Prandtl Meyer Angle Calculator

Prandtl-Meyer Function:

\[ \nu = \sqrt{\frac{\gamma+1}{\gamma-1}} \arctan\left(\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}\right) - \arctan\left(\sqrt{M^2-1}\right) \]

Prandtl-Meyer Angle (ν):

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1. What is the Prandtl-Meyer Angle?

The Prandtl-Meyer angle is the maximum angle through which a supersonic flow can turn isentropically. It represents the angle change required to accelerate flow from Mach 1 to a given supersonic Mach number.

2. How Does the Calculator Work?

The calculator uses the Prandtl-Meyer function:

\[ \nu = \sqrt{\frac{\gamma+1}{\gamma-1}} \arctan\left(\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}\right) - \arctan\left(\sqrt{M^2-1}\right) \]

Where:

\( \nu \) — Prandtl-Meyer angle (degrees)
\( \gamma \) — Heat capacity ratio (dimensionless)
\( M \) — Mach number (dimensionless)

Explanation: The function calculates the angle change required for isentropic expansion from Mach 1 to the specified Mach number.

3. Importance of Prandtl-Meyer Angle

Details: The Prandtl-Meyer angle is crucial in supersonic aerodynamics for designing expansion fans, analyzing flow around corners, and understanding shock-expansion theory.

4. Using the Calculator

Tips: Enter the heat capacity ratio (γ) and Mach number (M). Typical γ values: 1.4 for air, 1.3 for combustion products. Mach must be ≥1.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical meaning of the Prandtl-Meyer angle?
A: It represents the angle through which a supersonic flow must turn to achieve a given Mach number via isentropic expansion.

Q2: What are typical values for γ?
A: For air at standard conditions, γ ≈ 1.4. For combustion products, γ ≈ 1.2-1.3. For monatomic gases, γ ≈ 1.67.

Q3: What happens at Mach 1?
A: At Mach 1, the Prandtl-Meyer angle is zero, as this is the reference condition for the expansion.

Q4: How is this used in nozzle design?
A: The function helps determine the required expansion angle for supersonic nozzles to achieve desired exit Mach numbers.

Q5: What's the relationship with shock waves?
A: Prandtl-Meyer expansions are the isentropic counterpart to oblique shock waves, both being mechanisms for supersonic flow turning.