Simple prediction of any sailboat speed at given wind; VMG has been added

The formulas presented at my page

http://yoavraz.googlepages.com/sailingboatspeedvs.windspeed

that can be easily used on any computer on board (and off...) have been augmented with a formula for computing VMG up and down wind. I suspect these VMGs are often more reliable than the VMGs on many polar diagrams for speeds.

Please have a look and comment.

 

could you tell us some more on how you deived the equation linking real and aparent wind direction, wind speed and the term eta. Also how you moved from equation 1 to equation 2?

 

Some elaboration on formulas

farjoe: I see I have to be more prompt with a detailed short paper I intend to attach to my web page and possibly here. Hopefully happens in not too long.

Meanwhile some comments about the formulas.

The main (first) formula results from equating sail-forward-force with drag-force. The first force is derived the usual straightforward way of force decomposition, assuming the original force of wind is perpendicular to a planar sail and approximately proportional to a proper wind speed component squared. Sail at a/2 for max force (approximation). Drag is both from travel medium (water, land, ice) and from air (apparent wind). Proportional to square of relative speeds between boat and media. eta is a combination of the proportion coefficients. The resulting functions of boat velocity for constant etas are close in shape to those of real polar diagrams, and provide the possibility to emulate a boat behavior quite accurately by fitting an eta function for the boat. The eta function is quite flat and can be approximated compactly, without getting into the extremely complicated physics mechanisms that take part in a boat's behavior. Thus it does not provide any insight into a boat's physics but rather a practical tool to predict behavior. It may ease preparing polar diagrams for a given boat.

The speeds formulas result from the vector equation

apparent_velocity = wind_velocity - boat_velocity

and using the sines (2nd equation) and cosines (3rd) formulas for triangles.

 

See more formula details

See

http://yoavraz.googlepages.com/sailboatspeedvs.windspeed

which is pointed to from

http://yoavraz.googlepages.com/sailingboatspeedvs.windspeed

in the comment before the first formula.


Also, see attached resulting speed polar diagram example for eta=0.001 and real wind speeds of 10, 20, and 30 mph. This represents a very fast water sailboat, close in eta to the current speed holder.