Compressible Flow

When density changes matter — speed of sound, Mach number and flow regimes, isentropic relations, stagnation properties, area–velocity relation and nozzles, normal shock basics, and why civil-engineering water flows are treated as incompressible.

Compressible Flow

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📑 Contents (6 sections)

Last reviewed 4 Nov 2026 · 4 min read

1. When is a flow compressible?

A flow is compressible when its density changes appreciably with pressure. Liquids almost never qualify (water's bulk modulus is about 2.2 GPa), and gases at low speed behave as incompressible too. The measure is the Mach number.

DefinitionMach number

M = \frac{V}{c}

where $V$ is the flow velocity and $c$ the local speed of sound. Density changes stay below about 5 % for $M < 0.3$ , so such flows are treated as incompressible.

Regime	Mach number
Incompressible (practically)	$M < 0.3$
Subsonic	$M < 1$
Transonic	about $0.8 < M < 1.2$
Sonic	$M = 1$
Supersonic	$1 < M < 5$
Hypersonic	$M > 5$

2. Speed of sound

Sound is a small pressure wave; its speed depends on how "stiff" the medium is.

FormulaSpeed of sound

c = \frac{K}{ρ}

For a perfect gas undergoing an isentropic change ( $K = γ p$ ):

c = γ R T

Worked ExampleSpeed of sound in air at 15 °C

$γ = 1.4$ , $R = 287$ J/kg·K, $T = 288$ K. $c = 1.4 \times 287 \times 288 = 115, 718 \approx 340$ m/s.

In water, $c = 2.2 \times 1 0^{9} /1000 \approx 1, 480$ m/s — the speed with which a water-hammer pressure wave travels in a rigid pipe.

Mach cone

A source moving faster than sound leaves a cone of disturbance; its half-angle (the Mach angle) is

sin α = \frac{1}{M}

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