The isentropic flow calculator above gives the complete set of compressible flow ratios at a chosen Mach number: how much higher the stagnation pressure is than the static pressure, the same for temperature and density, the nozzle area needed relative to the sonic throat, and the Mach angle where one exists. These are the numbers that fill the isentropic flow tables at the back of every gas dynamics textbook, computed for whatever ratio of specific heats your gas actually has rather than only for air.
Arb Digital publishes free engineering calculators that state the range in which their relations hold. Isentropic means adiabatic and reversible: no heat added or removed, no friction, no shock waves. Real nozzles approach it closely enough to be designed this way, but the assumption fails hard across a shock, and the page says where and why.
What This Isentropic Flow Calculator Does
When a gas accelerates, its static pressure and temperature fall, and the energy shows up as kinetic energy instead. Stagnation conditions are what you would recover if you brought the flow to rest without loss, so the ratio between stagnation and static values is a direct measure of how fast the gas is moving, expressed in Mach numbers rather than metres per second.
The area ratio is the other half of the story. A converging duct accelerates subsonic flow and a diverging duct decelerates it, but above Mach 1 both reverse: supersonic flow accelerates in a diverging duct. The consequence is that a nozzle producing supersonic exhaust must converge to a sonic throat and then diverge, and the area ratio between the exit and that throat determines the exit Mach number entirely.
The hero figure is the stagnation-to-static pressure ratio. The grid gives the temperature ratio, the density ratio, the area ratio, and the static pressure that follows from the reservoir pressure you supplied.
How to Use It
- Choose your input. Enter a Mach number directly, or switch to the pressure ratio mode if you have a Pitot-static reading and want the Mach number back.
- Set γ for your gas. Air at ordinary temperatures is 1.4. Rocket exhaust and other hot combustion products run nearer 1.2 to 1.3, and using 1.4 there gives visibly wrong area ratios.
- Add reservoir conditions if you want real units. Stagnation pressure and temperature turn the dimensionless ratios into actual static values at the station you are examining.
- Enter a throat area for nozzle work. Multiplying the area ratio by the throat area gives the physical area needed at that Mach number.
- Check the regime note. The tool says whether you are subsonic, sonic or supersonic, and reminds you where the isentropic assumption stops being safe.
The Formula: How the Isentropic Ratios Are Derived
All four ratios come from one grouping. Write b = 1 + (γ − 1)/2 × M². Then the temperature ratio is T0/T = b, which follows directly from conservation of stagnation enthalpy. The pressure ratio is p0/p = bγ/(γ−1) and the density ratio is ρ0/ρ = b1/(γ−1), both obtained by applying the isentropic relation between pressure, density and temperature.
The area ratio comes from mass conservation between the sonic throat and the station of interest: A/A* = (1/M)[(2/(γ+1)) b](γ+1)/(2(γ−1)). The Mach angle, defined only above Mach 1, is μ = arcsin(1/M). NASA's Glenn Research Center publishes these same relations in its isentropic flow equations reference, with the area relation set out separately under compressible area ratio.
Work the defaults through by hand at M = 2 with γ = 1.4. Here (γ − 1)/2 = 0.2, so b = 1 + 0.2 × 4 = 1.8, and that is the temperature ratio immediately. The pressure exponent is 1.4/0.4 = 3.5, giving 1.83.5 = 7.824. The density exponent is 1/0.4 = 2.5, giving 1.82.5 = 4.347. For the area ratio, (2/2.4) × 1.8 = 1.5, the exponent is 2.4/0.8 = 3, and 1.5³ = 3.375, divided by M = 2 to give exactly 1.6875 — the value printed in every standard isentropic table at Mach 2. The Mach angle is arcsin(0.5) = 30°. With a 500 kPa reservoir the static pressure is 500 ÷ 7.824 = 63.9 kPa, and a 500 K reservoir gives a static temperature of 500 ÷ 1.8 = 277.8 K.
Why the Area Ratio Has Two Answers
This is the trap that catches people building their first converging-diverging nozzle. The area ratio is not a one-to-one function of Mach number. It reaches its minimum of exactly 1 at Mach 1 and rises on both sides, so any area ratio above 1 corresponds to two Mach numbers: one subsonic and one supersonic. An area ratio of 2 in air is satisfied at Mach 0.31 and at Mach 2.20.
Which one a real nozzle produces depends on the back pressure, not on the geometry alone. With only a modest pressure drop the flow accelerates in the converging section, decelerates in the diverging section, and stays subsonic throughout — a venturi rather than a nozzle. Drop the back pressure far enough and the throat chokes, the diverging section goes supersonic, and the exit Mach number is fixed by the area ratio. Between those two states sits a range where a normal shock stands inside the diverging section, and downstream of that shock the flow is no longer isentropic at all.
That is why a rocket nozzle is designed for one altitude. The area ratio fixes the exit Mach number and therefore the exit pressure, and the nozzle is only perfectly expanded where ambient pressure matches it. At sea level an upper-stage nozzle is over-expanded and flow can separate from the wall; high in the atmosphere a sea-level nozzle is under-expanded and wastes energy in a plume that keeps spreading.
