Free Prandtl-Meyer Expansion Calculator

Understanding Supersonic Expansion Waves

When a supersonic flow encounters a convex corner (a surface that folds outward), it does not produce a shock—instead, a continuous fan of expansion waves forms, turning the flow and increasing its Mach number. This behavior is central to nozzle design, external aerodynamics, and high‑speed propulsion. The Prandtl‑Meyer expansion calculator—a free online expansion wave calculator—lets you rapidly obtain the supersonic flow Mach number and, if desired, the complete set of downstream flow properties after such an expansion.

Depending on the nozzle exit condition, three situations arise:

• Optimally expanded – the flow exits at the design pressure; no waves appear.
• Under‑expanded – the nozzle pressure is too high; oblique shock waves form.
• Over‑expanded – the nozzle pressure is too low; an expansion wave helps the flow adjust to the back pressure.

An expansion wave is therefore a key feature of over‑expanded nozzles and any external corner in a supersonic stream.

The Prandtl‑Meyer Theory

The foundation for analyzing expansion waves was laid by Ludwig Prandtl (1907) and his student Theodor Meyer (1908). The theory links the deflection angle to the Mach numbers upstream and downstream of the wave through the Prandtl‑Meyer function $u(M)$. For a calorically perfect gas with specific heat ratio $\gamma$:

$$u(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \arctan\!\left(\sqrt{\frac{\gamma-1}{\gamma+1}\left(M^{2}-1\right)}\right) - \arctan\!\left(\sqrt{M^{2}-1}\right)$$

The function is monotonic with Mach number, so given an upstream Mach number $M_{1}$ and an outward deflection angle $\theta$, the downstream function value is:

$$u(M_{2}) = u(M_{1}) + \theta$$

Solving this transcendental equation for $M_{2}$ (the supersonic flow Mach number after the wave) requires either an iterative method or a pre‑computed $u$-$M$ table. The calculator performs this step automatically, saving significant time.

Why the Expansion Is Isentropic

Because the expansion wave is a continuous, non‑discontinous process, there is no dissipative loss. As a result:

• Total pressure and total temperature remain constant.
• Entropy does not change—the process is isentropic.

With $M_{2}$ known, the downstream static properties can be obtained using the standard isentropic expansion relations:

\begin{aligned} \frac{P_{2}}{P_{1}} &= \frac{\left(1 + \frac{\gamma-1}{2}M_{1}^{2}\right)^{\frac{\gamma}{\gamma-1}}}{\left(1 + \frac{\gamma-1}{2}M_{2}^{2}\right)^{\frac{\gamma}{\gamma-1}}} \$$6pt] \frac{T_{2}}{T_{1}} &= \frac{1 + \frac{\gamma-1}{2}M_{1}^{2}}{1 + \frac{\gamma-1}{2}M_{2}^{2}} \$$6pt] \frac{\rho_{2}}{\rho_{1}} &= \frac{T_{1}}{T_{2}} \cdot \frac{P_{2}}{P_{1}} \end{aligned}

These equations make the tool a complete downstream flow properties calculator for expansion fans.

How to Use the Expansion Wave Calculator

The expansion wave calculator provides two modes:

1. Basic mode – Input only the upstream Mach number $M_{1}$ and the deflection angle $\theta$. The tool outputs the downstream Mach number $M_{2}$, the forward and rearward Mach angles ($\mu_{1}$, $\mu_{2}$), and the Prandtl‑Meyer function evaluations ($u_{1}$, $u_{2}$).

2. Full mode – In addition, provide the upstream static temperature $T_{1}$, pressure $P_{1}$, and density $\rho_{1}$. The calculator then also returns the full set of downstream flow properties ($P_{2}$, $T_{2}$, $\rho_{2}$).

The default specific heat ratio is $\gamma = 1.4$ (air), but you can change it if your working fluid differs.

Worked Example

Consider a supersonic flow with the following upstream conditions:

• $M_{1} = 1.5$
• $P_{1} = 1 \, \text{atm}$
• $T_{1} = 288 \, \text{K}$
• $\rho_{1} = 1.22586 \, \text{kg/m}^{3}$
• Deflection angle $\theta = 15^{\circ}$

Selecting full mode (YES for flow properties) and $\gamma = 1.4$, the Prandtl‑Meyer expansion calculator yields:

Parameter	Symbol	Value
Downstream Mach number	$M_{2}$	2.0
Downstream pressure	$P_{2}$	0.469 atm
Downstream temperature	$T_{2}$	230.0 K
Downstream density	$\rho_{2}$	0.69896 kg/m³
Forward Mach angle	$\mu_{1}$	41.81°
Rearward Mach angle	$\mu_{2}$	29.69°
Prandtl‑Meyer function for $M_{1}$	$
u_{1} $	11.91°
Prandtl‑Meyer function for $M_{2}$	$
u_{2} $	26.91°
Rearward Mach line angle from horizontal	$\mu_{2} - \theta$	14.69°

The results confirm that the flow accelerates and the static pressure, temperature, and density all decrease—behaviour typical of an isentropic expansion. This expansion wave calculator thus offers a fast, reliable way to perform Prandtl‑Meyer analysis without manual iteration.