APPENDIX THREE – CONTRIBUTION OF A RUDDER OR FIN TO THE VELOCITY DERIVATIVES
The lateral velocity at a point x on the x-axis is v + rx. This movement of the hull leads to a movement of the water beneath the keel in the opposite direction, varying with depth z, of magnitude say (v + rx)g(z).
So if a vertical surface such as a rudder or fin is fixed below the keel, the relative lateral velocity between it and the water at a depth z is:
rV = (v + rx)(1 + g(z)) pz where the extra term pz arises from any roll-rate.
If the below-water section of the hull is a semi-ellipse with waterline semi-beam yw and keel depth zk, then from equipotential flow theory:
g(z) = zk yw zk 1 z √(z2 zk 2 + yw 2)
For simplicity we take an average value g, averaged for z varying from the keel at zk to the tip of the rudder etc at zt:
_ _ g = zk zt zk 1 + zt √(zt 2 zk yw zk
If the vessel has a forward velocity u0, and the rudder is turned through an angle δ, the water flow will meet it at an incidence:
= δ rV/u0 (δ = 0 for a fixed fin) We assume the rudder/fin has an area A and a lift
coefficient CL = dCL/d. Then the lateral force on it is approximately: Y = CL ½u0
2A = ½A dCL/d {u0 2δ u0[(v + rx)(1 + g) pz]} _
The contributions of this surface to the rudder and velocity Y-derivatives are thus the derivatives of the above expression with respect to δ, v, r and p respectively:
Yδ = ½A dCL/d u0 2
Yv = ½A dCL/d (1 + g) u0 Yr = ½A dCL/d (1 + g)x u0 etc
_ _
For the N-derivatives the contribution has to be multiplied by the surface’s x-coordinate, and for the L-
2 + yw 2)
derivatives by z, z being the depth of the hydrodynamic centre of the surface.
If the propulsion arrangements are similar to those shown in Figure 4 or 5, their contribution to the derivatives is not negligible. The propeller-shaft support is effectively another fin, and the propeller behaves like one too. The shaft is a cylinder, for which typical characteristics have been published [17].
©2007: Royal Institution of Naval Architects
B-11
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