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Issue No 35, 10 July 2023
By: Anthony O. Ives
Different types of drag has been looked at in some previous articles, [1] and [2]. This time we are looking at the main source of lift induced drag in fixed wing aircraft which is trailing vortex drag. Of course lift can cause increases in other types of drag due to changes in pressure distribution, etc. While trailing vortex drag is mainly associated with fixed wing aircraft the same effect causes an increase in helicopter rotor power known as induced power.
Trailing vortex drag is the result of high pressure on bottom wing surface and low pressure on the top wing surface causing to air to flow from the bottom wing surface to the top wing surface which induces a vortex which trails behind the aircraft. An illustration of trailing vortex drag is given in the picture below.
Trailing vortices can cause a hazardous flight condition for aircraft taking off immediately behind another aircraft particularly if there is a difference in their weight. For example a light aircraft taking of behind a heavy aircraft could cause the light aircraft to lose control if it flies in the vortices produced by the heavy aircraft. To prevent this happening aircraft usually wait before taking off for between 2 and 4 minutes if an aircraft is just taking off in front of them. Typical delays in take off times is given in the following table.
| Take Off Sequence | Time Separation |
|---|---|
| Medium aircraft behind a Heavy Aircraft | 2 Minutes |
| Heavy aircraft behind a Super Aircraft | 2 Minutes |
| Light aircraft behind a Heavy or Medium Aircraft | 3 Minutes |
| Medium aircraft behind a Super Aircraft | 3 Minutes |
| Light aircraft behind a Super Aircraft | 4 Minutes |
Helicopters are as equally exceptable to the effect of trailing vortices known as wake turbulence. The mathematical expression for trailing vortex drag coefficient has already been introduced in references [1] and [2]. The expression is given below:
\[C_D(Vortex) = k C_L^2\]
Where k is the drag factor and CL is the lift coefficient. The drag factor is given by the expression below:
\[k = \frac{1}{\pi A_r}\]
Where Ar is the wing aspect ratio. Wings with a high aspect ratio that is a large wingspan and a small chord produce less trailing vortex drag which explains why airliners and gliders have these types of wings as they designed for range. High aspect ratio wings however, are limited by structural strength considerations hence why some aircraft such as fighter jets, etc have low aspect ratio wings.
There are various other things that you can do to reduce vortex drag such as tapering the wing chord, essentially having a larger chord at the wing root tapering to a smaller chord at the wing tip. Tapered wings are used to approximately replicate elliptical wings. Wings with an elliptical chord distribution are the most efficient wings however, they are awkward and expensive to manufacture. The spitfire is the most famous aircraft to have an elliptical wing planform however, it was not for aerodynamic reasons, it was the only shape that could accommodate more guns. The spitfire could carry more guns in it wings take other aircraft of the day such as the Messerschmitt Me 109. More modern ways to reduce trailing vortex drag are winglets at the wing tips.
Vortices also occur at the tips of helicopter rotor blades and they increase power losses [3], the same principle applies to reduce power losses, that is to make the rotor blade with a high aspect ratio however, as with wings how high you can make the aspect ratio is limited by strength of the rotor blade material. The tip shape design can also be tweaked to give some reductions in power losses.
References [4] gives the full mathematical derivation of the equations for trailing vortex drag, reference [5] gives derivations for rotor power loss due to tip vortices and other effects of tip vortices.
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Mathematics
Helicopters
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References:
[1] http://www.eiteog.com/EiteogBLOG/No2EiteogBlogDragCD.html
[2] http://www.eiteog.com/EiteogBLOG/No29EiteogBlogDrag.html
[3] http://www.eiteog.com/EiteogBLOG/No16EiteogBlogThrust.html
[4] Fundamentals of Aerodynamics, John D. Anderson Jr., 3rd Edition, 2001, McGraw Hill
[5] Helicopter Theory, Wayne Johnson, 1980, Dover Publications
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