Isentropic Flow Calculator — Mach Number, Pressure, Temperature & Area Ratios

Isentropic flow is the idealized model of a compressible gas flow that is both adiabatic (no heat transfer) and reversible (no friction, no shocks), so entropy stays constant. Under these assumptions the static properties (T, p, ρ) at any point are linked to the stagnation properties (T₀, p₀, ρ₀) purely by the local Mach number and the specific-heat ratio γ — the relations this calculator evaluates. It is the baseline model for nozzle, diffuser, wind-tunnel and turbomachinery analysis.

The four core relations are: T₀/T = 1 + (γ−1)/2 · M²; p₀/p = (T₀/T)^γ/(γ−1); ρ₀/ρ = (T₀/T)^1/(γ−1); and the area–Mach relation A/A* = (1/M)·[(2/(γ+1))·(T₀/T)]^{(γ+1)/(2(γ−1))}. They assume steady, one-dimensional flow of an ideal gas with constant γ. Enter any Mach number in the calculator above to evaluate all four instantly.

Pick your gas (γ = 1.4 for air), square the Mach number, and apply T₀/T = 1 + 0.2·M². Example at M = 2: T₀/T = 1.8, then p₀/p = 1.8^3.5 ≈ 7.824 and ρ₀/ρ = 1.8^2.5 ≈ 4.347. The forward tab of this calculator does exactly this, and also returns A/A* = 1.6875, the Mach angle μ = 30° and the Prandtl–Meyer angle ν ≈ 26.38° for the same input.

The area–Mach relation cannot be inverted in closed form, so this calculator solves it numerically with bisection: it brackets the root on the subsonic interval (0, 1] or the supersonic interval [1, 50) and halves the interval ~300 times until the width is below 10⁻¹³. Use the "Find Mach" tab, enter your A/A* value, choose the subsonic or supersonic branch, and read off M. Inversions from T₀/T, p₀/p or ρ₀/ρ are exact closed-form rearrangements.

A/A* has a minimum of exactly 1 at M = 1 and grows toward infinity both as M → 0 and as M → ∞, so every value above 1 is hit twice — once subsonically, once supersonically. Example with air: A/A* = 2.0 gives M ≈ 0.306 (subsonic) or M ≈ 2.197 (supersonic). A converging–diverging nozzle with that area ratio can operate on either solution depending on the back pressure, which is why the inverse solver asks you to select a branch.

Choked flow occurs when the Mach number reaches 1 at the minimum-area section (throat) of a nozzle or orifice. At that point the mass flow rate is at its maximum for the given upstream stagnation conditions, and reducing the downstream pressure further cannot increase it — pressure disturbances cannot propagate upstream past a sonic throat. For air, choking begins when the downstream-to-stagnation pressure ratio falls below the critical value p*/p₀ = 0.5283.

The critical (sonic, M = 1) ratios are the isentropic relations evaluated at M = 1: T*/T₀ = 2/(γ+1), p*/p₀ = [2/(γ+1)]^γ/(γ−1), ρ*/ρ₀ = [2/(γ+1)]^1/(γ−1). For air (γ = 1.4): T*/T₀ = 0.8333, p*/p₀ = 0.5283, ρ*/ρ₀ = 0.6339. The "Choking & nozzle guide" tab computes them live for any γ you select.

The Mach angle μ = sin⁻¹(1/M) is the half-angle of the cone of weak pressure disturbances trailing a body moving supersonically — the familiar Mach cone. At M = 2 in air, μ = 30°; as M → 1 the cone opens to 90°, and the concept only applies for M > 1. The calculator displays μ automatically whenever your Mach number is supersonic.

The Prandtl–Meyer function ν(M) gives the angle through which a supersonic flow must turn to expand isentropically from M = 1 to Mach M: ν(M) = √[(γ+1)/(γ−1)]·tan⁻¹√[(γ−1)/(γ+1)·(M²−1)] − tan⁻¹√(M²−1). It is the key quantity for designing supersonic expansion corners and the diverging section of nozzles. For air at M = 2, ν ≈ 26.38°; at M = 3, ν ≈ 49.76°.

Use γ = 1.4 for air, nitrogen and oxygen; γ ≈ 1.31 for methane/natural gas; γ ≈ 1.289 for carbon dioxide; γ = 1.66 for helium and other monatomic gases. The preset buttons cover these, and you can type any custom γ between 1.01 and 1.67. Note the isentropic formulas assume γ is constant — for very high temperatures where vibrational modes excite, treat results as approximate.

The relations are exact only for steady, one-dimensional, adiabatic, reversible flow of an ideal gas with constant specific heats. They break down across shock waves (entropy jumps — use normal/oblique shock relations instead), in regions with strong friction or heat transfer (Fanno/Rayleigh flow), and at very low Reynolds numbers. For smooth nozzles, diffusers and wind-tunnel cores away from boundary layers, they are typically accurate within a few percent.

Isentropic relations describe smooth, reversible compression or expansion where stagnation pressure is conserved (p₀ constant). A normal shock is an irreversible, near-discontinuous compression: static pressure, temperature and density jump up, Mach drops below 1, and stagnation pressure is lost (entropy rises). This tool covers only the isentropic side; shock tables are a separate topic with their own relations.

Static properties (T, p, ρ) are what a probe moving with the fluid would measure. Stagnation properties (T₀, p₀, ρ₀) are the values the gas would reach if brought to rest isentropically — they represent the total energy content of the flow. Stagnation temperature is conserved in adiabatic flow, which is why every ratio here is anchored to the stagnation state. The calculator reports stagnation-over-static ratios (always ≥ 1).

Yes — it is completely free, runs 100% in your browser with no sign-up, and makes zero network requests after the page loads, so it works offline once opened. All calculations (forward relations, bisection inverse solver, the full isentropic table and the Prandtl–Meyer function) execute locally in JavaScript. It is intended as an educational and preliminary-design aid; validate safety-critical designs with qualified analysis.