This Prandtl-Meyer expansion calculator handles the case where a supersonic stream meets a corner that turns away from it. The flow does not shock; it fans out through a continuous band of Mach waves, speeding up and dropping in pressure and temperature as it goes. The Prandtl-Meyer function is the bookkeeping that makes this tractable: it assigns every supersonic Mach number a single angle, and turning the flow by θ simply adds θ to that angle.
Arb Digital builds free physics calculators that each own one regime cleanly. This page is the expansion side of supersonic corner flow. The compression side — a corner that turns into the flow, generating an attached shock with a total-pressure loss — belongs to the oblique shock calculator. The two are mirror images in geometry and opposites in thermodynamics, and confusing them is the fastest way to get a supersonic problem badly wrong.
What This Prandtl-Meyer Expansion Calculator Does
You give it the upstream Mach number, the angle the wall turns through and the ratio of specific heats. It computes the Prandtl-Meyer angle of the incoming flow, adds your deflection to get the outgoing angle, then inverts the function numerically to recover the downstream Mach number.
Because the process is isentropic, the total (stagnation) pressure and total temperature are unchanged across the fan. That lets the calculator convert the Mach numbers straight into static pressure and static temperature ratios using the ordinary isentropic relations, and report absolute downstream values when you supply upstream ones.
It also reports the two Mach angles that bound the fan. The leading wave sits at μ₁ = arcsin(1 ÷ M₁) to the incoming flow direction and the trailing wave at μ₂ = arcsin(1 ÷ M₂) to the outgoing direction, so the fan opens between them. Those angles tell you where the expansion physically is, which matters if you are placing a probe or reading a schlieren image.
How to Use It
- Enter the upstream Mach number. It must be greater than one. Prandtl-Meyer theory has no meaning in subsonic flow, where a convex corner simply accelerates the stream smoothly with no wave structure at all.
- Enter the turn angle. This is the angle between the upstream and downstream wall directions, measured as a positive number for an expansion.
- Set gamma to match your gas. Air at moderate temperature is 1.4. Rocket exhaust nearer 1.2 expands further for the same turn, which is why the choice matters.
- Add pressure and temperature if you want absolute numbers. Leave them alone and you still get the ratios, which are the physics.
- Check the Mach angles. If the fan is wide, the assumption that the flow downstream is uniform only holds some distance from the corner.
The Formula: How the Prandtl-Meyer Angle Is Calculated
The Prandtl-Meyer function is ν(M) = √[(γ+1)÷(γ−1)] × arctan√[(γ−1)(M²−1)÷(γ+1)] − arctan√(M²−1), with the arctangents in radians and the result usually quoted in degrees. NASA Glenn Research Center's Prandtl-Meyer angle page defines ν as the angle through which a sonic flow must expand to reach a given Mach number, and notes that the function is derived from conservation of mass, momentum and energy applied to differentially small deflections.
Because ν is defined that way, the turn rule is simply ν₂ = ν₁ + θ. Inverting ν to get M₂ has no closed form, so this calculator uses a bracketed numerical search. NASA Glenn's page on the centered expansion fan sets out the contrast that makes the whole method work: across an expansion the Mach number rises, static pressure falls and total pressure is conserved, whereas across a shock the Mach number falls and total pressure is lost.
The isentropic relations do the rest. With T₀ and p₀ constant, T₂÷T₁ = [1 + (γ−1)M₁²÷2] ÷ [1 + (γ−1)M₂²÷2] and p₂÷p₁ is the same bracketed ratio raised to the power γ÷(γ−1).
Work the defaults by hand. At M₁ = 2 and γ = 1.4, √(2.4÷0.4) = 2.44949, and √[(0.4÷2.4) × 3] = √0.5 = 0.707107, whose arctangent is 0.615480 rad. Multiplying gives 1.507614 rad. Subtracting arctan√3 = 1.047198 rad leaves ν₁ = 0.460417 rad = 26.380°. Adding a 10° turn gives ν₂ = 36.380°, and inverting the function returns M₂ = 2.3849. Then the bracket ratio is (1 + 0.2 × 4) ÷ (1 + 0.2 × 5.6877) = 1.8 ÷ 2.13754 = 0.842086, so T₂ = 300 × 0.842086 = 252.63 K and p₂ = 100 × 0.8420863.5 = 54.80 kPa. The Mach angles are arcsin(0.5) = 30.00° and arcsin(1÷2.3849) = 24.79°.
Why an Expansion Is Free and a Shock Is Not
An oblique shock is a discontinuity: properties jump across a surface thinner than a few mean free paths, entropy rises, and total pressure is permanently lost. An expansion fan is the opposite. It spreads the turn across an infinite number of infinitesimal Mach waves, each one carrying a vanishingly small change, so entropy does not rise and total pressure survives intact.
The asymmetry is geometric, not arbitrary. Compression Mach waves converge, because each one raises the local sound speed and steepens the next; they coalesce into a shock. Expansion waves diverge, because each one lowers the local sound speed and the next trails further behind. Nothing ever piles up, so nothing ever becomes a discontinuity.
