Compute the stagnation-to-static ratios of temperature, pressure, and density for a compressible flow at a given Mach number and ratio of specific heats. Enter the local static temperature and pressure and the calculator also returns the absolute stagnation (total) temperature and pressure — the foundation of nozzle, diffuser, and pitot-tube analysis.
When a compressible gas accelerates or decelerates without friction or heat transfer, the process is isentropic — entropy stays constant — and simple algebraic relations link the local (static) properties to their stagnation (total) values through the Mach number alone. These isentropic relations are the everyday tools of nozzle design, wind-tunnel calibration, pitot-static airspeed measurement, and gas-turbine analysis.
Static properties (T, p, ρ) are what a thermometer or pressure tap moving with the flow would read. Stagnation (total) properties (T₀, p₀, ρ₀) are what the flow would reach if brought to rest isentropically — converting all its kinetic energy into thermal energy and pressure. The difference between total and static grows with Mach number, which is exactly why a pitot tube can infer speed from the pressure rise at its stagnation point.
For a calorically perfect gas:
T₀/T = 1 + (γ−1)/2 · M² p₀/p = (T₀/T)^(γ/(γ−1)) ρ₀/ρ = (T₀/T)^(1/(γ−1))
The temperature ratio comes from energy conservation, and the pressure and density ratios follow from the isentropic relation p ∝ ρ^γ. With γ = 1.4 for air, M = 1 already gives T₀/T = 1.2 and p₀/p ≈ 1.893, so compressibility is far from negligible at sonic conditions.
These relations require adiabatic (no heat transfer) and reversible (no friction or shocks) flow. Across a shock wave entropy rises, so stagnation pressure is lost and the isentropic p₀/p no longer holds — though stagnation temperature is still conserved. In real nozzles and diffusers, small losses are folded in through an isentropic efficiency. For smooth, shock-free, attached flow, the isentropic assumption is an excellent engineering approximation.
In a converging-diverging nozzle, the same relations (combined with the area-Mach relation) set the temperature, pressure, and density at every station from the chamber to the exit, and predict the exhaust velocity that produces thrust. For airspeed, a pitot-static probe measures p₀ and p; inverting the compressible p₀/p relation yields the true Mach number and, with temperature, the true airspeed — the basis of high-speed air-data computers.
γ (gamma) is the ratio of the specific heat at constant pressure to that at constant volume, cp/cv. For diatomic gases like air at moderate temperatures it is 1.4; for monatomic gases like helium or argon it is about 1.67, and for combustion products or high-temperature air it drops toward 1.2–1.3. It sets how strongly temperature, pressure, and density couple in compressible flow.
It holds when the flow is adiabatic and reversible — no significant heat transfer, friction, or shock waves. This describes smooth, attached, shock-free flow through well-designed nozzles, diffusers, and over streamlined bodies at subsonic and moderate supersonic speeds. Across shocks or in highly viscous or separated regions the relations break down and you must account for entropy generation.
A shock is adiabatic, so no energy leaves the flow and the total (stagnation) temperature is unchanged. However, a shock is irreversible — entropy increases — and that entropy rise shows up as a loss of stagnation pressure. So T₀ is the same on both sides of a shock while p₀ always drops, which is central to inlet and diffuser design.
A pitot-static probe reads the stagnation pressure p₀ at its nose and the static pressure p at side ports. At low speed Bernoulli suffices, but in compressible flow you invert p₀/p = (1 + (γ−1)/2·M²)^(γ/(γ−1)) to solve for Mach number, then combine with the measured or assumed temperature to get true airspeed. This is exactly the calculator run in reverse.
Yes — the isentropic relations themselves are valid for any Mach number as long as the flow remains isentropic (shock-free). In a converging-diverging nozzle the supersonic branch is fully isentropic. But if a shock appears (for example in an over-expanded nozzle or ahead of a blunt body), you must apply normal- or oblique-shock relations across it and then resume the isentropic relations downstream.