Relative Flows
Opening Note: I never got around to issues regarding relative velocity in my propulsion series. So, why bother now? Good question; I have no answer. For those of you with absolutely no appetite for Math, skip to the Summary So Far section, if you’re so inclined.
Consider some stationary frame of reference. In this frame the water particle velocity will be given by,
[indent]u = ux i + uy j[/indent]and surfboard velocity by,
[indent]v = vx i + vy j + vz k[/indent], see figure 1.
As experience by a surfboard moving with velocity v, the water particle velocity will be given by,
[indent]w = wx i + wy j + wz k[/indent] where,
[indent]wx = (ux - vx)
wy = (uy - vy)
wz = - vz[/indent]Consider the special case when the board has the same forward velocity as that of the water particles, [indent] vx = ux [/indent] and isn’t climbing or dropping on the wave, [indent] vy = 0 [/indent] Then we have, [indent]w = uy j - vz k[/indent]See figure 2.
Now v, is generally modeled as the motion of the curl – i.e. where the surfer is most likely to be. Of course he need not be sitting in the curl, or some nearby region, but for modeling this is convenient. If this assumption is made, than it becomes apparent that the faster the curl, the more dominant the role of - vz k possibly becomes. Or, conversely, the slower the curl, the more important the term the other terms possibly become.
It is possible to put this is a form which allows for some quick calculations of angles, particularly the angle at which the flow will be hitting the board bottom, just divide [indent]w = wx i + wy j + wz k[/indent] by |w| = w, which will allow for the calculation of the cosine angles. To keep things simple lets take the no climbing or dropping case. [indent]n = uy/w j - vz/w k[/indent]Where n is a unit vector pointing in the direction of w and here, [indent]w = (uy* uy + vz*vz)[/indent]Then you simple take the arccosine as follows,[indent] arcCos(uy/w)[/indent][indent]arcCos(- vz/w)[/indent] Which will give you the respective angles from the y-axis and z-axis.
Summary So Far?
I plan on plugging in some numbers in my next post. But for now, there are a few things worth pointing out. Firstly, for this special case - of moving with forward water particle speed and no climbing or dropping - the flow as experience by the bottom of the surfboard seems to happen in one plane, the yz-plane. Second there is a upward component, the importance of which seems to depend on how the magnitude and direction of v.
To put this in perspective, here I’ve made the assumption that the board is pointing in the direction of v, which is not vz, vz is just the z-component of v. This of course, need not be the case, but if it is, than the flow interaction with the fins is, in the very least, not which most might visualize it to be. In particular, you’ve also got the upward component smacking the bottom of the board and fins - especially if they’re toe-canted. Worse, that upward component become more important the slower the curl, e.g. the smaller the vz component. But even if the vz is dominating, unless it also dominates or is the largest component of v, you’ve got a nice rail-to-rail kind of flow impacting the bottom and fins.
Though I plan of continuing this is subsequent posts, for now you’ve just got to wonder what toe-cant are doing. I haven’t dealt with changing angle of the bottom or fins with respect to this flow, but its not that hard to visual how a given change in board and fin orientation might change the interaction - (…gets even stranger.)
Am I Correct?
Hell, I’m open to correction. I know this, from a lot of the comments that have been made in this forum, as well as those made on the Net (fin manufacturer sites, fin research sites, various manufacturer sites, etc.) such a treatment seems to be somewhat off the mark. The flow past fins seems to be viewed in a manner similar to the flow past a submerged wing traveling in the opposite direction to the flow, but more or less parallel to it - kind of a classical view of how things might be working. Some authors have given consideration to the angle of attack for toe-cant, but still the flow seems to be seen as coming virtually head on - which is not necessarily the case as presented here. Then again, the case primarily described above is limited -i.e. a trim, or quasi-steady-state sort of case. It’s kind of an understatement, but if you start making turns - things do change.
kc
PS
Somebody please check my Math, and hopefully, reasoning.
Request for Example Data
If anyone has some cant, toe, water particle, or surfboard velocities they like to suggest as examples please let me know. Right now, I’m inclined to go with a range of cant from (off the vertical) 0, 5, 7, 9 degrees (9 degrees being unlikely in practice, but possibly interesting), a range of toe from (off the center-line) 0, 2, 4, 5, 7 degrees (7 degrees being unlikely in practice, but possibly interesting), and as for surfboard velocity (or curl velocity) whatever I can dig up in the archives or literature, same for water particle velocity.
