SAILING VESSEL DYNAMICS:
INVESTIGATIONS INTO AERO-HYDRODYNAMIC COUPLING
by
GRAHAM TABER SKINNER
Brown University
(1975)
AB.
B.M.E., The Catholic University of America
(1980)
Submitted in partial fulfillment of the
requirements for the degrees of
Master of Science
in
Naval Architecture and Marine Engineering
and in
Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1982
f, Graham T. Skinner
The author hereby grats
to distribute
copies
of
to
.I.T, permission to reproduce and
nii'doqumenV/X whole
or in part.
Signature of Author
DOpartment of O0ean Engineering
Certified By
J-tip
,wIn
Accepted
By
E. Kerwp, Thesis Supervisor
R
enovl-Thesis Reader
A. Dougtas Carmichael, Chairman
Departmental Committee on Graduate Students,
Department of Ocean Engineering
-la-
SAILING VESSEL DYNAMICS:
INVESTIGATIONS INTO AERO-HYDRODYNAMIC COUPLING
by
GRAHAM TABER SKINNER
Submitted to the Department of Ocean Engineering and Mechanical Engineering, June 1982 in partial fulfillment of the requirements for the
degrees of Master of Science in Naval Architecture and Marine Engineering,
and Master of Science in Mechanical Engineering.
ABSTRACT
An investigation was undertaken of the extent to which the
motions and added resistance of a vessel sailing in waves
could be attenuated by the aerodynamic damping of sails.
The analysis used quasi-steady inviscid lifting line and
lifting surface approaches to the aerodynamic problems.
The primary focus was the attenuation of added resistance
by aerodynamic pitch damping, which was 14% for an example.
The value depended strongly on the viscous drag. In addition,
it was shown that pitching can increase the driving force
slightly and that rolling is overdamped by aerodynamics.
Thesis Supervisor:
Title:
Justin E. Kerwin
Professor of Naval Architecture
-2-
ACKNOWLEDGMENTS
The author wishes to thank Professor M. Triantafyllou
for his helpful hints and Professor J.E. Kerwin and
Professor J. Milgram for providing the direction which
was needed to conclude this investigation. Thanks also
to Professor Hedrick for his advice and encouragement.
I am grateful for the excellent typing service provided
by Jessa Frankel, Joanne Sullivan and Marilyn Staruch.
-3-
'CONTENTS
.....
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-9
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Abstract . . . . . . . . . . . . . . . . . . . . .
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Title
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Acknowledgements
Contents . . . . . .
Nomenclature . . . . . . . . .
I
I'I
II'I
IIV
V
fI
VIII
VI]I
IX
X
. .
Introduction
.
.
. .
. .
Formulation of the Problem . .
.
8
11
The Effect of Vessel Motions on the Apparent Wind
Derivation of the Linearized Aerodynamic Pitch
Damping Coefficient . . . . . . .
.
.
.
.
.....
.
12
Approximate Calculation of the Aerodynamic Pitch Damp ing
Coefficient Using Lifting Line Theory . . .. . . .
15
. . .
17
a··
24
Numerical Modeling of Sail . .
Seakeeping
Reduced Frequencies
.
.
. .
. .
. . . . . .
. . . . . . .
·
·
·
28
The Effect of Aerodynamic Pitch Damping on Added
. . . . . . . .
.
Resistance Due to Pitching
30
. .
32
The Effect of Pitching on the Average Drive Force
. . .
XI
Other Motions:
Roll ...
XII
Other Motions:
Surge, Sway, Heave, Yaw
XIII
Other Motions:
Pitch Coupling
. . .
·
·
0
0
·
0
*
0
·
0
·
·
·
e*
*
XIV
7
0
.
References
. . . . . . .. . . . . .
*
0
*
0
Appendices:
39
40
0
0
. . . . . ..
Conclusions
33
.
*
*
8
· o·
· ·
·
· ·
·
·
·
·
41
43
·
1.
Rough Calculation of Aero Pitch Damping
56
2.
Explanation of Program Cyclone
58
3.
Transformation
·
·
of PSF2 Output
63
·
o·
4.
PSF2 Output .
.
.
. . .
65
·
5.
MIT 5-D Output
. . . . . . . .
·
·
·
86
-4-
LIST OF FIGURES
Page
..
. .
45
1.
Sstems ..
Vessel-ixedCoordinate
2.
Wind Triangle
3.
Sailplan for Lifting Line Analysis
4.
Geometry for Lifting Surface Analysis . . .
.
......
. . .
..
.
....
46
47
48
-5-
NOMENCLATURE
a
Wave amplitude
Aj
Added mass coefficients
A
Aspect ratio s2/S
c
Chord
CD = D/
Cji
-
SVA
Drag coefficient
Hydrostatic restoring force coefficients
CL = L/-
p SV~
Lift coefficient
d
Vertical distance between hub surface and center of lateral
resistance
D
Drag Force
Fji
Exciting force (or moment) in jth direction due to motion in
in ith direction,
i, j
=
1,2..6
Fn
Froude number v/vji
FR
Driving force
Gj.
Element of the generalized mass matrix
GM
Metacentric height
GRAD
Vertical true wind gradient, sec
k
Reduced frequency
L
Lift force
n
Propeller rpm
1
L,M,N
Aerodynamic moments corresponding to i = 4,5,6
p,q,r
Aerodynamic notation equivalent to
RHUB
Radius of Propeller HUB
s
Span
S
Sail planform area
t
Time
x4, x5, x6
-6-
T
Period of oscillation
VA
Apparent wind velocity
VB
Steady (average) vessel velocity
XCAM
Ratio of Maximum camber of section to chord
~X
Displacement in jth direction (surge, sway, heave, roll, pitch, yaw)
;
YVessel velocity
in ith direction
a
Apparent wind angle to vessel centerline
A
Displacement
AF1
Added resistance
Y
True wind angle to vessel centerline
X
Leeway angle between course and vessel centerline
P
Density of fluid
e
Pitch angle
(lbs.)
