Prandtl Meyer Speed Calculator

Prandtl-Meyer Function:

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

Mach Number (M):

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1. What is the Prandtl-Meyer Function?

The Prandtl-Meyer function describes the angle through which a supersonic flow turns as it expands around a corner. It relates the Mach number of the flow to the maximum possible turning angle for a given specific heat ratio.

2. How Does the Calculator Work?

The calculator uses the inverse Prandtl-Meyer function:

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

Where:

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

Explanation: The calculator uses an iterative numerical method (Newton-Raphson) to solve for Mach number given the Prandtl-Meyer angle.

3. Importance of Mach Number Calculation

Details: The Mach number is crucial in aerodynamics for determining flow characteristics, shock wave formation, and expansion fan properties in supersonic flows.

4. Using the Calculator

Tips: Enter the Prandtl-Meyer angle in degrees and the specific heat ratio (typically 1.4 for air). The calculator will determine the corresponding Mach number.

5. Frequently Asked Questions (FAQ)

Q1: What is a typical value for γ?
A: For air at standard conditions, γ ≈ 1.4. For other gases: monatomic (1.67), diatomic (1.4), triatomic (1.33).

Q2: What is the maximum turning angle?
A: The maximum occurs when M→∞, giving ν_max = 130.5° for γ=1.4.

Q3: When is this calculation important?
A: For designing supersonic nozzles, analyzing expansion fans, and studying supersonic flow around corners.

Q4: What are the limitations?
A: Only valid for isentropic, steady, 2D supersonic flow. Not applicable for subsonic flow or flows with shocks.

Q5: How accurate is the numerical method?
A: The Newton-Raphson method provides high accuracy (typically within 0.1%) when properly implemented.