Pick a scenario, then watch where the operating point sits on the torque plane.
Annual energy of the 5-turbine row (Weibull k = 2 winds) for spacings 3–10 D with the same Jensen/Katic model as the 3D page.
The time-domain simulation (inertia, PI pitch loop, rate limit) is run to steady state and compared with the closed-form algebraic operating point.
The science
What this 2D view adds. The 3D page shows the machine; here you see the momentum stream tube, the torque balance that sets the rotor speed, the full power and Cp curves, and the wake field of the farm from above. All numbers come from the same physics module as the 3D page.
Wind power and Betz.
P_wind = ½ ρ A v³ A = πR²
P_rotor = P_wind · C_p(λ,β) λ = ωR / v
C_p ≤ 16/27 ≈ 0.593 (Betz limit)
Cp(λ,β) is Heier's empirical fit for a 3-blade rotor (peak 0.48 near λ = 8.1 at β = 0): a typical shape, not the data of a real machine.
Stream tube. Momentum theory gives Cp = 4a(1−a)² and Ct = 4a(1−a). The wind slows to (1−a)v at the disc and (1−2a)v far downstream; mass conservation makes the tube widen by √((1−a)/(1−2a)). The tube drawing is to scale in these ratios.
Torque plane (phase plane).
J dω/dt = Q_aero(ω) − Q_fric − T_g(ω)
Q_aero = ½ρA v³ C_p(ωR/v, β) / ω
Region 2: T_g = K ω², K = ½ρπR⁵C_p,max/λ³_opt
Where the aerodynamic curve crosses the generator curve the rotor is in equilibrium. Above the crossing the aerodynamic torque is larger and the rotor accelerates; below it, it slows. Above rated wind the controller pitches the blades so the aerodynamic curve slides down and crosses at rated speed.
Losses. Gearbox (50/10 × 54/12 × 48/12 = 90:1), generator and converter each lose a fixed + linear + quadratic share of rated power: illustrative typical values, not a specific product.
Yaw error. Only the wind component normal to the rotor does work, so power ∝ cos³γ (ideal).
Wake (Jensen / Katic).
deficit = (1 − √(1−C_t)) (R/(R+kx))² · overlap
k = 0.075 v_in = v · (1 − √Σ deficit²)
A top-hat wake with no turbulence build-up, deflection or terrain; the map colours each point with the same formula. It shows the trend (more spacing, less loss), but real farm losses need measurements or CFD.
Simplifications, honestly. Steady wind except the gust button, rigid blades and tower, no turbulence, shear, tower shadow or grid faults. Gear rotation in the cross-section is drawn 10× slower than real. This is an independent educational model, not affiliated with any turbine maker.
Independent educational model. Canvas 2D, no external libraries.