If the ships being attacked are unarmed then an optimal engagement range might be something like 200 yards. If the ships being attacked have strong armament (20mm and up), then the optimal range might be 500 yards or more.
Barges with gun tied onto them are not strongly armed, 20mm is not strongly armed anyway
The Sandhurst naval results against the second wave were 1,500 barges sunk by 75 warships with no warships sunk by air attack. That's a tempo of 20 barges sunk per warship. Is that a realistic result for an unarmed invasion fleet with no escorts or air support? Probably. Is that realistic for an armed invasion fleet with escorts and heavy air support? Probably not.
What armed invasion fleet (see above) armed doesn't just mean has guns, are they usable guns, are they usable when you rolling on seas you not designed for at 2-4knts and firing at actual warships travelling at 15-20knts.
Also what escorts, what heavy air support. or rather you seem to have forgotten the if there is the LW around then there is likely the RAF around as well, similarly if you now talking about the KM running escort than well so will the RN larger ships.
Also you forgetting or ignoring that you talking a bout a flotilla travelling 2-4knts, this impacts on everything, an actual KM warship trying to escort this will either be travelling at 2-4km itself (and thus a sitting duck) or will have to be running loops around the flotilla at higher speed. Similarly the LW will have to escort the same slow moving fleet which means they will have to loiter over the channel for 2 days (or rather they will in fact stagger themselves to provide continuous cover thus diluting thier force and as per the ships above will be sitting ducks)
(also didn't the sandhurst war games just have the RN not turn up or turn up very late?)
Case 1. A DD vs. 10 barques in daylight and none of the barques are armed.
Case 2. A DD vs. 10 ships in daylight armed with: 6 x 20mm, 4x40mm, 4x50mm, 4x75mm.
Question - what is the optimal engagement range for the destroyer in Case 1? What is the optimal engagement range for the DD in Case 2?
case 1. Barques are a lot faster and generally more seaworthy than river barges again you keep ignoring this, it is huge factor, so not relevent to sealion
case 2 "ships"means a lot of efferent things and that the most important thing, 10 barges that for some reason have 6 x 20mm, 4x40mm, 4x50mm, 4x75mm. attached to them is very much not the same thing as 10 actual sea going ships like converted trawlers with 6 x 20mm, 4x40mm, 4x50mm, 4x75mm.
(Also a barque is a rigged ship, not sure that's what you mean , however if the German can some get enough of these and enough crew they might be better than the river barges
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You keep doing the same thing over and over and ignore the whole contest so a river bat with a 75mm bolted to it becomes a combat ship with 75mm mounting. The LW are now in attendance en masses, but there's now RAF
and most of all you ignore, and ignore and ignore that this barge flotilla is crawling along at 2-4 knt totally at the mercy of seas they're not fit for.