Expanding Around a Corner

When a supersonic flow turns away from itself, the speed must increase to satisfy the requirement that information propagates within the expanding region. Unlike compression, which concentrates disturbances into shocks, an expansion spreads changes smoothly through a fan of infinitesimal waves known as a Prandtl–Meyer expansion. Each wave makes a tiny adjustment to velocity and direction, and together they produce an isentropic transition from the initial Mach number to a higher one. This calculator evaluates the Prandtl–Meyer function to determine how a given turn angle modifies the Mach number and thermodynamic properties of a perfect gas.

Ludwig Prandtl and Theodor Meyer analyzed these expansion fans in the early twentieth century while building the foundations of compressible flow theory. Their names now identify a key relation connecting Mach number \(M\) and the flow turning angle \(\nu\). For a calorically perfect gas with ratio of specific heats \(\gamma\), the Prandtl–Meyer function is

where \(\nu\) is measured in radians. The function increases monotonically with \(M\), making it invertible. To relate two states, subtract their Prandtl–Meyer angles:

Given an upstream Mach number and a desired turning angle \(\theta\), one can compute the required downstream Mach number \(M_2\) by inverting \(\nu\). This tool accomplishes the inversion through a simple bisection routine, providing accurate results without external libraries.

Isentropic Property Changes

Because the expansion is isentropic, total pressure and total temperature remain constant across the fan. Static quantities, however, change dramatically as velocity rises. The temperature ratio obeys

and the pressure ratio follows from the isentropic relation \(p/p_0=(1+\frac{\gamma-1}{2}M^2)^{-\gamma/(\gamma-1)}\). Combining these gives

The density ratio simply divides the pressure and temperature ratios. These formulas allow quick evaluation of how static conditions fall as the flow accelerates through the expansion fan. Even modest turning angles can reduce pressure substantially, which is why nozzle designers carefully manage expansion to maximize thrust without overexpanding the flow.

Expansion Fan Insight

Prandtl–Meyer expansions occur when a supersonic jet exits a nozzle into a lower-pressure environment, when flow turns around a convex corner, or within overexpanded rocket nozzles that contain complex patterns of shocks and expansions. In high-speed intakes, controlled expansions can help align the flow with downstream components while preserving total pressure. The absence of shocks means entropy remains constant, making these expansions far more efficient than compression shocks.

The expansion waves emanate from the corner like rays of sunlight, each wave turning and accelerating the flow by an infinitesimal amount. At the fan’s outer edge the Mach number reaches its new value and the flow has rotated by the full angle \(\theta\). Because information travels at the local speed of sound, characteristics of the governing equations reveal that the expansion must spread over a finite region rather than a discontinuity. This subtle structure makes expansion fans more challenging to visualize than shocks, yet schlieren photography and computational simulations capture the graceful spreading of the flow.

To illustrate typical behavior, the table below shows how a 15° expansion affects a Mach 2 stream of air. The resulting Mach number climbs appreciably, while static pressure and temperature drop.

Quantity	Value
Initial Mach M₁	2.0
Deflection angle θ	15°
Final Mach M₂	2.56
p₂/p₁	0.57
T₂/T₁	0.79
ρ₂/ρ₁	0.72

Such expansions help explain the diamond-shaped shock-cell patterns seen in photographs of rocket plumes. Where the jet pressure exceeds ambient, the flow first expands, overshoots, and then recompresses through oblique shocks. The alternating sequence repeats until the jet equilibrates with the surroundings. Designers exploit these features to tailor thrust and reduce acoustic loads.

Use this calculator by supplying an upstream Mach number, a turning angle in degrees, and a heat capacity ratio. The script computes the downstream Mach number and the isentropic property ratios, providing immediate insight into how strongly an expansion accelerates a supersonic stream. By adjusting \(\theta\) you can explore the limits: as the angle increases, the required Mach number approaches a maximum where the Prandtl–Meyer function reaches \(\pi/2\) radians. Beyond that, the flow would theoretically expand to infinite Mach number, though real gases and finite nozzle geometry impose constraints.

While the theory assumes a perfect gas and steady, inviscid conditions, the Prandtl–Meyer relations nonetheless capture the essential behavior of many real flows. They underpin the design of supersonic wind tunnels, converging-diverging nozzles, and high-speed exhaust systems. Moreover, the concept connects to modern computational methods; numerical solvers often verify their accuracy by reproducing the analytic Prandtl–Meyer solution. Armed with this tool, students and engineers can deepen their understanding of supersonic expansions and appreciate the elegance with which mathematics describes high-speed aerodynamics.