Frequency of oscillation
X
Angle of incidence of waves (for head seas
- 108° )
Damping ration between actual damping coefficient and damping
coefficient required to just damp out oscillations
Superscripts & Subscripts
-A
Aerodynamic
-H
Hydrodynamic
-n
Natural (frequency or period)
-p
Derivative with respect to p
-q
Derivative with respect to q
-7-
I.
INTRODUCTION
Analysis of sailing vessel performance in a seaway is complicated
by the coupling of hydrodynamic and aerodynamic forces and motions.
Performance of a vessel sailing in calm water has been analyzed by
Kerwin 3.
Behavior of a yacht hull moving through waves has been analyzed
by Gerritsma and Moeyes 4 .
A program has been written to combine these
analyses for performance prediction for a vessel sailing in a seaway 5 ,
but neglects the contribution of the rig to the equations of motion.
For downwind sailing results have been reported for studies
hull-sail
of
interaction, such as the coupling between wave induced roll and
wind induced sway moment, which can lead to the instability known as
broaching 6 '7
For upwind sailing the possibility exists that aerodynamics provides
substantial damping of wave excited motions.
If so, then added resistance
due to wave radiation should be less than that predicted by models which
neglect this coupling.
No investigations of this phenomenum are known to have been reported.
Such an investigation was the primary purpose of this work.
A corollary result of this work was the development of a program to
facilitate the analysis of a single sail using numerical lifting surface
techniques.
Use of such techniques for sail design has been reported 1 ,
and such techniques have been used for wing and propeller blade analysis 8
However, no previous use of this technique for sail analysis is known.
-8-
II.
FORMULATION OF THE PROBLEM
Observation of the motions of vessels sailing in waves shows that
pitch is the most severe motion and probably contributes most to added
resistance due to wave radiation.
These observations are supported by the
results of the seakeeping analysis of a sailing vessel hull moving through
waves, (see later chapter).
Since pitch motions seem to make the largest contribution to added
resistance, the initial and primary focus of this investigation was in:terms
or the aerodynamic contribution to pitch damping.
It is easiest to model this flow using quasi-steady inviscid theory 9
First, however the relative importance of viscous and unsteady effects must
be evaluated.
The most significant effect of viscosity is separation of the boundary
layer from the sail surface.
When this condition is severe the sail has
-"stalled", and the effective mean camberline of the sail is so different
from the actual sail shape that inviscid flow modelling based on the actual
shape is invalid.
Experiments have shown that stall occurs when the lift coefficient
exceeds a value of about 1.9.2
For a sail of fixed shape the lift coeffi-
cient varies with angle of attack.
Then as long as the pitch induced change
in angle of attack is small, separation is insignificant.
Unsteady effects include periodic changes in sail shape.
Because the
sail is a flexible membrane, its shape is pressure distribution dependent.
-9-
When the angle of attack is "ideal" the fluid velocity at the leading edge
is directed along the mean camberline at the leading edge.
Below ideal
angle of attack the pressure drop across the fabric at the leading edge
will tend to deform (or "luff") the sail.
For small pitch induced changes
in angle of attack not much less than ideal, the change in sail shape due to
luffing can be neglected.
Unsteady effects also include periodic changes in the inviscid flow
alone.
Outside the boundary layer the flow is essentially inviscid.
Here
the vorticity must be constant so that vorticity must be shed into the flow
as the lift coefficient varies.
As this vorticity is convected downstream
it changes the effective angle of attack.
large with respect to the pitch rate,
If the rate of convection is
then this effect can be neglected.
A measure of the importance of this unsteady effect is the reduced
frequency,
k
TVA
For small k unsteadiness of the flow can be neglected.
The magnitude of all the effects considered above is proportional to
the magnitude of the change of angle of attack due to pitch.
small in high winds.
This change is
Since high winds generate the large amplitude waves
which excite large pitch motions, the simplification of the problem to a
quasi-steady inviscid analysis should be valid.
Further assumptions, which will be repeated later as they are applied,
are
-1 0-
(1) Heel, leeway are negligible for hydrodynamic motions, and
(2) Groundplane, heel, boundary layer effects, aspect ratio,
sail interaction, optimal sail trim are second order aerodynamic considerations.
Two different approaches were taken to the evaluation of both
the
aerodynamic and hydrodynamic pitch damping coefficients.
The first approach consisted of simple rough estimates.
The second approach consisted of very accurate evaluations using
numerical methods.
-11-
III.
THE EFFECT
OF VESSEL MOTIONS ON THE APPARENT WIND
Two equivalent vessel fixed coordinate systems are given in Figure
1.
The one without subscripts is usually used for analysis, the other
for comparison of results.
Sail interaction is neglected.
Consider the effect of pitch and roll on the apparent wind.
Assume
axes of motion pass through center of waterplane area, (which is at
least consistent with hydrostatic theory).
Define the true and apparent
wind angles y,B with respect to the vessel centerline.
Then the local
instaneous apparent wind is given by
VA(pqz)
= [(VTsiny - VBsinx - pz) + (VTcosy + VB
q)]1/2
(1)
and the instaneous apparent wind angle is
o(p,q,z) = arctan[(VTsiny
- VBsinx - pz)/(VTcosy + VBcosX + qz)]
(2)
-12-
IV.
DERIVATION OF THE LINEARIZED AERODYNAMIC PITCH
DAMPING COEFFICIENT.
Consider the linearized single degree of freedom equation for pitch
motions:
(G5 5 + A55 )q + B5 5 q + C 55 X5 = F5 sinwt
(3)
It can easily be shown that this is the truncated Taylor series expansion
18
about the equilibrium condition
= q
X=5
0 . The aerodynamic contribution,
BA ,to the damping coefficient is to be determined.
