Quitty,
Over the past several years I worked out a scientific way to scale aircraft parameters in order to build a sensibly realistic model.
The outcome is: there IS NO hard and fast rule. It depends upon what weight you choose to build and that determines how fast it will fly. Once you choose a weight everything else falls into place mathematically.
All this eventually appeared as a series of magazine articles in the issues June to September 2009 of the British magazine R/C Model World, published by Traplet. The first of the four articles had all the theory and equations to prove my main point - there is no such thing as Scale Speed. There is one weight that gives a model that can fly its own length in the same time as full size (but most modellers would say it is too light and floaty). At quite a different weight the model will fly turns and loops whose radius is in scale (but for practical purposes they are a bit heavy).
So we compromise by choosing a weight in between.
Don't worry Quitty, I'm getting to your answer
The final (fourth) part of my series in September '09 R/C Model World featured a spreadsheet that runs in Excel, with instructions how to use it. You input basic information about a full size and it spits out useful data in a whole range of scales. You can download the spreadsheet from the R/C Model World website
http://www.rcmodelworld.com/ (click "Features", sixth item down)
or from my club's website (under Alasdair's Aerodynamics)
http://www20.brinkster.com/gvmac/
ANSWERS
Eventually. I decided that making a model bigger or smaller is no different to making a model of a full size so I tried the Spreadsheet and it works. You have to choose a new WIX (Weight Index, a scaling term I made up) and Scale Factors - that's the number you divide by to get the model's wingspan (4 for quarter scale).
To make a twice size model you need a scale factor SF = 0.5
And from experience I found that a WIX of 2.5, give or take a little, gives realistic results. By that I mean models of a different size that fly pretty much like the original, and are practical to build and fly.
I input figures from one of my own models, and with a WIX of 2.5 if you
DOUBLE the wingspan, you get
Four times the Area,
5.65 times the weight,
1.415 times the wing loading,
1.18 times the flying speed
and you need 5.9 times the power
All the figures relate to each other on the premise that you are flying at the same lift coefficient with both sizes of model.
As has been pointed out, increasing the Reynolds Number gives a little performance bonus.
You asked for the mathematical reasoning, well it's all in the articles and if you download the spreadsheet you can check out the math and play with the answers. Adjust the numbers in blue to see "what if..."