Rick, " There is essentially no wave drag or induced drag associated with dynamic lift.  It is all viscous drag."

Dynamic lift can and does produce huge wave and drag forces. You just don't see them and the vertical wave is so much longer than the horizontal waves that there is almost no wavelength limit.

Why do you think Subs are four times faster than surface displacement vessels?

The LCS Independence is a stabilised monohull operating in displacement mode at 40kts

Show me a credible reference of any submarine that will do 120kts.

"That is why there is a specific shape for a displacement hull that gives minimum resistance for a particular speed and displacement.  A slender hull has lower wave drag but more viscous drag than a wider hull while the wider hull has more wave drag and less viscous drag than the slender." 

You are making my point that wave drag and AOA are bigger drags than viscous drag.

Not on a slender hull.  Viscous drag dominates meaning drag is a squared function of speed so power is a cubed function.  The Crouch formula is not applicable to this situation - that is the point.  Crouch formula applied to a proa with round sections is nonsense.

 

"A simple example - two hulls of rectangular section with draft half of beam so same section shape.

Short hull 2m long 2m beam 1m draft gives volume of 4Cu.m and wetted surface of 12sq.m

Long hull 16m long 0.5m beam and 0.25m draft volume of 4 Cu.m and wetted surface of 16.25sq.m

Viscous drag on the longer hull is 35% higher - certainly not trivial."

An elliptical shape has less wetted surface than either, and a round shape has the greatest drag of all (except a flat plate moving perpendicularly). That is why you have to compare similar shapes.

I can do the same analysis for an elliptical section and show that the wetted surface increases as the length increases for a hull of the same displacement.  

But back to your example, what is important is the flat plate drag or frontal area. In your first example the frontal (and rear) area is 4 square meters and in your second example the frontal (and rear) area is .25 square meters. With the same surface finish, everyone knows which shape is faster and has less drag.

What is important is that a longer hull has more wetted surface than a shorter hull of the same section.  The effects of form and beam will be a function of speed.  I expect the 4X4 box would have lower drag at 0.1knot than the 16X0.7 box.  At that speed the viscous drag will dominate form drag.  The speed is a function of the driving force.  For any driving force and displacement there is a particular length and shape that gives lowest overall drag - longer is not necessarily better.  The gain in reduced wave making of a long slender hull is offset at some increasing length by the increasing wetted surface.  That is why there is a specific length for a displacement hull that give the least drag for a given speed and displacement. 

The long slender shape is almost on plane at rest. If it is actually on the surface it really is on plane.  That is why weight is so important with the HarryProa.

No boat can be almost on plane at rest.  Dynamic lift is a function of speed squared and AoA until the weight is fully supported by the dynamic forces.  The statement is akin to saying a glider is almost flying while sitting in a hanger.

A curved underwater hull decreases wetted area and viscous drag, but increases wave formation and has much higher drag. 

The section shape of the hulls on the 18m proa that you are applying Crouches planing formula on are semi-circular.  Semi-circular section offers the lowest wetted surface and at low speed to displacement is the lowest drag form.  That is why rowing shells have near semi-circular sections. Flat bottom sections can offer some drag reduction where low weight and higher speed will promote some dynamic lift. There can also be benefit with flatter sections at moderate speed to displacement ratio where the reduced beam plays a role in reducing wave drag.

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