# Thoughts of resistance and leeway

Discussion in 'Hydrodynamics and Aerodynamics' started by TeddyDiver, Aug 10, 2014.

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### TeddyDiverGollywobbler

Bit of brainstorming about the added resistance of a sailboat when making leeway.
Haven't found anything written except occasional comments about it, nothing from Marchaj and Fossatti mentions it briefly. Davidson did studies 80yrs ago but I haven't found anything from him.
Anyway this is how I figured this:
We have longitudinal and transversal components of resistance. The longitudinal component we wan't to keep as low as possible. Transversal component is equal to transversal aerodynamic forces subtracted with the transversal lifting forces of the submerged appendices (keel and rudder).
My conclusion is to have as much transversal resistance as possible without increase of the longitudinal component. IE the equlibirum of the tranversal forces with a minimum of leeway.
I know there are flaw's in my thinking but please correct me

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### baeckmoHydrodynamics

This boils down to the maximum available lift/drag ratio in the horizontal plane of the combination of the wetted shapes, ie hull plus fin/keel plus rudder etc? Or do I misunderstand your q?

If I recall Marchaj correctly, he has devoted quite some text to the different L/D characteristics of varying aspect ratios under extreme loads (in particular the critical stalling sequence). The Fastnet disaster was used for exemplifying.

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### TeddyDiverGollywobbler

I'm thinking if there's some advantage to increase wave making resistance transversaly (towing sideways). Less beam and deeper draft instead of more "trendy" hull forms which have practicly no transverse plane to speak about.

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### RurudyneSenior Member

That sounds like, in the more extreme forms, it would tend to produce either a plank on an edge design -- though perchance one with a fully cut away forefoot and a rudder separated from the keel (or at least partly so) -- or a fine bowed scow with a relatively long external keel (I believe one of the America's Cup boats had such a hull).

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### philSweetSenior Member

Teddy, your head is stuck in the wrong frame of reference. In order to balance forces and moments on a boat, you need an inertial reference frame. This is usually taken as earth location. There is a velocity field of air relative to the earth, and perhaps a velocity field of the water as well.

The boat's course, not its heading, is the important reference axis in earth coordinates. To describe a boat's state, you can use Euler angles. Conventionally, You start with the boat level with heading = course. You then roll the hull, then pitch the hull, then yaw the hull to get the boat in its actual state. (this corresponds to the usual x,y,z axes, taken in order, with positive x going from y towards z, positive y going from z towards x, etc.) You now have hull axes.

Next you have to describe velocities. There are three translation and three rotation velocities. Most dynamic (ie inertial, acceleration, stability) calculations have to be carried out in the inertial frame, but the forces are generated based on the hull and appendage geometry which are defined in their respective body axes.

So what you do is transform the air and water velocity fields into the body axis system, calculate the forces for each component in each of their body coordinates (for instance, a rudder will have it's own coordinates), then transform all the forces back into the inertial frame. There they can be summed and accelerations computed, six new velocities calculated, and the process repeated.

There really aren't any good shortcuts for a new design. From a designer's standpoint, leeway is a design option. It isn't good or bad, It's just something to play with. It's one of three Euler angles that describe the hull's orientation. It has a direct bearing on jib sheeting angles on a stayed rig and it effects the aerodynamic forces on the hull as well as the hydrodynamic forces.

As far as wavemaking is concerned, wavemaking is bad. Its very nature is such that it only applies a force in one direction - opposite to velocity. You can't increase sideways wavemaking. It's just the waves the boat is making as it travels on its course in a particular hull orientation (which is described by a state vector which lists the value of all the things that determine a vessel's performance).

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### TeddyDiverGollywobbler

Yup! Thou didn't didn't think so extreme as plank on edge, but that's a good comparison.
To go a bit further wonder if there would some relevance btw towing tank testing both longitudinal and transversal resistance separately vs real world sailing with leeway.

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### TeddyDiverGollywobbler

My point is to question if wavemaking transversally is actually a good thing. Thinking about a boat adrift for example. Adding velocity forward bit by bit. Is there a point when tranversial resistance becomes a liability in the big picture.

I'm not trying to invent anything, just some excise for the grey matter..

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### RurudyneSenior Member

Per one graph I've seen the benefits of narrowness for displacement hulls start in the 5:1 neighborhood, and improve thereafter.

In a 40' LWL boat having an 8' beam is more comfortable inside than a 6'8" beam to be sure.

But there appears to be a trade off between beam and draft that probably really hurt those planks on edge, in that it seems like the lower the beam to draft ratio the worse the wave making. Now, to be sure, this probably has less effect if the keel really is, as in modern practice, much narrower than the hull. There is, on page 127 (the article starts on p117), of the article I'll link to hereafter visual representation of wave making for different set ups and you'll see that higher BDR coincides with reduced wave making.

This has led me to wonder, for example: contrary to what seems normal for stabilized monohull designs, if a power boat with sorta narrow hull (5:1) that has shallow draft wouldn't achieve low wave resistance and be better for cruising at moderate speeds and provide greater ease of laying out the interior (for reason of a multitude of weak knees and also backs in my family a lot of steps would be annoying).

