An Explanation of Aspect Ratio and how it affects sailingbyIvor G. Slater, P. Eng. |
To begin, let's establish
some basics. First is the idea of Apparent Wind (Wa). Most sailors will
be familiar with the idea that if True Wind is south 10 knots, and you
are sailing at Boat velocity (Vb) 10 knots East, then the apparent Wind
(Wa) is from the South East at about 14.1 knots (10 times root 2.) This
is the wind that is driving
the boat. Even though we
will find later that the air flow around the rig has been redirected locally,
all sail forces are referenced to this direction [i.e. the apparent wind
determined by boat velocity (speed in her track) and true wind.] If you
are tracking East, 90 degrees, by making 5 degrees of leeway, we don't
care that your boat's heading is 95 degrees. You are sailing 90 and generating
an apparent wind 45 degrees from that track. That's all that matters at
this stage. Doing it this way will save untold complication and handwringing.
Now we can consider forces
being developed by the apparent wind coming in 45 degrees off the track
ahead.
First, for completeness,
we should note that there is a parasitic drag on the above-water hull and
rigging, pointing in the direction of the port quarter. (We say "parasitic"
because it is not associated with, or caused by, the development of any
useful force.) This drag depends on above-water hull shapes and excrescences,
exposed spars and wire lengths
and diameters and wind speeds at various heights. We don't need to go into
those complications here.
But the idea of a force
at an angle to the boat's track is an important one that we should look
at next. Later, we will use the ideas in several ways.
Please bear in mind that
all numbers for speeds, angles, directions and forces are chosen merely
for convenience of explanation and arithmetic. They should not be interpreted
as representing what some
particular real vessel would
experience.
Suppose for sake of argument
and simple numbers that the parasitic drag is 14.1 pounds. Again, it is
headed down wind. This is of no interest to us directly. We are concerned
with: forces aiding or impeding the boat's motion in her track (not her
heading); and forces across her track that have to be overcome to keep
her on that
track. What we do is resolve
that quartering 14.1-pound force into 10 pounds downtrack (meaning towards
the stern) and 10 pounds across the track. [For anyone not familiar with
the idea of resolving and combining forces, the subject is covered a few
paragraphs down.] The ten-pound downtrack force is a direct charge on the
sails. They must provide that much force in the direction of motion to
overcome it and maintain speed.
Now, what happens to the crosstrack force? It must be overcome by the lateral resistance of the hull and its appendages. In other words, they must generate 10 pounds of hydrodynamic Lift (cross- waterflow force) towards the upwind (starboard) side of the track. Associated with that underwater lift are certain charges, involving lifting area, flow velocity, sectional shapes, and the dreaded Aspect Ratio. Only for completeness, I'll mention that these charges against the lifting force are called Profile Drag, and Induced Drag. We'll see more of them and reach an understanding as we go.
To round out and complete this introduction to the handling of forces, I'll mention that decent hydrodynamic surfaces can develop a very favoUrable (sorry guys, I'm Canadian and that's how I spell) ratio of Lift to Drag in most circumstances. So, the ten-pound side force caused by the parasitic drag of the rig might be overcome by the underwater hydrodynamics of the hull at a cost of 1 pound of drag. So the total demand against sail Thrust of the 14.1-pound quartering parasitic rig drag would be 10 pounds direct, plus the 1-pound hydrodynamic (underwater) cost of the cross-track Lift, for a total of 11 pounds.
Now it's time to see the sail forces and how they resolve into Thrust and Side Force. Facing north, let's get our heads up over our ship and look down on her. Apparently she must be moving, because there is a wake behind her. But we don't see her motion because we are moving with her. We feel a 14.1-knot wind behind our right ears. This is all that we, and she, are aware of. [What is going on underwater is another subject.] Her rig is a single unit that may be composed of several elements. That unit is living as a motionless object in that apparent wind like a test device in a wind tunnel. Intuition suggests correctly that the rig must be developing a downwind force (northwest) that we call Drag. Also, unless our sailmaker has given us cause to hang him, (when we couldn't be sailing east anyway) there is a cross-wind force (northeast) called Lift. Notice that we don't care about the heading of the boat, the angle of the boom(s), or the twist of the sails here. All that matters is that there is a downwind Drag and a cross-wind Lift referenced to the direction of the apparent wind. [We don't call the components of this Drag "parasitic" because they are asssociated with the lifting surfaces and the development of lift.]