Taking the partial
-55
derivative with respect to q yields
BA
-a5M-
Mq
(4)
Now the moment of the aerodynamic force can be written
M(q) =dM(q,z)
(5)
where for rigs of high aspect ratio and neglecting heel, the moment increment
can be approximated by
dM(q,z) = -z[dD(q,z)cosa(q,z)-dl(q,z)sin(q,z)]
(6)
where
dD(q,z)
= : P[VA(q,z)]2
c(z) dz C(q,,z
)
(7)
and
dL(q,z) =
Pp[VA(q,z)]2 c(z) dz C(q,z)
(8)
-13-
Combining
and (5) yields
(4)
BA5
=;(dM)q
(9)
S
Combining
(6),
dM =
implicitly
(8) yields
(7),
Z
zV
2
c dz(CDcos
ADL
(10)
- CLsinB)
+
dependent on q, z.
Taking the derivative
(dM)q
with respect to q:
= -p
-z
V c dz(Cqcos
-CLqsin
This expression
- CLsinS)
z VA VAq c dz(CDcosS
can be simplified.
- CDsinB
q +
- CLcosS
q)
Note
that
(12)
z cosB
VAq
(11)
and
-z sinB
q
at
= 0.
(13)
VA
Combining (11), (12),
(13) and substituting
into (9) yields for the
linearized aerodynamic damping coefficient
A
- PVA
55
S
cz2 [2CDCos2 3 + (CD + CLB)sin2B-
2(c L
2 ]dz
+ CDB)sin
(14)
-14Here VA and
are found from the steady state and CD, CL, CL,
in general vary with spanwise position,
CDS may
-15-
V.
APPROXIMATE CALCULATION OF THE AERODYNAMIC PITCH
DAMPING COEFFICIENT USING LIFTING LINE THEORY
Equation (14) was used to get a rough idea of the order of magnitude
of the pitch damping coefficient for a 12-meter yacht.
This example was
chosen because data was already available in a useful form for calculation
of the hydrodynamic coefficients of the 12-meter yacht "Valiant".
Gross approximations to the lift and drag coefficients were made on
the basis of diagrams in a handy reference11.
Although no constraint was
placed on the heeling moment, the lift coefficient was set low.
the lift and drag coefficients were assumed to be spanwise
Accordingly,
constants,
The lift coefficients were assumed to be
CCL
=
0,7
and
(15)
CL
LO
= 211 AA
A+2
which would be correct for an elliptically loaded foil.
To account for
viscous drag the lift to drag ratio was assumed to be fairly low:
L/D = 6
(16)
whence
CD
= .1167
and4
CD8:
Da
CL
TL
(17)
-1 6-
The assumed sailing conditions throughout are
VT = 20 knots
VB =
8 knots
'y
=
360
X
=
40
(18)
Then for steady state from (1) and (2)
VA = 45
B
fps
= 29 °
(19)
The damping coefficients were calculated for the typical
mainsail and genoa jib shown in Figu
The result obtained (in Appendix
1 2-meter
Sail interaction was neglected.
) for the overall aerodynamic pitch
damping was
BA555 = 79,000 (ft-lb)/(rad/sec)
-17-
VI.
NUMERICAL MODELLING OF SAIL
An alternative and more accurate method than lifting line theory
was desired for evaluation of aerodynamic pitch damping.
A set of computer
programs 1 0 for propeller analysis were available, and could be adapted to
the analysis of a single sail by specifying a one bladed propeller.
A
very large ratio of hub radius to blade (sail) span was specified so that
the helical nature of the geometry and wake was linearized on the sail and
in the neighboring flow.
The spanwise distributions of chord, camber, and twist, (see Figure 4),
were specified.
Precise relative displacements of the sections in the
y, z directions (corresponding to
dihedral and sweep) were neglected on
the assumption that these caused only higher order effects.
Heel was not
modelled, on the assumption that the effect of heel is also analogous to
sweep.
The ground plane (water surface) was not modelled either.
The
programs only model lifting surfaces (not the propeller hub for example).
These assumptions were justified by experience from modelling and testing
propellers.
Sail Specification and Design
For the 12-meter used as an example, a sail was specified with the same
span and area as a typical mainsail and jib combination so as to provide
approximately the same forces and moments.
For simplicity a sail with an untwisted planform equivalent to an
isoceles triangle was selected.
a
=
0.8 mean camberlines2,
It was also convenient to specify N.A.C.A.
since they could be generated automatically
by the program.
The following sectional properties remained to be determined:
(1) camber ratio, defined as the ratio of maximum camber to
chord length, and
(2) twist defined as the angle between the chord of the sail and
the centerline of the vessel,
so as to optimize the driving force subject to the constraint of maximum
allowable heeling moment equal to 150,000 ft-lb.
This limit was chosen
because it corresponded to a maximum allowable heel angle of 30° and a
5,000 ft-lb righting moment per degree of heel.
(This is the rate of
righting moment generated by a 65,000 lb vessel with a GM = 4.4 ft).
At this point it might have been convenient to design the optimum sail
shape for these specifications by using the design program of Milgram1 .
Instead an iterative approach using the analytical package was undertaken.
This was a very expensive and time consuming procedure.
As a practical approach to the problem, optimization of the driving
force was interpreted as maximization of the lift coefficient.
Also, the independent variables were subject to various constraints.
-19-
Twist was assumed to vary linearly with height.
Observations1l
that camber ratio, as well as twist, increases with height.
indicated
Furthermore,
the package included a spline curve fitting routine which required a smoothly
specified spanwise distribution of camber ratio.
Finally, it was recognized that large pressure differentials across
the leading edge could not be tolerated in the real flow, as indicated in
Chapter
I.
The iterative procedure was complicated by (1) the presence of downwash,
whose effect on the vertical distribution of load was hard to predict, and
by (2) the extreme sensitivity of the leading edge pressure distribution to
18
the angle of twist.
