Isentropic flow calculator

Pick γ and one value you know. The tool finds the Mach number, then every other ratio. It runs entirely in your browser: nothing you type is sent anywhere.

The equations this tool uses

These hold for a calorically perfect gas (γ stays constant) flowing isentropically: no heat added or lost, no friction, and no shock waves. Subscript 0 means stagnation (total) conditions, the values the gas would have if you slowed it to rest without losses (NASA's pages write T_t, p_t for T₀, p₀). $A^*$ is the area where the flow would be exactly sonic (the throat of a choked nozzle).

Temperature

The energy equation links the stagnation temperature to the local temperature and Mach number:

\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2}\,M^2

Pressure and density

Along an isentropic path, pressure and density follow temperature with fixed exponents that depend only on γ:

\frac{p}{p_0} = \left(\frac{T}{T_0}\right)^{\gamma/(\gamma-1)}

\frac{\rho}{\rho_0} = \left(\frac{T}{T_0}\right)^{1/(\gamma-1)}

Area ratio

Mass flow is the same through every cross-section. Writing that out with the ratios above gives the area needed at Mach M, compared with the sonic area:

\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{\gamma+1}\left(1 + \frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}

This curve has its minimum, exactly 1, at M = 1. Any larger area ratio matches one subsonic and one supersonic Mach number, which is why you pick a branch. The tool finds that Mach number by bisection: it keeps halving an interval that is known to contain the answer on the chosen branch, until the interval is narrower than about one part in 10¹⁴ of the answer. It halves on a logarithmic scale, so very small subsonic Mach numbers (large area ratios) are found just as precisely as large ones. After solving, the tool puts the Mach number back into the formula and checks that it reproduces your input to within one part in 10⁹; if it doesn't, you get an error instead of a result.

Mach angle and Prandtl-Meyer angle

Both are defined for M ≥ 1. At M = 1 the Mach angle is 90° and the Prandtl-Meyer angle is 0° (its reference point); below M = 1 neither exists. The Mach angle is the half-angle of the cone that small disturbances spread along. The Prandtl-Meyer angle is how far a sonic flow has to turn, through a smooth expansion, to speed up to Mach M. Angles are in degrees.

\mu = \arcsin\frac{1}{M} \qquad (M \ge 1)

\nu = \sqrt{\frac{\gamma+1}{\gamma-1}}\,\arctan\sqrt{\frac{\gamma-1}{\gamma+1}\left(M^2-1\right)} \;-\; \arctan\sqrt{M^2-1} \qquad (M \ge 1)

When you enter p/p₀, T/T₀ or ρ/ρ₀, the tool solves these equations for M directly (no iteration). A Prandtl-Meyer angle input is solved by bisection, like the area ratio. If the solver can't find an interval that contains the answer, it shows an error instead of a result.

Sources

The relations are standard gas dynamics, written and coded for this site. We checked them against these NASA Glenn Research Center pages (Beginner's Guide to Aeronautics, read 2026-10-28):

NASA Glenn Research Center, Beginner's Guide to Aeronautics, “Isentropic Flow Equations,” www.grc.nasa.gov/www/k-12/airplane/isentrop.html
NASA Glenn Research Center, Beginner's Guide to Aeronautics, “Mach Angle,” www.grc.nasa.gov/www/k-12/airplane/machang.html
NASA Glenn Research Center, Beginner's Guide to Aeronautics, “Prandtl-Meyer Angle,” www.grc.nasa.gov/www/k-12/airplane/pranmyer.html
NASA Glenn Research Center, Beginner's Guide to Aeronautics, “Specific Heats, Calorically Imperfect Gas,” www.grc.nasa.gov/www/BGH/realspec.html

About the real-gas caveat: NASA's Specific Heats page supports only the vibrational-excitation part (air's specific heats change once the vibrational modes of N₂ and O₂ are excited, at low hypersonic conditions) and its γ(T) model. The ~800 K figure is computed from that vibrational model, not stated on the page. The dissociation temperatures, liquefaction and the monatomic-gas remark are standard gas-dynamics values (approximate, pressure-dependent) and are not from that page.