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Design > Scale Speed for Small Aircraft
[Courtesy of Ib Therkelsen, September 1998]
Some observations on the requirements for scale models to fly at
scale speed.
Slightly intrigued by the opinion in model aeroplane litterature, that scale
models more often than not are regarded as "difficult", I began to search for
the troubeling aerodynamic rules connected with having small scale models fly at scale
speed. Vague hints in the direction of wing loading figures was all I ever managed to
locate during a somewhat concentrated effort, whereas the more specific hardcore
information I was out to hunt always seemed to evade any book or article I came across.
Finally, curiosity made me sharpen my pencil and figure it out for myself. The results do
not pretend to be mindblowing news by any measure (some of them are indeed trivial), and
no doubt they have all been presented elsewhere by numerous other persons many times in
the past. Of the doublesided problem that I find it to be, namely one of weight
and one of aerofoil behaviour in the low Reynolds regime, the weight part is
relatively simple and may easily be done with, while the second part is substantially more
elusive, in worst cases uncontrollable to the point of overturning all efforts towards
very small scale model feasibility.
Experimental Nieuport17 built to 1/8 scale from white foam sheets of 5 mm thickness.
Wingspan 1 m, weight 135 gram (including 20 gram noseballast). Glides at about 3 m/s.
What everyone knows
Time is fixed. One second is one second, come what may.
Distance is 1Dimensional, hence velocity is also 1D.
Area is 2D.
Volume is 3D, hence mass is also 3D.
Let r be the scale (reduction) factor.
For r = 12, the model is in scale 1/12
Scaling down, the distance shrinks to 1/r, the area shrinks to 1/r^{2},^{ }the
volume to 1/r^{3}
When we say, for instance, that a model is in scale 1/12, we refer to the 1D properties
(distances). From this fact follows that areas must be 1/ (12·12) and volumes must be 1/
(12·12·12).
What scalemodelers should know
In order not to bore readers more than necessary, the outcome is presented
first, the details and comments later for those who might want a closer look. In
everything that follows, the subscript 1 stands for some property of the full size
aeroplane, while subscript 2 is reserved for the model.
Rule 1 z_{2} = z_{1}/r
Wingloading, z, resulting from simple downscaling.
Rule 2 v_{2} = v_{1}·sqrt(1/r)
Velocity, v, resulting from simple downscaling.
Rule 2a v_{2} = v_{1}/r
True scale velocity.
Rule 3 w_{2} = w_{1}/r^{4}
Mass, w,
required to fly at true scale speed.
Rule 4 svz = z_{1}/r^{2}
Scale
velocity wing loading, svz, resulting from Rule 3
Remark 1 Concerning weight ( mass, to be
correct ), there seems to be no principle which in itself prevents very small models from
flying at scale speed. The limitation lies alone with our own abilities of very light
construction.
Remark 2 Concerning aerofoil performance at very small
Reynolds numbers, odd problems may considerably jeopardize very small scale model
feasibility. More efficient nonscale wing profiles might be brought into play, only
somewhere there will be an insurmountable barrier to the gain that can be attained this
way.
Rule 1
A property, often referred to when comparing aircraft, is the wing loading,
z, i.e. total mass divided by wingarea. If we scale down, the wingarea diminishes by the
square, whereas the volume, and hence mass, diminishes by the cube, that is, the mass
shrinks faster than the area, which in turn means that the model always will have
less wingloading than the full size plane. This principle of "lost" wingloading
is also known  at least in the model aeroplane folklore  as the "squarecube"
law. Quite clearly it will only be valid on the condition that the materials used for the
model are of roughly the same density as those for the full size machine, and that the two
structures are fairly similar.
