6.5 Prandtl-Meyer expansion waves

Prandtl-Meyer Expansion Waves

Prandtl-Meyer expansion waves describe how supersonic flow behaves when it turns around a convex corner. The flow expands, accelerates, and drops in pressure, density, and temperature. This is one of the core building blocks for analyzing and designing anything that operates at supersonic speeds: nozzles, airfoils, intakes, and more.

The central tool here is the Prandtl-Meyer function, which connects a flow's Mach number to its turning angle. Once you understand how this function works, you can predict the downstream conditions after an expansion with just a few steps of calculation.

Definition of Expansion Waves

An expansion wave is a fan of Mach waves that spreads out from a convex corner in supersonic flow. Unlike oblique shock waves (which compress the flow and increase entropy), expansion waves do the opposite: they decrease pressure, density, and temperature while increasing velocity and Mach number.

The expansion process is isentropic. That means no heat is added or removed, and entropy stays constant. Total (stagnation) pressure and total temperature are preserved across the wave. This is a major difference from shocks, where total pressure always drops.

Assumptions in Prandtl-Meyer Theory

The classical Prandtl-Meyer analysis rests on several simplifying assumptions:

• The flow is steady, inviscid (no viscosity), and adiabatic (no heat transfer)
• The gas is ideal and calorically perfect, meaning constant specific heats (constant $\gamma$)
• Effects of boundary layers, shock waves, and flow separation are neglected

These assumptions work well for the core of the flow away from walls. In real applications, boundary layer effects near surfaces can modify the picture, but the inviscid Prandtl-Meyer solution remains the starting point for design.

The Prandtl-Meyer Function

The Prandtl-Meyer function $\nu(M)$ gives the total turning angle a flow has undergone in expanding from Mach 1 to Mach number $M$. It is defined as:

$\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \tan^{-1} \sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \tan^{-1} \sqrt{M^2-1}$

where $\gamma$ is the ratio of specific heats and $M$ is the Mach number. At $M = 1$, $\nu = 0$. As $M \to \infty$, $\nu$ approaches a finite maximum value:

$\nu_{\max} = \frac{\pi}{2}\left(\sqrt{\frac{\gamma+1}{\gamma-1}} - 1\right)$

For air ($\gamma = 1.4$), this maximum is about 130.45°. That's the theoretical upper limit on how much a supersonic flow can turn through expansion.

Relating Mach Number to Flow Deflection

When supersonic flow at Mach number $M_1$ turns through a convex corner of angle $\theta$, the downstream Mach number $M_2$ is found by:

$\nu(M_2) = \nu(M_1) + \theta$

Here's the step-by-step process:

1. Look up (or calculate) $\nu(M_1)$ for the upstream Mach number
2. Add the wall deflection angle $\theta$ to get $\nu(M_2)$
3. Invert the Prandtl-Meyer function to find $M_2$ (use tables, charts, or a numerical solver since the function can't be inverted analytically)
4. With $M_2$ known, use isentropic relations to find downstream pressure, temperature, and density ratios

Because the process is isentropic, the stagnation properties don't change. You can write the static property ratios purely in terms of $M_1$ and $M_2$:

$\frac{p_2}{p_1} = \frac{\left(1 + \frac{\gamma-1}{2}M_1^2\right)^{\gamma/(\gamma-1)}}{\left(1 + \frac{\gamma-1}{2}M_2^2\right)^{\gamma/(\gamma-1)}}$

The same pattern applies for temperature and density using the standard isentropic relations.

Expansion Wave Geometry

Expansion waves originate at the corner where the wall turns. Each individual Mach wave in the fan is inclined at the local Mach angle:

$\mu = \sin^{-1}\left(\frac{1}{M}\right)$

The first Mach wave sits at angle $\mu_1 = \sin^{-1}(1/M_1)$ relative to the upstream flow direction. The last Mach wave sits at $\mu_2 = \sin^{-1}(1/M_2)$ relative to the downstream flow direction. Since $M_2 > M_1$, the downstream Mach angle is smaller, and the fan spreads out.

Between these two bounding waves, the flow properties change continuously and smoothly. Outside the fan, the flow is uniform and unaffected.

Centered Expansion Waves

When the wall has a sharp corner (an abrupt change in direction), all the Mach waves originate from a single point. This produces a centered expansion fan with a radial, fan-like structure.

