Ducted Fan Performance: Method

Ducted Fan Performance: Blade Element Method and Conventions

The Ducted Fan Optimization tool answers the design question: which chord and pitch distribution gives the most thrust for a given shaft power. This tool answers the analysis question for a blade that already exists: what thrust, torque and power does it produce at a given rotor speed, climb or inlet velocity and altitude, and how do they change when one of those is varied.

Blade geometry

The blade is entered as stations from hub to tip: radius, chord and pitch angle θ measured from the plane of rotation to the chord line. Between stations the chord and pitch are interpolated linearly; outside the first and last station the end values are held. The optimizer's element table can be pasted in directly (every row, or a few of them), or a measured blade can be entered.

Blade element momentum theory

The blade is divided into elements between the hub radius and the tip radius. At each element the air arrives with the axial velocity V_c + v_i (inlet velocity plus the induced velocity at the disk) and the tangential velocity Ω r. The inflow angle is φ = atan((V_c + v_i) / (Ω r)), the angle of attack α = θ − φ, and the section lift and drag follow from the airfoil polynomials. The thrust of the element per unit radius is dT/dr = N_b (dL cos φ − dD sin φ) and its in-plane force dF_t/dr = N_b (dL sin φ + dD cos φ), from which torque and power follow with the radius and the rotor speed.

The induced velocity is the one that makes this blade-element thrust equal to the momentum thrust of the same annulus, dT/dr = 4 π ρ r (V_c + v_i) v_i. The tool solves that balance exactly at every element by bisection. The optimizer uses instead the closed-form small-angle solution of the same balance with a local linear lift model; near the hub, where the inflow angle is large, that solution leaves the balance open by up to 20 percent and its thrust comes out a few percent higher. The difference for the optimizer's own 48-inch rotor is about 2 percent in thrust.

As in the optimizer, no tip-loss factor is applied (the duct suppresses the tip vortex), the swirl in the wake is not recovered, and the duct's own thrust is not included: the thrust reported is the blade thrust. The ducted ideal power and figure of merit are given alongside the open-rotor ones so the blade can be judged against either reference.

Airfoil data and corrections

Lift and drag are 6th-order polynomials in the angle of attack in degrees, the same form as the optimizer. The Reynolds-number correction scales the coefficients by the optimizer's fits referenced to Re = 10⁶ at 7°, and the Mach correction by its fits referenced to M = 0. Elements whose angle of attack falls outside the fitted range are marked with an asterisk in the element table and counted in the notes: the polynomials are extrapolated there and the loads of those elements should not be trusted.

Figures of merit

The ideal power of an open rotor is T^3/2 / √(2 ρ A), its wake contracting to half the disk area. A duct with exit area σ_d A has the ideal power T^3/2 / √(4 σ_d ρ A): for a cylindrical duct (σ_d = 1) that is 1/√2 of the open rotor's, which is the classic halving of the induced velocity by the duct. The figure of merit is the ideal power divided by the shaft power; both references are reported. The optimizer prints √2 times the open-rotor value and calls 1.414 its maximum, which is the same quantity scaled.

Sweeps

With the blade fixed, the rotor speed, the inlet velocity or a collective pitch offset (added to every station's pitch) can be swept between two limits. Thrust, shaft power and figure of merit are charted against the swept variable with the operating point marked, and the sweep table lists the values with the number of stalled elements at each point. Thrust rises roughly with the square of the rotor speed and power with its cube until the sections stall; an inlet velocity unloads the blade and reduces both.

Atmosphere

Density, temperature and speed of sound come from the International Standard Atmosphere at the entered pressure altitude (troposphere lapse of 6.5 K/km to 11 km, isothermal above); viscosity from Sutherland's law. At sea level the density is 1.225 kg/m³ (0.002377 slug/ft³) and the speed of sound 340.3 m/s (1,116 ft/s).