Duct geometry
Operating point
Validity flags
Meridional section — streamlines, static-pressure field & axial distributions
Cp = (p−p∞)/(½ρud²)
−1.3 · 0 · +0.4
· stations: 1 inlet lip · 2 rotor disk · 3 diffuser entry · e exit
Performance
Thrust breakdown
Thrust augmentation vs diffuser half-angle
with losses
ideal (lossless duct)
transitory-stall regime
Data table
Efficiency vs expansion ratio σ = Ae/Athroat
system η = ΔKE/Pshaft
diffuser ηd
Data table
Augmentation & propulsive efficiency vs flight speed
T/Topen (same power & disk)
ηp = TV∞/Pshaft
Data table
Governing equations
ṁ = ρAdud = ρAeue, σ = Ae/Adcontinuity; σ is the disk-referenced expansion ratio
T = ṁ(ue − V∞)global momentum; exit at ambient static pressure; ram (momentum) drag included; V∞=0 → static
Δprotor = ½ρ(Λue² − V∞²), Λ = 1 + (1−ηd)(σ²−1) + kinσ², Trotor = AdΔprotorrotor jump = jet head + diffuser + inlet total-pressure losses
Tdiff = ṁue(1−σ) − p₃Ad, p₃ = ½ρue²[1+(1−ηd)(σ²−1)−σ²]diffuser-wall axial force (a drag for σ>1)
Tlip/cowl = T − Trotor − Tdiff − Tleakinlet-CV remainder; static limit ½ρAdud²(1−kin); in flight includes pre-entry (additive) terms
Pshaft = ṁΔprotor/ρ = ½ṁ(Λue² − V∞²), ηp = TV∞/Pshaft, ηsys = (ue²−V∞²)/(Λue²−V∞²)
Cp,diff = ηd(1−σ−2)static-pressure recovery; loss = (1−ηd)(1−σ−2)·½ρud²
Open rotor (baseline): To = 2ρAdv(V∞+v), Po = To(V∞+v)ideal induced power; solved numerically for v at same P; static limit T/Topen = ηtip(2σ)⅓/Λ⅔, Küchemann–Weber ideal (2σ)⅓
ISA: ρ = 1.225(1 − 2.2558·10⁻⁵h)4.2559, a = √(γR(288.15−0.0065h))troposphere, h ≤ 6 km
Assumptions & where 1-D theory breaks down
- Rigorous part: incompressible, steady, uniform actuator disk; no swirl (assumes stator/contra-rotation recovery); jet exits at ambient static pressure; ram drag included in flight.
- Flight mode caveat: external nacelle/cowl drag and duct external aerodynamics are not modeled — thrust ratios vs the open rotor are optimistic at speed. The lip-loss and tip-gap correlations are hover-derived and assumed speed-independent.
- ηd(θ) is an empirical fit (logistic, peak ≈ 0.87 at small angles, floor ≈ 0.30) to conical-diffuser effectiveness data (Sovran–Klomp type). Real ηd also depends on L/R, inlet blockage, and Reynolds number.
- Separation: beyond θ ≈ 12–14° half-angle (2θ ≈ 24–28°) conical diffusers enter transitory stall — flow detaches, the exit area is no longer filled, and 1-D momentum theory is unreliable, not just lossy. The shaded recirculation zone is a sketch, not a prediction.
- Lip loss kin(rlip/R) is empirical (0.30 for a sharp edge → ≈0 above r/R ≈ 10%). 1-D theory cannot predict lip separation; a sharp lip can lose most of the computed lip suction.
- Tip gap: linear thrust knock-down of 3% per 1% gap/R (hover-test range 2–4%). The leakage vortex is a 3-D effect outside momentum theory.
- Compressibility ignored — results degrade above exit Mach ≈ 0.3.
- Drawing: blade planform, hub/spinner and blade count in the views are illustrative; the model is an actuator disk. In-duct pressure and velocity are station-averaged (uniform over each cross-section, per 1-D theory).