Scale Time
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Distance is a linear dimension (factor S), and scale speed has a factor square root of S. Thus scale time has a factor Scale/square root of scale = square root of Scale (or the same factor as scale speed).
I would rather argue it depends on what form of similarity is viable (see below).
Not my theory but that of Ron Warring who was pretty much an expert on this sort of thing and I'm happy to go along with his arguments.
I'm afraid, I don't know who Ron Warring was.
(just have googled this name and got a link to a tribute/obituary page about a guy who worked for the Aeromodeller magazine; is it him you refer to?)
But I'm pretty sure, as far as (scientific) ship modelling is concerned, that the kudos is due to
William Froude who was the first to discover a viable law of ship-model similitude
and from there to devise an ingenious method to predict the power required to propel a ship at a given speed from model tests alone.
By the Froude method roughly the ship's total resistance is devided into a frictional and a wavemaking component.
By towing fully submerged plates Froude could derive a frictional correlation line from which he could extrapolate the frictional resistance of the full scale ship.
The remaining part of the total resistance (which mainly consists of the potential or displacement resistance) was called residual resistance and measured by towing the ship model at Froude similarity,
i.e. at a speed resulting from identical
Froude Nos. for model and ship.
If you equate both Froude Nos. for model and ship and cancel out the gravitational acceleration you indeed arrive at the mentioned square root of scale relation,
i.e.
Vm = Vs / sqrt(scale)But this is only half of the story because we also have to account for effects caused by viscosity.
And here comes a second dimesionless parameter/number into play, i.e. the
Reynolds No. named after
Osborne Reynolds.
Froude and Reynolds Nos. can also both be derived from dimensional analysis of the
Navier-Stokes equations which are the mathematical model describing fluid flow.
I would say they can be thought of as a formulation of Newton's Second Law applied to fluids.
Because the NSE must hold for model and ship as well, to achieve full similitude one would have to adhere to identical Froude and Reynolds Nos. for the model at the same time,
which is impossible to do.
For once the model would have to swim in e.g. mercury rather than water.
But what is even more prohibitive or impractical is the resulting model speed.
If you equate the Reynolds Nos. for model and ship and cancel out the kinematic viscosity you arrive at this scaling relation.
Vm = Vs * scaleI'm not sure if Froude knew about the works of Reynolds, but he devised a very clever and practical method to the shipbuilders' rescue from this dilemma.
In fact his method is still used today by ship model basins/towing tanks around the world to predict the required power to propel a ship,
as can be seen
here and
here.
The latter is the by the proposed standard ITTC'78 procedure.
Thus, I would argue that the scaling of time depends on what similarity you want to achieve.