This is why supersonic nozzles are designed with gently expanding contours and why the exhaust from an under-expanded nozzle fans outward at the lip rather than shocking there. It is also why a supersonic inlet is hard: you have to compress, and compression always costs total pressure unless you split it across many weak shocks.
The Maximum Turn and What Happens Beyond It
The Prandtl-Meyer function has a ceiling. As M tends to infinity, ν approaches (π÷2)(√[(γ+1)÷(γ−1)] − 1), which for γ = 1.4 is 130.45°. A sonic flow can therefore turn through at most 130.45° before it would need infinite Mach number, and a flow already at M = 2 has only 130.45 − 26.38 = 104.07° left.
Ask for more than that and the calculator says so rather than returning a number. Physically, the gas cannot expand further: static pressure and temperature have gone to zero and the flow has separated from the surface entirely. Real gases reach this limit even sooner, because the perfect-gas assumption with fixed γ fails once the temperature drops far enough for condensation, or rises far enough for vibrational excitation and dissociation.
The ceiling also falls with γ. For a rocket exhaust at γ = 1.2, the maximum turn is about 208°, so a low-γ gas can expand much further — which is precisely why exhaust plumes from high-altitude engines fan out so dramatically.
Where the Simple Answer Stops Being Right
Everything here assumes steady, two-dimensional, inviscid, perfect-gas flow with a sharp corner. A real corner has a radius, which spreads the fan's origin over a small region but changes little else. Viscosity matters more: the boundary layer thickens rapidly through an expansion, and if the turn is severe the layer can separate, in which case the wall no longer sets the flow direction and the calculation stops describing reality.
Three-dimensional effects break it more thoroughly. Around a swept edge or a body of revolution, the flow relieves in a second direction and the effective turn is not the geometric one. And when the expansion is followed downstream by a compression — the usual case on an airfoil, where the flow expands over the shoulder and then must return to the free-stream direction — the waves interact and a full method-of-characteristics solution replaces the simple sum.
Where This Sits Next to the Other Supersonic Tools
Use the Mach number calculator to get M₁ from a speed and a temperature, and the speed of sound calculator for the sound speed itself. The isentropic flow calculator covers the stagnation-to-static relations on their own, without any turn. For the compression case use the oblique shock calculator. On the vehicle side, the drag force calculator and the lift coefficient calculator take the pressure field you build from these waves and turn it into loads, and the rocket thrust calculator handles the nozzle exit condition.
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
- Applying it to a compression corner — turning into the flow makes a shock, not a fan. Total pressure is lost and the Mach number falls.
- Using degrees inside the arctangents — the function is built in radians. Convert only at the end.
- Forgetting that ν is measured from M = 1 — ν is not the turn angle. The turn is the difference between two ν values, which is why ν₁ appears at all.
- Assuming total pressure changes — it does not across an isentropic fan. If your downstream total pressure differs, a shock is hiding somewhere in the problem.
- Ignoring the maximum turn — past the ceiling there is no solution, and a calculator that returns one anyway is wrong.
Related Free Tools From Arb Digital
Start with the Mach number calculator and the speed of sound calculator to fix the upstream state. Use the isentropic flow calculator for stagnation ratios and the oblique shock calculator for the compression counterpart. Downstream, the rocket thrust calculator, the drag force calculator and the lift coefficient calculator turn flow conditions into forces. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is the angle through which a sonic flow would have to expand to reach a given Mach number. Because it is defined from M equals one, the turn between any two supersonic states is simply the difference between their Prandtl-Meyer angles.
No. The fan is isentropic, so total pressure and total temperature are both conserved across it. That is the fundamental difference from a shock wave, which raises entropy and destroys total pressure permanently.
For a sonic flow with gamma of 1.4 the ceiling is about 130.45 degrees, reached only as the Mach number tends to infinity. A flow already at Mach 2 has used 26.38 degrees of that, so it has about 104 degrees left.
The Prandtl-Meyer function cannot be inverted in closed form. Given a target angle, the Mach number has to be found by iteration, and this page uses a bracketed search that converges to well under a thousandth of a Mach number.
An oblique shock forms when the wall turns into the flow, compressing it: Mach number falls, static pressure rises and total pressure is lost. An expansion fan forms when the wall turns away: Mach number rises, static pressure falls and total pressure is preserved.
Yes, in both places. It appears inside the Prandtl-Meyer function and again in the isentropic pressure exponent. Hot rocket exhaust near 1.2 expands considerably further for the same turn than air at 1.4 does.
Approximately. A rounded corner spreads the origin of the fan over the curved section but gives nearly the same downstream state, provided the flow stays attached. Severe turns can separate the boundary layer, and then none of this applies.
This tool is provided for educational and study use. It assumes steady, two-dimensional, inviscid, calorically perfect flow with a constant ratio of specific heats, and does not model viscosity, boundary-layer separation, real-gas effects, three-dimensional relief or wave interaction. Treat its output as a physics result rather than a design-grade aerodynamic value.
