Scaling laws are really interesting for modelers...it's a part of virtually everything we do. Often it's assumed that we're scaling from a full-scale aircraft but with so many variances in sizes these days we can apply scaling techniques within the confines of the models we fly.

The scaling of any general object must follow the "square-cube" law which defines (using my nomenclature) a change in scale by a change in a characteristic linear dimension.

If L1 is the length of the original object and L2 is the scaled version of that object then the scaling factor k is defined as:

k=L2/L1

You can derive the square-cube law by applying this technique to a simple geometry like a cube. You will see that the surface area of the cube will grow with k^2 and the volume of the cube will grow with k^3 such that:

S2=k^2*S1 (area)
Vol2=k^3*Vol1 (Volume) (mass, weight and force also scale with k^3)

If we apply this to our wing loading problem we start with a particular airplane with some characteristic length (typically the wingspan) which I'll call b. k is set by the ratio of the wingspans

k=b2/b1
likewise the wing area will grow with k^2 and the weight will grow with k^3 such that
S2=k^2*S1
W2=K^3*W1

If we define wing loading as W/S then we have k^3/k^2 = k so we know from the square cube law that :

(W/S)2=k*(W/S)1 This simply means that if you double the scale you can double the wing loading.

The reason for the so-called "cubic" wing loading also comes from the square-cube law. Modelers don't like to have to remember what wing loading is appropriate for their scale model so they find a figure of merit that works for all scales. This happens to be W/S^1.5 If you look at this using what we've already discussed you get that k^3/(k^2)^1.5 = k^3/k^3 = 1 so it's independent of scale.


You can play this game all over the place...for instance take the level flight linear velocity. The standard equation for lift is L=W=CL*q*S=CL*rho/2*V^2*S...assuming that air density and CL are independent of scale (which they aren't exactly...this is where Reynold number, Mach number and Froude number come into play) you find that V is proportional to (W/S)^0.5

Therefore velocity is proportional to k^0.5...this means that if you double the scale of the airplane the level flight speed increases by the square root of 2 or 41%.

Since we actually see scale speed as body lengths per second we can note that the time it takes for us to see the distance of one body length covered is given by

t=BL/V (body-length/velocity) we know that this is k/k^0.5 or k^0.5 so now we can see that

t2=k^0.5*t1 or stated differently, the model that's double in scale will appear to take 41% longer to cover this characteristic length.

This is quite interesting because the true linear velocity of the double-scale airplane is in fact flying 41% faster but appears to fly 41% slower.

You can also use scaling laws on power. If you say that the thrust required for level flight is approximately equal to D and that the power required is Drag*Velocity (D*V) you get the following:

P2=k^3*k^0.5 = k^(7/2) or k^3.5 This means that power required grows at a little faster rate than volume.

You can test this by scaling a 40% airplane down to foamy size. if the 40% has a span of 120" and uses a 16HP motor how much HP would an airplane of 30" require to have similar performance?

k=30/120 = 0.25 P2=0.25^3.5*16HP = 0.125HP or 93 watts (this is close to what we get running 12V at 8 amps.


You'll also notice that things like moments will scale with k^4 and moments of inertia will scale with k^5, RPM will scale as k^0.5 .....on and on.

The one thing to remember is the square cube-law knows nothing about aerodynamics and in many cases the aerodynamics can't keep up at the lower Reynolds numbers...in this case things like wing area have to grow to obtain similar performance.

George Hicks