Predicting sailboat's speed at a given wind  The sailboat transform
A previous version of this page, with minor editing, appeared (with my permission) as an article in
Catalyst  Journal of the Amateur Yacht Research Society
Number 34, Page 21, April 2009
Polar diagrams: Sailing boat speeds for several water boat types at wind speed of 10 knots.
From the book The symmetry of sailing by Ross Garrett, Sheridan House Inc., 1996.
Tornado class catamaran 18ft Skiff class boat
Introduction
Sailboat's speed predictions based on its geometry and physics are complicated and pose a substantial challenge. The simple approximation method presented here allows quite accurate predictions for almost all boat situations (e.g., not when the speed in a displacement hull gets close to and above its hull speed) based on several measurements. It also allows approximate quick predictions based on a single measurement set consisting of boat speed, wind speed, wind direction, and possibly other relevant parameters. It can be used as well for adhoc corrections of existing polar diagrams based on current conditions and measurements. In this case error may increase the more variables deviate from the measured point, but not necessarily. Several sets of such measurements across the boat's variable space allow quite accurate predictions for the boat's speeds across the entire variable space, comparable with complete polar diagrams for sailboat speed. The method is much simpler, with less needed measurements than direct speed interpolations. While the method bypasses most of the sailboat's specific details, it is based on approximating the two major factors that determine sailboat behavior: sail interaction with wind (specifically, momentum transfer due to wind deflection by a sail) and boat interaction with travel medium (water, land , or ice). By this it defines and utilizes mathematical functions that are very close in shape to typical speed polar diagrams, and are natural for describing sailboat performance. Basically, the method provides a transform (and its inverse) from sailboat speed to a quite flat, slowly changing, easy to approximate function, eta (comprising a combination of the boat's unknown constants and coefficients), which transforms back to accurate speeds at given winds and conditions. Even no change in eta can describe large changes in boat speed based on wind speeds and directions. The sailboat transform seems to be the common denominator of (modern) sailboats while sailboat's specific eta determines the boat's (including gear) inherent performance.
Content:
The mathematical relationship  The sailboat transform
Using the method  A detailed example
Best VMG  Zigzag both upwind and downwind
Water sailboats
Ice and land sailboats
The mathematical relationship  The sailboat transform
Several years ago I sent friends an article about attempts to reach speed of 200 mph with an Ice Yacht, at wind speed of ~60 mph. They wondered how possible, and I have tried to explain by applying the qualitative explanation I got as a kid about sailing boat speed, and later taught, together with a little more Physics. The major factor is wind ("regions/elements/particles") momentumchange by the sail (which is approximately considered here) and forces of interaction between the boat and the travelmedium (water, ice, or land; the force of drag/friction along the travel direction is calculated). A sail's goal is to generate maximal forward force. The efficiency of a given sail at a desired setting is affected mainly by its ability to maintain around it mostly laminar wind flow (vs. turbulent which results in efficiency losses) regardless of its type and material. I have reached the following equation for relatively fast (i.e., "normal" and faster) sailboats:
At steady state Sails forward force = Drag backwards force implies approximately the following equation (at approximated optimal sails' setting)
The sailboat transform
Where
a0 – real wind direction relatively to boat (0 when wind from front)
a – apparent wind direction relatively to boat
VW – real wind velocity (VW > 1)
eta (VW, a, optional parameters) – sailboat's slowly changing function of many parameters (1/velocity units), which incorporates most of the boat's and rigging's specific Physics. eta is a measure of the overall boat's resistance to increased speed, and in correlation with the (overall) drag coefficient. The smaller eta is, the faster the boat at a given wind velocity. It depends on many changing parameters like heeling
angle, flows turbulence levels, sea waves conditions, etc., and even may depend on a0 and a, depending on boat architecture.
It may have relatively large jumps when conditions change abruptly, like when moving to planing mode, or lifting a spinnaker, but usually not in order of magnitude. An accurate eta function, which is quite difficult to determine by the boat's physics alone, but can be determined quite accurately by few measurements across the parameters' space, makes the above formula very accurate.
However, even an average eta, a constant, usually provides a rough approximation for a given boat for a quite wide range of conditions (range of the eta function parameters) and wind velocities, since eta typically does not change very much  for some water boats a change factor of up to 4 over the entire applicable wind velocity range has been observed  and typically not in order of magnitude. Thus, such average eta is a good characterization of the overall sailboat's performance, and helpful in rough comparison of different boat designs. Furthermore, even a coarse approximation of the eta function (with very few constants) for a given boat, based on few measurements, provides quite good predictions with the formula above.
