Hi Lee –
Sorry for the length of following but this is really complicated. Rather than respond in bits and pieces, here is the whole explanation.
You are exactly correct. The flow under the bottom of the board is the vector sum of the water flowing up the face of the wave (and mostly across the board), the axial flow due to the forward motion of the board, and the presence of the board of course also affects the flow field. It’s easier to think of the inflow being comprised of two flow fields, one flow field aligned with the centerline of the board, and a transverse flow field perpendicular to the centerline of the board with the classic flat plate planning theory applicable to both flow fields. (One of the fundamental premises of potential flow (no viscosity) theory & computer programs is that the potential flow fields are linearly additive). Since the angle of attack of the bottom of the board is higher relative to the transverse component of the flow, this component of the flow is slowed down more than the axial flow component and much of the lift is generated by the transverse flow. For instance if the board were not moving forward, it could still support the riders weight by planning on the flow moving up the face of the wave just like the guys riding on those stationary wave rides. Ie you can take the fins off and ride your board sideways into the beach…
The flow velocity relative to the board bottom in the axial flow direction is much higher than the flow velocity in the transverse direction, and that is why the streamers are mostly aligned with the centerline of the board except near the rails and along the stagnation line just behind the spray root. Even though the axial flow velocity is higher, the trim angle of the board is lower relative to this component of the flow, and therefore the effect of the board on this axial component of the flow is typically lower and the lift generated may be less than the lift associated with the transverse component of the flow when riding across the wave face.
One of the classic planning theories is 2D + time where a planing boat is represented by a 2D wedge being quickly immersed. Ie where the keel line intersects the water is where the Vee shaped wedge is just contacting the water surface, and the transom of the boat is where the wedge has plunged down to the depth of the keel at the transom. The total pressure on the boat bottom is calculated by using the pressure associated with the transverse flow field generated by immersion of the wedge at each instant in time as it is immersed, and the axial flow is basically ignored. A surfboard bottom can also be thought of a single sided wedge being immersed into the wave face with the forward rail to wave intersection being analogous to the forward keel line intersection, and the tail of the board the transom.
On a surfboard, the flow angles, velocities, relative magnitude of the two components of lift etc. are obviously varying continuously with the direction and speed of the surfer/board and the flow field associated with the wave. However the angle of the contact line across the bottom of the board is in general an indication of the relative importance of the two components of flow in generating lift. The more parallel the contact line is in relation to the centerline of the board, the more the lift is associated with the transverse flow. Conversely the more perpendicular the contact line is relative to centerline, the more the lift is associated with the axial flow. When you are going straight down the face of the wave the contact line is perpendicular to centerline, and the all of the lift is associated with the axial flow. When taking a high line in the pocket of a steep wave, the lift is mostly due to transverse flow.
The concave concentrated along the rail of the board is designed to exploit the transverse component of the flow, and generate higher pressures on the bottom of the board and therefore increased lift. Increasing lift associated with the transverse flow, results in reduced lift and reduced trim angles associated with the axial flow for a given total amount of lift/rider weight. Since the pressure on the bottom of the board acts perpendicular to the surface of the board, lower axial trim angle results in less drag and a faster board.
Computer simulations I have done indicated that even when the flow is 100% axial, this bottom shape generated about 5% more lift, and with moderate amounts of transverse flow about 15% more lift. The increased lift with 100% axial flow allows you to take off earlier, and the increased lift/reduced drag with some transverse flow gives you more speed on the face of the wave to make sections etc.
