9.5 Compressibility effects

Compressibility in Fluid Dynamics

Compressibility effects describe what happens when a fluid's density changes significantly in response to pressure changes. At low speeds, you can usually ignore density changes and treat air as incompressible. But as flow velocity climbs toward and beyond the speed of sound, density, pressure, and temperature all become tightly coupled, producing phenomena like shock waves and choked flow that are central to aerospace engineering.

The Mach number is the single most important parameter for deciding whether compressibility matters. This guide covers how it defines flow regimes, how the speed of sound works, how flow properties change in compressible flow, and what all of this means for aerodynamic and propulsion design.

Mach Number

Definition of Mach Number

The Mach number ($M$) is a dimensionless ratio of flow velocity to the local speed of sound:

$M = \frac{v}{a}$

where $v$ is the flow velocity and $a$ is the local speed of sound. Because $a$ depends on local temperature, the same aircraft at the same airspeed can have different Mach numbers at different altitudes.

Mach Number Regimes

Each regime has distinct physical behavior:

• Subsonic ($M < 1$): Flow velocity is below the speed of sound. Pressure disturbances can propagate upstream, so the flow "knows" about obstacles ahead of time. No shock waves form.
• Transonic ($M \approx 0.8 - 1.2$): The freestream may be subsonic, but local acceleration over curved surfaces can push regions of the flow past $M = 1$. Both subsonic and supersonic regions coexist, and shock waves first appear on the body surface.
• Supersonic ($1 < M < 5$): The entire flow field exceeds the speed of sound. Shock waves and expansion fans are the dominant features. Pressure disturbances cannot travel upstream.
• Hypersonic ($M > 5$): Extremely strong compressibility effects. Shock layers become very thin and temperatures become high enough to cause real-gas effects like molecular dissociation and ionization.

Mach Number and the Onset of Compressibility

Compressibility effects don't switch on at a single threshold; they grow gradually:

• Below about $M = 0.3$, density changes are less than ~5%, so treating the flow as incompressible introduces negligible error.
• Between $M = 0.3$ and $M = 0.8$, density variations grow and should be accounted for in accurate analyses.
• Above $M = 0.8$, compressibility dominates. Density, pressure, and temperature are all strongly coupled to velocity changes.

Speed of Sound

Definition

The speed of sound ($a$) is the speed at which small pressure disturbances propagate through a fluid. For an ideal gas:

$a = \sqrt{\gamma R T}$

where $\gamma$ is the ratio of specific heats (1.4 for air), $R$ is the specific gas constant (287 J/(kg·K) for air), and $T$ is the absolute temperature in Kelvin.

This means the speed of sound depends only on temperature for a given gas. At sea level on a standard day ($T \approx 288$ K), the speed of sound in air is about 340 m/s. At cruise altitude where temperatures drop to around 220 K, it falls to roughly 295 m/s.

Factors Affecting Speed of Sound

• Temperature: Higher temperature means faster molecular motion, so sound propagates faster. This is the dominant factor in gases.
• Gas composition: Different gases have different $\gamma$ and $R$ values. Helium, for example, has a much higher speed of sound than air because of its low molecular mass (high $R$).
• Phase of matter: Sound travels faster in liquids and solids than in gases because of their much higher stiffness relative to density. In water at room temperature, $a \approx 1{,}480$ m/s, roughly four times faster than in air.

Compressible vs. Incompressible Flow

Incompressible Flow

Incompressible flow assumes density stays constant ($\rho = \text{const}$). This decouples the energy equation from the continuity and momentum equations, making the math much simpler. It's a valid approximation for:

• Most liquid flows (water in pipes, open-channel flow)
• Gas flows at $M < 0.3$ (low-speed airflow around cars, buildings, small drones)

Compressible Flow

In compressible flow, density varies significantly and is coupled to pressure, temperature, and velocity. You must solve the full set of conservation equations together with an equation of state. Compressible flow features include:

• Shock waves: Thin regions where properties change nearly discontinuously
• Expansion fans: Gradual, continuous regions where supersonic flow accelerates and pressure drops
• Choked flow: A condition where the mass flow rate through a restriction reaches its maximum at $M = 1$ at the throat

Typical compressible flow situations: high-speed flight, rocket nozzles, gas turbine internals, and high-pressure gas pipelines.

