Prandtl-Meyer Expansion Fan Calculator

A Prandtl-Meyer expansion fan is a centered fan of Mach waves that forms when a supersonic flow turns around a convex corner. Unlike a shock, the expansion is isentropic (reversible), so the flow accelerates smoothly, and Mach number rises while pressure, temperature, and density all fall continuously through the fan.

The Prandtl-Meyer function nu(M) mixes M inside two arctan terms and a square root, and there is no algebraic way to isolate M once nu is known. Every practical calculator, including this one, solves nu(M2) = nu(M1) + theta for M2 with a numeric method such as bisection or Newton-Raphson.

As Mach number approaches infinity, the Prandtl-Meyer function nu(M) approaches a finite limit, nu_max = (pi/2) times (sqrt((gamma+1)/(gamma-1)) minus 1), about 130.45 degrees for air. Since nu(M2) = nu(M1) + theta can never exceed nu_max, theta is capped at nu_max minus nu(M1) for any M1, no matter how large M2 is allowed to grow.

An expansion fan accelerates the flow while keeping total pressure and total temperature constant (isentropic flow), so as Mach number rises the static pressure, temperature, and density, each a decreasing function of Mach number at fixed stagnation conditions, must all drop. This is the direct opposite of a compression shock, where the flow decelerates and those same properties rise.

A shock wave is a sudden, irreversible, entropy-generating compression where total pressure drops. A Prandtl-Meyer expansion fan is a smooth, reversible, isentropic expansion where total pressure and total temperature stay exactly constant. Use a shock calculator when flow compresses and slows down, and this calculator when flow expands around a convex corner and speeds up.

The Mach angle mu equals asin(1/M) and marks the angle of the Mach wave (the edge of the disturbance cone) relative to the local flow direction. This calculator reports mu1 at the leading edge of the fan (using M1) and mu2 at the trailing edge (using M2), together they mark the two boundary Mach lines of the fan.

Yes. The maximum turn angle nu_max = (pi/2) times (sqrt((gamma+1)/(gamma-1)) minus 1) depends only on gamma. For air with gamma = 1.4, nu_max is about 130.45 degrees. Gases with a gamma closer to 1, such as many polyatomic gases, have a larger nu_max, while gases with higher gamma have a smaller one.

A negative theta would represent a compression turn rather than an expansion, which cannot be handled isentropically by the Prandtl-Meyer relation alone, a compression corner instead produces an oblique shock. This calculator only accepts a positive theta, representing the flow expanding away from itself around a convex corner.

Expansion fans appear at the trailing edge of supersonic airfoils, at the corners of supersonic nozzle contours, and inside over-expanded or under-expanded rocket exhaust plumes wherever a supersonic flow needs to turn around a convex corner. They are one of the two fundamental building blocks of supersonic aerodynamics, the other being the oblique shock.

If the requested turn angle theta would require nu(M2) to exceed nu_max, no finite M2 satisfies the Prandtl-Meyer relation, so the calculator rejects the input and reports the maximum theta actually achievable for the entered M1 and gamma instead of returning an invalid or infinite result.

The calculator narrows the bracket between the upstream Mach number and a generous upper bound over 100 iterations, which drives the error in nu(M2) to well below 1e-10 radians in every practical case, giving a downstream Mach number accurate to more decimal places than are shown in the results.