Prandtl-Meyer Expansion Fans

When a supersonic flow encounters a convex corner, it doesn’t form a shockwave; instead, it expands smoothly and continuously through a fan of Mach waves. This phenomenon, the Prandtl-Meyer expansion fan, is fundamental to designing efficient supersonic inlets, nozzles, and wings. Understanding how to calculate the precise turning and property changes of the flow is key to predicting aerodynamic forces and preventing performance losses in high-speed aircraft and spacecraft.

The Nature of a Supersonic Expansion

A supersonic flow is characterized by its Mach number $M>1$, meaning disturbances propagate more slowly than the flow itself. When such a flow turns around a smooth, convex surface, it must accelerate to navigate the increased area. This turning occurs not instantaneously, but through a series of infinitely weak expansion waves that emanate from the corner. Unlike shock waves, these waves are isentropic, meaning they involve no loss in total pressure and are therefore thermodynamically reversible. The entire region encompassing these waves is called an expansion fan. Its existence is a direct consequence of the supersonic flow’s inability to "sense" the corner upstream, requiring a gradual adjustment process.

Deriving the Prandtl-Meyer Function

The cornerstone of analyzing this expansion is the Prandtl-Meyer function, denoted as $\nu (M)$. It provides an exact relationship between the local Mach number of a supersonic flow and the maximum angle through which it could theoretically be turned, from Mach 1 to its current Mach, isentropically. The derivation begins by considering the flow deflection across a single, weak Mach wave—a characteristic line. Using the governing equations of motion for a steady, two-dimensional, isentropic flow of a perfect gas, one arrives at a fundamental relation between the turning angle $d\theta$ and the change in Mach number $dM$.

Integrating this relation from a reference state (Mach 1, where $\nu =0$) to an arbitrary Mach number $M$ yields the Prandtl-Meyer function: $$\nu (M)=\sqrt{\frac{\gamma +1}{\gamma -1}}\cdot {\tan }^{-1}\left(\sqrt{\frac{\gamma -1}{\gamma +1}(M^2-1)}\right)-{\tan }^{-1}\left(\sqrt{M^2-1}\right)$$ Here, $\gamma$ is the ratio of specific heats for the gas (e.g., $\gamma =1.4$ for air). The function $\nu (M)$ is monotonically increasing; a higher Mach number corresponds to a larger cumulative turning angle from the sonic condition. In practice, for a flow turning around a corner of angle $\theta$, the final Mach number $M_2$ is found by solving $\nu (M_2)=\nu (M_1)+\theta$, where $M_1$ is the initial upstream Mach number and $\theta$ is the wall turning angle.

Geometry of the Expansion Fan

The expansion fan is not a single wave but a centered, straight-line fan originating at the convex corner. The first Mach wave of the fan is defined by the Mach angle ${\mu }_1$ of the upstream flow, where $\mu ={\sin }^{-1}(1/M)$. The last Mach wave is defined by the Mach angle ${\mu }_2$ of the downstream flow. Therefore, the fan's angular spread is ${\mu }_2-{\mu }_1+\theta$. Within this fan, flow properties like Mach number, pressure, and temperature change continuously from their upstream to their downstream values along any streamline. The flow direction also changes smoothly from its initial to its final angle. This contrasts sharply with an oblique shock, where properties change discontinuously across a single line.

Property Changes Across the Fan

Because the expansion process is isentropic, the total (or stagnation) temperature $T_0$ and total pressure $p_0$ remain constant throughout. This is the defining advantage over shock waves, which incur losses. The changes in static properties are determined using the standard isentropic flow relations in conjunction with the Mach number change calculated via the Prandtl-Meyer function.

