Prandtl Meyer Expansion Calculator
Calculate supersonic flow properties after expansion waves using Prandtl-Meyer theory
Expansion Wave Calculations
Typical values: Air γ = 1.4, Diatomic gas γ = 1.4
Must be supersonic (M₁ > 1)
Surface turning angle (positive for expansion)
Upstream Flow Properties
Expansion Wave Results
Mach Numbers and Angles
Prandtl-Meyer Functions
Prandtl-Meyer function: ν(M) = √((γ+1)/(γ-1)) × arctan(√((γ-1)(M²-1)/(γ+1))) - arctan(√(M²-1))
Expansion relation: ν₂ = ν₁ + θ
Flow type: Isentropic expansion through Prandtl-Meyer fan
Validation
Example Calculation
Supersonic Flow Expansion
Initial conditions:
• Upstream Mach number: M₁ = 1.5
• Deflection angle: θ = 15°
• Specific heat ratio: γ = 1.4
• Pressure: P₁ = 1 atm
• Temperature: T₁ = 288 K
Solution Steps
1. Calculate ν₁ = 11.91° (upstream Prandtl-Meyer function)
2. Calculate ν₂ = ν₁ + θ = 11.91° + 15° = 26.91°
3. Solve for M₂ using inverse Prandtl-Meyer function: M₂ = 2.0
4. Calculate flow properties using isentropic relations
Result: M₂ = 2.0, P₂ = 0.469 atm, T₂ = 230 K
Key Concepts
Expansion waves form when supersonic flow turns away from itself
Prandtl-Meyer fan consists of infinite Mach lines
Flow accelerates and pressure decreases through expansion
Process is isentropic (entropy constant)
Total temperature and pressure remain constant
Typical Values
Understanding Prandtl-Meyer Expansion Theory
What are Expansion Waves?
Expansion waves occur when a supersonic flow encounters a surface that turns away from the flow direction. Unlike shock waves, expansion waves are continuous and isentropic, meaning the entropy remains constant throughout the process.
Key Characteristics
- •Flow accelerates (Mach number increases)
- •Pressure and temperature decrease
- •Density decreases
- •Total properties remain constant
Prandtl-Meyer Function
ν(M) = √((γ+1)/(γ-1)) × arctan(√((γ-1)(M²-1)/(γ+1))) - arctan(√(M²-1))
ν₂ = ν₁ + θ
- ν(M): Prandtl-Meyer function (radians or degrees)
- M: Mach number
- γ: Specific heat ratio
- θ: Deflection angle
Note: The Prandtl-Meyer function is only valid for supersonic flows (M > 1). The maximum deflection angle depends on the specific heat ratio.