An attempt to design a suitable sail with a camber ratio
of 0.11 was
abandoned due to indications that such a sail could not meet the heel
constraint without excessive luffing.
-20-
Finally a sail was found which came close to meeting all the design
requirements.
It should give a fairly good idea of the quasi-steady inviscid
pitch damping.
The Designer's Choice:
(Also see Appendix
)
Geometry:
Area = 2320 ft2
Span = 80 ft.
Camber Ratio = .075 to .095
Twist Angle
= 11.5 ° to 29.5°
Operating conditions as in (18), plus wind gradient
Grad = 0.05 (ft/sec)/ft
Performance:
CL = 0.68
CD = 0.167 (including viscous drag coeff = 0.1)
L = 159,000
ft-lb
-21-
Damping
Coefficients:
Damping coefficients were calculated using a simple finite difference
method.
For the design sail, approach velocities were perturbed to correspond
to small changes from equilibrium in the rates of pitch, roll, and sway.
Then the damping coefficients for the jth
-[F(
+
- F(i
Bji =
i)
force due
th motion were
)]
Ak.
Following linear theory these coefficients were evaluated for xi = 0,
and Axi
The velocity increment Axi was considered small enough if
small.
(1) Sectional lift coefficients were low enough for unseparated flow,
(i.e. less than 1.9)2.
(2) Bji(-Axi)
Bji!(xi).
Results:
The damping coefficients for the designed sail were, for
Pitch:
BA 5 = 80,000
(ft - lb)/(rad/sec),
BA
(ft - lb)/(rad/sec),
55
Roll:
= 700,000
-22-
Sway:
A
B22 = 300
lb/(ft/sec),
Pitch due to roll:
BA
= -350,000
(ft - lb)/(rad/sec),
Pitch due to sway:
BA2 = 7,000 (ft - lb)/(ft/sec).
Summary of Results of Numerical Investigation of Aerodynamics:
The results of the numerical analysis using vortex lattice techniques
for realistic sail shapes confirmed the results of the approximation which
assumed constant spanwise coefficients of lift and drag.
damping coefficients were obtained.
roll
is probably
the largest
Additional
These indicated that roll damping of
aerodynamic
contribution
to damping.
It is
suprising to note that the damping of pitch by roll is very negative.
However, this unusual coupling is not significant since roll is heavily
damped.
-23-
In addition it was found (see Appendix 4
) that when the viscous
drag coefficient was decreased by 10% from 0.1 to 0.09 then the aerodynamic pitch damping coefficient was decreased by 20%.
Next the hydrodynamic characteristics of this vessel are investigated, making it possible to determine (1) the relative values of the
aerodynamics and hydrodynamic damping, and (2) the absolute value of
the total damping.
.
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-24-
VII.
SEAKEEPING
To quickly obtain a very rough idea of the order of magnitude of the
hydrodynamic pitch damping coefficient published data14 was used to
calculate the hydrodynamic pitch damping coefficient for a merchant vessel
hullform.
For a series 60 hullform of block coefficient 0.6, with water-
line length of 50 feet, it was calculated that the hydrodynamic pitch damping
coefficient would be about 500,000 (ft-lb)/(rad/sec).
To obtain a more accurate value, the M.I.T. 5-D seakeeping program 15 was
run using available data for the hull of the 12-meter "Valiant".
theoretical
1
basis of the program can be found in Salveson, et a1
Gerritsma and Beukelman
7
.
The
6
and in
The program calculates the frequency dependent
added mass and damping coefficients, and the Froude-Kriloff and sectional
diffraction force coefficients using two parameter Lewis forms.
The program
can use these results to compute various responses such as motions in
irregular waves and added resistance due to waves.
The input and output are presented in Appendix
5.
The operating
conditions correspond to the example used throughout, except that it is not
possible to specify a nonzero leeway angle to the program.
Therefore the
angle of incidence of the waves was set at
= 180
-
+ X/2 = 1420
as a compromise between accurate modelling of the encounter frequency and
the angle of incidence.
-25-
The program assumes the vessel has port-starboard symmetry, i.e. it
neglects the effects of a mean heel angle.
Gerritsma and Moeyes have
The vessel
and centers
Experiments reported
by
4
indicated that this may be acceptable4
geometry is specified by the sectional area coefficients
of buoyancy.
For motions not in the vertical plane there is
also a multiple choice of sectional shapes.
Viscous roll damping due to
twin bilge keels can be calculated by a stripwise method.
To model a yacht
keel it is necessary to specify an equivalent pair of bilge keels.
This
was not done because pitch response was the primary focus of this investigation, and the program assumes pitch is uncoupled from roll.
Results of the Seakeeping Program
Displacement,
The
A:
displacement
is
calculated
tonnes divided by gravitational
from
the shape and appears in units of
acceleration
G22 = 1.04318
so that in yacht sized units
A = 75,000
lb.
This is high, but in the rightrangel3
Natural Frequency of Pitch,
5n:
Since the added mass is frequency and speed dependent the natural
-26-
frequency in pitch was found by interpolation to be equal to the encounter
frenquency,
we, at which the following equation was satisfied
e
+A5n
G55
Here C5 5 is the pitch restoring coefficient (tonne-ft/radian); Grr is the
pitch moment of inertia
2
G 22 (RYY)
G55=
where RYY is the specified pitch gyradius; and A 5 5 is the added moment of
inertia for pitch.
m5n
The result was
2.1 rad/sec.
=
This corresponds to a natural period of pitch for these conditions
T
-
2
2.9 sec.
which is somewhat high for a 12-meter but also in the right range
3
.
Since
these results were fairly accurate it was assumed that the results for the
hydrodynamic damping coefficients were also within the correct range.
At the natural frequency the pitch damping coefficient was (by
interpolation):
B55 = 378
55
tonne-ft-sec
-27-
or, in yacht-sized
units:
H
B5 5 = 847,000 lb-ft-sec
The damping coefficient decreased moderately with frequency of encounter to
about 255,000 at
= 3.0, we = 5.97.
