Compressibility and Mach number

For liquids such as water, density can often be treated as nearly constant. In gases such as air, increasing flow speed can couple changes in pressure, temperature and density strongly to the motion itself. This is the essence of compressible flow.

The key parameter is the Mach number M = V/a. At low Mach number density changes are often small, but as M approaches one, compressibility cannot be neglected and local supersonic regions or shock waves may appear [1,2].

Compressibility is the tendency of density to change when pressure changes. Gases are generally much more compressible than liquids, and in high-speed gas flows kinetic and internal energy are coupled through changes in temperature, pressure and density.

A gas does not always require a compressible-flow model. If density variations have negligible influence on the phenomenon, an incompressible approximation can still be useful. Pressure waves, nozzles, shock waves and high-speed vehicles, however, inherently involve density variation.

Incompressible approximation

Density changes are neglected for the purpose of the flow model.

Compressible flow

Density must be coupled to pressure, temperature or other thermodynamic variables.

The Mach number M is defined as the ratio of local flow speed V to local sound speed a [1,2]. Sound speed is the propagation speed of a small pressure disturbance, so M compares fluid motion with the propagation of pressure information.

A speed of 300 m/s does not correspond to one universal Mach number. Sound speed depends on the gas and its local thermodynamic state, especially temperature.

For a calorically perfect gas, sound speed a is obtained from the specific-heat ratio γ, specific gas constant R and absolute temperature T [1]. For air over ordinary temperature ranges, γ ≈ 1.4 and R ≈ 287 J/(kg·K) are common representative values.

Equation (2) shows that sound speed rises with the square root of absolute temperature for a fixed gas. T must be entered on an absolute temperature scale.

Mach number is dimensionless. Flow speed and sound speed must use consistent units, and absolute temperature is used in the sound-speed equation [4].

M < 1 is subsonic, M = 1 is sonic and M > 1 is supersonic. Near M = 1, transonic flows can contain both subsonic and locally supersonic regions, with shock waves forming in some configurations [2].

At still higher Mach number, aerodynamic heating and real-gas effects become increasingly important. The label supersonic alone therefore does not uniquely determine the required physical model.

M < 1

Subsonic. Small pressure disturbances can propagate upstream relative to the body.

M ≈ 1

Transonic. Local supersonic regions and shocks can occur; compressibility is strong.

M > 1

Supersonic. Disturbance propagation is directionally constrained and shock/expansion waves become central.

M > 5

Commonly called hypersonic; high-temperature and real-gas effects may become important [2].

For a small isentropic change, the momentum relation and definition of sound speed lead to Equation (3), linking relative density and velocity changes.

The relation shows that, for a comparable fractional change in velocity, the magnitude of fractional density change scales with M². Density changes therefore tend to be small at low M and become progressively harder to neglect as M approaches unity.

Treat air at 20 °C as a calorically perfect gas with T = 293.15 K, γ = 1.40 and R = 287 J/(kg·K). Find the sound speed and Mach number for V = 250 m/s.

Equation (2) gives a ≈ 343 m/s. Substitution into Equation (1) gives M ≈ 0.729. The flow is subsonic, but it is not a very-low-Mach flow, so compressibility should be assessed before density variation is neglected.

Equation (2) is the sound-speed relation for a calorically perfect gas. In high-temperature hypersonic flow, temperature-dependent specific heats and real-gas effects may be required.

Mach number measures the importance of compressibility. Viscous effects require Reynolds number, while other phenomena involve other similarity parameters. Engineering models often require several dimensionless groups together.

When calculating Mach number, check local flow speed, local gas temperature and gas properties together.

Mach number has velocity units

M = V/a, so it is dimensionless.

Sound speed is always 340 m/s

Sound speed depends on gas composition and temperature.

M < 1 means incompressible

Compressibility can be significant in subsonic flow as M approaches one.

Use T = 20 in Eq. (2)

T is absolute temperature: 20 °C corresponds to 293.15 K.

Compressibility and Mach number are entry points to aircraft, rockets, gas turbines, compressors, nozzles, high-speed piping, shock waves and acoustics.

Review density, pressure and viscosity together with the Euler and Navier–Stokes equations. Natural next topics are isentropic flow, stagnation quantities, nozzles and shock waves.

Prerequisites

Density, pressure & viscosity; Euler equation; Navier–Stokes equation

Next

Isentropic flow, stagnation temperature and pressure, nozzles, shock waves