• Flow properties (pressure, density, temperature, velocity direction) vary continuously from the first wave to the last
• The solution depends only on the angular position within the fan, not on distance from the corner
• This is the classic textbook case and the one the Prandtl-Meyer function directly solves

If the wall curves gradually instead of turning sharply, the expansion waves are distributed along the surface rather than emanating from one point. The same physics applies, but the waves don't all originate from the same location.

Weak vs. Strong Expansion Waves

The terms "weak" and "strong" here refer to the size of the deflection angle $\theta$:

• Weak expansions involve small $\theta$. The change in flow properties is gradual, and each Mach wave produces only a tiny perturbation. Linear (small-disturbance) theory can approximate these well.
• Strong expansions involve large $\theta$. The Mach number increase is substantial, and the full nonlinear Prandtl-Meyer function is needed.

For very large turning angles, the static temperature and pressure can drop dramatically. If the required $\nu(M_2)$ exceeds $\nu_{\max}$, the flow cannot turn that far through expansion alone, and the analysis breaks down (physically, a vacuum region would form).

Supersonic Flow over Convex Corners

This is the most direct application of the theory. When supersonic flow hits a convex corner (the wall turns away from the flow), an expansion fan forms at the corner. The result:

• Mach number increases
• Static pressure, density, and temperature all decrease
• The flow direction changes to follow the new wall angle
• Total pressure and total temperature are unchanged (isentropic process)

On a supersonic airfoil, the upper and lower surfaces typically have regions where the wall curves away from the flow. Each of these regions generates expansion waves that must be accounted for in the overall pressure distribution and force calculation.

Prandtl-Meyer Expansion in Nozzles

In a converging-diverging (de Laval) nozzle, the diverging section accelerates the flow beyond Mach 1. You can think of the diverging wall as a series of small convex turns, each producing a small expansion.

• The nozzle contour is designed so these expansions combine to produce a uniform, parallel flow at the desired exit Mach number
• The method of characteristics is the standard technique for designing the nozzle wall shape. It traces Mach waves (characteristic lines) through the flow to ensure waves cancel properly and the exit flow is uniform
• A poorly designed contour can leave residual waves in the exit flow, degrading performance

Rocket nozzles are a prime example: the bell-shaped contour of a rocket nozzle is essentially a Prandtl-Meyer expansion surface optimized for maximum thrust at the design altitude.

Method of Characteristics and Compressible Flow Analysis

The method of characteristics is closely tied to Prandtl-Meyer theory. It exploits the fact that in steady, supersonic, 2D flow, information propagates along characteristic lines (Mach waves).

The approach works as follows:

1. Identify the characteristic directions at each point (the left-running and right-running Mach waves)
2. Along each characteristic, a specific combination of flow angle and Prandtl-Meyer function is constant (these are the Riemann invariants)
3. By tracing a network of characteristics through the flow, you can solve for the Mach number and flow direction everywhere

For a simple expansion fan, this reduces to the straightforward $\nu(M_2) = \nu(M_1) + \theta$ relation. For more complex geometries with multiple waves interacting, the full characteristic network is needed.

Prandtl-Meyer Function Tables and Charts

Since $\nu(M)$ can't be inverted in closed form, tables and charts are standard tools. A typical table lists $M$, $\nu(M)$, and $\mu(M)$ for a given $\gamma$.

For example, with $\gamma = 1.4$:

These tables let you solve expansion problems quickly without evaluating the function each time. Most compressible flow textbooks include extensive tables, and many online calculators are available for quick lookups.

Applications of Prandtl-Meyer Expansions

• Supersonic airfoil design: The pressure distribution on a supersonic wing is determined by a combination of oblique shocks (compression) and Prandtl-Meyer expansions. Shaping the airfoil surfaces controls where each occurs, directly affecting lift and wave drag.
• Rocket nozzles: The diverging section is designed using Prandtl-Meyer expansion theory (via the method of characteristics) to produce the target exit Mach number and uniform exit flow.
• Supersonic wind tunnels: The test section Mach number is set by expansion in the nozzle upstream. Prandtl-Meyer theory guides the nozzle design to ensure clean, uniform flow over test models.
• Supersonic inlets and diffusers: Understanding expansion waves is necessary for designing inlets that efficiently decelerate supersonic flow into an engine, often in combination with oblique shocks.