Let w_{1 }and w_{2} be the masses of the prototype and model respectively,
and let S_{1} and S_{2} be the areas of the wing surfaces:
w_{2} = w_{1}/r^{3 }( by simple
downscaling )
S_{2}= S_{1}/r^{2} ( by
simple downscaling )
z_{1} = w_{1}/S_{1 }( wingloading,
full size )
z_{2} = w_{2}/S_{2} = (w_{1}/r^{3})·(r^{2}/S_{1})
= w_{1}/ (r·S_{1}) = z_{1}/r
Wingloading for model ( by simple downscaling ) = full size wingloading / r
Rule 2
Of special interest for aeroplanes is the Lift Equation: L = C_{L}·q·S
L = w·g = Newton ( N )
w = aeroplane total mass ( kg )
g = acceleration due to gravity = 9.81 m/s^{2 } (
32.2 ft/s^{2} )
C_{L} = Lift Coefficient ( a dimensionless value to indicate the lift capacity of
a certain shape )
q = d/2·v^{2
}d = density of air = 1.225 kg/m^{3 }(^{
}0.0765 lb/ft^{3} or, to be correct, 0.002377 slugs/ft^{3} )
v = velocity ( m/s )
S = surface area of wing ( m^{2} )
S = c·b
c = mean wing chord ( m )
b = wingspan ( m )
The lift equation in full: w·g = C_{L}·d/2·v^{2}·S
or, upon rearrangement: 2·w·g = C_{L}·d·v^{2}·S
Rearranging for velocity squared: v^{2} = 2·w·g / ( C_{L}·d·S)
Comparing model velocity to full size velocity:
v_{2}^{2}/v_{1}^{2} = (2·w_{2}·g·C_{L}·d·S_{1})
/ (C_{L}·d·S_{2}·2·w_{1}·g) = (w_{2}·S_{1})_{
}/ (S_{2}·w_{1}) = z_{2}/z_{1
}Substituting z_{2} by Rule 1 leads to:
v_{2}^{2}/v_{1}^{2} = z_{1}/(r·z_{1}) =
1/r
v_{2}^{2} = v_{1}^{2} ·1/r
v_{2} = v_{1}·sqrt(1/r)
Velocity for model ( by simple downscaling ) = full size velocity · squareroot of
downscaling
Of likely interest may also be to compare the original velocity, v_{2}, with the
resultant velocity, v_{3}, in case the model mass ( and hence wingloading ) is
changed:
v_{3} = v_{2}·sqrt(w_{3}/w_{2})
or the
equivalent: v_{3} = v_{2}·sqrt(z_{3}/z_{2})
Calculated example: Spitfire 1/12 scale.
Full size aircraft: v_{1} = 58 m/s ( estimated takeoff ), S_{1 }= 22.5 m^{2},
w_{1} = 3000 kg.
C_{L} = (3000 · 9.81 · 2) / (1.225 · 22.5 · 58^{2}) = 0.63
( total aeroplane )
Full size wingloading, z_{1} = 3000/22.5 = 133 kg/m^{2
}Model: r = 12, leads to:
S_{2} = 22.5/(12·12) = 0.156 m^{2
}(model wing area )
w_{2 }= 3000/(12·12·12) = 1.74 kg ( by simple downscaling )
z_{2 }= 1.74/0.156 = 11 kg/m^{2
}( or by Rule 1: 133/12 =
11 kg/m^{2} )
v_{2 }= 58·sqrt(1/12) =16.7 m/s ( by Rule 2 )
...as opposed to true scale velocity: 58/12 = 4.8 m/s.
It is interesting to note that the velocity 16.7 m/s is not to scale, which has to
do with velocity appearing as squared in the lift equation. If we insist on flying our
model at true scale speed, obviously one or more of the other parameters in the lift
equation will have to be changed in order to compensate for the squared velocity. As it
turns out, only C_{L} and w are free for manipulation, of which the C_{L}
is not easily adjustable, so clearly the candidate for tampering with must be w, the model
mass. But before solving the problem, let's take a look at an expert's opinion:
In his book "Rubber powered airplane models", Don Ross gives some valuable
numbers for the optimal wingloadings of small flying models. The unit chosen, gram/in^{2},
is an unusual example of bad taste, but fortunately the conversion is easy enough.
gram/in^{2 }·1.55 = kg/m^{2
}kg/m^{2 }·3.28 = oz/ft^{2
}oz/ft^{2} ·1/16 = lb/ft^{2}
Ross says that for a small duration model ( e.g. Peanut scale, 13" wingspan ( 33
cm )), the best wingloading should be about 0.33 gram/in^{2}, and for a medium
size plane ( about 30" ( 76 cm )), it should be somewhere around 0.5 gram/in^{2}.