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### TeddyDiverGollywobbler

Thanks Rurydyne, interesting paper. There might be something more to achieve in this regard with sailing multihulls than mono's. That is of course if the tranverse resistance works as imagined in the OP.

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### philSweetSenior Member

I've been wondering how to respond to this statement (and hoping somebody else would).

Waves are related to pressures, once free of their source, they are just a pressure field that is felt in all directions. They can carry momentum with them, or not. The solution to the wave equation in any odd dimensioned domain is an expanding shell in which all the energy and momentum is contained within that shell and the interior is quiescent. Think of a shock wave following an airburst explosion. The solution to the wave equation in any even dimensioned domain such as a gravitational free surface involves Bessel functions and the energy and momentum remain distributed throughout the region within the ever expanding wave front. The waves you see are a history of earlier pressures on the hull. They represent both a time and space integral of forces on the hull.

If you increase the lateral resistance of the boat by causing it to produce larger waves when it is towed sideways, two arguments come to mind.

1. You are treating the lateral and longitudinal results as if they were linear and independent and can be manipulated like vectors. Waves don't work like that. You have to combine them using something like Michell's intergral. Waves don't decompose conveniently in force vector components.

2. Even if you could decompose a wave into spacial vector components, adding resistance, no matter how you do it, is going to slow the boat down. That means the leeway angle is going to increase if the sway velocity is constant. And if the speed is decreased, the sway velocity may even have to increase in order to maintain a fixed sideforce.

I know these have been roundabout ways of addressing your original question, but the trouble is the question itself, and the confounded physics that led to it.

Again, forces usually can be calculated in the body frame after the flow field has been transformed into those coordinates. Then those force components are walked back to the inertial frame and cached. When you have all the force components from all the different bodies cached in the inertial frame, then you can sum forces and calculate moments, apply mass and inertia (inertia has to be transformed into the inertial frame since it is calculated in body frame), and calculate accelerations. You can do this for a boat that has roll, pitch, and yaw velocities (and accelerations)as well as translation velocities. This is the approach that works. The computer doesn't care how tedious and repetitive it is. Hopefully, reading this will help orient your intuition a bit.

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### Leo LazauskasSenior Member

I agree, in general.

The velocity potential and velocity components aren't too difficult to
calculate in thin-ship, linear wave theory. For example, here are some plots
of the y-wise and z-wise perturbation velocities on z=0 for a catamaran.

http://www.boatdesign.net/forums/at...91237d1400320783t-flotilla-6-2-released-v.png
http://www.boatdesign.net/forums/at...91236d1400320773t-flotilla-6-2-released-z.png

The x-wise velocity is just the wave elevation multiplied by -g/U, where U is
ship speed.

For multihulls there is the added complication that the waves from one hull
impact on BL development on the other hull and that can also increase
resistance (and squat).

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### tspeerSenior Member

Oh boy. Your post mixes up a lot of different concepts! It is true that leeway is something that is picked by the designer - the smaller the area of the keel/board, the greater the leeway will be. It's also true that leeway, by itself, does not hurt performance at the design stage If the keel/board has excess area, then increasing leeway by trimming the chord of the board can improve performance through reduced wetted area. However leeway is not an Euler angle.

Euler angles are used to represent the orientation of the hull's attitude. The standard Euler angles, going from a North-East-Down (NED) local level Earth axis system to the body axes are (in this order), heading, pitch, and roll. Heading is the rotation about the down axis system, positive when the bow moves to starboard. Pitch is the rotation about a transverse horizontal axis that is rotated from East by the heading angle, positive bow up. Roll is rotation about the boat's longitudinal axis, positive when rolling to starboard.

Leeway is the angle between the boat's heading and its course through the water. Leeway is not an Euler angle, however. Leeway can change without any rotation of the boat's heading at all. You've probably experienced that very thing when you maintained heading on a close-hauled direction and stalled out in a bad tack. The boat sagged to leeward, with a large leeway angle, and then started to make forward progress with the leeway angle reducing as the boat picked up speed.

Actually, the most convenient axis system for performance analysis is not an inertial axis system, but a velocity-aligned axis system. The reason is because forces tangent to the velocity vector change the speed of the boat but not the direction, while forces at right angles to the velocity vector change the direction but not the speed. Lift and drag are force components in the velocity axis system. It doesn't matter what the heading of the boat is - if the hydrodynamic and aerodynamic lift and drag components are the same, the performance will be the same.

If apparent wind angle, beta, is measured between the boat's course through the water and the apparent wind vector (instead of between the boat's centerline and the apparent wind vector), the fundamental sailing performance relationship is:

Vb = Vt*sin(gamma-beta)/sin(beta)

Vb = boat speed through the water
Vt = true wind speed relative to the water
gamma = boat's point of sail (course relative to the true wind), with gamma=0 being head to wind
beta = apparent wind angle measured between the course through the water and the apparent wind vector

This expression is nothing more than the law of sines applied to the wind triangle, and is an exact relationship that holds for all times and all boats. This simple relationship is why a velocity axis system is so useful for performance analysis.

