Now you are taking me back to the dark recesses of my past, when I was young and silly and more interested in drinking beer and chasing girls*, rather than listening to my aeronautics lecturers. (Though as we speak of flying boats, perhaps it was more disinterest in doing extra work for optional credits.)
Surface friction on boat-hull and wing lift - two quite separate though obviously linked and related subjects.
Part of the take-off technique is to get the boat-hull up on the "step" (use this photo-link for reference https://history.nasa...P-468/p166a.jpg) as soon as possible, it reduces the "sticky" force on the hull-afterbody. In the reference photo of the PBM Mariner, that's about a 40% reduction in "stick-down" force. You can then build up to take-off speed, while planing on the forward hull (forebody). For wing weight efficiency, I guess you would design between this take-off load or your maximum take-off weight climb out load, whichever is higher.
On calm days, when the sea is still, getting the flying boat "up on the step" during take off is a big problem. This indicates to me, that for maximum wing weight-lift efficiency - most designers don't try to come up with a wing that can lift the max take-off weight of aircraft AND generate enough extra force to break the surface friction on a hull forebody and afterbody. From my reading over the years, techniques to aid take off during calm seas included 1) using the supporting marine launches to generate wake disturbance across the take-off path; or 2) if no launches available, to do a high speed taxi in a circle so that the take-off run cuts across the flying boats own wake.
Taxiing across the wake disturbance was generally enough to break the surface friction force, and get the hull afterbody out of the water. The rest of take off then proceeds normally.
Flaps are always a trade-off between high-lift and high-drag - can't avoid it. Dropping the flaps changes the wing profile (and increases the C(L) in the Bernoulli equation previously discussed). This has the effect , though, of increasing the C(d) - Coefficient of Drag - in a similarly related Bernoulli equation. Increasing the drag, decreases available force in forward-motion, essentially decreasing the speed - results in a decrease in lift force (in proportion to the speed-squared). There's a band of tolerance in which the use of flaps increases lift before the drag over-compensates and decreases lift. Simply putting bigger flaps on the wing, doesn't always work to your benefit.
The PBY wing is sized correctly to fly efficiently without needing flaps, but even it can't get off the water on a calm sea without using the maritime take off techniques. It's plus benefit, is that the PBY can carry a relatively large payload with minimal drag increase.
Aeronautical design is typically a number of compromises between competing tasks - aerodynamics, structure, power, weight-balance, operational cg shift, client requirements. Sometimes you get it right - Spitfire, Mustang, PBY, etc. Sometimes you get it wrong.
Additional option for extra credit.
Check out the PBY's bigger brother - PB2Y Coronado. The wing is long and slender, as opposed to the short squat wing of the PBY. Known as the Davis wing, the design was similar to the laminar wing profiles of the P-51 Mustang - deeper at the 40% chord rather than at the 25% chord popular in the day. This meant more efficient lift/drag ratio and deeper wing for storage of fuel tanks, weapons, etc. The Davis wing would be further refined on the B-24 Liberator, and the B-32 Dominator. One of the key selling points of the Davis wing was improved lift at small angles-of-attack (angle to horizontal). Ideal for the flying boat take-off run and transition "on to the step".
* (I'm older and wiser now, and have learnt to drink beer, talk aeroplanes and chase girls at the same time. Though the only girl is my daughter, and I still can't catch her running the obstacle course through the house...despite her laughing herself breathless at the time.)
Edited by bearoutwest, 04 February 2018 - 03:25 AM.