Prandtl Meyer Expansion Fan Calculator

Prandtl-Meyer Function:

\[ \theta = \nu(M_2) - \nu(M_1) \] \[ \nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \tan^{-1}\left(\sqrt{\frac{\gamma - 1}{\gamma + 1}(M^2 - 1)}\right) - \tan^{-1}\left(\sqrt{M^2 - 1}\right) \]

Initial Mach Number (M₁):

Mach

Final Mach Number (M₂):

Mach

Specific Heat Ratio (γ):

(dimensionless)

Expansion Fan Turn Angle (θ):

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1. What is the Prandtl-Meyer Expansion Fan?

The Prandtl-Meyer expansion fan describes the gradual turn of supersonic flow around a convex corner. The flow accelerates through a fan of Mach waves, and the turn angle θ is calculated using the Prandtl-Meyer function ν(M).

2. How Does the Calculator Work?

The calculator uses the Prandtl-Meyer function:

\[ \theta = \nu(M_2) - \nu(M_1) \] \[ \nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \tan^{-1}\left(\sqrt{\frac{\gamma - 1}{\gamma + 1}(M^2 - 1)}\right) - \tan^{-1}\left(\sqrt{M^2 - 1}\right) \]

Where:

\( \theta \) — Turn angle (degrees)
\( \nu(M) \) — Prandtl-Meyer function (radians)
\( M \) — Mach number
\( \gamma \) — Specific heat ratio (1.4 for air)

Explanation: The function relates the Mach number to the maximum possible turn angle for an expansion fan in supersonic flow.

3. Importance of Expansion Fan Calculations

Details: These calculations are essential in aerospace engineering for designing supersonic nozzles, wings, and other components where flow expansion occurs.

4. Using the Calculator

Tips: Enter the initial and final Mach numbers (must be ≥1) and the specific heat ratio (γ=1.4 for air). The calculator will determine the turn angle between these two Mach numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical meaning of the Prandtl-Meyer function?
A: It represents the maximum angle through which a sonic flow (M=1) must turn to reach a given Mach number.

Q2: What happens if the turn angle exceeds the maximum?
A: The flow cannot turn through angles larger than ν(M) from M=1. This would require additional expansion waves or other flow phenomena.

Q3: What's the typical γ value for air?
A: For air at standard conditions, γ ≈ 1.4. For other gases: monatomic (γ=1.67), diatomic (γ≈1.4), polyatomic (γ≈1.3).

Q4: Can this be used for compression turns?
A: No, this is only for expansion fans. Compression turns involve oblique shock waves and different calculations.

Q5: What are typical applications?
A: Supersonic nozzle design, aerodynamic surfaces in supersonic flight, and analysis of supersonic flow around corners.