Prandtl Meyer Function Calculator

Prandtl-Meyer Function:

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

Prandtl-Meyer Angle (ν):

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1. What is the Prandtl-Meyer Function?

The Prandtl-Meyer function describes the angle through which a supersonic flow turns isentropically from Mach 1 to a given Mach number. It's fundamental in supersonic aerodynamics for analyzing expansion fans.

2. How Does the Calculator Work?

The calculator uses the Prandtl-Meyer function:

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

Where:

\( \nu \) — Prandtl-Meyer angle (radians)
\( M \) — Mach number (dimensionless, ≥1)
\( \gamma \) — Specific heat ratio (dimensionless, >1)

Explanation: The function calculates the maximum turning angle possible for an isentropic expansion from Mach 1 to the given Mach number.

3. Importance of Prandtl-Meyer Function

Details: This function is crucial for designing supersonic nozzles, analyzing expansion waves, and understanding supersonic flow around corners.

4. Using the Calculator

Tips: Enter Mach number (must be ≥1) and specific heat ratio (γ). For air at standard conditions, γ = 1.4.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical meaning of the Prandtl-Meyer angle?
A: It represents the maximum angle through which a supersonic flow can turn isentropically (without shocks) from Mach 1 to the given Mach number.

Q2: What are typical values for γ?
A: For air, γ ≈ 1.4. For monatomic gases (He, Ar), γ ≈ 1.67. For diatomic gases at high temp (N₂, O₂), γ can approach 1.2.

Q3: What happens at Mach 1?
A: The Prandtl-Meyer angle is zero at Mach 1, as this is the reference starting point.

Q4: Can this be used for oblique shock calculations?
A: Yes, the Prandtl-Meyer function is used in conjunction with oblique shock relations to solve flow deflection problems.

Q5: What are the limitations of this function?
A: It assumes isentropic flow, perfect gas behavior, and two-dimensional flow. It's not valid for subsonic Mach numbers (M < 1).