Foil Angle of Sweep
The optimum angle of sweep for a foil is a complicated problem to solve and understand, especially as there is not much published data to analyse. Figure 2.143 of the "Aero-hydrodynamic of sailing" shows how induced drag varies with angle of sweep. The following crude mathematical analysis is a development of an approach suggested in a letter to Seahorse (July 1993 issue) by Dave Hollom. By combining the two sources of information in the following way we shall hopefully increase our understanding of the problem and quantify the various effects that are involved.
V = free stream velocity
VN = component of V normal to the 1/4 cord line
VP = component of V parallel to the 1/4 cord line
h = distance along foil from its root to the centre of pressure
L = lift
CL = lift coefficient for foil (analogous to lift)
CDP = profile drag coefficient for foil (profile drag)
CM = moment coefficient for foil about X-X of lift (heeling moment)
L = angle of sweep of the 1/4 cord line
CDPN = drag coefficient component normal to 1/4 cord
CDPP = drag coefficient component parallel to 1/4 cord
VN = V cos L
VP = V sin L
CL µ VN2
CL µ cos2 L
Assuming Vp does not contribute to the lift produced by the foil.
CM µ L h cos L
CM µ cos3 L
Assuming that h is constant in that the centre of pressure does not move.
CDPN µ VN2
CDPP µ VP2
The addition of both components in the direction of the free stream will give the total profile drag coefficient CDP
CDP µ VN2
cos L + Vp2 sin L
CDP µ cos3 L + sin3 L
Therefore we have
CL (swept) / CL (unswept) = cos2 L
CM (swept) / CM (unswept) = cos3 L
CDP (swept) / CDP (unswept) = cos3 L + sin3 L
CDI (swept) / CDI (unswept) = taken from figure 2.143 "A-H. of sailing"
From figure 2.113 of "The Aero-hydrodynamics of sailing" it can be seen that induced drag is 50% greater than the profile drag at a reasonable angle of attack in that particular case of an elliptical main sail.
\CD = (1.5 CDI + CDP) / 2.5
Calculated values of CL and CD for a foil with sweep where induced drag is 50% larger than profile drag.
The graph shows that the optimum angle of sweep of the 1/4 cord line of the main sail is 3° . However, the following assumptions must be born in mind.
·Induced drag is 50% larger than profile drag.
Moth sailors tend to sail around with a mast rake of around 5° which indicates a 1/4 cord line angle of sweep of 3° . (Note: the difference between mast rake and the 1/4 cord rake angle has been measured at 2º on a Moth sail.) This value has been derived empirically so its validity is questionable but it does agree with the mathematical derivation of the optimum angle of sweep which helps validate both approaches.
A Moth should be rigged with 5° of mast rake to be optimum for under powered conditions up wind. This will define a set of conditions for which the centre of lateral resistance and centre of effort of the sail can be balanced. In over powered conditions the mast can be raked to decreases the heeling moment of the sail. This decreases the efficiency of he sail but a raked full sail is more efficient than an upright stalled sail. If the 1/4 cord line of the sail is raked by 15° to 18° the heeling moment will be reduced by 14%. There is no real practical limit to mast rake, only a control problem. If the boat is balanced with 5° of mast rake it will be unbalanced with 20° of rake. A degree of unbalance is not a problem but more will eventually result in loss of control.
Mainsail and jib combinations are harder to asses because the problem is far more complicated than a single sail or foil arrangement. There is a highly raked foil in front of an unraked foil that is affected by the presence of the preceding foil. Therefore the optimum rake of the rig is a compromise between the optimum rake of its two elements. The interactive affects between these elements means that they can not be dealt with in isolation.
This type of analysis can also be applied to rudders and centreboards. The specification for these foils is very different between foils and from that of the rig. But I feel it is reasonable to conclude that foils should be raked with their 1/4 cord line at approximately 3º (the angle at which they are most efficient). It would be relatively straight forward to do an exact analysis because section profile drag and induced drag can be quantified with good accuracy so long as the angle of attack of the foil is known as well as its section, area and aspect ratio.
To conclude raking a foil or sail makes it less efficient, however for a given limiting value of lift, the total drag from the foil can be reduced by rake. In the case of a centre board or rudder raking is only advantageous when it is necessary to move the centre of pressure or reduce foil area. A rig is designed to produce a fixed heeling moment in as wide a range of wind speeds as possible. Therefore raking is advantageous in strong winds because it reduces lift and drag.
You probably realised the conclusions of this article before you started to read it, but I hope you have found the analysis interesting.