Once again, to have any meaning for us, these Lift and Drag forces must be resolved into Thrust along the track, and cross-track forces. To make it easy, let's assume for a moment that Drag is zero. Further, let's assume that the Lift force is 141 pounds. This can be resolved into 100 pounds driving Thrust along the track, and 100 pounds side force that must be resisted by the underbody, and for which a price in underwater drag must be paid.
Now let's assume that the Drag generated by the sail(s), as part of their action in creating the Lift in these conditions is 42 pounds. [With apologies to those familiar with vectors, I'm going to explain them a little now for the benefit of those who are not.] We can combine the Lift and Drag forces to get what is called their Resultant, a single force equal to their combined effect. To do that (I'm doing it on a piece of squared paper as I sit here) do this: >From a beginning (the origin) draw a line off towards the right, representing the track. Number 15 squares towards the right from the origin. Using a separate piece of paper pick off a length equal to 14.1 squares. At a ten-to-one ratio this represents 141 pounds. From the origin, lay this length off at a 45 degree angle upwards to the right. (That is, along the intersections of the squares.) Put an arrow head at the upper right hand end. This is a vector representing the Lift force. Label it if you like. Now, using your separate piece of paper again, mark off a length of 4.2 squares (representing 42 pounds.) Starting from the arrowhead of the Lift vector, lay off your 42 pounds at 45 degrees upwards towards the left (downwind). Put an arrowhead at the upper left end of your Drag vector.
On this sketch that you have just made, lengths to the right from the origin represent Thrust, and distances upwards represent Side Force. You will notice that if you drop straight down from the end of your Lift vector to the base line, the Thrust is 100 pounds (square 10) and the distance from the base line upwards to its arrowhead is the same 100 pounds (as we discussed earlier.)
The line from the origin to the arrowhead end of your Drag vector is the Resultant of Lift and Drag. Put an arrowhead on that line also. To resolve that Sail Force Resultant into vessel Thrust, drop a vertical from the Resultant arrowhead onto the base line. You'll see that Thrust is now about 70 pounds. Going upwards, you'll see that Side force is about 130. These are the forces that interest us. We don't care about the magnitude of the Lift, Drag, or Resultant except as steps in finding Thrust and Side Force. Just as in our discussion of parasitic drag, the aerodynamic Side Force must be overcome by underwater hydrodyanamics for which a price in water drag must be paid, as another charge against the sail Thrust.
Now let's consider what happens when the apparent wind goes 45 degrees abaft the beam, still at 14.1 knots. If you care to plot it, you'll see that at ten knots boat speed, this requires a True Wind of about 22.2 knots, blowing from about 27 degrees off the line astern or 153 degrees clockwise from the line ahead. But those things don't interest us here. As before, the above-water boat is like a stationary object in the apparent wind 45 degrees abaft the track-beam. But there have been some changes.
Assuming the above water profile (of hull and rig) is more or less the same from the quarter as it is from the bow, the parasitic drag is still 14.1 pounds. But now it resolves into 10 pounds of THRUST and ten pounds of Side Force. Using the same assumptions we made before, if the hydrodynamic underbody cost of overcoming the 10 pound side force is still 1 pound, then parasitic wind drag of hull and upper gobbledegook been converted into 9 pounds of net Thrust.
Now what happens to the aerodynamic rig-sail forces? There must always be a Drag. It is pointing downwind (northeast.) Is there a Lift (southeast?) There may or may not be depending on the rig.
If there is no Lift, it is because the rig has "stalled" because what is called the angle of attack is too coarse. Then simple "profile" drag is the only force. Compared to forces that can be developed, it is not terribly large, and it is 45 degrees to port. It resolves into 70 percent Thrust, and 70 percent side force. The latter requires the development of a cross-track underwater Lift to resist it. That has its associated costs of water drag to pare off the already limited 70 percent Thrust.
If there is a Lift, it may combine with a large Drag (much larger than profile drag) to give a very large Resultant that in some circumstances may be nearly straight ahead (mostly Thrust) with very little side (or heeling) force. In that circumstance, there is little need for the hull or its appendages to generate side force, and so there is little development of underwater Induced Drag. (Sorry to throw this Induced Drag bit at you. We'll get into what it is and its measurement soon.0
Apect Ratio, which we'll begin to examine in AR-2, is one of the critical factors in whether the rig will generate Lift in these conditions.
Thought for the day: Have you ever heard talk of someone with a "hot" rig being burned by some rotten old schooner on a reach?