The hydrodynamic damping ratio
at the natural frequency was
H
=
B5 5
2(G55 +
= 0.24.
55)w5n
VIII.
REDUCED FREOUENCIES
......
_
.....
-
i~~~~~~~~~~~~~~~~~~~~~~~~~~
A typical reduced frequency along the span is
k(z) ¥(z)
-_c(z)
TVA
x 15
2.9 x45
T VA
A weighted average reduced frequency can be defined:
Jz. c dz
TV
T V
_ TA
~A
2
TAzdz
.T
k
This is quite high.
less.
841,500
=
=
:
65
+ 1,796,320
z'8 -6386
VA
z
-A
(9.1967 + 8.4753)
j
.4254
Therefore unsteady inviscid damping will be even
than the quasi-steady result, (still assuming steady geometry).
To see this intuitively, consider what happens while pitching forward.
Angle of attack is reduced.
Therefore circulation must be shed into
flow parallel to span, leaving upwash behind as it goes.
to increase the effective angle of attack temporarily.
This tends
The amplitude
of attack oscillation is inversely proportional to the frequency.
Consider now the aerodynamic terms in the analytic expression
for the pitch damping coefficient,
(14).
By far the most significant
factor here is the variation of lift coefficient with angle of attack.
-29-
(See also Appendix
I )o
This suggests that the damping coefficient
will decrease at high frequencies, since as we have just seen the
effective angle of attack changes less at high frequencies.
Thus, since the pitch damping coefficient is most highly dependent
on changing angle of attack, and since the angle of attack changes less
at high frequencies, it may be concluded that the pitch damping
coefficient decreases with frequency.
-30-
IX.
THE EFFECT OF AERODYNAMIC PITCH DAMPING
'ON ADDED RESISTANCE'DUE TO PITCHING
Given that the results of the hydrodynamic investigation and the
results of the aerodynamic investigation are correct, then the effect
of aerodynamic pitch damping on added resistance due to pitching can
be calculated.
For pitch alone 2 0 the transfer relation between pitch amplitude
and wave amplitude is
a5-
recalling the results
H
BA
55
=
0.24
= 0.1
.
H
55
A comparison can be made between the transfer relations with and without
aerodynamic damping.
05
The largest difference occurs at resonance,
W5n' where
-5
a
=
.927
a
aX
-31-
The pitch amplitude is proportional to the amplitude of the
radiated waves, at least within the range of validity of any linear
theory.
But the added resistance due to pitch is proportional to the
energy lost in the radiated waves, which is proportional to the square
of the wave height.
AF 1
f-
Therefore
aF 1 H
a
0.86: aF1
H
is the added resistance at resonance.
Thus the added resistance calculated by the seakeeping program
should be adjusted downwards by up to 14%.
e
-32-
THE EFFECT OF PITCHING ON THE AVERAGE DRIVE FORCE
X.
In a linear analysis the average driving force, FR, will equal the
steady state driving force.
there may be a difference.
But for nonvanishing pitch amplitudes
To get a rough idea of the difference,
the driving force was calculated (see Appendix
output for the three conditions
~ = 0,
+.1
-±0
)from the PSF program
rad/sec.
From the results
_
·. F R
0.1
830.3
0.0
1,000.1
-0.1
1,191.3
A FR
-175.8
---+185.2
it can be inferred that the average quasi-steady inviscid driving force
was higher than the steady state value by a fraction of a percent.
(The exact nonlinear quasi-steady inviscid average thrust requires
integration of the instaneous thrust throughout a half period of
oscillation.)
-33-
XI.
OTHER MOTIONS:' ROLL
..Natural Frequency of Roll, W4n:
From the output of the seakeeping program it should be possible
to calculate the natural frequency for roll,
4n, defined as that
frequency at which the following relation is satisfied
: e
44'
44.
G44 +A
44
:
4n
Recall, however that the displacement calculated by the program was
probably high, and that both the radius of gyration and the metacentric
height were somewhat arbitrary inputs to the program.
Therefore values
were used which were consistent with the assumptions made in the chapter
on numerical aerodynamics:
C4 4 = 5000 (ft-lb)/degree
= (Ryy)2
65,000
G5
slug-ft2
50,506
Then, (using the value of A 5 5 at the encounter frequency
without interpolating), the natural frequency in roll is
. .... ..
'4n =
5000
l0.4619x 2240 + 50,506
= 1.97 rad/sec
]/2
me = 1.78
The corresponding natural period is
.T4n
3.19 sec
:
Wn
4n
which seems high, although less so than if C4 4 and G44 had been taken
directly from the output.
From this data the damping ratio, 54 for uncoupled rolling can be
calculated:
........
*4 -(G4
2
.B
.
B44
+ A 4)
w4n
44 (ft-lb)/(rad/sec)
291,326
Roll Damping:
Linearized Aerodynamic
Here the approach is identical to that of the pitch damping
derivation.
For a chordwise strip of the sail the increment of rolling
moment is
dLA = z[dL cosB + dD sin.]
z c dz 2 VA [CLCCSs
+ C sin3].
Taking the derivative with respect to the roll rate p, using the
derivatives
VAp = -zsin.
VAp
-35-
and
p
VA
and integrating over the span yields for the linearized quasi-steady
invisicid aerodynamic
roll dampingcoefficient
BA
44
IdL
AP
-
SA
2
'PAVA
A2A X c z
[2
2
2
CDsin2
C+
LB) cos
2
A
1
2
For the elliptically
BA44 = 580,000
(CL+CD)sin2S]
dz
loaded example of (15), (17), (19), the value is
(ft-lb)/(rad/sec)
and the dampingratio is
A
= 1.99
which is very highly damped.
Aerodynamic Roll Damping by Numerical Lifting
Surface
Methods
Recall the results presented earlier:
BA = 700,000
(ft-lb)/rad/sec)
C
-36-
Then the aerodynamic roll damping ratio for the numerical sail model is
44 = 2.4
which is still well over damped.