After solving the equation above for a, the boat velocity VB is given by the well known formula
Comments:
1. The expression VW*eta where eta is a constant is a first approximation to a more complex unknown function with the sailboat's parameters eta1. It indicates the fact that the boat's resistance to speed increase increases with the speed.
2. When solving iteratively for a, a good initial value for a, to converge to the right solution, is slightly below a0. This emulates starting with low boat speed and converging to the final speed.
3. Alternatively, eta can be determined by the equations from VB, VW, and a0 (or, on board, from measured VB, VA = Apparent_wind_speed, and a, where now a0 is solved from the equation above, and VW is calculated by
VW := SQRT( VA*VA + VB*VB  2*VA*VB*cos(a) ) ).
The following is an example of resulting VB Vs. a0 plot for eta = 0.001 at wind speeds of 10, 20, and 30 mph, characterizing a very fast water sailboat (eta is close to that of the water speed record holder; see below):
An example of VB Vs. a0 polar diagrams resulting from the formulas above. eta = 0.001 (a land or very fast water sailboat) at 10, 20, and 30 mph winds. Close to 180 degrees the sail physics changes, and lack of numerical convergence is seen. It is also quite different with spinnaker at broad reach, which results in less accurate approximation with spinnaker, especially close to 180. In the range 0 to ~30 air fluctuations typically cause luffing in soft, conventional sails, and sail effectiveness diminishes.
This solution is a good approximation for polar diagrams given in the picture above at the top and alike, where every boat type with a given setting is fitted with a specific eta. It is applicable to both traditional flexible sails, stiff sails, and vertical wings. This solution together with rough eta approximations also provides a compact implementation for accurate Velocity Prediction Programs (VPPs) which become more and more common on board sailing boats, for tactical planning (e.g., for calculating the fastest path to destination given wind geographical distribution; the simplest well known calculation that uses polar diagrams is choosing an optimal a0 (fastest VMG) and gibing/tacking for a destination downwind/upwind in a wind with constant velocity along the path), and for boat design. For boats with a relatively heavy keel (e.g., yachts) VB is quite uniquely determined by VW and a0 in typical sea conditions and optimal tuning, since any deviations from the assumed averages of mechanical forces and moments effects created by the crew are typically negligeable. Thus a single polar chart for multiple VW values provides a good description of such boat's behavior. See a detailed example (polar charts) in the Polar diagram for sailboat speed page.
 Due to its slow and relatively small change eta can be approximated for a given sailing boat compactly (with few constants) and accurately across the entire parameter space by few measurements of VW, VB, and either a0 or a: After calculating eta for each measurement, eta can be approximated accurately across the parameter space from the accurate tuples [a0,VW,eta,any_desired_parameters]. In this form it is an explicit function eta(VW,a0,any_desired_parameters) where VWmin<VW<VWmax, and 0<a0<180. Then VB(VW,a0,any_desired_parameters) is calculated across the parameter space using the formulas above. This results in a considerably better compact approximation of VB, with less measurements needed than for approximating VB directly at the same quality.
 These formulas also provide simple good predictions of sailing boat's behavior change from existing current behavior when wind speed and direction change. From measured VB, and VW (or apparent wind speed on board) and a0 (or a on board) the current eta is calculated. Then this eta is used with the expected new VW and a0 to calculate the new VB.
For VW = 10 mph, and a0 = 1 radian ~= 57 degrees, the following results are calculated using the formulas above to demonstrate how eta affects the relationship between wind speed and boat speed:
eta 
0.1

0.01 
0.001 
0.0001 
0.00001 
VB 
4.01 
10.28 
24.21 
54.31 
119.15 
VB/VW (boat/wind) 
0.4 
1 
2.4 
5.4 
11.9 
Ranges of eta for various sailing boat types seem to be as follows:
Water boats Land boats Ice boats
eta (1/mph) .5  .0005 .005  .0001 .0005  .00005
Using the method  A detailed example
The sailboat speed (VB) prediction method based on eta approximation is demonstrated with the Aerodyne 43 yacht class in the Polar diagrams for sailboat speed page.
