Isentropic Flow Calculator
Enter a Mach number (or a known ratio) to get the complete set of isentropic flow relations: static-to-stagnation ratios for pressure, temperature, and density; the area ratio A/A*; Mach angle; and the Prandtl-Meyer expansion angle. Works for both subsonic and supersonic regimes with selectable specific-heat ratios for air, helium, or a custom gas.
What is isentropic flow?
Isentropic flow is compressible fluid flow in which entropy remains constant throughout. This occurs when there is no heat transfer with the surroundings (adiabatic) and no friction losses, so the process is both reversible and adiabatic. Real nozzle and diffuser flows are well approximated as isentropic away from boundary layers, making these relations the foundation of compressible aerodynamics and rocket propulsion. The core idea is that the total (stagnation) pressure, temperature, and density of the fluid are conserved along each streamline, so the static conditions at any point can be described purely as a function of local Mach number and the specific heat ratio gamma.
How to use this calculator
Select the quantity you already know from the "Solve from" menu: Mach number, pressure ratio P/P0, temperature ratio T/T0, density ratio rho/rho0, or area ratio A/A*. Enter its value, choose the specific heat ratio for your gas (1.4 for air, 5/3 for monatomic gases, lower values for hot combustion products), and the calculator instantly returns all five isentropic ratios along with the Mach angle and Prandtl-Meyer function. When solving from area ratio, specify whether your duct section is upstream (subsonic) or downstream (supersonic) of the throat, because each area ratio has two Mach solutions. The chart below shows how all three ratios vary continuously from M = 0 to M = 5 so you can read off neighbouring values at a glance.
The isentropic flow equations
The three fundamental isentropic relations link static properties to their stagnation (total) counterparts through the Mach number M and specific heat ratio gamma. Temperature ratio: T/T0 = 1 / (1 + (gamma-1)/2 * M^2). Pressure ratio: P/P0 = (T/T0)^(gamma/(gamma-1)). Density ratio: rho/rho0 = (T/T0)^(1/(gamma-1)). At the sonic throat (M = 1), these give the critical ratios: P*/P0 = 0.52828, T*/T0 = 0.83333, and rho*/rho0 = 0.63394 for air. The area-Mach relation A/A* = (1/M) * [(2/(gamma+1)) * (1 + (gamma-1)/2 * M^2)]^((gamma+1)/(2*(gamma-1))) determines the duct cross-section needed to achieve a given Mach number. For supersonic flow, the Mach cone half-angle is mu = arcsin(1/M), and the Prandtl-Meyer function nu(M) gives the total isentropic turning angle achievable from M = 1.
Applications in engineering and aerospace
Isentropic flow relations are used wherever compressible gas dynamics matter. In rocket nozzle design, the area ratio determines the exit Mach number and thrust coefficient. In wind tunnel operation, stagnation and static pressure measurements combined with the pressure ratio give the test-section Mach number directly. In gas turbine aerodynamics, the temperature ratio across a nozzle stage sets the blade relative flow angle and power extraction. For supersonic inlets, the Mach angle governs the oblique shock geometry, and the pressure recovery depends on how closely the flow can stay isentropic. Aviation pitot-static systems implicitly invert the pressure ratio to give indicated airspeed. Understanding whether a ratio corresponds to a subsonic or supersonic solution is critical: the same area ratio A/A* exists on both sides of the throat, but the physics are completely different.
Isentropic flow relations for air (γ = 1.4) - standard Mach table
| M | P/P₀ | T/T₀ | ρ/ρ₀ | A/A* | Regime |
|---|---|---|---|---|---|
| 0.1 | 0.99303 | 0.99800 | 0.99502 | 5.8218 | Subsonic |
| 0.3 | 0.93947 | 0.98232 | 0.95638 | 2.0351 | Subsonic |
| 0.5 | 0.84302 | 0.95238 | 0.88517 | 1.3398 | Subsonic |
| 0.7 | 0.72093 | 0.91075 | 0.79158 | 1.0944 | Subsonic |
| 0.9 | 0.59126 | 0.86058 | 0.68704 | 1.0089 | Subsonic |
| 1 | 0.52828 | 0.83333 | 0.63394 | 1.0000 | Sonic |
| 1.5 | 0.27240 | 0.68966 | 0.39498 | 1.1762 | Supersonic |
| 2 | 0.12780 | 0.55556 | 0.23005 | 1.6875 | Supersonic |
| 2.5 | 0.05853 | 0.44444 | 0.13169 | 2.6367 | Supersonic |
| 3 | 0.02722 | 0.35714 | 0.07623 | 4.2346 | Supersonic |
| 4 | 0.00659 | 0.23810 | 0.02766 | 10.719 | Supersonic |
| 5 | 0.00189 | 0.16667 | 0.01134 | 25.000 | Supersonic |
Key values of isentropic flow ratios at representative Mach numbers for air.
