Two-Deflector Model
The base model makes a sharp asymmetry: the sail is treated as a momentum deflector acting on a column of air, while the centreboard is given infinite lateral resistance — leeway is forbidden by assumption. That is equivalent to giving the water-foil infinite area.
This note removes that assumption. The centreboard is modelled as a finite lifting surface whose side-force grows with leeway angle, introducing leeway $\alpha$ as a second equilibrium variable alongside boat speed $v$.
Coordinate frame
| symbol | meaning |
|---|---|
| $x’$ | along the boat’s heading (positive = forward) |
| $y’$ | across the boat (positive = to leeward) |
| $\theta$ | heading angle from true wind |
| $\alpha$ | leeway angle (drift to leeward, $\alpha > 0$) |
The boat’s actual track over water is $\theta + \alpha$ from the wind.
Forces
Sail (momentum deflector — unchanged)
The sail presents area $a_s |\sin\theta|$ to the wind. Decomposing the full 2-D momentum-flux change gives both components:
\[F_{\text{sail},x'} = \rho_a \, a_s \, v_s^2 \, |\sin\theta|(D_s - \cos\theta)\] \[F_{\text{sail},y'} = \rho_a \, a_s \, v_s^2 \, \sin^2\theta \qquad \text{(leeward push)}\]The $y’$ component is what the base model discards by assuming infinite lateral resistance.
Centreboard (lifting surface)
Applying the pure deflector equation to the centreboard fails: it predicts no side-force until the leeway exceeds $\arccos(D_c) \approx 12°$, far more than real boats experience. A centreboard is an efficient low-drag foil that generates lift linearly with angle of attack — thin-flat-plate theory gives:
\[C_L = 2\pi\sin\alpha \approx 2\pi\alpha, \qquad C_D = \frac{C_L^2}{\pi \cdot \mathrm{AR}}\]where $\mathrm{AR} = \text{span}^2 / A_c$ is the aspect ratio ($\mathrm{AR} \approx 6$ for a dinghy centreboard). The dynamic pressure of the sideways water flow is $\tfrac{1}{2}\rho_w A_c v^2$, giving:
\[F_{\text{cb},y'} = \tfrac{1}{2}\rho_w A_c \cdot 2\pi\sin\alpha \cdot v^2 \qquad \text{(windward restoring force)}\] \[F_{\text{cb},x'} = -\tfrac{1}{2}\rho_w A_c \cdot \frac{(2\pi\sin\alpha)^2}{\pi \cdot \mathrm{AR}} \cdot v^2 \qquad \text{(induced drag)}\]Hull drag (unchanged)
\[F_{\text{hull}} = -(1-D_h)\,\rho_w\,A_h\,v^2\]Equilibrium
Setting net forward and lateral forces to zero:
\[\underbrace{\rho_a a_s v_s^2 \sin\theta(D_s-\cos\theta)}_{\text{sail drive}} - \underbrace{\frac{2\pi^2 \rho_w A_c \sin^2\!\alpha}{\mathrm{AR}}\,v^2}_{\text{CB induced drag}} = \underbrace{(1-D_h)\rho_w A_h v^2}_{\text{hull drag}} \tag{1}\] \[\underbrace{\rho_a a_s v_s^2 \sin^2\!\theta}_{\text{sail side-force}} = \underbrace{\pi \rho_w A_c \sin\alpha \cdot v^2}_{\text{CB lift}} \tag{2}\]Two equations in two unknowns $(v, \alpha)$ — solved numerically for each heading $\theta$.
Results for the Laser Pico
Left: heading $\theta$ (grey) vs actual track $\theta+\alpha$ (green) at the optimal angle. Right: full force breakdown — sail drive and leeward push (blue/purple), centreboard lift balancing the leeward push (green), combined drag opposing forward motion (red).
Boat speed and upwind component vs heading. The annotation shows the maximum upwind speed, the corresponding optimal heading, and the leeway angle at that point.
