| name | compressible-flow |
| description | Compressible flow — Mach number, isentropic relations, normal shocks, oblique shocks, Fanno flow, Rayleigh flow, choked flow, nozzle design, stagnation properties. |
| metadata | {"priority":6,"promptSignals":{"phrases":["compressible flow","Mach number","supersonic","shock wave","nozzle","choked flow","isentropic","stagnation"],"minScore":4}} |
Compressible Flow — Complete Skill
Key Relations
γ = c_p/c_v = 1.4 (air, diatomic), 1.67 (monatomic), 1.3 (combustion gases)
Speed of sound: a = √(γRT) [m/s], where R = 287 J/(kg·K) for air
At 20°C: a = √(1.4×287×293) = 343 m/s
Mach number: M = V/a
Isentropic Flow Relations
For steady, adiabatic, frictionless flow through varying area duct:
T₀/T = 1 + (γ-1)/2 × M²
p₀/p = (T₀/T)^(γ/(γ-1)) = (1 + (γ-1)/2 × M²)^(γ/(γ-1))
ρ₀/ρ = (T₀/T)^(1/(γ-1))
Stagnation (total) conditions: what fluid properties would be if brought to rest isentropically
T₀ = T + V²/(2c_p) [total temperature]
p₀ = pressure if flow decelerated isentropically to M=0
Critical (*) conditions at M=1:
T*/T₀ = 2/(γ+1) = 0.833 (air)
p*/p₀ = (2/(γ+1))^(γ/(γ-1)) = 0.528 (air)
ρ*/ρ₀ = (2/(γ+1))^(1/(γ-1)) = 0.634 (air)
a* = √(γRT*) = V* (velocity = local speed of sound at throat)
Area-Mach Relation (Convergent-Divergent Nozzle)
A/A* = (1/M) × ((2/(γ+1)) × (1 + (γ-1)/2 × M²))^((γ+1)/(2(γ-1)))
For γ=1.4:
M=0.5: A/A*=1.340, M=2.0: A/A*=1.688, M=3.0: A/A*=4.235, M=5.0: A/A*=25.0
Choked flow: when throat reaches M=1, mass flow rate is maximum:
ṁ_max = p₀×A*×√(γ/T₀) × √(2/(γ+1))^((γ+1)/(γ-1)) / √R
For air (γ=1.4, R=287):
ṁ/A* = 0.6847 × p₀/√T₀ [kg/(m²·s), p₀ in Pa, T₀ in K]
Normal Shock Relations
Across normal shock (M₁ > 1 → M₂ < 1):
M₂² = (M₁² + 2/(γ-1)) / (2γ/(γ-1) × M₁² - 1)
p₂/p₁ = 1 + 2γ/(γ+1) × (M₁²-1)
T₂/T₁ = (p₂/p₁) × (ρ₁/ρ₂) (using ideal gas)
ρ₂/ρ₁ = (γ+1)M₁² / (2+(γ-1)M₁²)
Total pressure loss (entropy production):
p₀₂/p₀₁ = (p₂/p₁)^(γ/(γ-1)) × (T₁/T₂)^(γ/(γ-1))
At M₁=1: no shock (isentropic)
At M₁→∞: M₂→√((γ-1)/(2γ)) = 0.378 (air), ρ₂/ρ₁→(γ+1)/(γ-1)=6 (air)
Oblique Shocks
Deflection angle θ and wave angle β (shock angle to flow):
tan(θ) = 2cot(β) × (M₁²sin²(β)-1)/(M₁²(γ+cos(2β))+2) [θ-β-M relation]
M₁n = M₁sin(β) [normal component for shock relations]
After shock: M₂sin(β-θ) = M₂n [normal component downstream]
Detached bow shock occurs when θ > θ_max (from chart/table)
Maximum deflection angle (air): θ_max ≈ 45° at M₁=∞
Fanno Flow (Friction in Adiabatic Duct)
Flow with friction → M approaches 1 (sonic) at exit
4fL*/D = (1-M²)/(γM²) + (γ+1)/(2γ) × ln((γ+1)M²/(2+(γ-1)M²))
L* = max duct length from M to M=1 choking
Flow direction:
Subsonic inlet → M increases toward 1
Supersonic inlet → M decreases toward 1
Rayleigh Flow (Heat Addition, Frictionless)
Heat addition drives M toward 1 from either side
T₀/T₀* = (2(γ+1)M²/(1+γM²)²) × (1+(γ-1)/2 × M²)
Choking by heat addition: Q_max = c_p(T₀*-T₀₁)
Thermal choking: if too much heat added → inlet conditions change
Converging Nozzle Operation
- p_back/p₀ > 0.528: subsonic, unchoked
- p_back/p₀ ≤ 0.528: choked (M=1 at throat, flow rate fixed)
- Further reducing p_back: no effect on ṁ (choked)
De Laval (C-D) Nozzle
- Converging section: accelerates to M<1
- Throat (A*): M=1 if choked
- Diverging section: can accelerate to M>1 (supersonic) OR decelerate (subsonic)
- Which path depends on pressure ratio
- Design pressure ratio for shock-free supersonic exit: p₀/p_e from isentropic table at M_e
Output
Provide: M₁ and M₂, T₀, p₀, ṁ, A/A* at key locations, shock properties if normal/oblique shock present, nozzle efficiency if applicable.
