Calculate isentropic flow relations: T/T₀, P/P₀, ρ/ρ₀, A/A*, velocity, and normal-shock properties for any Mach number and gas.
Isentropic flow relations describe how temperature, pressure, density, and area change in a frictionless, adiabatic (isentropic) compressible gas flow. These relations are the backbone of gas dynamics, nozzle design, supersonic wind tunnels, and jet-engine analysis.
Given the Mach number M, specific heat ratio γ, and stagnation conditions (T₀, P₀), this calculator computes the local static temperature, pressure, density, speed of sound, flow velocity, and the critical area ratio A/A*. For supersonic Mach numbers, it also provides normal-shock relationships (P₂/P₁, downstream Mach, total-pressure recovery) and the Prandtl-Meyer expansion angle.
Gas presets include air, helium, CO₂, nitrogen, and steam. Mach preset buttons let you quickly switch between subsonic and supersonic conditions, while the table and charts make it easier to compare nozzle states across the operating range. It is especially helpful when you need to move quickly from a Mach number target to the corresponding thermodynamic ratios and area requirement. That makes it a practical reference for first-pass nozzle sizing, wind-tunnel setup checks, and classroom gas-dynamics problems where the main need is a consistent set of ratios from one input state.
Use this tool to move from Mach number to pressure, temperature, density, velocity, and area ratio without flipping through printed gas-dynamics tables or re-deriving the relations by hand. It is particularly useful for nozzle and diffuser screening, where a quick check of the isentropic state and the shock consequences often answers the first design question.
T/T₀ = [1 + (γ-1)/2 · M²]⁻¹ P/P₀ = [1 + (γ-1)/2 · M²]^(-γ/(γ-1)) ρ/ρ₀ = [1 + (γ-1)/2 · M²]^(-1/(γ-1)) A/A* = (1/M) × [(2/(γ+1)) × (1 + (γ-1)/2 · M²)]^((γ+1)/(2(γ-1))) Normal shock: P₂/P₁ = 1 + 2γ/(γ+1) × (M²-1) Prandtl-Meyer: ν(M) = √((γ+1)/(γ-1)) atan(√((γ-1)/(γ+1)(M²-1))) − atan(√(M²-1))
Result: T = 166.7 K, P = 12,780 Pa, V = 517 m/s, A/A* = 1.6875
At M = 2 with γ = 1.4: T/T₀ = 1/1.8 = 0.556, P/P₀ = 0.1278. T = 300 × 0.556 = 166.7 K. P = 101325 × 0.1278 = 12,949 Pa. a = √(1.4 × 287 × 166.7) = 258.8 m/s. V = 2 × 258.8 = 517.6 m/s.
These equations assume adiabatic, reversible flow with no shaft work. They work well for ideal nozzle and diffuser estimates, wind-tunnel calculations, and first-pass engine-cycle analysis. Once friction, shocks, heat transfer, or strong chemistry matter, the simple ratios stop being exact.
A/A* is especially useful for nozzle work because it links geometry to Mach number. For values greater than 1, remember that there are two mathematical solutions: one subsonic and one supersonic. The surrounding boundary conditions determine which branch is physically possible.
If a normal shock forms, total pressure drops and the downstream state is no longer isentropic across the shock. At very high temperatures, the assumption of constant gamma also weakens. For serious design work, treat this calculator as a fast reference, then verify the final design with a more complete compressible-flow model.
Isentropic = constant entropy = no friction + no heat transfer. Real flows are never perfectly isentropic, but the relations are excellent approximations for well-designed nozzles and diffusers with smooth walls.
A* is the throat area where M = 1 (choked condition). A/A* tells you the duct area needed to achieve a given Mach number. Each A/A* > 1 corresponds to two solutions: one subsonic, one supersonic.
A normal shock occurs when supersonic flow is decelerated abruptly (e.g., at a blunt body or inside an over-expanded nozzle). It is irreversible — total pressure drops across the shock.
The Prandtl-Meyer function ν(M) gives the maximum turning angle for an isentropic expansion. If supersonic flow at M₁ turns through angle θ, the downstream Mach M₂ satisfies ν(M₂) = ν(M₁) + θ.
Yes, for real gases γ decreases at high temperatures as vibrational modes activate. For air below ~800 K, γ = 1.4 is an excellent approximation. At 2000 K, γ ≈ 1.3.
Choose the design Mach number, look up A/A* from the table, and multiply by the throat area. The exit area = A* × A/A*. The throat must be sized for the mass flow: ṁ = P₀A* × √(γ/(RT₀)) × (2/(γ+1))^((γ+1)/(2(γ-1))).