downstream appendages’ viscous resistance might be significant [3].


2.2(b) The wave resistance


When a body moves with forward speed at or near the free surface it generates waves that manifest themselves as drag, usually referred to as wave resistance. Not only is the hull producing wave resistance, but the addition of extra appendages will also increase the wave drag [4]. Adding a forward rudder will thus increase the wave resistance. A general expression for


the wave


resistance can be written [5]: ws


R aFn w  4


2 1


V A 2


In which a is an arbitrary constant. 2.2(c) The heel resistance


The heel resistance can also be subdivided in two parts associated with the


both the canoe body, and with the


appendages. The first is caused by a change in wetted surface area and a change in the three dimensional shape of the submerged part of the hull. These changes influence resistance.


appendages [6]. viscous resistance and wave


The heeling of the yacht appears to have no influence on the viscous resistance of its


However, the wave making drag of the appendages is strongly dependent on the heeling angle because the volume of the appendages is brought closer to the free surface. This


effect, which is not related to lift


production by the appendage, has been proved by experiments and has been reported by Beukelman and Keuning [7]. When the keel


is canted, a submerged


volume is brought even closer to the free surface and an additional change in resistance is to be expected.


2.2(d) The induced resistance


The induced resistance can be regarded as being resistance associated with the generation of lift by the hull and its appendages. The generation of waves on the free surface, caused by the low pressure zone at the windward side of the appendages when they are brought closer to the surface, is directly linked with the lift generation on these foils. On the basis of the above definition,


this wave pattern will interact with the


waves generated by the hull and will increase the (induced) resistance. For the present purpose, the latter part of the induced resistance defined in this way will be called the “free surface” induced resistance.


Induced resistance is more commonly associated with the induced angle of attack of a wing in a fluid flow. The induced angle of attack is a result of the downwash which is created by a lifting foil. It


©2007: Royal Institution of Naval Architects Figure 2: Distance between trailing vortices or the gap


This distance is also known as “the gap”. For a yacht, the gap can be approximated [1] using )


z s sin( downwash angle _ (4)


where s is the distance between the two rudders along the axis of symmetry and the downwash angle refers to the forward appendage


Equation (4) shows how the induced resistance can be lowered by reducing the interference between the two trailing vortices. This can be achieved by increasing the


should be B-41 (2)


appreciated that because of the downwash effect, the angle of attack of the appendages downstream of a lifting foil are reduced. The angle of attack could be reduced by as much as one half [8].


Claughton et al. [1] have reported that, as a first approximation, the canting keel with a twin rudder configuration can be regarded as a biplane whose induced resistance takes the form of equation (3). should be


It noted that this expression ignores the


interaction effects introduced by the canting keel and is only valid for ideal flows and elliptical rudder loading.


RI 


b L


2 1


2 1


   


4 2 2


b b L L


V


1 2 1 2


2


b L


2 2


2 2


(3)


in which: L1 is the lift produced by the forward rudder; L2 is the lift produced by the aft rudder; b1 is the span of the forward rudder; b2: is the span of the aft rudder; and, σ is the interference factor.


The interference factor σ depends on the span ratio ( and on


2


b b z


2 1 


trailing vortices of the two rudders in a direction perpendicular to the flow direction.


The following figure, based on one from Larsson [5] clarifies this point.


distance between trailing vortices


b b )


1 2


, in which z is the distance between the


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