The Expansion Fan Calculator is a specialized tool used in compressible fluid dynamics to analyze how supersonic flow behaves when it expands around a convex corner. This tool applies the Prandtl-Meyer expansion theory, which describes the smooth and continuous changes in flow properties such as Mach number, pressure, temperature, and density during this type of expansion.
This calculator is essential for aerospace engineers, nozzle designers, and students of aerodynamics who need to model flow turning in supersonic jets, rocket nozzles, and shock-expansion interactions. By automating complex mathematical computations, the tool helps users determine the downstream Mach number, turning angles, and resulting flow conditions using upstream flow data.
Formula of Expansion Fan Calculator
1. Prandtl-Meyer Function (ν(M))
This function calculates the angle needed to turn a sonic flow (M = 1) to a desired Mach number M:
ν(M) = √((γ + 1) / (γ − 1)) × arctan(√(((γ − 1) / (γ + 1)) × (M² − 1))) − arctan(√(M² − 1))
Where:
- ν(M) = Prandtl-Meyer function value at Mach number M (in radians)
- M = Local Mach number
- γ = Ratio of specific heats (e.g., 1.4 for air)
2. Total Turning Angle Across an Expansion Fan (Δθ)
Δθ = ν(M₂) − ν(M₁)
Where:
- Δθ = Angle the flow turns across the fan (in radians)
- M₁ = Upstream Mach number
- M₂ = Downstream Mach number
3. Isentropic Flow Relations (Static to Stagnation)
Since the expansion fan flow is isentropic, total pressure, temperature, and density remain constant. The following equations relate static to stagnation properties:
Pressure Ratio:
P / P₀ = (1 + ((γ − 1) / 2) × M²)^(−γ / (γ − 1))
Temperature Ratio:
T / T₀ = (1 + ((γ − 1) / 2) × M²)^(−1)
Density Ratio:
ρ / ρ₀ = (1 + ((γ − 1) / 2) × M²)^(−1 / (γ − 1))
Where:
- P = Static pressure
- T = Static temperature
- ρ = Static density
- P₀, T₀, ρ₀ = Stagnation (total) values
- γ = Ratio of specific heats
These calculations help track the full change in flow behavior during expansion around a corner in high-speed systems.
Common Expansion Fan Reference Table
Here’s a simplified table showing key flow changes for air (γ = 1.4) based on upstream Mach numbers and sample turning angles.
| M₁ (Upstream) | Turning Angle Δθ (°) | M₂ (Downstream) | P₂/P₁ | T₂/T₁ | ρ₂/ρ₁ |
|---|---|---|---|---|---|
| 2.0 | 5 | 2.25 | 0.874 | 0.938 | 0.931 |
| 2.5 | 10 | 2.97 | 0.749 | 0.883 | 0.849 |
| 3.0 | 15 | 3.79 | 0.650 | 0.840 | 0.774 |
| 2.2 | 7 | 2.55 | 0.814 | 0.911 | 0.894 |
| 1.8 | 3 | 1.94 | 0.936 | 0.972 | 0.963 |
Note: These values are based on isentropic flow and calculated using typical numerical inversion of the Prandtl-Meyer function.
Example of Expansion Fan Calculator
Suppose a supersonic nozzle has an upstream Mach number M₁ = 2.5, and the flow needs to turn around a 10° convex corner.
Step 1:
Calculate ν(M₁) using the Prandtl-Meyer function for M₁ = 2.5
ν(M₁) ≈ 0.700 radians (approximate value)
Step 2:
Add the turning angle Δθ = 10° = 0.175 radians
ν(M₂) = 0.700 + 0.175 = 0.875 radians
Step 3:
Solve numerically for M₂ using the inverse of the Prandtl-Meyer function:
M₂ ≈ 2.97
Step 4:
Use isentropic formulas to compute downstream pressure, temperature, and density ratios if desired.
This gives a full picture of how the flow transitions from M₁ to M₂ across the expansion fan and what changes occur in the physical properties of the gas.
Most Common FAQs
It is part of the compressible flow and supersonic aerodynamics calculators category. It is mainly used in aerospace, propulsion, and high-speed gas dynamics engineering.
No. Expansion fans only occur in supersonic flow turning through convex corners. Subsonic flow does not exhibit Prandtl-Meyer expansion behavior.
The function contains multiple nested arctangent terms and square roots, making analytical inversion nearly impossible. Numerical methods or lookup tables are used to find the downstream Mach number for a given ν(M).