Upwind component vs heading for centreboard aspect ratios AR ∈ {3, 4, 6, 8, 12} (centreboard area $A_c = 0.125$ m² fixed). Higher AR → less induced drag → slightly higher upwind speed and marginally narrower optimal heading.
Centreboard: $A_c = 0.125\ \text{m}^2$, $\mathrm{AR} = 6$.
| $\theta$ (°) | $v$ (m/s) | $\alpha$ (°) | $u = v\cos(\theta+\alpha)$ (m/s) |
|---|---|---|---|
| 33 | 0.76 | 7.7 | 0.57 |
| 45 | 1.92 | 2.0 | 1.31 |
| 56 | 2.85 | 1.2 | 1.53 |
| 62 | 3.29 | 1.0 | 1.49 |
| 73 | 4.12 | 0.8 | 1.11 |
Optimal heading: $\theta_\text{opt} = 57.0°$, leeway $\alpha = 1.2°$, effective track $58.2°$ from wind.
True upwind speed: 1.53 m/s (2.97 knots).
Compared with the one-deflector model ($\theta_\text{opt} = 56.8°$, $u_\text{max} = 1.59$ m/s), the optimal heading is virtually unchanged — confirming that the base model’s closed-form cubic gives the right answer to good accuracy. The upwind speed is slightly lower because centreboard induced drag adds a small forward-resistance penalty.
Sensitivity to centreboard size
| $A_c$ (m²) | $\theta_\text{opt}$ (°) | Leeway (°) | $u_\text{max}$ (m/s) |
|---|---|---|---|
| 0.05 | 57.3 | 3.1 | 1.43 |
| 0.10 | 57.1 | 1.5 | 1.51 |
| 0.125 | 57.0 | 1.2 | 1.53 |
| 0.20 | 57.0 | 0.8 | 1.55 |
| 0.30 | 57.0 | 0.5 | 1.57 |
| 1.00 | 56.8 | 0.2 | 1.59 |
As $A_c \to \infty$ the leeway $\to 0$ and the result converges to the one-deflector model, as expected. Smaller centreboard → more leeway → more induced drag → lower upwind speed and a very slightly wider optimal heading, consistent with the intuition that a boat with a poor centreboard should bear away slightly.
What the model still cannot do
- The sail side-force ($F_{\text{sail},y’}$) in this derivation assumes air exits axially — a rough approximation that over-states the leeward push at very large $\theta$. A full 2-D deflector would require tracking the exit direction of the air.
- Hull side-drag (the water resistance opposing leeway from the hull itself, not just the centreboard) is ignored.
- Heel angle and its effect on effective sail area and underwater body are ignored.
- The thin-plate $C_L = 2\pi\alpha$ formula holds only for small $\alpha$ and attached flow — reasonable for $\alpha < 5°$ but the model should not be trusted at larger leeway.
Code
The model lives in
sailing_upwind/two_deflector.py.
To run the full model and regenerate all plots and diagrams, use the CLI (config is managed by Hydra — any config key can be overridden on the command line):
# Default Laser Pico parameters
python -m sailing_upwind
# Try a stronger wind
python -m sailing_upwind wind.speed_ms=7
# Larger centreboard with higher aspect ratio
python -m sailing_upwind model.centreboard.area_m2=0.20 model.centreboard.aspect_ratio=8
To use the two-deflector model for the main upwind-speed plot:
python -m sailing_upwind model.mode=two_deflector
Or call the Python API directly:
from sailing_upwind.two_deflector import TwoDeflectorParams, optimal_angle
p = TwoDeflectorParams(
v_s=4.0, # wind speed m/s
a_s=5.1, # sail area m^2
rho_a=1.225,
D_s=0.895, # sail drag coefficient
A_c=0.125, # centreboard area m^2
AR_c=6.0, # centreboard aspect ratio
D_h=0.9, # hull drag coefficient
rho_w=1000.0,
A_h=0.0343, # hull frontal area m^2
)
theta_opt, v_opt, alpha_opt = optimal_angle(p)