The hydrodynamic lift is defined to be perpendicular to the yacht's course and the hydrodynamic drag is defined to be tangent to the yacht's course. The aerodynamic lift is defined to be perpendicular to the apparent wind direction and the aerodynamic drag is parallel to the apparent wind direction. Note that while the lift and drag may be affected by the boat's heading, they aren't aligned with the boat's heading.

Some basic geometry will show that beta is the sum of the aerodynamic "drag angle", arctan(aero drag/aero lift), and the hydrodynamic drag angle, arctan(hydro drag/hydro lift). The smaller beta is, the faster the boat will go, because of the sin(beta) term in the denominator. You make beta small by increasing the lift/drag ratio.

Leeway does locate the boat relative to the velocity vector. And the hydrodynamic lift and drag are dependent on the leeway angle. Leeway is important. But, if it weren't for leeway's impact on the lift and drag, it could be anything at all and the boat could be pointed in any direction. So leeway's influence is indirect.

If you want to analyze the boat's dynamics, then it is most convenient to work in body axes and inertial reference frames come into play.

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### philSweetSenior Member

It's early and will take me a while to work thorough your post, But I think we are talking past each other in some areas. You didn't actually say how you calculate leeway so I'm left to guess a bit.
Leeway is the angle between a PROJECTION of the boat's heading onto the surface plane (earth xy plane in my description). There are two reasonable choices of projection, either vertically in the plane defined by the boat's x axis and the earth's z axis, or "vertically" in the plane of the boat's x axis and the boat's z axis. I prefer the first one for calculation purposes, but I understand there are reasons to like the second one, and I believe that is what the lubber's line on a binnacle compass would do. The second point isn't a problem for the way I described it. If course changes and heading doesn't, leeway changes. I should have said leeway angle, but I used sway to indicate the velocity.

I'll dig through the rest tonight, I have a 12 hour day ahead of me.

<edit> It should read a "projection of the boat's longitudinal axis onto the surface plane". Heading is this projection.

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### philSweetSenior Member

Okay, I'll try to patch things up here.

I agree that leeway is not a Euler angle. Euler angle would be from North to heading, and leeway would be from course to heading. I may be forgiven for this because when I worked in the missile business, the earth frame was physically aligned downrange. That's how the instrument package sat in the silo. There wasn't any choice in the matter.

Regarding Roll, pitch, yaw, or yaw, pitch, roll - We are doing the same thing mathematically, I was using an extrinsic progression, you are using an intrinsic progression. This is quoted from wiki on Euler angles - "Any extrinsic rotation is equivalent to an intrinsic rotation by the same angles but with inverted order of elemental rotations, and vice-versa. For instance, the intrinsic rotations x-y’-z″ by angles α, β, γ are equivalent to the extrinsic rotations z-y-x by angles γ, β, α. Both are represented by a matrix." I find the extrinsic progression easier to visualize. The computation is identical.

Regarding velocity aligned axis system, I am guessing that you are referring to a further shift in axes once the flow vectors have been transformed to body coordinates. At that point, one can calculate an angle of attack and a sideslip angle of the flow relative to the body. Then you can run the lift and drag calcs relative to the flow by a further transform by either just the angle of attack, or by both AoA and Sideslip. I'm not sure what the difference between the two is as long as do the calcs correctly in the one you choose. The reason for using those axes is that supposedly, it lowers the H.O.T. errors in some of the linearized terms. I haven't looked into the details of how.

The last part of your post is probably in response to my quirky way of treating a keel's induced drag as an aero force. It doesn't change the important features of the diagram you posted, just some details of it. My diagram would have the induced hydro drag subtracted from the total hydro vector and added to the total aero vector. The result is the the two remain equal and opposite, but they are rotated beamwise somewhat - the sideforce to drive appears greater in my diagram. Again, this is purely for the convenience of my intuition, the velocity prediction doesn't change. What changes is how one estimates the effect of a change in RM, or cost, or weight, based on looking at the diagram. It provides a different view of the sensitivity of the craft to changes in sideforce, which is usually the biggest force component acting on a sailboat. I would argue that there is a minor physical justification for doing this as well. When looking at something like gust response, the forces that depend on the air velocity change quickly, but the forces that depend on the hydro velocity change more slowly. My way collects all force terms that depend on air velocity because it is subject to faster rates of change. This is true for slow leadmines like mine, but probably not true for very fast craft like the AC72. If I were trying to sort out a bearaway on a fast catamaran, I guess I would do things the other way around and collect all the terms that are functions of water velocity.

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### jlcongerJunior Member

Leeway approximation for keel boat

On a related subject, I found this approximation to estimate the leeway angle for a keel boat:

Leeway Angle = K * heel Angle / boat speed ^2

Where K is a boat-specific value (around 10) if speed is in knots and the leeway angle is in degrees.

I found this reference online:

http://sailboatinstruments.blogspot.com/2011/02/leeway-calibration.html

but no other information or derivation. Does anyone have a reference, derivation, or something similar that can be used when doing VMG calculations?

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