Quote:
Spuuut referenced some of your work in a prior post (at least I think it’s you.)
I’d like to understand the treatment that is being applied here (to hydrofoils.) Please, if you’re inclined, direct me to a reference, post, or URL of yours that explains your treatment? Also, is what I’ve diagrammed in figure 1, a reasonable start? And, are you willing to deal with a few more questions?
Thanks,
Kevin
(Edit: 09/21/06, Included legend for figure 1, KC)
Hi Kevin,
I’m not aware of anywhere on the web that I have discussed anything other than snippets of my treatment(s) of the motion of conventional or hydrofoil wave-craft on a wave. However (like the construction of my next version of my hydrofoil paipo board but with a lower priority) that is one of the projects that I am hoping to work on this coming winter.
In the meantime, and in reference to your figure…
Imagine looking down and from a fixed location videoing a surfer riding on a wave and who remains at the same position relative to the curl as the wave moves toward shore. You will find that his path (over the bottom) traces out a diagonal line consisting of the onshore and alongshore components of his motion. I’ll refer to his track as the “pathline” of his motion. Observation indicates that such a surfer, in trim, generates a path angle (the angle between his pathline and an axis paralleling the wave crest) that is typically between 30+ degrees and 45 degrees.
Now let’s look as some slopes. The slope of the face of the wave measured in a vertical plane perpendicular to the crest of the wave will depend on one’s location on the wave face. Observations suggest that for a surfer in steady-state trim, this slope will be about 1:1 (or 45 degrees). Just as a skier skiing diagonally across a ski slope experiences a slope that is less than if he went straight downhill, so too does the surfer experience a reduced slope along his pathline compared with the slope of the wave face. If the slope of the wave face at the surfer’s location on the wave is “Sw”, and the angle of his pathline relative to the crest of the wave is “theta”, then to a first approximation the slope along his pathline, “Sp”, will be:
Sp = Sw x sin(theta)
This slope directly relates to the force driving the surfer and board.
At the same time, there will be a transverse slope between the left and right sides of the board. To a first approximation, this slope, “St” will be:
St = Sw x cos(theta)
This slope can be important in determining the aspect ratio of the surfboard ( aspect ratio = max width of the wetted hull divided by the average wetted length of the hull). The aspect ratio plays an important role in determing the efficiency of a planing hull.
To see this, consider a rectangular flat plate measuring 6" in one dimension and 6’ in the other direction. If the plate is moved across the water with some non-zero angle-of-attack (AOA), water moving toward the plate will be deflected downward as it passes under the plate. Since this imparts a downward momentum to this water, the water (which had no downward momentum at the upstream end), by Newton’s Second Law (Force = time rate of change of momentum) and Third Law (equal and opposite reaction), there is a upward pressure force exerted on the plate (and perpendicular to its plane).
Since the pressure force is exerted normal to the bottom of the plate, and the plate is inclined relative to the horizontal (by the AOA), the pressure force has both a vertical and a horizontal component. The component perpendicular to the motion of the craft (and the sea surface) is the lift force; the component perpendicular to the lift force and directed downstream is the induced drag force.
FL = P x cos(AOA) (lift force)
FD = P x sin(AOA) (induced drag force)
…where the magnitude of P (the pressure force) is also proportional to the AOA.
The larger the AOA (within limits), the greater the pressure force, lift force, and the greater the induced drag.
However, not all the water moving toward the inclined plate need be deflected downward. Instead it can flow off to one side of the plate or the other instead. Since less downward momentum is now being imparted to the water approaching the plate, the pressure force exerted on the plate is diminished. In order to compensate (since some specific lift force is required to support the plate–e.g. the combined weight of the rider and board), the AOA of the plate must be increased to provide the required lift force. This, in turn, increases the incline of the pressure force, thus increasing the induced drag.
