The major point of this thread is to illustrate the nature of the elliptical paths that water particles follow when a waveform passes. Once understood, then how these moving water particles develop forces can be better appreciated.
The illustrations are highly simplified and if they could be applied to a real wave they might approximate a deep-water wave. The paths of these particles distort dramatically as a wave shoals –i.e. moves into shallower water. As a wave shoals it increases in height, and height is one of the determining factors in the actual path of the water particle. Nevertheless, the ‘picture’ is the same, albeit distorted.
In the illustrations, see figure 1 and 2, the wave is propagating in the direction indicated and a given water particle, indicated by the red dot, is followed in the sequence. Figure 2 is just a more detailed version of the leading portion of the waveform –i.e. the face.
It is understandable that most, when watching waves do not visualize the velocity vectors of water particles. I suspect most see the waveform as a ‘whole’ –i.e. the complete form, and therefore do not see any flow* at all. And if considering the whole form, they would be correct, for the net flow is small in deep water, and though it increases as the wave shoals, and it is not until the wave breaks that you can see any substantial net flow. The key term here is net. Luckily surfers aren’t interested in the net flow, just the flow on the face of the wave.
But water particles with velocity, amount to particles with momentum –i.e. momentum is mass times velocity, and objects that posses momentum can and will apply a force to objects they come in contact with.
Here an interesting exercise regarding something called stagnation pressure. In figure 3a a jet of water (density = 1000 kg/cubic-meter) is impacting on a plate. The velocity of the water is V. The impact of the water on the plate destroys the momentum it had prior to impact. Though a full development of the solution is too involved for purposes here, the force required to destroy the jets momentum is given by, (density)(area of jet)(velocity)(velocity) as indicated in the illustration (see any introductory text in Fluid Dynamics). The average pressure that the plate experiences, is given by (density)(area of jet)(velocity)(velocity)/(area of plate).
This is not exactly what’s happening under a surfboard, the stagnation pressure is sort of an extreme treatment, a useful estimate when dealing the possible requirements of a barrier. In the above illustration the barrier was completely at right angles to the jet. If the barrier was a an angle, then the momentum of the jet would not have been completely destroyed and the final solution would have to take into account the final momentum of the liquid, see figure 3b. All things being equal, the component of the force that would have to be applied parallel to the jet’s initial direction would be less in this case, but you’d now have to apply a component of the force in the upward direction (in the diagram.) In figure 3b, an important point to note is that because the plate is stiff it experiences a force that is normal to its plane of presentation to the jet. So you can control the force that the plate experiences by changing its plane of presentation to the jet.
So, The Wave Is A Jet?
Yes, at least for the moment. It reality is far more complicated (at least mathematically, but perhaps not intuitively), but such a simplistic illustration will hopefully serve to make the nature of propulsion in surfing clear. In figure 4a (see next post) a very small piece of the wave is considered. As you can see, in the very small time interval delta t the wave has moved –i.e. the water has moved, and this is the ‘jet’ if you like, see figure 4b. In figure 4c we put a little bit of the surfboard bottom in front of the jet. The force that this little bit of surfboard exerts on the jet is in red, and the reaction force, the force the little bit of surfboard experiences is in green.
In figure 4c the little bit of bottom is presented perpendicular to the moving water, but again it need not be the case, see Dynamics – The Trim Equation.
Wow, The Wave Is A Jet
No, the above model of a little jet hitting the bottom of surfboard is not the best model, in fact it leaves a whole lot out. For one thing the water that impacts the bottom has to then go somewhere, worse the board moves as a result of the impact, virtually moving across the surface of the ‘jet’. Mathematically modeling this interaction in three dimensions is not an easy task. Luckily, it’s only necessary to appreciate the forces involved and how they are developed and manipulated through bottom presentation, (and of course the presentation of bottom contours.)
Nevertheless, there is the matter of how the flow once it impacts the plate will divide itself. Again, the principles which govern this can be found in an introductory text on Fluid Dynamics, but you probably can sense that how a flow might partition itself given the illustrations in figures 3b, 3c and 3d.
In figure 3c, the treatment is two-dimensional. In the concave illustration, because the water is literally sent off in the opposite direction you will have to provide a greater force parallel to the direction of the jet’s initial flow, hence experiencing a greater reaction force. In the case of a convex or vee (or belly?), all things being the same you will in general have to provide less of a force, hence experiencing less of reaction force, and in this case less of a lift than with a concave. (See, Dynamics – The New Spoon Experiment.)
In figure 3d, I’ve attempted to illustrate what might be happening in three-dimensions. In fact this is really a dangerous drawing for the I put a bit of concave in the rear of this board. (I wouldn’t design a board like this. It’s for illustration purposes.) Here if somehow the board was presenting itself to the flow in the wave as illustrated, because of the concave and rocker the direction of the flow off the board would be beneficial to forward motion, and you’d get a little lift too. Does this actually occur? I suspect to some degree, it depends on a lot, in particular for what conditions the board is being designed for and then on how its being used. In general different solutions exist for different conditions. However, quite often shapers will throw a little vee right at the end of the board, especially on longboards. The vee is nice for turning, but my guess is that’s also a plus for propulsion. A great exercise is to take a peak at the bonzer on Eaton’s page, http://www.eatonsurf.com/Bonzer.htm . Though I tend not to agree with his explanation, I’m inclined to believe under the right conditions the board ‘goes fast.’
The point which I keep coming back to however, is that designing for propulsion is one thing, designing for the motion produced from propulsion is another. Too much concave and you tend to just add ‘fin’ or the rudder function of a fin, same for convex shapes. When and how these cross design considerations will impact the functioning of the design will depend on a lot of different things. For example, in slower or small conditions the popular solution is a fish strategy. I believe the fish strategy tends to be more desirable for it addresses both propulsion and lift through an increase in surface area given the reduced water particle velocity. Of course there is a slew or compromises that can be made.
But in general tampering with the ‘apparent’ or relative flow that is produced will cost you. Perhaps you might think of it this way. You’re in a motorboat zooming along. You look over the side and see all this water rushing by. You get the bright idea that if you put a propeller into the rushing water, you could use it to turn a generator, generating some electricity as a result. You figure you could then use the electricity generated to propel the boat! Try it, if it works the world will surely beat a path to your door. When it comes to contours or other design elements that address the relative motion, the question is whether or not the cost is worth it.
Disclaimer
Nothing has been proved. Nothing has been disproved.
Some Notes For Modelers
The motion of the surfboard is virtually perpendicular to what I have referred to as the propulsive flow.
Kevin
*, Here flow is the movement of a volume per unit time, but since we are dealing with water, which is a virtually incompressible liquid for the conditions with which we are interested, a flow virtually amounts to mass per unit time. Incompressible means that there is no change in volume with pressure. A gas is compressible. Liquids generally are not, or at treated as if there weren’t.
Good Lord ! I just saw the “jet impacting bottom” diagram again, it’s BUNKUM!!! .