Best VMG  Zigzag both upwind and downwind (see equations at Sailing  Best VMG )
An exact formula for optimal VMGs, up and down wind, has been derived from the formulas above. It is assumed that eta is a constant, or at most very little changing in and around these points. The optimal VMGs are calculated by finding the points in the polar diagrams above at the top where the tangent line is horizontal. If VB is described in a Cartesian system of coordinates where x is the horizontal axis, then dVB/dx=0 exists at these points (the quite involved resulting equation that should be solved simultaneously together with the equation at the top (for both unknowns a0 and a) is given here). The solutions provide numerical results as follows:
VW * eta 
0.1 
0.01 
0.001 
0.0001 
VMG upwind a0 
43.8 
41.9 
40.6 
39.9 
a 
23.7 
13.7 
7.1 
3.5 
VB/VW 
0.85 
2 
4.49 
9.85 
Best_VMG/VW 
0.61 
1.49 
3.41 
7.56 
VMG downwind a0 
153.5 
145.1 
142.5 
141.5 
a 
60.8 
21.1 
8.6 
3.8 
VB/VW 
1.14 
2.31 
4.8 
10.16 
Best_VMG/VW 
1.02 
1.89 
3.81 
7.95 
Thus for best VMG, picking a0=~4445 degrees upwind, and a0=153155 downwind seem to be the best choices for most keel boats in all their applicable winds (and knowing accurate etas in these directions determines a0 very accurately). These results are very close to values found on polar diagrams of yachts. For land and ice boats the a0 values for best VMG are a little smaller (~40 and ~143). Very fast water boats are inbetween (~4143 and ~144153). When a spinnaker is used deviations occur downwind, typically pushing the best VMG angle towards the 180 side as wind speed increases.
Water sailboats
Racing in fog
From The Physics of Sailing , by Bryon D. Anderson, physicstoday, February 2008
See also the book The Physics of Sailing Explained, by Bryon D. Anderson, Sheridan House Inc., 2003.
Comment: The above approximation is inaccurate for a displacement hull (with no planing capabilities) in speeds close to and above the hull speed (which are typically out of the range of polar diagrams). I.e., it should be used only within a circle around the origin in the diagram, with a radius that is a little smaller than the hull speed.
The following calculations (using the formulas above and constant etas) roughly emulate a Tornado (eta = 0.008)
and a 18ft Skiff (eta = 0.003):

VW > 35 for Tornado and Skiff is quite impractical; it is shown for formulas demonstration.
At higher wind speeds the speed amplification (boat/wind speeds ratio) reduces due to increased drag.
For boats with a jib typically eta increases slightly with a0 (speed decreases) for back winds, due to jib reduced efficiency and contributing interaction with mainsail. Also wind turbulence initiation and increase, as a is getting relatively large, increases eta (a0 is getting closer to 180; this turbulence can be observed with telltails on the main's leech). Thus picking an average eta, the above results are slightly overoptimistic for back winds.
In October 1993 the speed record for water was broken by Yellow Pages Endeavor (YPE) which reached ~52 mph at wind of ~21 mph (resulting eta ~= 0.0008 at optimal wind direction and sail setting). Since then the record was broken in 2004 by a windsurfer and in October 2008 by a kitesurfer who at winds of ~45 mph reached speeds of ~56 mph.
Yellow Pages Endeavor (YPE). Contenders are: Macquariespeedsailing
thus expected here to pass the ~56 mph record
at wind above ~21.5 mph),
and Sailrocket See a December 3, 2008 video: "Lift" reclaims its original meaning...
(Peaked 58.33 mph at 26.2 wind; eta = .00086 if reached steadystate, and consistent with previous eta calculations. Unfortunately flew in air, capsized, and probably not qualified yet for official speed record along 500m distance)
Ice and land sailboats
Ice and land sailboats/vehicles typically operate at higher speeds than water sailboat, and the aerodynamic drag becomes the bigger factor. Also typically the travel medium (ice or land) drag is much smaller than that of water (and the typical hulllength effect does not exist). These facts and the less complicated interaction between the boat and the travelmedium (here travelsurface) make the boat transform above even more accurate for ice and land.
Modern iceboats are documented to reach maximum
amplification of wind speed of factor 810 at light winds. At
1520 mph winds they reach their top
reasonably measured and documented speeds of 8090 mph. Based on these
data iceboats have very low drag coefficient and eta (< 0.0001), so
theoretically (by the above formulas) they can pass the 200 mph barrier
in 4060 mph winds. Though I have no detailed ice yachts speed data,
eta seems to hardly change near the optimal truewind direction for
such yachts (85 < a0 < 115) with a singlesail (or a single
vertical wing), with good aerodynamics (of both body and sail/vertical
wing; low drag and turbulence at highspeed apparent winds are crucial
at such speeds) and proper blades on a uniform ice surface at a range
of temperatures. In this case, with a properly measured eta, the above
formulas predictions are expected to be quite accurate.
The following calculations (using the formulas above) approximately emulate an Ice yacht (eta = 0.00005):
0.00005 < eta < 0.00010, which imply reduced speeds, but at the same range of magnitude, seem to be more realistic etas for ice yachts, but still, speculative.