Hydrodynamic Roll Damping from the Seakeeping Program
At an encounter frequency close to the natural frequency, the
seakeeping program output is
B4
= 1.796 tonnes/(rad/sec)
= 4023 (ft-lb)/(rad/sec)
The value changes only moderately with frequency.
The corresponding damping ratio is
H
= 0.014
However observations
indicate that hydrodynamic damping is much higher.
As :aforementioned, better modeling of roll in the seakeeping
program requires the specification of an equivalent pair of bilge
keels.
Recall also that leeway is neglected.
To improve the results
here, revision of the FORTRAN code may be necessary.
Hydrodynamic Roll Damping by an Approximate Lifting Line Approach
Here the same approach is used as previously for the aerodynamics
of pitch and roll.
The underbody is modeled as a lifting line rather than a slender
body.
-37-
The differential cohtribution to the hydrodynamic rolling moment
along the span is
dLH =
zC-dz LV
(CLcosX
+
CDsinx)
where CD includes all hydrodynamic drag.
Differentiating with respect to the roll rate, p, using the
derivative forms
VBp
zsinx
and
_ zcos
p
V
and integrating over the span yields for the quasi-steady inviscid
hydrodynamic roll damping coefficient
H
1
44 ==2
P PH
H V;
B44
IciBc
2
2
nz"
f sin
DLx
LD
2
z2[2CDsin2z
+ (CD+CL)COSx
+ (CL+CDxd
dz
Assuming elliptical loading, constant chord of 14 ft, and span of 9 ft,
and neglecting all bracketed terms except CLA cos2x, a very rough
estimate
is
BH4 44
92,000
The corresponding damping ratio is
4 = 0.3
'Summary of Results of Investigation into Aero-Hydrodynamics
of Rolling
Both the lifting line and lifting surface approaches indicate that
roll is aerodynamically overdamped.
improved analysis techniques.
The hydrodynamic roll damping needs
XII.
OTHER MOTIONS:
'SURGE; SWAY, HEAVE; YAW
No investigations were made of these motions, except that it
was possible to calculate from the available data that for-sway
B2 A = 300 lb/(ft/sec)
and
H
B2 2
=
4,000 lb/(ft/sec)
depending somewhat on operating conditions.
Surge is not modeled by the seakeeping program.
Heave could be modeled by the lifting surface program but was
neglected as uninteresting.
Presumably quasi-steady inviscid heave is
similar to variable sweepback since in both cases the angle of attack
is constant, but the angle of the streamlines to the midchordline
changes.
Yaw could not be modeled by the lifting surface program unless
the sail was rotated 900 about its y-axis.
was not essayed.
Such geometrical magic
-40-
OTHER MOTIONS:
PITCH COUPLING
The seakeeping program does not indicate any pitch-roll coupling,
but coupling arises due to the asymmetry of the aerodynamic surface.
Recall the results from the numerical sail analysis:
B54
=
B52
=
-350,000 (ft-lb)/(rad/sec)
7,000 (ft-lb)/(rad/sec)
The pitch damping due to sway, B52, ought to be negligible except
under conditions of extreme sidewise motion.
damping due to roll is surprising.
The (negative) pitch
However its effect is likely to
be small since, as it has been shown, roll is very heavily damped.
-41-
XIV.
Observations
CONCLUSIONS
indicate that for a vessel sailing in waves, pitch is
a large periodic motion which generates significant added resistance.
This was confirmed by results of the M.I.T.
5-D Seakeeping Program.
An investigation was undertaken of the extent to which the added
resistance in waves could be attenuated by the aerodynamic damping of
sails.
Quasi-steady, inviscid lifting line and lifting surface models
were used to solve the problem for a 12-meter yacht sailing to windward
in a twenty knot breeze.
For this problem it was found that aerodynamic
pitch damping attenuated the added resistance by 14%
when the waves
were encountered at the natural pitching frequency.
It was found that a 10% decrease in the viscous drag coefficient
caused a 20% decrease in the aerodynamic pitch damping coefficient.
The quasi-steady assumption was not particularly good.
Unsteady
effects probably diminish the significance of aerodynamic pitch damping.
According to the linearized analysis, there are six contributions
to an aerodynamic damping coefficient:
to lift.
three due to drag and three due
For both drag and lift the three contributions are due to
(1)
Changing apparent wind speed,
(2)
Changing lift or drag coefficient, and
(3)
Changing apparent wind angle.
The sign and magnitude of each contribution depends on the motions, sail
geometry, and operating conditions.
-42-
For example aerodynamic damping of roll appeared to be very highly
over-damped.
Some motions were not investigated due to lack of interest or
modeling difficulties.
It was found that pitch motions could increase the average driving
force by at least a fraction of a percent.
The development of an aero-hydro-integrated
performance prediction
program for vessels sailing in waves could be developed.
In the process,
further investigation into coupling should consider the remaining motions
Such as the aerodynamics of yaw and the hydrodynamics of rolling and
sideslipping.
Another exciting development which could be undertaken is the coding
of a versatile program for analysis of sailing vessel aerodynamics using
numerical lifting surface methods.
-43-
.REFERENCES
1.
Milgram, Jerome H., "The Analytical Design of Yacht Sails", S.N.A.M.E.
Trans., 1968.
2.
Milgram, "Sail Force Coefficients for Systematic Rig Variations",
S.N.A.M.E. T&R Report R-10.
3.
Kerwin, Justin E., "A Velocity Prediction Program for Ocean Racing
Yachts Revised to June, 1978," M.I.T.
4.
Gerritsma, J., and Moeyes, G., "The Seakeeping Performance and Steering
Properties of Sailing Yachts", H.I.S.W.A., 1973.
5.
Warrick, Alexander, Jr., "Racing Yacht Performance in a Seaway",
M.I.T.