Frequently asked questions
What is the difference between static and stagnation (total) conditions?
Static conditions are the thermodynamic properties of the fluid as seen by a particle moving with the flow - what a tiny thermometer carried along in the stream would measure. Stagnation (total) conditions are what you would measure if you brought the flow isentropically to rest, converting all kinetic energy back into pressure and temperature. The stagnation pressure P0 and temperature T0 are conserved along a streamline in isentropic flow, so they serve as reference values. The static pressure P and temperature T are always lower than their stagnation counterparts once the flow is moving (M > 0).
Why does the area ratio A/A* have two Mach solutions?
The area-Mach relation A/A* = f(M) has a minimum at M = 1 where A = A* (the sonic throat). For any area ratio greater than 1, there is one subsonic solution (M < 1) and one supersonic solution (M > 1). Which one exists physically depends on the boundary conditions: a converging-only duct can only reach M = 1 at its exit; a converging-diverging nozzle reaches M = 1 at the throat and then either returns to subsonic (diffuser) or continues to supersonic (nozzle) in the diverging section depending on the downstream pressure.
What value of gamma should I use?
For standard air at ambient and moderate temperatures use gamma = 1.4. Monatomic gases such as helium and argon use gamma = 5/3 (about 1.667). Hot combustion products from hydrocarbon fuels typically range from 1.15 to 1.30 depending on temperature and equivalence ratio. As temperature rises, vibrational modes become active and gamma decreases. For high-altitude hypersonic flow, real-gas effects (dissociation, ionization) make gamma drop further still, and the isentropic relations must be supplemented with equilibrium chemistry data.
At what Mach number does flow become supersonic or hypersonic?
By convention, flow is subsonic when M < 1, transonic when approximately 0.8 < M < 1.2 (wave effects become significant), supersonic when M > 1, and hypersonic when M >= 5. In hypersonic flow, real-gas effects such as vibrational excitation and dissociation become important, and the simple isentropic relations with constant gamma become increasingly inaccurate. The Mach angle mu = arcsin(1/M) decreases from 90 degrees at M = 1 toward 0 degrees as M increases, producing a narrow Mach cone in hypersonic flight.
What is the Prandtl-Meyer function and how is it used?
The Prandtl-Meyer function nu(M) is the total angle through which a supersonic flow can turn isentropically from M = 1. For example, at M = 2 in air (gamma = 1.4), nu = 26.38 degrees - meaning the flow can turn up to 26.38 degrees around a convex corner and remain isentropic. To find the Mach number after an expansion around a corner of turning angle theta, add theta to the incoming nu(M1) and then look up (or solve for) the M2 that gives nu(M2) = nu(M1) + theta. The calculator shows nu at the entered Mach number; run two calculations to solve the expansion problem.
Can I use these relations for liquids?
No. Isentropic flow relations in this form apply to calorically perfect ideal gases only. Liquids are nearly incompressible, so density changes with pressure are negligible and the Mach number based compressibility effects treated here do not apply. Liquid flows are analyzed with Bernoulli equation methods. Water hammer and cavitation problems in liquid pipe systems require separate compressibility treatment based on the liquid bulk modulus rather than gamma.
How do I find stagnation pressure from a pitot tube reading?
A pitot tube measures stagnation pressure P0 directly. If you also measure static pressure P (from a wall tap or static port), the ratio P/P0 gives the Mach number via the isentropic pressure relation. Enter P/P0 into this calculator with "Solve from: Pressure ratio" and you get M instantly. Note that at M > 1 a normal shock stands ahead of the pitot tube, so the measured pressure is the stagnation pressure behind the shock, not the freestream P0. Use the Rayleigh Pitot tube formula for supersonic pitot measurements.