Where the Isentropic Assumption Breaks
Isentropic means no entropy change: no friction, no heat transfer, no shocks. Three things break it in practice.
Shock waves are the sharpest violation. Across a normal shock the flow goes abruptly from supersonic to subsonic, static pressure and temperature jump, and stagnation pressure falls irreversibly. Stagnation temperature is conserved because no heat was added, but stagnation pressure is not, so the loss shows up precisely in the quantity this page treats as constant. A supersonic Pitot probe therefore cannot use the subsonic relation at all, because a bow shock forms ahead of it; the Rayleigh Pitot formula is needed instead, and NASA's normal shock wave equations page covers the jump conditions.
Boundary layer friction is the quiet one. It always raises entropy, so real nozzles deliver slightly less than the ideal figures, and the discrepancy grows with wetted area and falls with Reynolds number. Nozzle discharge coefficients of 0.97 to 0.99 exist to absorb it.
Finally, the perfect-gas assumption with constant γ fails at high temperature. In rocket exhaust, vibrational excitation, dissociation and recombination all change the effective γ along the nozzle, and serious work uses a chemical equilibrium code rather than a single value.
Where This Sits Next to the Other Flow Tools
This page assumes you already have a Mach number and gives the compressible ratios that follow from it. Getting the Mach number in the first place from a speed and an air temperature is the job of the Mach number calculator, which returns the Mach number and the local speed of sound and stops there. The speed of sound calculator covers the acoustic velocity in its own right, and the ideal gas law calculator handles the equation of state that all of this rests on. For incompressible problems the Reynolds number calculator and the drag force calculator are the relevant pages, and the pressure converter and temperature converter handle unit changes on the reservoir conditions.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using γ = 1.4 for combustion products — hot exhaust runs nearer 1.2, and the area ratio at a given Mach number is noticeably different.
- Assuming an area ratio fixes the Mach number — every ratio above 1 has both a subsonic and a supersonic solution, and the back pressure decides which one occurs.
- Applying these relations downstream of a shock — stagnation pressure drops irreversibly across a shock, so the stagnation reference on one side is not the reference on the other.
- Using the subsonic Pitot relation above Mach 1 — a bow shock forms ahead of the probe, and the Rayleigh supersonic Pitot formula is required instead.
- Reading the temperature ratio as ambient — stagnation temperature is what a probe recovers, and it is well above ambient at high Mach numbers, which is why kinetic heating matters.
Related Free Tools From Arb Digital
Start with the Mach number calculator to get the Mach number from a speed and an air temperature, then bring it here for the compressible ratios. The speed of sound calculator gives the acoustic velocity directly, the ideal gas law calculator handles the equation of state, and the Reynolds number calculator tells you whether viscous effects deserve attention. Use the pressure converter and the temperature converter for reservoir conditions quoted in unfamiliar units, and the drag force calculator for the low-speed aerodynamic case. Everything Arb Digital publishes is on the free online tools hub.
Frequently Asked Questions
Adiabatic and reversible at the same time: no heat crosses the boundary and no entropy is generated internally. In practice that means no friction, no heat transfer and no shock waves. Real nozzles and inlets come close enough for design purposes over most of their operating range, but the assumption fails completely across a shock.
Because the ratio is at its minimum of one at Mach 1 and rises on both sides. A converging-diverging duct can either accelerate flow to supersonic speeds or behave as a venturi that returns to subsonic, and both satisfy the same geometry. Which one occurs depends on the back pressure, not on the shape alone.
Air and other diatomic gases at ordinary temperatures are close to 1.4. Monatomic gases such as helium and argon are about 1.667. Hot combustion products typically fall between 1.2 and 1.3 because vibrational modes are excited. Gamma decreases as temperature rises, and using an air value for rocket exhaust produces visibly wrong area ratios.
No. A shock is irreversible, so stagnation pressure falls across it and the stagnation reference on the upstream side is not the same as on the downstream side. Stagnation temperature is preserved because no heat is added, but pressure and density ratios must come from the normal or oblique shock relations instead.
It is the temperature the gas would reach if brought to rest without loss, and it is what a temperature probe in the flow largely recovers. It rises rapidly with Mach number: at Mach 2 it is 1.8 times the static value. This is the origin of kinetic heating, and it is why high-speed aircraft structures face a thermal problem as much as an aerodynamic one.
Choose the exit Mach number you want, read the area ratio, and multiply it by your throat area to get the exit area. The exit static pressure follows from the chamber pressure and the pressure ratio. The nozzle is perfectly expanded only where ambient pressure equals that exit pressure, which is why a nozzle is optimised for one altitude and compromised everywhere else.
Because it does not exist there. The Mach angle describes the cone of pressure waves left behind by an object moving faster than the disturbances it creates. Below Mach 1 those waves outrun the source and spread ahead of it, so no such cone forms and the arcsine of one over the Mach number has no real solution.
This tool is provided for educational and preliminary design use. It assumes a calorically perfect gas with constant specific heats undergoing adiabatic, reversible, one-dimensional flow, and does not model shocks, friction, heat transfer, real gas effects or chemical dissociation. Verify designs against a validated compressible flow code and test data.
