Euler, Reynolds, Froude, and Strouhal

Hydrodynamics are fun. Euler, Reynolds, Froude, and Strouhal believed so and left their initials in the Navier-Stokes Equation. The best I can do is leave mine in the water.

Navier-Stokes Equation

The Navier-Stokes equations with the continuity equation are the accepted model for describing fluid flow. The equations are usually normalized by replacing dimensional forms with dimensionless forms. This lets us discuss "similar" situations. Some of the terms are then deemed small allowing us to ignore those terms and produce solutions to the equations.

The following table gives the usual normalization, the dimensional form forces divided by the steady-flow inertia force.
 
Characteristic Number name 
(& the force)		 Dimensional Form Dimensionless Form
steady-flow inertia force    ro V2 / L    1
Euler Number 
pressure force		    Po / L    E = Po /ro V2
Reynolds Number 
viscous force		    mV / L2    1/Re = m / ro V L
Froude Number 
gravity force		     ro g    1/F = g L / V2
Strouhal Number 
transient inertial force		    ro V / T    S = L / T V

The dimensionless forms appear in the Navier-Stokes and continuity equations as shown in the following table. All of the variables including S, E, Re, and F are dimensionless. dt is the derivative wrt time, and L is the Laplacian.
 
Dimensionless Navier-Stokes Equation S r dtV + r V . LV = - E p + 1/Re (v2V + 1/3L(L. V)) - 1/F r L h
Dimensionless Continuity Equation S  dt r + L . (r V) = 0

Observations

Free Lunch

There is no free lunch: the forces will be with you.

Our Power

We don't like the word resistance. We prefer power. Power is resistance times speed. The following table presents a summary of power requirements for 64 production kayaks built by others and 2 kayaks, one a 17.2' touring kayak; the other an 18.1' racing kayak, built by us.
 
 
4 knots
4.5
5.0
5.5
6.0
6.5
Average
119 calories/hr
189
309
464
638
845
our hulls
111 (112)
166 (165) 
256 (248)
379 (363)
524 (500)
752 (708)
1st and 2nd best
105 (105)
160 (163)
235 (255)
335 (380)
465 (531)
629 (710)

Our hulls look pretty good.

Power outputs for a strong local paddler are about 120 calories/hr; for a strong local racer are about 180 calories/hr.  For touring 4 or 4.5 knots appears to be a reasonable upper limit on speed. There is not much difference at those speeds.

Above 5 knots we are talking about racing. While most of the kayaks had 300 pound loads (not real race conditions but ...), some had only 250 pound loads. That is why the "1st best" looks so good at high speed.

Accuracy

One needs to be careful with free advice. Taylor and Gertler engineering is the basis for power computations. The following graphs shows the relationship between velocity and resistance and power requirements for three different formulations of Taylor and Gertler engineering. Taylor and Gertler engineering for large boats is shown in red. This formulation overestimates the drag on small boats by 12% at the top of the scale. J. Winters' and M. Broze's attempt to correct for the overestimation is shown in black. Their correction introduces 2 nasty steps of 5%. These steps cause significant errors when comparing hulls of slightly different lengths. This poor formulation is used by several others including Sea Kayaker Magazine. We implement a proper correction in blue. It does not introduce any steps. It allows us to do proper resistance/power analysis.

Aside from that correction Taylor and Gertler engineering ignores the power required to keep a kayak on course. The resistance and power requirements of most kayaks is underestimated. Often by 10%. Our kayaks are designed with passive course correction devices and there is no underestimation of the resistance or power requirements.

The graphs are for a 17.2' touring kayak with a 180 pound total load. The "X" axis is speed scaled to V/L2 (1.3 = 5.4 knots, 1.7 = 7 knots). These are racing speeds.  The "Y" axis is pounds of drag (Resistance graph) and calories burned per hour (Power graph).