Thus, at this parameter range any drag coefficient (and eta) reduction is critical to breaking the 200 mph barrier. Windtunnel experiments with a scaled down model and turbulence visualization at ~200 mph equivalent and respective apparent wind flow directions (a = ~ 15 degrees) are desired for designing a proper vehicle.
The
above also suggests that at these speeds when the real wind direction
is a0 = ~90 degrees, and apparent wind = ~15 degrees, if the blades'
direction is 0, then the aerodynamic fuselage should point to ~15
degrees (against apparent wind for minimum fuselage drag), and
the aerodynamic vertical wing direction (of a certain well defined
average plane in the wing) should be set to ~7.5 degrees from the
blades' direction for maximal drive in the blades' direction. At this
positioning the total force measured on a successful vehicle (including
a correction for the blades' drag/friction), in the direction of the
blades, should be 0.
The Debutante
"The most spectacular claim is that by John D. Buckstaff, who in 1938, apparently clocked 143 mph (230 km/h), in a 72 mph wind on Lake Winnebago in Wisconsin (USA). His craft was a stern steerer "Debutante", pictured above. Little is known about how it was timed or who witnessed it. The consensus amongst most modern ice sailors is that this sort of speed would have been impossible in such a craft."
Comment: Resulting eta ~=.00045 at optimal setting looks reasonable for such vehicle
See there also The windjet project, Reality and folklore, Ice record competitors and Windjet Videos
Miss Wisconsin Insanity (see linked page bottom)
The latest Windjet land craft. Looks like a sail boat?
Miss Wisconsin on land Windjet ice craft
Greenbird land craft. Greenbird ice craft
August 25, 2008
Attempts to break the world record for land vehicle are now underway
by Greenbird at the salty Lake Lefroy in West Australia.
Published data, 90 mph at 20  25 mph wind for the land craft, imply (at optimum) .00015 < eta < .0002.
If eta does not change much (e.g., due to travel surface moisture and texture change, increased turbulence and
friction coefficients due to increased speed) expected speed (optimal a0 ~=98; a ~=16)
at 30 mph wind is 104  115 mph.
At 40 mph wind it is 125  140 mph.
At 50 mph wind it is 140  160 mph.
To break the current record of 116.7 mph by Bob Schumacher in the Iron Duck on March 20, 1999
(at probable 25  30 mph wind (see link below) implying .0001 < eta < .00014 at optimal setting)
the expected needed wind for such Greenbird's etas (.00015 to .00020) is above 31  36 mph.
September 4, 2008
Attempts stopped (never really started) due to surface flooding by rains and lack of proper winds.
Good luck on the next attempts!
March 26, 2009
The Greenbird broke the world record on land (confirmed), measuring average speed of 126.19 mph over 3 seconds and 500+ feet distance at wind speeds of 3040 mph with measured peak gust of 47 mph (an average between 35 to 40 mph looks a fair estimate). The Greenbird was operated by Richard Jenkins, the main force behind the project, on Ivanpah Dry Lake, California.
Congratulations!!!
See more, including a NALSA approval report, in http://www.greenbird.co.uk/ .
The result is well within the predictions above (though maybe some vehicle modifications affecting eta have been made since the predictions, and travel surface conditions, also possibly affecting eta, have changed)
Vestas Sailrocket 2 water recods
Thu, 26 Jun 14
VestasSailrocket 2 World record holder 65.45 knots/500m
On the 16/11/12 VESTAS Sailrocket 2 set out on it's 13th run for this
session Walvis Bay Namibia. In moderate conditions with around 25 knots
of wind average... VSR2 smashed the previous Outright world record by
the biggest margin in the records history*. Paul larsen averaged 59.23
knots over the record 500
https://www.youtube.com/watch?v=wnjyusAgk8I#t=167
Records:
The 500m Record
Year  Boat  Speed 

2012  Vestas Sailrocket 2 
65.45 kts 
2012  Vestas Sailrocket 2 
59.23 kts 
2010  Kiteboard  55.65 kts 
2010  Kiteboard  55.49 kts 
2010  Kiteboard  54.10 kts 
2009  Hydroptere  51.36 kts 
2008  Kiteboard  50.57 kts 
2008  Kiteboard  50.26 kts 
2008  Kiteboard  49.84 kts 
2008  Windsurfer  49.09 kts 
2005  Windsurfer  48.70 kts 
2004  Windsurfer  46.82 kts 
1993  Yellow Pages  46.52 kts 
...  
1980  Crossbow II  36.00 kts 
...  
1972  Crossbow  26.30 kts 
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