Converting these two figures to more familiar units:
Small: 0.33 gram/in^{2} = 0.50 kg/m^{2}
(1.64 oz/ft^{2 })
Medium: 0.50 gram/in^{2 }= 0.78 kg/m^{2 }(2.56 oz/ft^{2})
These values, to be sure, are for duration models. From other sources I have found
that R/C people are not particularly unhappy about quite large wingloadings, as long as
they do not exceed 20 oz/ft^{2} = 6.1 kg/m^{2 }(a crude average extract
from several Newsgroup posts). So our Spitfire of 11 kg/m^{2} apparently has a
severe weight problem for rubber duration and R/C alike, besides the fact that it's
totally out of range of true scale speed. Of course, the 1.74 kg is a worst case example,
and we would no doubt have little difficulty in building the model a lot lighter than
that, only we still don't know just exactly how light it needs to be. What I will
show you in a moment, is that we can actually develop a simple rule to determine the mass
of the model so that it should fly at scale velocity, everything else being equal (which
may not be the case). I shall give you the details shortly, but for now I can reveal that
should our model be able to fly at 4.8 m/s, it would require that the mass be 0.145 kg,
dramatically different from the 1.74 kg, and, as it were, exactly 1/12 of 1.74. The
resultant wingloading would be 0.145/0.156 = 0.93 kg/m^{2}. Now take a second look
at the value Ross gives for medium sized models. We are quite close.
Rule 3
Rearranging the lift equation for mass: w = C_{L}·d·v^{2}·S /
(2·g)
Under the somewhat bold assumption that C_{L} remains constant for the full size
aeroplane and the model alike, we may write:
w_{1} = C_{L}·d·v_{1}^{2}·S_{1 }/ (2·g)
( mass of full size )
w_{2} = C_{L}·d·v_{2}^{2}·S_{2 }/ (2·g)
( model mass )
Comparing the two:
w_{2}/w_{1} = (C_{L}·d·v_{2}^{2}·S_{2}·2·g)
/ (2·g·C_{L}·d·v_{1}^{2}·S_{1}) = (v_{2}^{2}·S_{2})_{
}/ (v_{1}^{2}·S_{1})
v_{2} = v_{1}/r ( true scale
velocity )
S_{2} = S_{1}/r^{2 } (
simple downscaling )
Inserting these expressions for v_{2} and S_{2} we get:
w_{2}/w_{1} = (v_{1}/r)^{2}·(S_{1}/r^{2})
/ (v_{1}^{2}·S_{1}) = 1/r^{4
}w_{2} = w_{1}/r^{4
}Model mass required to fly at true scale speed = full size mass / r^{4}
( This is the piece of information I originally went out to find ).
Table of scale effect, resulting from Rule 3. Original mass = 3000 kg, original
vel. = 58 m/s
Scale Mass (gram) Wingloading (kg/m^{2})
Scale speed (m/s) Re (description below)
1/4 11719
8.33
14.5
555000
1/6 2315
3.70
9.7
247000
1/8 732
2.08
7.3
139000
1/10 300
1.33
5.8
89000
1/12 145
0.93
4.8
62000
1/16
46
0.52 3.6 35000
1/24 9
0.23
2.4
15000
1/32
3 0.13 1.8
8700
1/48 0.6
0.058
1.2
3800
1/64 0.2
0.033
0.9 2100
1/72 0.1
0.026
0.8
1700
With regards to mass alone it appears that in principle there is no problem with
scaling down and remain flying at scale speed. The practical construction of such
extremely lightweight devices as required in the small scale regime may however not always
come easy. As to our imaginary 1/12 scale Spitfire model above, we could easily find
ourselves in a situation where we do not have sufficiently lightweight components at hand
to get away with anything more than a crude freeflight version.
The finest successful example I have heard of so far (Walter Scholl, Ezone sep. 1997) is
a Blériot XI, 1/10 scale model of 115 gram mass, electric engine and micro R/C included,
and flying at scale speed 2 m/s. The wingloading is 0.46 kg/m^{2}, even less than
Ross' smallest figure.