For a flow turning an angle $\theta$, you first find $M_2$ using $\nu (M_2)=\nu (M_1)+\theta$. Then, property ratios are computed:

• Pressure: $p_2/p_1={\left[\frac{1+\frac{\gamma -1}{2}{M}_1^2}{1+\frac{\gamma -1}{2}{M}_2^2}\right]}^{\gamma /(\gamma -1)}$
• Temperature: $T_2/T_1=\frac{1+\frac{\gamma -1}{2}{M}_1^2}{1+\frac{\gamma -1}{2}{M}_2^2}$
• Density: ${\rho }_2/{\rho }_1={\left[\frac{1+\frac{\gamma -1}{2}{M}_1^2}{1+\frac{\gamma -1}{2}{M}_2^2}\right]}^{1/(\gamma -1)}$

Across an expansion fan, the Mach number increases ($M_2>M_1$), while static pressure, temperature, and density all decrease. The flow accelerates.

Applications: Nozzle Design and Shock-Expansion Theory

The Prandtl-Meyer expansion principle is critical in supersonic nozzle design, particularly for rocket engines and wind tunnels. The diverging section of a supersonic nozzle is designed to guide the flow’s expansion smoothly. If the nozzle walls turn too sharply, an expansion fan originates at the corner, potentially leading to non-uniform flow or separation if it reflects from the opposite wall. Designers use the Prandtl-Meyer function to contour the nozzle for a desired, uniform exit Mach number.

For supersonic airfoil analysis, shock-expansion theory provides a powerful, first-order method for calculating lift and pressure drag. The theory models the flow over a diamond-shaped or curved airfoil by patching together solutions from oblique shocks (on surfaces facing into the flow) and Prandtl-Meyer expansions (on surfaces facing away from the flow). For example, on a flat plate at an angle of attack in a supersonic stream, the lower surface experiences an oblique shock (increased pressure), while the upper surface experiences an expansion fan (decreased pressure). The net pressure difference yields lift. Integrating the pressure distributions calculated from shock and expansion relations over the entire airfoil surface gives the aerodynamic forces.

Common Pitfalls

1. Confusing Expansion Fans with Shock Waves: The most fundamental error is treating a convex turn as generating a shock. Remember: concave turns (flow into itself) create compression shocks; convex turns (flow away from itself) create isentropic expansion fans. The property changes are opposite: shocks increase pressure and temperature and decrease Mach number, while expansion fans decrease pressure and temperature and increase Mach number.
2. Misapplying the Turning Angle Equation: A common calculation mistake is misplacing the angle in the relation $\nu (M_2)=\nu (M_1)+\theta$. The turning angle $\theta$ is added to the initial Prandtl-Meyer function to find the final one. For an expansion, $\theta$ is positive. If you accidentally subtract it, you will incorrectly calculate a lower final Mach number, implying a compression.
3. Assuming Total Pressure Loss: Because the process is isentropic, the total pressure remains perfectly constant. Do not apply any loss coefficient or efficiency factor when calculating downstream total pressure from upstream conditions. Any total pressure loss in a real supersonic flow is due to viscous effects or the presence of shock waves elsewhere, not the expansion fan itself.
4. Ignoring the Maximum Turning Limit: The Prandtl-Meyer function $\nu (M)$ represents the maximum possible turning from sonic conditions. For a given initial $M_1$, there is a theoretical maximum turning angle ${\theta }_{max}={\nu }_{max}-\nu (M_1)$, where ${\nu }_{max}=\frac{\pi }{2}(\sqrt{\frac{\gamma +1}{\gamma -1}}-1)$ (e.g., ~130.5° for air). Attempting to turn the flow more than this would require expanding to infinite Mach number and zero pressure, which is physically impossible and indicates a vacuum region will form.

Summary

• A Prandtl-Meyer expansion fan is an isentropic, continuous wave structure that guides a supersonic flow around a convex corner, causing it to accelerate.
• The Prandtl-Meyer function $\nu (M)$ quantifies the relationship between Mach number and turning angle, allowing precise calculation of the final flow state using $\nu (M_2)=\nu (M_1)+\theta$.
• Across the expansion fan, Mach number increases while static pressure, temperature, and density all decrease; crucially, total pressure and temperature remain constant.
• Key applications include designing the diverging section of supersonic nozzles and calculating forces on airfoils using shock-expansion theory, which combines solutions from oblique shocks and expansion fans.