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Saturday, September 29, 2007

Get a Load of this! Glide Angle, Wing Loading & Span Loading

You wanna fly model planes, so why bother with all this glide angle and loading mumbo jumbo? Well, because it is helpful and, as Mr Spock famously said, "fascinating". Glide angle is a fundamental concept. Wing Loading and Span Loading are important design considerations that affect the model's performance. In this post, I will explain the terminology and try to describe what it means. There is some maths, but don't let that put you off!

GLIDE ANGLE

Figure 1. Showing a plane in steady glide and how the lift force and drag force relate to the glide angle.

L is the lift force generated by the wing.

D is the drag force experienced by the model.

W is the model's flying weight.

h is the height above ground.

x is the distance travelled.

Glide Angle or L/D (pronounced "ell over dee") is an indicator of a glider's performance. From the geometry of the forces acting on a glider in steady flight, you can see that the glide angle relates directly to the ratio of the lift force over the drag force, regardless of the weight of the plane. The maths is set out in Figure 1 (above). High lift and low drag means a flatter (more horizontal) glide. The plane flies forwards, not down like a brick!

WING LOADING & SPAN LOADING

Wing Loading is simply the weight of the plane divided by the wing area. Span Loading comes in two "flavours". The first is the weight of the plane divided by the span. The second is the weight of the plane divided by the span squared. Unfortunately, both are referred to as "span loading". For convenience, I will call these Span Loading v.1 and Span Loading v.2 respectively.
Referring to Figure 2 (below):
  • Wing Loading or WL = W / S
  • Span Loading v.1 = W / b
  • Span Loading v.2 = W / b^2
Figure 2. Showing the important areas for Wing Loading and Span Loading
where:
W is the flying weight of the model.
S is the area of the wing (in blue in Figure 2).
b is the span, and S = ab, where a is the mean chord of the wing. The red area in Figure 2 above is b^2.

Qualitatively speaking, lower Wing Loading means that the model:
  • climbs better, both under power and when gliding: it has a "floatier" glide
  • flies slower with respect to the air
  • requires a shorter take off and landing for rise off ground models
  • is more prone to being bumped around by turbulence
Lower Span Loading v.1 means:
  • less drag at lower flying speeds, that is less "induced drag
Lower Span Loading v.2 means:
  • better glide performance, that is, a flatter more horizontal glide angle
Taken all together, the above indicates that a light weight plane with big span would be ideal. Just consider a typical full-size soaring glider (or 'sailplane' for readers in the USA). For example, the famous and beautiful Duo Discus. I've flown one of these and it was absolutely lovely. In these glass ships, when you push the stick forward, the plane just whooshes forwards (not downwards!).

Note that you can also reduce the Wing Loading (WL) of a given design by increasing the wing area while trying to keep the weight increase to a minimum. However, if you do this by increasing the wing chord alone while keeping the span the same this may not result in a flatter glide. Hopefully, these intricacies will become clearer after you've read the rest of this post.

LEARNING FROM MATHEMATICS!

I suggest that there are three equations worth considering, playing with and understanding. This is quite rewarding and helps to develop a feel for some basics of aerodynamics generally. They are:

Or, when limited by my keyboard:
L = q S CL ......(1)

Di = q S CDi ......(2)

CDi = (CL^2)/(pi AR E) .....(3)
where:

L is the lift force. For a decent model in a steady glide it is essentially equal to the weight.

q is 1/2(rho v^2), also called the "dynamic pressure". Rho is the air density, v is the airspeed.

CL is the coefficient of lift (which depends on angle of attack of the wing).

S is the area of the wing (see above).

Di is the induced drag force. The total drag is this plus the drag from other sources (skin friction and form drag). At lower speeds, the induced drag dominates.

CDi is the induced drag coefficient.

pi is its usual 3.14...

AR is the aspect ratio, that is the span over chord: AR = b/a = b^2/S.

E is a factor relating to the efficiency of the wing. It depends on the design and shape of the wing, 1 for perfect shape, otherwise less than 1, 0.7 is typical for a rectangular planform.

Eeek! What does all this mean?

First, there are some basic ideas contained in these three relationships.

The lift force depends directly on the wing area. Bigger wing, more lift. When the plane goes faster, the lift increases as the square of the airspeed. That's why jet airliners look as if they have smallish wings in relation to the size of the whole plane. Lift from the wing is greater at sea level than in the mountains, where density is lower. So your model may not fly well if you move to a flying location at a different altitude. The drag increases if you decrease the aspect ratio while keeping all else the same (which goes part of the way in explaining why increasing chord to reduce Wing Loading as discussed above, may not lead to a flatter glide).

Substituting (1) and (2) into (3) gives the induced drag:

Di = L^2 / (q pi E S AR) = (L/b)^2 / (q pi E)

This tells you that the induced drag depends on the quantity L/b squared. As mentioned above, in a steady glide, L is essentially the weight of the plane, W, so L/b is the same as Span Loading v.1. A small increase in Span Loading v.1 increases the induced drag significantly because of the squared relationship.

Dividing both sides by L gives the very important Di/L ratio (often referred to the other way round as L/Di):

Di/L = (L/b^2) / (q pi E)

From the geometry of the forces acting on a gliding aircraft, it is a measure of the glide angle (please see Figure 1 to see why). The above relationship shows that Span Loading v.2 is a key determinant of the glide angle (again using the relationship that L ~ W in a steady glide). The smaller Span Loading v.2, the flatter (more horizontal) the glide angle. Also, the more efficient the wing design (E gets closer to 1.0) and the flatter the glide angle.

Another thing to note from the above equations is that Span Loading v.2 = W/b^2 is just the same as WL/AR (just divide top and bottom by wing area S). This explains the point that I made above that reducing the Wing Loading by increasing the chord alone and not the span, may not flatten the glide because although the Wing Loading reduces, the AR decreases as well. We can now see that it will not flatten the glide unless the ratio WL/AR decreases overall.

CONCLUSION

Summing up, if you want to improve the Glide Angle of your model, then concentrate on reducing Span Loading v.2 and on improving the wing's design and efficiency (airfoil, planform, etc). In free flight, good models also tend to have low Wing Loading, at least for calm conditions.

I hope this gives a flavour of how powerful this stuff can be!

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