Master Thesis, O.E., 1976.
6.
Newman, J. Nicholas, "Stability and Control in Quartering Seas",
Symposium on Sailing Yacht Research,
7.
M.I.T., November, 1966.
Letcher, John S., Jr., "Steering Qualities off the Wind", A.I.A.A.
Symposium "The Ancient Interface", 1971.
8.
Greeley, David S., and Kerwin, Justin E., "Numerical Methods for
Propeller Design and Analysis in Steady Flow", to be published.
9.
10.
Newman, J. Nicholas, "Marine Hydrodynamics", M.I.T. Press, 1978.
Greely, David S., "Introduction to Propeller Steady Flow Computer
Programs", unpublished.
11.
Marchaj, C.A., "Sailing Theory and Practice", Dodd, Mead, & Co., 1964.
12.
Abbott, I.H., and Von Doenhoff, A.E., "Theory of Wing Sections",
Dover, N.Y., 1959.
-44-
13.
Spens, P.G., DeSaix, P., and Brown, P. Ward, "Some Further Experi-
mental Studies of the Sailing Yacht", Stevens
Institute of Technology,
Hoboken, N.J., 1967.
14.
"Principles of
Naval Architecture",
Comstock, J.P., Editor, S.N.A.M.E.,
1967.
15.
User's Manual, M.I.T. 5-D Seakeeping Program,
Design Laboratory,
Department of Ocean Engineering, M.I.T.
16.
Salveson, N., Tuck, E.O.,Faltinsen, O., "Ship Motions and Sea Loads",
S.N.A.M.E. Trans.,
17.
Gerritsma,
1970.
J3. and Buekelman, W., "Analysis
in Waves of a Fast Cargo Ship", International
of the Resistance Increase
Shipbuilding Progress,
Vol. 14, February, 1967.
18.
Marchaj, C.A., "Aerohydrodynamics of Sailing", Dodd & Mead, N.Y. 1979.
19.
Hedrick, J. Karl, Paynter, Henry M., "Nonlinear System Analysis and
Synthesis:
20.
Vol. 1 - Fundamental Principles", A.S.M.E., 1978.
Thomson, William T., "Theory of Vibration and Applications",
1972.
-45-
-
X3·F
3
X 1 F, :
.
X2
z, Z,w
N,r
Mq
yl Y
v
VESSEL-FIXED COORDINATE SYSTEMS
Figure
1
J,
XP
I
-46-
p2
WIND TRIANGLE.
Figure
2
-47-
rf'
"""
o
7
I
X
"
_ _ _ _ _ _ ___
SAILPLAN FOR LIFTING LINE ANALYSIS
Figure
3
-48-
Figure
4
LIFTING SURFACE ANALYSIS: GEOMETRY
WITH WAKE FOLLOW4ING FREESTREAM
(a) Sail, Hub, and Flow
-49-
Ily
___
_
_
_
__I
_
(b) Planform view
__i
-50L
__
__ __
_1___
______
___
i
II
I
I__
I
_
_
I
__
A
It
I
y
~ ~
Y
__
i
~
__
m
_
lI i½
i'll
.am
I
___
___I__
I
V
_
__
-
-
-
zs-
--
ii~
_
i
i
I
--
\.1
---
\-
-\ --
(c)
I
C--·~--LIII
Front view
·
11-*---
-
--_
--
-51 -
)
00
t-
LL
-J
ct,
Li
XZ
0
I-
Li
C-)
ul
LL
-I
LLJ
0-0
-52-
k,~~~~~Z
N--t1
··~~~~~~~~
(e) Pitching forward
-53-
(f) Pitching aft
-54-
(g) Rolling
-55-
(h) Swaying
-56-
APPENDIX
ROUGH CALCULATION
OF AERODYNAMIC PITCH
DAMPING
(14)
Recall Equation
BA P2VAjcz2[2CCDcos2
+ (CD + CL)sin
-
(CL + CD) sin2¢]dz
S
Assume bracketed portion is spanwise constant (as for an elliptical load
distribution)9
Recall (15), (17), (19):
0.7
CL
CL_
211
A
4C
CD= .1167
C
D
A-+2
45
VA
p = 0.00234
8
= 25
From Figure 3:
s2
A
S
jib
602
-
Amai n = 5.39
2
= 3.53
=>
60 34
=>
jib
A
A-
+
=
i
0. 7294
main
Hence
jib
CL
main
4.0087
4.5838
.5063
.3789
_1
CD B
= 0.638
A+2
-57-
then
main
jib
2 C COS2 5
=
.1906
.1906
(CD + CL )Sin2B =
.7367
.8394
=
-(-CL +CDS) sin2~
-.4620
-.4132
.4653
.6168
and the moments of area are
34(65 z3
cz2 dz
jCZ
dz=
4
65
= 841,500
*0
5
jib
I
cz2 dz= 27 (86
main
main
z3
80
= 1,796,320
then
BA
55
=
78,950
z4486
836
7)6
2~(402
4+ 40
2
[
- 462)
3
3
92 z 4
+4
5
5
86
1
6
-58-
APPENDIX2
·EXPLANATION OF PROGRAM CYCLONE
This program takes a sail and whips it into a propeller.
This program reads a datafile (with filetype DATA), containing
generalized coordinates for a sail's geometry and operating conditions.
Finally, it
Then it transforms this DATA into propeller coordinates.
writes the converted geometry into two files (filetypes INPGEO and ADIINt!)
for analysis using the PSF2 package of lifting surface programs.
For the most part the program is easily understood but note the
following:
* Geometry can be specified at spanwise stations.
Since splines
will be fit to the specified geometry, it must be smooth!
- Alternatively, geometry can be specified by polynomial functions
using the dummy variable "crew" which increases with height from
its midspan origin.
* The propeller must rotate(?).
An RPM of 1 or less is recommended
for ease of output interpretation and formating.
: Heel cannot be accomodated at this time.
