The rudder is most effective at moderate angles. As the helm angle increases, the lift (normal force) on the rudder rises up to about 30-35 degrees, beyond which the flow separates from the rudder surface, the lift falls off and the drag rises sharply (stall). The rudder force also produces a heeling moment and increases resistance. Beyond about 35 degrees the additional turning effect is negligible or negative, while the drag and heeling moment increase, so there is no benefit in a larger angle. Hence the maximum rudder angle is limited to about 35 degrees (SOLAS requires the rudder to be capable of being put hard over to 35 degrees in 28 seconds), giving the best turning performance with acceptable drag and heel.

Data: displacement 12,000 t; rudder area A = 15 m2; rudder centre 5 m below the waterline; GM = 0.3 m; centre of buoyancy 3.3 m below the waterline; speed 20 knots; helm angle 30 deg; force Fn = 577 A v2 sin(alpha); allow 20% race effect.

Speed: v = 20 x 0.5144 = 10.29 m/s, v2 = 105.84.

Fn = 577 x 15 x 105.84 x 0.5 = 458,035 N.

Race effect 20%: F = 1.2 x 458,035 = 549,642 N.

Heeling moment = F x lever = 549,642 x 5 = 2,748,210 Nm (lever = 5 m, the rudder centre below the waterline).

tan(theta) = (F x lever)/(W x GM) = 2,748,210/(12000 x 9810 x 0.3) = 2,748,210/35,316,000 = 0.07782.

theta = atan(0.07782) = 4.45 deg.

Answer: initial angle of heel about 4.5 deg.