Isentropic Flow Relations

For isentropic flow (adiabatic and reversible, meaning no shock waves or friction), the relationships between local properties and their stagnation (total) values are functions of Mach number alone. Stagnation properties (subscript 0) are the values the flow would have if brought to rest isentropically.

Temperature ratio:

$\frac{T}{T_0} = \left(1 + \frac{\gamma - 1}{2}M^2\right)^{-1}$

Pressure ratio:

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

Density ratio:

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

As $M$ increases, static temperature, pressure, and density all decrease relative to their stagnation values. The kinetic energy of the flow increases at the expense of internal energy.

Normal Shock Relations

A normal shock wave is a very thin region (on the order of a few mean free paths) oriented perpendicular to the flow, across which properties change abruptly. Shocks are irreversible, so entropy increases and stagnation pressure decreases across them. The flow upstream is supersonic; the flow downstream is always subsonic.

Density ratio:

$\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2}{(\gamma - 1)M_1^2 + 2}$

Pressure ratio:

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

Temperature ratio:

$\frac{T_2}{T_1} = \frac{\left[2 + (\gamma - 1)M_1^2\right]\left[2\gamma M_1^2 - (\gamma - 1)\right]}{(\gamma + 1)^2 M_1^2}$

Here, subscript 1 is upstream and subscript 2 is downstream. Across a normal shock:

• Pressure, density, and temperature all increase
• Velocity decreases (from supersonic to subsonic)
• Stagnation pressure decreases (entropy is generated)
• Stagnation temperature remains constant (for a calorically perfect gas)

Density, Pressure, and Temperature Effects on Flow

Density, pressure, and temperature are linked through the ideal gas equation of state:

$p = \rho R T$

In compressible flow, a change in any one of these quantities forces changes in the others. Some key consequences:

• Mass flow rate through a cross-section is $\dot{m} = \rho v A$. Because density varies, you can't just track velocity and area; all three matter.
• Dynamic pressure ($\frac{1}{2}\rho v^2$) doesn't capture the full picture in compressible flow the way it does in incompressible flow. The compressible form of Bernoulli's equation involves enthalpy rather than just pressure and velocity.
• Speed of sound changes locally because $a = \sqrt{\gamma R T}$. As the flow accelerates and static temperature drops, the local speed of sound drops too, which means the local Mach number rises even faster than velocity alone would suggest.
• At very high temperatures (hypersonic regime), air molecules begin to dissociate and ionize. The calorically perfect gas assumption ($\gamma = \text{const}$) breaks down, and you need real-gas models.

Compressible Flow Equations

The governing equations for compressible flow are the conservation of mass, momentum, and energy, supplemented by an equation of state.

Continuity (Conservation of Mass)

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$

For steady, one-dimensional flow this simplifies to:

$\rho_1 v_1 A_1 = \rho_2 v_2 A_2$

Unlike incompressible flow where $v_1 A_1 = v_2 A_2$, here density must be tracked because it changes along the flow.

Momentum (Conservation of Momentum)

$\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \mathbf{f}$

where $\boldsymbol{\tau}$ is the viscous stress tensor and $\mathbf{f}$ represents body forces. For inviscid, steady, one-dimensional flow:

$\rho v \frac{dv}{dx} = -\frac{dp}{dx}$

This shows that pressure gradients directly drive velocity changes, and vice versa.

Energy (Conservation of Energy)

$\rho \frac{D}{Dt}\left(e + \frac{v^2}{2}\right) = -\nabla \cdot (p\mathbf{v}) + \nabla \cdot (k \nabla T) + \Phi$

where $e$ is internal energy per unit mass, $k$ is thermal conductivity, and $\Phi$ is the viscous dissipation function. For steady, adiabatic, inviscid flow, this reduces to the statement that total enthalpy is constant along a streamline:

$h_0 = h + \frac{v^2}{2} = \text{const}$

This is the compressible equivalent of Bernoulli's equation and is one of the most useful relations in compressible aerodynamics.

