…to continue from my previous post (my comments in […]):
[From Schlichting (1960) for laminar flow:
R* = 1/Reynolds number
T99 = thickness of the boundary layer at a distance
L from the leading edge of a flat plate (approximation to a surfboard)
v = kinematic viscosity of the fluid
T99/L = 5.0 x square-root(R*)
[But, as Schlichting notes:]
"It is impossible to indicate a boundary layer thickness in an unambiguous way, because the influence of viscosity in the boundary layer decreases asymptotically outwards. The parallel component of u [the along-plate component of the flow] tends asymptotically to U [the parallel component well forward of the beginning of the plate] of the potential flow. [We define a boundary layer thickness, T99, as the thickness] …for which u = 0.99 x U.
[Alternatively] A physically meaningful measure for the boundary layer thickness is the ‘displacement-thickness’, … which is that distance [TD] by which the external potential field of flow is displaced outwards as a consequence of the decrease in velocity in the boundary layer…
[The resulting boundary layer thickness, TD, is:]
TD/L = 1.72 x square-root(R*)
[or about 1/3 the thickness of the T99 boundary layer.]
[To be complete] We may at this point evaluate the ‘momentum thickness’ [TM, which is related to] …the loss of momentum in the boundary layer as compared with potential flow…[this calculation] gives:
TM/L = 0.664 x square-root(R*)
[At some Reynolds Number (i.e. at some point downstream from the leading edge, the laminar flow will change to turbulent. For a flat plate this transition can occur somewhere between a Reynolds Number of 2 x 10^5 and 10^6, depending on the smoothness of the plate and the degree of ambient turbulence in the flow. Once the boundary layer has change from laminar to turbulent, its thickness is related to the downstream distance and the Reynolds Number by the equation:]
TB99/L = 0.37 x (R*)^0.2
[Since the surf zone is virtually certain to have some degree of turbulence, let us assume that the boundary layer changes from laminar to turbulent at a Reynolds Number (based on the distance from the leading edge of the wetted area of the board) of 2 x 10^5. ‘Over the bottom’ speed measurements indicate typical speeds (in my area) of 15-20 mph (22-29 ft/sec). ‘Through the water’ speeds are likely to be less, so let’s assume a speed of 20 ft/sec (610 cm/sec). In that case, the downstream distance from the leading edge of the wetted surface of the board to the transition from laminar to turbulent flow will be about 3.3 cm (1.3) inches, and the T99 boundary layer thickness will be about 0.037 cm (and the displacement thickness, TD, of the boundary layer will be about 0.012 inches).
At a speed of 5 ft/sec (~paddling speed) the transition from laminar to turbulent flow will occur about 5 inches from the leading edge, and the boundary layer thickness (T99) at the transition point to turbulent flow will be about 0.05 inches from the leading edge.
Thus it would appear that boundary layer flow on the bottom of a board is generally turbulent–except, perhaps, in the outer region of slender (‘high’ aspect ratio) fins moving through the water at slow speeds.
Now let’s estimate the turbulent boundary layer thickness at the tail end of a board traveling at 20 ft/sec. Assuming a wetted length of 4’, or 490 cm (from the entry point on the ‘wave side’ of the board to the tail), the Reynolds Number will be about 30 x 10^6, and the thickness of the turbulent boundary layer at the tail of the board will be about 5.8 cm (or 2.3 inches)–about 1.2 percent of the wetted length).]