Trans RINA, Vol 152, Part B1, Intl J Small Craft Tech, 2010 Jan-Jun
the sum is small for small values of t and t0 and the equation simplifies to
C CLopt (β ε2 AR e′
Di = π ())2 2 + + tt0
2 eff ) 1+c t t (
C CLopt (β ε2 eff )
Di = πAR ⎛1
⎜ ⎝
e c t t (
+ ′ + tt2 ) 2 0 t
2 0
⎟ ⎠ ⎞
, . (23)
Substituting equation (22) into equation (23) then gives 2
(24)
where ct′=ct/e′. For small values of t0 and for e around one, e′ is of the order of one and ct′≈ct.
An additional input parameter for the centre of effort height at which the induced drag is minimal (zCoEminDi) is required to define t0. Assuming semi-elliptical loading with the water surface as the mirror plane as the ideal fully powered-up loading distribution, zCoEminDi can be calculated from lifting line theory to be 4/(3π)≈42% of the mast height (Hansen [13]). Figure 4 shows t0 obtained from zCoEopt measured in the wind tunnel and from the B- spline curve of zCoEopt modelled in FS-Equilibrium. The
1.2 1 0.8 0.6 0.4 0.2 0 0 0.2
(a) No twist expression 1.2
1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 (c) Twist expression with t0 CD (measured) [-] 0.6 0.8 1 1.2
EWA 20° EWA 25° EWA 30° EWA 40° EWA 60° EWA 90° No error
0.4 CD (measured) [-] 0.6 0.8 1 1.2
EWA 20° EWA 25° EWA 30° EWA 40° EWA 60° EWA 90° No error
0.1 0.2 0.3
-0.1 0
0 15
304560 βeff [°]
7590
Figure 4: Base twist (t0) for fully powered-up mainsail and jib vs. βeff assuming zCoEminDi is 42% of mast height
positive t0 values confirm that a centre of effort height corresponding to semi-elliptical loading is not achieved for the fully powered-up sails in the wind tunnel.
Mainsail & jib (wind tunnel) Mainsail & jib (modelled)
1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 CD (measured) [-] 0.6
(b) Jackson’s twist expression 1.2
1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 CD (measured) [-] 0.6 0.8 1 1.2 (d) Twist expression with t0 and CDp proportional to (CL/CLopt)2
Figure 5: Calculated CD plotted against CD measured in the wind tunnel with the Real-Time VPP for sails at different de-powering levels for a range of effective wind angles (EWA)
©2010: The Royal Institution of Naval Architects B-15
EWA 20° EWA 25° EWA 30° EWA 40° EWA 60° EWA 90° No error
0.8 1 1.2
EWA 20° EWA 25° EWA 30° EWA 40° EWA 60° EWA 90° No error
CD (calculated) [-]
CD (calculated) [-]
CD (calculated) [-]
CD (calculated) [-]
t 0 [-]
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