· Bend is defined in the projection of the sail on the x-z plane as
the displacement of the leading edge from vertical.
The bend
at the tack is zero.
· Sag is defined in the projection of the sail on the x-y plane as
the y-displacement
of the leading edge from vertical.
at the tack is zero.
The sag
-59-
· Twist is defined as the angle between the chordline and the
x-z plane
· Chordwise thickness and camber distributions must be specified
by editing an existing INPGEO file.
* Use a large ratio of propeller diameter to sailspan!
* x skew and x-rake are currently set to zero.
· Due to an uncorrected error at this time, specifying positive
sway velocity yields results for negative velocity.
* z prime is a spanwise coordinate which increases with height
from its origin at the foot.
* Sailing vessel coordinates are
= -x of propeller
x
positive ahead
y
positive to port = z of propeller
2
positive up = y of propeller
Origin is at center of waterplane area.
Vessel sails directly upstream on starboard tack, "semisubmerged"
with sailfoot at hub surface.
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APPENDIX 3
the forces and momentsacting on the sail
From the output of PSF2FOR
are as follows:
Heeling Moment:
L = TORQUE -
KFZ
600
d)
(RHUB
- 688,072,320 KFZ
TORQUE
-
Lift Coefficient:
[KFZ - KFX tan (-X)]
CL =
1
2
=
V
PS VA
A
pn 2 D4 cos(-A)
3600
222(KFZ - 0.4663 KFX)
Drag Coefficient:
CD = 222(0.4663 KFZ + KFX)
Pitch Moment:
n2 5
3600
=
KMZ - THRUST(RHUB - 5)
1,617,410 KMZ - 515 THRUST
Sway Force:
=
pn2 D5
3600
KFZ
-64-
Driving Force:
FR = THRUST
cos
+ p n2D4
KFZ cos (90-x)
3600
= 0.9976
THRUST + 9,4020.5
KFZ
-65-
APPENDIX
4
PSF2 OUTPUT
CONTENTS:
PAGE
Geometry
Analysis:
66
Steady State
DATA File for X 5
±= 0.1
66
69
Analysis:
Pitching Forward
70
Analysis:
Pitching Aft
73
with 10% Change in Viscous Drag
76
DATA File for Roll and Sway
77
ADMIN File for Roll and Sway
78
Analysis:
79
Roll
with 10% Change in Viscous Drag
Analysis:
Sway
82
83
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-86-
APPENDIX 5
MIT 5-D OUTPUT
CONTENTS:
PAGE
INPUT DATA
87
REGULAP WAVE RESULTS
90
IRREGULAR WAVE RESULTS
134
-87-
VALIANT:8,0 KTS. 142,180 D. HEADINGS. BUZZARDS TOWER SPECTRUM, 20 KTS
NSTA NROMS NENC
22
9
2
NVL NMOT
2
0
NSP
0
NSEA
NS
IO NADR NEVT
2
NWX NSOMS
1
1
12
1
BEAM
11.2700
DRAFT
9.5800
RYY
15.2000
RXX
5.0000
RZZ
15.2000
XZI
0.0
UNIT
ORIGIN
ZETAA
0.0
1.0000
XI(I)
25.9600
23.6000
21.2400
18.8800
16.5200
14.1600
11.8000
9.4400
7.0800
4.7200
2.3600
0.0
-2.3600
-4.7200
-7.0800
-9.4400
-11.8000
-14.1600
-16.5200
-18.8800
-21.2400
-23.6000
STATION I
1
2
3
4
5
6
7
IWBK(I)
4
1
i
1
1
1
1
1
1
1
1
3
3
3
3
3
3
3
3
3
3
YM(I)
1.8700
3.5600
4.8900
6.0800
7.2000
8.2600
9.1500
9.8500
10.4300
10.8700
1i.1000
11.2700
11.2000
11.0500
10.7000
10.2400
9.5900
8.7500
7.7000
6.1800
3.7800
0.1500
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
.
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
NFQ
0
NFR
0
GRAV
32.1740
XCG
0.1290
VCG
-1.0000
GM
3.0000
RHO
XRHO
NU
0.0008890 1.9898996 0.0000128
WSURFA
1000.00
ALFA
0.0
ZM(I) SIGMA(I)
0.4300
0.6467
1.0800
0.6814
1.7000
0.6881
2.3200
0.6948
2.9700
0.6902
3.6400
0.6732
4.4200
0.6350
5.2500
0.5952
6.1500
0.5509
7.0500
0.5110
7.8500
0.4797
8.5300
0.4485
9.0500
0.4227
9.4000
0.3924
9.5800
0.3610
4.9000
0.6314
4.6000
0.5885
4.2500
0.5566
3.8300
0.5290
3.4100
0.4669
2.9600
0.2824
2.8000
0.5238
BKRAD(I) BILRAD(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
NPCH
1
0
XLBP
49.5600
STATION I
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
20
21
22
12
13
14
15
16
17
18
19
20
21
NSPC
2
MP NTURB HASBK N NBEND NWT
6
1
0
0
0
0
CB
0.2193
0
8
9
10
11
20
NP
10
BKGIR(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
ZCB(I)
0.1400
0.3600
0.5670
0.7700
0.9900
1.2000
1.5000
1.7500
2.0500
2.3000
2.6000
2.8000
3.0000
3.1000
3.2000
1.6000
1.5000
1.4000
1.3000
1.1000
0.1000
0.9000
GIRTH(I)
2.0000
4.1600
5.9500
7.6500
9.3000
11.0000
12.7000
14.4000
16.0000
17.8000
19.2000
20.4000
21.2000
21.8000
22.0000
14.1700
13.3000
12.2000
10.9000
9.2000
7.0000
5.6000
RIFLR(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
ALPH(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0'
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
BKWID(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
PHI(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
PSI(I)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
LIWO(I)
0.0
0.0
0.0
0.0
0.0'
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
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