Prandtl Meyer Expansion Calculator

When supersonic flow encounters a smooth convex corner, it undergoes an expansion process composed of an infinite number of infinitesimal Mach waves. This isentropic process is governed by the Prandtl-Meyer function, denoted as nu (nu). The function depends on the upstream Mach number (M1) and the specific heat ratio (gamma) of the gas. The calculation begins by determining intermediate parameters A and B, where A is a function of gamma and B relates to the initial Mach number: A = sqrt((gamma + 1) / (gamma - 1)) and B = sqrt(M1^2 - 1). The initial Prandtl-Meyer angle nu1 is then calculated as nu(M1) = A * atan(B / A) - atan(B). When the flow turns through an angle theta, the downstream Prandtl-Meyer angle becomes nu2 = nu1 + theta. Using nu2, the tool numerically solves for the downstream Mach number (M2) and subsequently computes static pressure (P2), temperature (T2), and density (rho2) using isentropic relations.

Consider a supersonic flow of air with an initial Mach number (M1) of 2.0, an upstream static pressure (P1) of 101.3 kPa, and an upstream temperature (T1) of 288 K. The specific heat ratio for air is gamma = 1.4. The flow encounters a convex corner turning angle (theta) of 15 degrees (0.2618 radians). First, compute parameter A = sqrt(2.4 / 0.4) = 2.4495 and B = sqrt(2.0^2 - 1) = 1.7321. Next, calculate the initial Prandtl-Meyer angle nu1 = 2.4495 * atan(1.7321 / 2.4495) - atan(1.7321) = 26.38 degrees. Adding the turn angle theta of 15 degrees yields the downstream Prandtl-Meyer angle nu2 = 41.38 degrees. Solving for the downstream Mach number M2 yields approximately 2.54. Finally, applying the isentropic temperature and pressure ratios results in a downstream temperature T2 of roughly 215 K and a downstream static pressure P2 of approximately 39.2 kPa, demonstrating how pressure and temperature drop significantly across the expansion fan.

Always ensure your initial Mach number is strictly greater than 1.0, as Prandtl-Meyer expansion only applies to supersonic and hypersonic regimes. Double-check your units for angles—ensure you are consistent between degrees and radians depending on trigonometric function inputs. Keep in mind that while static pressure, temperature, and density drop across an expansion fan, the total (stagnation) pressure and total temperature remain constant because the expansion is assumed to be ideal and isentropic.