Alarmingly primitive Bf 109 F prototype in mixed foam/balsa, built to 1/12 scale.
Although the artistic appearance of this my first foam job ever is nothing to be proud of,
the aerodynamic properties are not bad at all. Wingspan 83 cm, Clark Yish aerofoil.
Total weight 128 gram (including 34 gram nose ballast).
Glides at 4.5 m/s. Rate of descent is about 0.75 m/s.
Rule 4
w_{2} = w_{1}/r^{4} ( using Rule 3
)
S_{2} = S_{1}/r^{2 } (
by simple downscaling )
z_{2} = w_{2}/S_{2} = (w_{1}/r^{4})·(r^{2}/S_{1})
= w_{1}/ (S_{1}·r^{2}) = z_{1}/r^{2
}Scale velocity wing loading = full size wing loading / r^{2}
Reynolds number ( Re )
A quite different matter which has to be faced in the process of scaling down, is
the aerodynamic behaviour of wingprofiles (aerofoils) operated at small dimensions and/or
velocities, as would be required for small models.
To begin with the Reynolds number:
Viscosity, µ, may be expressed in Pascal·second, Pa·s = kg/(m·s)
µ (air) = 1.8 ·10^{5} Pa·s [ For
comparison, µ (water) = 1.0 ·10^{3 }Pa·s; 55 times more viscous ]
Reynolds number (Re) = c·v·d/µ
( c, v and d as defined earlier in details of Rule 2; d/µ has the dimension
second/m^{2 })
With c in meter, and v in meter/second, Re = approximately 68000 · c · v
With c in feet, and v in feet/second, Re will be approx. 6410 · c · v
Full size aeroplanes have Re in the million class, whereas indoor models may come as low
as 10000 (microfilm). Tables or plots of lift and drag coefficients as a function of angle
of attack (AoA, or alpha) and different Reynolds numbers are valuable when comparing
aerofoils, as well as an indication of which dimensions and/or velocities better to be
avoided for a specific wing profile. Data to this effect are most often obtained from
windtunnel tests, but also theoretical tools are capable of calculating fair predictions,
as long as we make sure to stay above Re 100000. One such tool by Martin Hepperle is given
in a link at the end of this webpage. The book by Martin Simons, also listed below,
devotes several chapters to the presentation and discussion of aerofoil and windtunnel
data.
In the notes to Rule 3 it was mentioned in passing that lift coefficients for the
full size plane and the model might not be quite the same value. Although it was a
prerequisite for developing Rule 3, and may well be close enough to the truth for
large models like 1/4 or 1/6 scale, it's not likely to be an equally safe assumption for
smaller models, say 1/10 and less. The problem here is the almost total absence of
linearity between the lifting capacity of two aerofoils of equal proportions but different
size. One cannot safely assume, without windtunnel evidence or practical field
experiments, that a downscaled version of some specific wing profile will have any
relationship with the full size version in terms of efficiency; in fact an extensive
amount of published data strongly indicates that the smaller of the two will show a
markedly inferior performance, and that the trouble usually aggravates as we move towards
the very low Reynolds regime. What happens here is that inertial forces to some extent
loose their grip in a struggle against viscous forces. Let me give you a taste of what's
to be expected, by a few examples, which, with the exception of EJ 85, are all drawn
from Simons, more or less at random:
N60, AoA = 3°
NACA 4412, AoA = 3°
Göttingen 801, AoA = 3°
Re
cl Re
cl
Re cl
168000 0.94
3000000
0.75
170000 0.88
126000 0.93
250000
0.70
100000
0.88
105000 0.90
75000
0.70
75000 0.83 +
84000 0.87
60000 0.50
75000
0.65 +
63000 0.50
45000
0.38 63000
0.55
42000 0.44 30000
0.31
42000
0.47
21000 0.42
20000 0.27
21000
0.40
Göttingen 417b, AoA = 3° HK 8556 [T],
AoA = 3° EJ 85, AoA =
3°
Re
cl Re
cl Re
cl
189000 1.16
120000 0.71
83000
0.95
84000 0.86
60000
0.71
60000
0.95
63000 0.65
30000
0.71
40000
0.93
42000 0.59
20000
0.65
20000
0.70
21000 0.76
14000 0.90
A few notes:
Göttingen 801: Suspicious behaviour around Re 75000 ( marked with + ), as
part of a socalled hysteresis loop, with the effect that in a certain Re range,
and specific AoA, the profile gives a better performance in going from higher to lower
speed than in going from lower to higher. This unpleasant behaviour is shown by many
profiles, including N60 and Gö 417b (at higher AoA than 3°), whereas this
is not the case with NACA 4412, HK 8556 [T] and EJ 85. See Simons for
details if you are interested.
