Now on the naïve principle that honesty is the best policy, I’m going to confess that mathematically uninclined readers might find this particular blog pretty hard going.
This is not only because of its reckless descent into a tedious mire of aerodynamic explanation and calculation, but also because it touches on those banes of your schooldays, sines and cosines. What can I say but sorry? Anyway, assuming that warning doesn’t induce you to depart and see what’s new and exciting on YouTube, or at least cut to the chase of my third-from-last paragraph below, here goes:
So in my last blog I took a look at how, when it comes to darts, aerodynamic drag is basically a puny, insignificant kind of a force which can only cower abjectly when its much more extrovert rival, lift, conceptually swaggers up and kicks sand in its face.
But, although they might appear mismatched rivals for your attention when it comes to affecting your darts, drag and lift are not entirely unrelated. In fact, for a dart flying along at an angle of yaw, lift can generously stop being a bully and help to compensate for drag’s innate lack of machismo. This is partly due to trigonometry, but mostly due to a woeful lack of scientific precision in my terminology.
Drag and lift are properly defined as mutually perpendicular, one going back down the line of flight and the other going at right angles to it. Strictly speaking, therefore, neither can contribute to the other, but they can be transformed into a slightly different pair of mutually perpendicular forces. For symmetric projectiles, these are called the axial and normal forces, with the former being reassuringly along the axis of the barrel and the latter at right angles (that’s “normal” in maths-speak) to it. At very low yaw, axial force is thus closely allied to drag and lift to normal force. As the yaw increases, though, the first relationship is in the divorce courts a long time before the second!
For a typical dart thrown at 6m/s, the axial force (and drag) at zero yaw will be around 0.15gms (approximately 0.0015 Newtons for the SI units devotees amongst you – either way I told you it was puny!). With no yaw, of course, a perfectly straight dart won’t have any normal force or lift, but for, say, 15 degrees of yaw and standard size flights, the normal force will be around 1gm, nearly seven times the zero yaw drag.
Now that level of yaw will hardly alter the axial force, and similarly the lift will be only fractionally less (about 7% - as those of you who are into those pesky sines and cosines and have a frankly weird obsession for “resolving” forces can testify) than the normal force. The drag, on the other hand, will rise meaningfully from its beach towel and draw itself up to an almost non-puny height of 0.4gms, getting on for three times its zero yaw value. From there it can at least stare lift in its midriff rather than its knee!
Although this increase in drag is caused by a component of normal force and not of lift, as we’ve seen, at moderate yaw angles these two forces are numerically not very different. The additional drag is thus somewhat carelessly referred to as “lift-induced”, maybe because this is punchier (and better to hyphenate) than “normal force-induced”.
Thus the drag of a dart in flight will be the sum of two components, the zero yaw drag and the “yaw drag”, which can be loosely regarded as mainly due to lift. At moderate yaw angles, the lift and the component in the drag direction of its normal force associate are both usually pretty much proportional to the yaw (an assumption known as linearization), which means that the yaw drag tends to vary as yaw squared.
Using this principle, by a bit of mathematical jiggery-pokery (integrating sine squared, if you must know) it can be shown that, neglecting yaw damping for a moment, a dart which has a not unreasonable peak yaw of 30degs will, averaged over a yaw cycle, have the same drag as one travelling at a constant 15degs. For the example above, this means it will have nearly triple the drag it had at zero yaw and its fractionally longer time of flight will cause it to drop 2 to 3mm more at the board due to gravity.
This effect may still be significantly less than the possible deviation due to lift, but, unlike straightforward zero yaw drag, it too will be dependent on any vagaries in the yawing motion and can thus contribute a little to inaccuracy.
So there you are – at least one component of drag does matter (a bit) in darts after all, but that component is only really lift in disguise! That’s why darts with very big flights can seem too “draggy” and why no amount of skin friction mitigation or any other form of straightforward drag reduction would make much difference to that.
Phew! At last our journey to reaching that passingly-interesting conclusion is complete. Please accept my abject apologies for any boredom you’ve had to suffer along the way - I fear that the motivation for the title of this blog will by now have become distressingly clear!
That’s all for now folks – bet you can’t wait for my next exciting instalment!