Figures 1, 2 and 3 set up the model. The interesting nonsense is illustrated figure 4.
In all the illustrations everything is drawn pretty flat. I've done three dimension illustrations, but they're very time consuming and, well, it just ain't in me at the moment. Nevertheless, hopefully the basic idea will be clear.
As can be seen in figure 2, the velocity of the flow relative to the board is not exactly nose-to-tail. Again, the velocity vectors should have been drawn in 3-dimensions, but hopefully the basic idea is conveyed. In figure 2, I've also indicated the region of interest, in particular a typical slice of the board.
Figure 3 is planing as applied to surfing, a rotated diagram taken from Savistsky's treatment of hydroplaning in watercraft. The direction of gravity is indicated.
Which takes us to figure 4. Below, though I've been somewhat brief in my explanation (...yippie!) of exactly what is going on, but my guess is that a picture, here figure 4 is worth a lot of words.
In figure 4 I attempted to account for the curvature of the wave face, so the flow is slightly different than the flow illustrated in figure 3. As illustrated, as you move deeper into the wave towards the wave-side rail there is a mixing of flows of different velocities. Here, the idealized model that I've shown would suggest that the velocities of the flows are not that different in their respective magnitudes, but only in their directions. That is the velocity changes in its sense of direction once it interacts with the bottom of the board or with other flows which have already interacted with the bottom. It is an idealization, but not a completely unreasonable starting point.
These impacting flows as they mix with flows that have already impacted the bottom, make a progressively diminishing contribution to the force of planing. The reason being is because unlike that first flow which impacts the bottom of the board, the next little strip of flow must also deal with mixing with the prior flow. The net effect is that its effective angle-of-attack is diminished, that is the velocity of the flow after mixing with the flow already moving along the bottom of the board takes on some that flow's trajectory, before fully making its contribution to planing. And as its effective angle-of-attack is diminished, its contribution to the force of planing is also diminished - at least that's one way of thinking about it.
I actually don't know if this is what is going on, but it seems to jive with Savistsky's conclusions, which he based on his empirical work. Of course I don't believe Savistsky went this far, he was likely too intelligent a fellow. But it has been a while since I read his work and perhaps he hinted to as much – a rash moment no doubt, but then again we all have them.
How deep is this mixing? I don't know, along with a whole lot of other questions.
But even with this simple model you can start to envision the impact of single, double, etc. concaves, as well as convexes like vees, etc. and maybe even the funky Zinger bottom of Eaton's bonzoid(?). What the various bottom contours are doing is playing with the contribution to planing, not so much in terms of total magnitude, but more in terms of direction, which really isn't illustrated to any great degree in any of the figures. Just remember that the force will be developed perpendicular to the surface with which the flow is interacting.
Then of course there are the fins... don't worry, I won't go there... I'm trying to keep this short.
There is also one other important note. Savistsky was concerned with planing on a horizontal flat surface. His force (actually pressure) profile reflects this. However, in surfing, things change, and it's not just a matter of worrying about that initial angle-of-attack at the surface of the wave face. As a result, its likely that the force profile, though progressively diminishing after that initial impact zone, does not diminish so rapidly, as it is likely to still be contributing substantially even at the wave-side rail.
Anyway, it was kind of a neat treatment, and the fact that it seemed to jive with Savistsky's pressure profile was also kind of neat. Doesn't mean it's correct... just neat.
kc