Obviously if the plate were now rotated 90 degrees, so that the distance between the upstream and downstream wetted edges were 6", instead of 6’, and the span of the plate perpendicular to the motion were 6’ instead of 6", a much greater percentage of the water approaching the plate would pass under it and not around it. Hence the downward momentum transmitted to the water approaching the plate would be increased, and the induced drag would be reduced in comparison with the first example (but still not as low as if the width of the plate were even greater (e.g. infinity).
If the aspect ratio is “AR”, then the reduction factor for the lift created by a plate with a finite wetted width “ARF”, relative to a width approaching infinity (but with the same wetted area) is approximately given by:
ARF = (AR)/(2 + square-root(4 + AR x AR))
Let’s get an idea of how big this factor might be for a surfboard by assuming that the wetted area when traversing across the face of a wave is approximated by a triangle with a max wetted width of 18 inches and an average wetted length of 24 inches (4’ max wetted length). The aspect ratio is then 18/24 = 0.75 and the aspect ratio factor is 0.18. Hence for a given wetted area, the AOA of the surfboard would have to be about 5.5 times greater than for the same wetted area of a very wide, but very short, shape. Thus the surfboard would have a high induced drag compared with a (very) wide board.
A possible example of the effect of width on induced drag is the speed potential of a fish, which is typically much wider than say the typical thruster for the same length.
But there is a complicating factor in that to take full advantage of the lower drag associated with a wide board, that width has to be wetted (not counting spray). But if one rolls the board toward shore so as to match the transverse slope angle (along the board’s pathline) in order to wet the full width, the pressure vector now also has a component perpendicular to the pathline and directed somewhat toward shore. This reduces the magnitude of the lift component (by the cosine of the transverse slope angle) and, if not balanced in some other way, will accelerate the rider and board away from the face of the wave and toward shore (resulting in him progressively getting lower on the wave). Lower on the wave means less wave face slope, which in turn means less pathline slope, which in turn means a reduced driving force. These opposing considerations are probably (intuitively) one reason for the limited banking of the typical surfboard in roll when traversing across the face of a wave. The steeper the wave face slope, the greater the transverse slope, and the lower the aspect ratio of the wetted area of the surfboard. Too steep a slope and the reduction in hydrodynamic efficiency associated with aspect ratio more than offsets the increased driving force otherwise associated with an increased wave face slope. This balance and trade-off may relate to where skilled surfers choose to trim for maximum speed on the face of a wave.
Now…as to a hydrofoil…
A typical hydrofoil is characterized by a chord (leading edge to trailing edge) dimension that is substantially smaller than it’s span. So the aspect ratio is increased, and the induced drag can be substantially reduced. In addition, for a fully submerged foil, there is no need to bank the board in roll to increase the wetted span (it’s already wet). Once again this decreases the induced drag relative to that of a typical surfboard. This is part of the motivation for choosing a fully-submerged main foil on my hydrofoil paipo board.
However, that’s only part of the total equation. Nothing has been said about parasitic drag. Within limits a conventional surfboard can, to some extent, be trimmed to vary the wetted area and hence the friction (parasitic) drag. In contrast, the wetted area of a fully-submerged foil (and the wetted area of the supporting struts) is fixed…and what is optimal at some speed, may be excessive at some faster speed. Hence, depending on how the hydrofoil is “optimized” as to function (e.g. speed vs maneuverability) it is possible that the hydrofoil will be the more efficient at some speeds, while at other speeds the planing hull of a conventional surfboard may be the more efficient.
A surface piercing foil lies somewhere inbetween. It’s lift generating capability is less than that of an equivalent fully-submerged foil due to ventilation, wave-generation, and reduction in pressure force when within two chord lengths of the sea surface (but may still be improved over the typical planing hull). But they can also be designed so that the wetted area decreases as speed increases, thus potentially reducing the parasitic drag relative to a fully-submerged foil.
Since my design goal was enhanced maneuverability (which includes carrying more speed through a maneuver), the greater efficiency (with regard to induced drag) of a fully-submerged foil, relative to that of a surface-piercing foil, is the motivation for the main foil on my hydrofoil paipo board supporting 90-100 percent, and the canard foil 0-10 percent, of the combined weight of the rider and board.