Compressibility Effects on Aerodynamics

Lift and Drag

Compressibility changes how lift and drag scale with speed:

• At low subsonic Mach numbers, lift and drag coefficients are roughly constant. The Prandtl-Glauert correction approximates the compressibility effect on pressure coefficient: $C_p = \frac{C_{p,\text{inc}}}{\sqrt{1 - M^2}}$, where $C_{p,\text{inc}}$ is the incompressible value. This predicts that pressure differences (and therefore lift) grow as $M$ increases.
• The critical Mach number ($M_{\text{cr}}$) is the freestream Mach number at which the flow first reaches $M = 1$ somewhere on the body. Beyond this point, local supersonic regions and shock waves appear.
• The drag divergence Mach number is slightly above $M_{\text{cr}}$. Here, shock waves on the surface become strong enough to cause boundary-layer separation, and drag rises sharply. This rapid drag increase was historically called the "sound barrier."
• In fully supersonic flow, wave drag is an additional drag component that exists even for inviscid flow, caused by the energy carried away by shock waves.

Flow Patterns

• In subsonic flow, streamlines smoothly adjust ahead of a body because pressure disturbances propagate in all directions.
• In supersonic flow, disturbances are confined within a Mach cone of half-angle $\mu = \arcsin(1/M)$. The flow upstream of the cone is completely unaffected.
• Shock-boundary layer interaction is a critical phenomenon in transonic and supersonic flow. A shock wave impinging on a boundary layer can cause it to separate, leading to large increases in drag and potential loss of control effectiveness.

Aircraft Design Strategies

Designers use several techniques to manage compressibility effects at high speeds:

• Swept wings: Sweeping the wing back reduces the effective Mach number component perpendicular to the leading edge, delaying the onset of transonic drag rise. Most commercial jets use sweep angles of 25°–35°.
• Thin airfoil sections: Thinner profiles produce weaker shocks at transonic speeds, reducing wave drag.
• Area ruling (Whitcomb area rule): The total cross-sectional area of the aircraft should vary smoothly along its length. Abrupt area changes produce stronger shocks. The characteristic "wasp waist" fuselage shape of some supersonic aircraft comes from area ruling.
• Supercritical airfoils: These profiles are designed to keep the flow over the upper surface closer to $M = 1$ over a longer region with a weaker terminating shock, reducing wave drag in the transonic regime.

Compressibility Effects in Propulsion Systems

Jet Engines

Compressibility is a factor throughout a jet engine:

• Compressor stages: Each blade row accelerates and decelerates the flow, and at high pressure ratios the relative Mach numbers at blade tips can approach or exceed 1. This limits the pressure ratio achievable per stage (typically 1.3–1.5 for subsonic stages, up to ~2 for transonic fan stages) and is why modern engines need many compressor stages.
• Turbine stages: The flow expands through the turbine at high velocities. Shock waves can form between blade passages, reducing efficiency and causing vibration that can lead to structural fatigue.
• Careful blade profiling and stage matching are required to manage these compressibility effects across the engine's operating range.

Rocket Nozzles

Rocket nozzles are a textbook application of compressible flow:

1. High-pressure, high-temperature combustion gases enter the converging section, where the flow accelerates toward $M = 1$.
2. The flow reaches exactly $M = 1$ at the throat (the narrowest cross-section). This is the choked flow condition, and it sets the maximum mass flow rate for given upstream conditions.
3. In the diverging section, the supersonic flow continues to accelerate as the area increases. Pressure, temperature, and density all drop as kinetic energy increases.

This converging-diverging geometry is called a De Laval nozzle. The ratio of exit area to throat area determines the exit Mach number. Nozzle designers optimize this ratio along with the expansion ratio to maximize thrust and specific impulse for the intended operating altitude. If the nozzle is over-expanded or under-expanded relative to ambient pressure, shock waves or expansion fans form at the exit, reducing efficiency.

Engine Performance

Compressibility affects overall propulsion performance in several ways:

• Ram compression at high flight speeds: As flight Mach number increases, the inlet decelerates the air, converting kinetic energy to pressure. At $M = 3$, ram compression alone can produce a pressure ratio of about 37:1, which is why ramjets need no mechanical compressor.
• Inlet design must manage shock waves to decelerate supersonic freestream air to subsonic speeds at the compressor face with minimal stagnation pressure loss. Supersonic inlets use carefully positioned oblique shocks followed by a normal shock.
• At hypersonic speeds ($M > 5$), decelerating the flow to subsonic speeds would produce unacceptably high temperatures. Scramjets (supersonic combustion ramjets) solve this by keeping the flow supersonic through the combustor, which introduces its own set of compressibility challenges for fuel mixing and flame stabilization.