Trans RINA, Vol 152, Part B1, Intl J Small Craft Tech, 2010 Jan-Jun
In this idealisation the maximum increase in sectional lift coefficient occurs at the symmetry plane so that the water surface is used as a reference point instead of the bottom of the sails. The increase in sectional lift coefficient at the symmetry plane (Cl(0)/ClminDi(0)) is found to be related to the reduction in the centre of effort height (zCoE/zCoEminDi) by C
plane. C
lminDi l
(0) 1 (0)
= + ⎛ − c tL ⎜1
⎜ ⎝
z
CoEminDi CoE
z ⎞
⎟ ⎠
⎟ = +1 ()0 c t + t tL , (26)
where ctL is the twist weight constant for lift. The value of ctL can be derived from lifting line theory to be 5/3≈1.667 (Hansen [13]). From this relationship it can also be deduced that Clopt(0)/ClminDi(0)=1+ctLt0 so that the change in sectional lift coefficient relative to the fully powered-up sectional lift coefficient (Clopt) is given by C
C
lopt l
(0) 1 1 (0)
= + + tL 0
c t c t
tL . (27)
For the sectional lift coefficient not to increase when twisting off the sails, CL needs to be reduced by easing the sails. The trim parameter ease (ε) models a linear reduction in lift due to the reduction in α so that ease due to twist (εt) can be defined as the inverse of equation (27) to give C
ε = t
lopt (0) C (0)
l = 1 1 1 + + tL 0
c t c t
tL . (28)
The expression for εt can be included in the ‘twist and ease’ model and ctL becomes a new input parameter. Equation (9) is then rewritten as .
C C εε= L Lopt t Since CDi, CDp and CDs are all functions of CL C CLopt ( C CDpOptε εt + c CS Lopt
Di = D =
2 2
2 β εε 1 2 2
eff ) πAR t ⎛
⎜ ⎝
e c t t (
+ ′ + tt2 ) 2 0 ⎞ ,
⎟ ⎠
2 εεt + C Di . 2 2
7.5 SUMMARY OF THE ‘TWIST AND EASE’ MODEL
In this model de-powering of the sails is achieved by using two trim parameters, which can be adjusted by the optimisation algorithm of a VPP to maximise the boat speed:
1) Twist, t, which models any change in trim resulting in a lowering of the centre of effort relative to that chosen when fully powered-up. In the fully powered- up situation t=0.
2) Ease, ε, which models changes in trim resulting in a reduction in lift without altering the centre of effort height. In the fully powered-up situation ε=1.0.
2, εt
(29) 2 is
therefore included in equations (24) and (25) so that they can be rewritten as
(30) (31)
These parameters modify the lift and drag coefficients and the centre of effort height through:
C C εε= C CLopt
L
C CDpOptε εt + c CS Lopt ,
Di = D =
2 2 zCoE = zCoEopt (1 )t − Lopt t ,
πAR e c t t (
2 εε 1 2 2
t ⎛
⎜ ⎝
+ ′ + tt2 ) 2 0 ⎞ ,
⎟ ⎠
2 εεt + C Di , 2 2
(32) (33) (34)
(35)
where CLopt, CDpopt and zCoEopt are all functions of the effective wind angle βeff and are determined by finding the optimal trim when the yacht is fully powered-up. In most cases even the fully powered-up load distribution will have a centre of effort height lower than the ideal semi-elliptic distribution.
Hence, even when t 0 = −1 z z
CoEminDi CoEopt
, where zCoEminDi is 42% of the mast height.
The model also includes an allowance for easing the sails whenever the centre of effort is lowered by twisting, where
ε = t 1 1 1 + + tL 0
c t c t
tL . (37)
The suggested model constants are ct′=8 and ctL=1.667. Kerwin [1] further suggests cS=0.0016 for upwind sailing conditions and 0.0019
Hochkirch [10] determined e=1.1 for Dyna from full- scale measurements.
Compared to the ‘reef and flat’ model, the ‘twist and ease’ model requires three additional input parameters: zCoEminDi, ct′ and ctL. The model has been constructed in such a way that these parameters can be
derived
analytically from lifting line theory and are independent of rig characteristics such as size, aspect ratio and efficiency. Therefore the derived
parameter values
summarised in Table 1 can universally be used in the ‘twist and ease’ model. In addition the presence of these parameters gives the user the capability to tweak the model by adjusting the values based on experimental, numerical or real-life observations.
Table 1: Additionally required input parameters and
associated analytically derived and universally applicable values for ‘twist and ease’ model
Symbol Description
zCoEminDi Centre of effort height for minimum induced drag [m]
ct′ ctL
Twist weight constant for lift [-]
©2010: The Royal Institution of Naval Architects Value
42% of mast height
Twist weight constant [-] 8 1.667 fully
powered-up the distribution is already partially twisted, such that
(36)
for downwind cases and
B-17
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