Transcript Hull Speed

The Physics of Sailing
Outline
•
Hulls
•
Keels
•
Sails
Hulls
•
•
•
•
“Hull Speed”
Resistance
Shape
Stability
Hull Speed
• Hull speed is determined by the length of the
boat.
• Water waves are dispersive, i.e., their speeds
depend on the wavelength of the wave; long
wavelengths are faster.
• Boats generate a wave at the bow. The speed
of this wave must equal the speed of the boat.
(This is the speed with which the crest is being
forced to advance.)
Hull Speed
• At first, the boat moves slowly and the bow
waves generated have short length; several
waves are seen along the side of the boat.
• As the boat moves faster, the wavelength
increases, until it equals the length of the boat.
• When the wavelength becomes longer than the
boat, the stern begins to fall into the trough of
the wave and the boat is ploughing “uphill” on
the bow wave.
• The resistance increases dramatically.
HULL SPEED FORMULA
Change in Potential Energy = Change in Kinetic Energy
m
mgh  mg (2 A)  (v  u ) 2  (v  u ) 2 
2
 2mvu
which yields
gA  vu
.
(1)
Now we need a relationship between v and u. We can obtain this
by noting that a wave can be described by a sine function.
y  A sin(
2 x

) , where λ = wavelength of wave.
Near the origin, where x is small,
y  A sin(
2 x

)
A(
2 x

)
and the ratio of y to x is then:
y
2
A
x

.
(2)
Now the ratio of the vertical to horizontal displacements near the
origin is the same as the ratio of the vertical to horizontal
velocities, u/v. Hence, we have
u y
2
 A
.
v x

Since u and v are each constant, we can use this relationship to
substitute for u in (1). This yields
gA  A
v
g
g

2
2
2
v2 ,

and solving for v,
  1.34  ( ft ) (v in knots).
Table 1.1
Wave/Hull Speeds
Wavelength
(feet)
Speed
(ft/sec)
2.3
5.0
7.1
10.1
12.4
16.0
19.5
22.6
31.9
39.1
1
5
10
20
30
50
75
100
200
300
Speed
(Mph)
1.6
3.4
4.8
6.9
8.5
10.0
13.3
15.4
21.8
26.7
Speed
(Knots)
1.4
3.0
4.2
6.0
7.4
9.5
11.6
13.4
18.9
23.2
HULL RESISTANCE
• Surface Resistance
Shearing
• Turbulence
Reynolds No.
• Eddies
Separation
• Shape
Friction: Intermolecular forces
REYNOLDS NUMBER AND TURBULENCE
Lv
/
R
L = length
v = velocity
 = viscosity
 = density
Viscosity is a measure of the force necessary to shear a fluid:
 v 
   
 y 
 = stress (force/area)
y = direction perpendicular to flow
The Reynolds number is the ratio of inertial forces (vρ) to
v
i
s
c
o
u
s
f
o
r
c
e
s
(
μ
/
L
)
.
R
e
y
n
o
l
d
s
o
b
s
e
r
v
e
d
t
h
a
t
l
a
m
i
n
a
r
f
l
o
w
becomes turbulent for R ≈ 106 .
For water:
 = 1.0  10-3 N·sec/m2 and  = 103 kg/m3 , which yields
R = L v  106 .
So that turbulence will begin when L v  1 .
5 knots = 2.5 m/sec = v, so L v = 1, when L = 0.4 m !
Roughness
• Hull should be “smooth”.
Bumps will introduce turbulence sooner
and/or will produce larger turbulence.
• “Polishing” does not help very much.
Shearing must take place!
Hull Shape (Form Resistance)
• Hull shape determines how fast a boat can
accelerate and how fast it can go in “light’
winds.
• Generally speaking, narrower, shallower
hulls are faster, but less stable and hold
less “cargo”.
• Exact shape for fastest hull is still a
subject of debate.
Modern Racing Hull Design
• Narrow, sleek bow
• Shallow, flat bottom toward stern
• Square stern, normally above water line
• Able to plane under certain conditions
Keels
• Keels are necessary to provide resistance
against “side-slipping”, and to provide counter
balance for sideways force of wind on sails.
• A large keel adds a lot of surface resistance.
• Want a balance between positive keel action
and negative keel resistance.
Wing theory
• Keels and sails act like airplane wings; i.e.,
they can provide “lift”.
• Proper design helps a lot!
Lift (Bernoulli’s Principle)
↑
Sail and Keel Lift
Fluid flow around wing
Typical Cruising Keel
Racing Keel
Shallow draft keel with wing
Keels and Stability
Sails
• Sails provide the power.
• Sails act like wings and provide lift and
generate vortices.
• Ideal sail shape is different for downwind
and upwind:
Downwind sails should be square-shaped
(low aspect ratio).
Upwind sails should be tall (high aspect
ratio) to minimize vortex generation.
Wind Power for Sailing
Moving air has kinetic energy, which is transferred in
part, to a sail. Using the Work-Energy Theorem:
Work = Force × Distance = Kinetic Energy
W = F × d = KE
Now,
KE = ½ M v2 ,
where
M=ρAvt,
where ρ is the density of the air, A is the area of the sail, v
is the velocity of the air with respect to the sail, and t is an
arbitrary time. If we take d as the distance the air travels
in the time t, then d = v t , and we have
F = KE/d = ½ ρ v2 A .
Not all of the air stops in the sail; some deflects and some
goes around the sail. This is usually taken into account in
an empirical way by writing this as
F = C (½ ρ v2 A) ,
where the C is a coefficient that is found from an empirical
look-up table and depends on the geometry of the sail and
the direction of motion relative to the wind direction.
Velocity Prediction Program
One can try to predict the speed and direction
(velocity) of a sailboat for different wind speeds and points
of sail (direction with respect to the wind direction).
Fdrive = RTotal
MHeel = MRight
Fdrive = L sin β
FHeel = L cos β + D sin β
-
β
=
e
f
f
e
c
t
i
v
e
a
p
p
D
a
c
r
e
n
o
t
s
β
w
i
n
d
a
n
g
l
e
L = “Lift” = CL ½( ρ v A )
D = “Drag” = CD ( ½ ρ v2 A )
RTotal = RF + RW + RS + RI + RH + RR
RF = Frictional resistance RW = Wave resistance
RS = Shape resistance
RI = Induced resistance
RH = Heeling resistance
RR = Residual resistance
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