Prandtl Meyer Function at Upstream Mach Number Calculator

Prandtl Meyer Function Formula:

\[ v(M_1) = \sqrt{\frac{\gamma_e + 1}{\gamma_e - 1}} \cdot \tan^{-1}\left(\sqrt{\frac{(\gamma_e - 1)(M_{e1}^2 - 1)}{\gamma_e + 1}}\right) - \tan^{-1}\left(\sqrt{M_{e1}^2 - 1}\right) \]

Prandtl Meyer Function at Upstream Mach no.:

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1. What is the Prandtl Meyer Function?

The Prandtl Meyer Function describes the maximum turning angle that a supersonic flow can undergo through an expansion wave. It is a fundamental concept in gas dynamics and compressible flow theory, named after Ludwig Prandtl and Theodor Meyer.

2. How Does the Calculator Work?

The calculator uses the Prandtl Meyer Function formula:

\[ v(M_1) = \sqrt{\frac{\gamma_e + 1}{\gamma_e - 1}} \cdot \tan^{-1}\left(\sqrt{\frac{(\gamma_e - 1)(M_{e1}^2 - 1)}{\gamma_e + 1}}\right) - \tan^{-1}\left(\sqrt{M_{e1}^2 - 1}\right) \]

Where:

\( v(M_1) \) — Prandtl Meyer Function at upstream Mach number (Radian)
\( \gamma_e \) — Specific Heat Ratio Expansion Wave
\( M_{e1} \) — Mach Number Ahead of Expansion Fan

Explanation: The function calculates the maximum turning angle for a supersonic flow through an expansion fan, considering the specific heat ratio and upstream Mach number.

3. Importance of Prandtl Meyer Function Calculation

Details: Accurate calculation of the Prandtl Meyer Function is crucial for designing supersonic nozzles, analyzing expansion waves in compressible flows, and understanding gas dynamics in aerospace applications.

4. Using the Calculator

Tips: Enter the specific heat ratio (typically 1.4 for air) and the upstream Mach number (must be greater than 1). All values must be valid (γe > 1, Me1 > 1).

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the Prandtl Meyer Function?
A: It represents the maximum angle through which a supersonic flow can turn isentropically through an expansion wave.

Q2: What are typical values for specific heat ratio?
A: For air at standard conditions, γ = 1.4. For monatomic gases like helium, γ = 1.67. For diatomic gases, it typically ranges from 1.3 to 1.4.

Q3: Why must the Mach number be greater than 1?
A: The Prandtl Meyer expansion theory applies only to supersonic flows. Subsonic flows cannot produce expansion waves in the same manner.

Q4: What are the units of the Prandtl Meyer Function?
A: The function is measured in radians, though it is often converted to degrees for practical applications (1 radian = 57.2958 degrees).

Q5: Are there limitations to this function?
A: The function assumes isentropic flow, perfect gas behavior, and two-dimensional flow. It may not accurately represent real-world conditions with significant viscous effects or three-dimensional complexities.