Göttingen 417b, curved plate: Surprises at Re 21000 and 14000, although
accompanied by large increases in profile drag (not shown).
HK 8556 [T] Turbulator, and EJ 85: These profiles seem to be
extraordinarily forgiving at all velocities and/or dimensions. EJ 85 is a Jedelsky
profile.
As seen in the previous few tables, it's evident that the highest lift coefficients
coincide with the highest Reynolds numbers, and also that above a certain Re the gain in
lift capacity with a further increase in Re is extremely limited, if any at all. In the
low end of the Re scale, for most profiles the lift coefficients are more widely scattered
with changes in Re. Apart from that, any broad generalisation is difficult to make because
of exceptions and surprises, thus rendering predictions a little hazy. For real flying
devices, matters are slightly more complicated, as most often we need to know the amount
of power required for flight and/or the distance covered in a glide from a certain height.
Various drag components must be lumped together and compared with the lift coefficients,
in order to make a realistic estimate of the rate of descent and hence the usefulness of
some particular wing in connection with the rest of the aeroplane. All this,
however, is so much better described in textbooks like the ones by Simons and Stinton.
At present, it is believed that for very small models, the best job will be done by a
thin, highly cambered wing profile, even such seemingly simple ones as the broken plate
variation of a Jedelsky profile (see Don Ross). This again indicates that perfect scale
appearence  for all but pre 1920 aeroplanes  may have to be sacrificed to some extent.
In Aeromodeller, sep / oct 1997, vol 62, No 742 and 743, two articles "Foam
at last!" by David Deadman, describe the construction of lightweight and
extraordinarily realistic peanut scale models. I have been informed that one of the
author's planes, a Lavochkin La7, scale approx 1/30 and weight 13 gram, flies at about
4.8 m/s, corresponding to 80 % of scale maximum speed. According to Rule 3 above,
the "ideal" weight should rather have been close to 4 gram, but the La7 model
features a curved plate wing with a lift capacity large enough to compensate for the extra
weight. Had the original wing profile been retained for strict scale appearence, the
resulting model speed would have been in the vicinity of 9 m/s. Mr. Deadman's La7
perfectly illustrates the tradeoff between weight and lift capacity that has to be
accepted when rather small scale models are to fly at scale speed.
From the observations accumulated here, I should think that from about scale 1/8 and
down, weight will be the primary problem to solve, while from approx. 1/10 and downwards,
aerodynamic trouble will begin to mix heavily in and pile up at a dramatic rate as we move
towards smaller dimensions. Lower lift coefficients translate into larger velocities (by
the lift equation) and will have to be either compensated for by a more or less drastic
change of wing profile (as far as it goes), or counterbalanced by an even further weight
reduction in a model which may already be stripped to the limit.
Litterature:
Martin Simons: Model aircraft aerodynamics. (1994)
Darrol Stinton: The design of the aeroplane. (1983)
Don Ross: Rubber powered model airplanes (1988)
Links:
Walter Scholl (EZone article):
http://ezonemag.com/articles/wiw/091997/wiw0997.htm
Martin Hepperle:
http://beadec1.ea.bs.dlr.de/Airfoils/calcfoil.htm
Bob Boucher. A different view of scale flight, taking scale maneuvers into
account:
http://www.astroflight.com/astroflight/scalespeed.html
Ib Therkelsen
Last update: sep 1998.
Public domain article.
Questions, comments or error notification to:
ib "at" nmr.ku.dk
