area duct using the isentropic relations plus conservation of mass.

Mass flow rate

The mass flow rate at any cross-section is:

$\dot{m} = \rho V A$

Conservation of mass requires $\dot{m}$ to be constant along the duct. Since $\rho$ and $V$ both change with Mach number, the area must adjust to keep their product times $A$ constant. This is what drives the shape of nozzles and diffusers.

Maximum mass flow

The maximum mass flow through a duct occurs when the flow is choked ($M = 1$ at the minimum area). This is a hard upper limit set by the stagnation conditions and throat geometry.

To increase the maximum mass flow, you have three options:

• Increase stagnation pressure $p_0$ (compresses more mass into the same volume)
• Decrease stagnation temperature $T_0$ (increases density for the same pressure)
• Increase the throat area $A^*$

Sonic flow conditions

Sonic conditions ($M = 1$) can only occur at a location where the area is a local minimum (a throat) or at a location with a normal shock. You cannot have $M = 1$ at an arbitrary point in a smoothly varying duct.

At sonic conditions, the flow velocity equals the local speed of sound, and the mass flow per unit area reaches its maximum possible value. This is the physical reason behind choking: no more mass can be pushed through that cross-section at those stagnation conditions.

Compressible flow tables

Before computers, engineers relied on pre-tabulated values to solve compressible flow problems. These tables are still widely used because they make problem-solving fast and build physical intuition about how properties scale with Mach number.

Isentropic flow tables

These are the most commonly used tables in this unit. For a given Mach number and $\gamma$, they list:

• $T/T_0$, $p/p_0$, $\rho/\rho_0$ (static-to-stagnation ratios)
• $A/A^*$ (area ratio relative to the sonic throat)

How to use them: If you know the Mach number, look up the ratios directly. If you know an area ratio or pressure ratio, find the corresponding Mach number (remembering that area ratios have two solutions: one subsonic, one supersonic).

Normal shock tables

These give property changes across a normal shock as a function of the upstream Mach number $M_1$:

• Downstream Mach number $M_2$
• Pressure ratio $p_2/p_1$, temperature ratio $T_2/T_1$, density ratio $\rho_2/\rho_1$
• Total pressure ratio $p_{02}/p_{01}$ (always less than 1, reflecting the entropy increase)

Rayleigh flow tables

These apply to frictionless flow with heat addition or removal in a constant-area duct. They're used for analyzing combustion chambers and heat exchangers. Rayleigh flow is not isentropic (heat transfer changes entropy), but the tables are often introduced alongside isentropic tables for comparison.

Fanno flow tables

These apply to adiabatic flow with friction in a constant-area duct. They're used for analyzing flow in long pipes and ducts where wall friction is significant. Like Rayleigh flow, Fanno flow is not isentropic, but it's a useful companion model.

Applications of isentropic flow

Wind tunnels

Supersonic wind tunnels use CD nozzles to accelerate flow to the desired test Mach number. The area ratio $A_{\text{test}}/A^*$ is set by the isentropic area-Mach relation to produce the target Mach number in the test section. Compressible flow tables are used to determine the required nozzle contour and the stagnation conditions needed to achieve the correct Mach and Reynolds numbers simultaneously.

Rocket nozzles

Rocket nozzles are classic CD nozzles designed to expand combustion gases from high-pressure, low-velocity conditions in the chamber to high-velocity, low-pressure conditions at the exit. The exit Mach number is set by the area ratio $A_e/A^*$, and the thrust depends on how well the exit pressure matches the ambient pressure. Overexpansion (exit pressure below ambient) and underexpansion (exit pressure above ambient) both reduce efficiency compared to the ideal matched condition.

Jet engines

Isentropic relations are used throughout jet engine design:

• Compressor stages: Each stage increases pressure roughly isentropically; real efficiency is compared against the isentropic ideal.
• Turbine stages: Each stage extracts work; isentropic efficiency quantifies how close the real expansion is to the ideal.
• Exhaust nozzle: Designed as a converging (subsonic exit) or CD nozzle (supersonic exit) depending on the engine type and flight regime.

Supersonic inlets

Supersonic inlets must slow incoming air from the flight Mach number to low subsonic speeds before it enters the compressor. The design combines isentropic compression (through carefully angled ramp surfaces that produce oblique shocks) with a final normal shock and subsonic diffuser. The ramp angles and throat area are optimized using isentropic relations and shock tables to maximize total pressure recovery across the full range of operating Mach numbers.