Compressible Aerodynamics of Prandtl-Meyer Expansion Fans
In supersonic aerodynamics, when a supersonic fluid flow encounters a concave boundary (deflecting into the flow), compressive streamlines converge to produce an oblique shock wave, raising pressure and density abruptly while generating irreversible entropy. Conversely, when a supersonic flow encounters a convex corner or curved outward surface (turning away from the flow), the streamlines diverge. Because disturbances in supersonic flow can only propagate along characteristic Mach lines, the flow expands smoothly through a centered continuous family of infinitesimal acoustic waves known as a Prandtl-Meyer Expansion Fan.
Formulated by Ludwig Prandtl and Theodor Meyer in 1908 at Göttingen, the Prandtl-Meyer expansion is a cornerstone of supersonic aerodynamic design, governing supersonic airfoil suction surfaces, rocket engine contour nozzles, scramjet inlet expansions, and overexpanded jet plumes.
1. The Prandtl-Meyer Function $nu(M)$
Consider a supersonic streamline turning through an infinitesimal convex angle $dtheta$. The velocity vector accelerates by $dV$, while static pressure drops by $dp$. By projecting the equations of continuity, momentum, and acoustic wave angle onto the characteristic direction, one derives the differential relation:
Integrating this differential equation from sonic speed ($M = 1$, where $theta = 0$) up to an arbitrary supersonic Mach number $M$ yields the classical Prandtl-Meyer Function:
The value of $nu(M)$ has the physical meaning of the total angle through which a sonic flow ($M=1$) must expand to accelerate to Mach number $M$.
For a flow entering at supersonic Mach number $M_1$ and turning through a finite convex angle $theta$, the downstream Prandtl-Meyer function value $nu(M_2)$ is simply:
Inverting this algebraic relationship yields the exact downstream Mach number $M_2$.
2. Geometry of the Expansion Fan and Mach Waves
The expansion fan is bounded by two limiting characteristic Mach lines emanating from the corner vertex:
- Forward Mach Line: Marks the upstream boundary of the fan. It is inclined relative to the upstream horizontal wall at the upstream Mach angle:
mu_1 = arcsinleft(frac{1}{M_1}right)
- Rearward (Terminal) Mach Line: Marks the downstream exit boundary of the expansion fan. It is inclined relative to the downstream deflected wall at the downstream Mach angle:
mu_2 = arcsinleft(frac{1}{M_2}right)Relative to the initial upstream horizontal axis, the terminal line angle is $mu_2 - theta$.
The total angular sector occupied by the expansion fan (the fan spread $Delta phi$) is:
3. Isentropic Thermodynamic State Evolution
Unlike a shock wave—which is non-isentropic due to viscous dissipation across a finite thickness—an expansion fan consists of an infinite continuum of infinitely weak acoustic Mach waves. The entropy generation across each wave is $ds propto (dp)^3 to 0$. Therefore:
Because the expansion is both adiabatic and reversible, the stagnation total pressure and total temperature are perfectly conserved across the entire expansion:
The static temperature, pressure, and density ratios between state 1 and state 2 are calculated using standard isentropic compressible relations:
4. Maximum Theoretical Expansion Angle ($theta_{max}$)
If the wall deflects through an extreme angle, the gas accelerates until static temperature and pressure asymptotically approach absolute zero ($M_2 to infty$, expansion into vacuum). In this theoretical limit, the Prandtl-Meyer function approaches a finite maximum value:
For standard air ($gamma = 1.40$):
Therefore, the maximum physical deflection angle before a supersonic flow expands into vacuum is:
Step-by-Step Worked Aerodynamic Example
Air at $M_1 = 1.50$, static pressure $p_1 = 100.0 text{kPa}$, and static temperature $T_1 = 300.0 text{K}$ expands around a convex corner with deflection angle $theta = 20.0^circ$ ($gamma = 1.40$).
Step 1: Compute Initial Prandtl-Meyer Function $nu(M_1)$
Step 2: Compute Downstream $nu(M_2)$ and Invert for $M_2$
Using Newton-Raphson iteration on $nu(M) = 31.91^circ$:
The flow has accelerated from $M_1 = 1.50$ to $M_2 = 2.21$!
Step 3: Evaluate Isentropic State Ratios
Step 4: Compute Mach Angles and Fan Spread
Comparison: Oblique Shock vs. Prandtl-Meyer Expansion
| Aerodynamic Feature | Oblique Shock Wave (Compression) | Prandtl-Meyer Expansion Fan (Divergence) |
|---|---|---|
| Geometry | Concave corner (turns into the flow) | Convex corner (turns away from the flow) |
| Mach Number Change | Decelerates ($M_2 < M_1$) | Accelerates ($M_2 > M_1$) |
| Static Pressure ($p_2/p_1$) | Sharp compression jump ($p_2 > p_1$) | Smooth continuous drop ($p_2 < p_1$) |
| Static Temperature ($T_2/T_1$) | Temperature increases ($T_2 > T_1$) | Rapid cooling ($T_2 < T_1$) |
| Stagnation Pressure ($p_{02}/p_{01}$) | Loss of total pressure ($p_{02} < p_{01}$) | Conserved ($p_{02} = p_{01}$, isentropic) |
| Thermodynamic Nature | Irreversible ($Delta s > 0$) | Reversible ($Delta s = 0$) |