Compressible Flow Calculator

Calculate stagnation properties, speed of sound, and flow parameters at different Mach numbers.

<1 subsonic, >1 supersonic
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Mach Number, Stagnation State, and Nozzle Area

Compressible flow becomes important when density changes can no longer be ignored, and Mach number M=V/a compares flow speed with the local speed of sound. For a calorically perfect gas, a=√(γRT). Because sound speed depends on temperature and gas properties, Mach 1 does not correspond to one universal velocity.

In adiabatic, reversible deceleration to rest, kinetic energy is converted into internal energy. The resulting stagnation temperature satisfies T0/T=1+(γ−1)M2/2. For isentropic flow, stagnation pressure follows p0/p=(T0/T)γ/(γ−1). These stagnation quantities represent the state reached by bringing the flow to rest ideally, not the same as ordinary static temperature and pressure measured while the fluid is moving.

M=V/a,   a=√(γRT),   T0/T=1+((γ−1)/2)M2
SymbolMeaningWhy it appears / units
MMach numberDimensionless ratio of flow speed to local sound speed.
aLocal speed of soundm/s; depends on γ, R, and static temperature.
T, pStatic temperature and pressureLocal thermodynamic state of the moving gas.
T0, p0Stagnation quantitiesIdeal rest-state values for isentropic deceleration.
A/A*Area ratioRatio to the sonic critical area A* for one-dimensional isentropic flow.

The area-Mach relation has two branches for A/A*>1: one subsonic and one supersonic. In a converging passage, subsonic flow accelerates toward M=1. To accelerate an already sonic flow to supersonic speed, a diverging section is required under suitable pressure conditions, which is the operating principle of a converging-diverging nozzle.

Worked Examples

Example 1: Speed of sound in air at 20°C (293K)
a=√(1.4×287×293)
Result: 343 m/s = 1235 km/h
Standard value at sea level
Example 2: Stagnation T: T=250K, M=0.8, air
T₀=250×(1+0.2×0.64)
Result: T₀=282K — 32K rise due to deceleration
Pitot tube measures stagnation pressure
Example 3: Supersonic air at Mach 2
T=220K, M=2, γ=1.4 → T0=T(1+0.2M2)
Result: T0=396K, p0/p≈7.82
The large stagnation rise reflects conversion of substantial kinetic energy into internal energy during ideal deceleration.
Example 4: Area ratio at Mach 2
Air, γ=1.4, M=2 → isentropic area-Mach relation
Result: A/A*≈1.6875
The same area ratio also has a subsonic solution, so area alone does not uniquely determine Mach number.

Common Mistakes

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Treating Mach 1 as one fixed speed

The speed of sound changes with gas and temperature. Mach number must use the local sound speed under the local thermodynamic conditions.

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Confusing static and stagnation temperature

Static temperature belongs to the moving gas state. Stagnation temperature is the ideal temperature after bringing that flow to rest adiabatically without losses.

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Assuming A/A* gives a unique Mach number

For an isentropic nozzle with A/A*>1, there is generally one subsonic branch and one supersonic branch. Boundary conditions determine which branch is physically realized.

Frequently Asked Questions

Choked flow?
At a choked minimum-area throat, Mach number is 1 and the mass flow cannot be increased further by lowering downstream pressure while the upstream state and geometry remain fixed. For ideal air with γ=1.4, the critical static-to-stagnation pressure ratio is about 0.528. Choking does not make the throat supersonic.
de Laval nozzle?
Converging section accelerates flow to M=1 at throat. Diverging section: subsonic→decelerates (exhaust diffuser), supersonic→accelerates (rocket nozzle). Shape controls exit Mach number for given area ratio.
At what Mach number do compressibility effects matter?
There is no perfectly sharp boundary, but gas-flow density changes are often small enough to neglect below about Mach 0.3 for many engineering estimates. As Mach number rises, compressibility increasingly affects pressure, temperature, density, and aerodynamic behavior.
Why does stagnation temperature increase with Mach number?
A faster flow contains more kinetic energy per unit mass. If the gas is brought to rest adiabatically without work extraction, that kinetic energy becomes internal energy, raising the stagnation temperature above the static temperature.
Does choked flow mean the entire downstream flow is supersonic?
No. Choking means the controlling minimum-area section reaches Mach 1 and the mass flow cannot be increased by further lowering downstream pressure under the same upstream state and geometry. Supersonic flow downstream requires the appropriate nozzle geometry and pressure conditions.
Why can a diverging duct accelerate supersonic flow?
Compressible-flow area behavior reverses across Mach 1. A diverging area decelerates subsonic isentropic flow but accelerates supersonic isentropic flow. This is why a de Laval nozzle first converges to a sonic throat and then diverges for supersonic acceleration.

Formula Explorer connections

Interpretation: This relationship connects pressure, velocity, density, viscosity, geometry or transport in a fluid system. Assumption: Check whether flow is steady, incompressible, laminar, fully developed or one-dimensional. Reynolds and Mach regimes determine whether simplified formulas are valid.

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