Aerodynamic Drag

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

causes & factors

The force on an object that resists its motion through a fluid is called drag. When the fluid is a gas like air, it is called aerodynamic drag (or air resistance). When the fluid is a liquid like water it is called hydrodynamic drag (but never "water resistance"). Drag is a complicated phenomena and explaining it from a theory based entirely on fundamental principles is exceptionally difficult. Drag is a topic that is best explored experimentally. Since this is a book (of sorts) and not a laboratory, our experiments will have to stay mental. Let's begin by identifying the causes of drag and the factors affecting it.

Fluids are characterized by their ability to flow. In semi-technical language, a fluid is any material that can't resist a shear force for any appreciable length of time. This makes them hard to hold but easy to pour, stir, mix, and spread. As a result, fluids have no definite shape but take on the shape of their container. Fluids are unusual in that they yield their space relatively easy to other material things; at least when compared to solids. A fluid will get out of your way if you ask it. A solid has to be told.

Fluids may not be solid, but they are most certainly material. The essential property of being material (in the classical sense) is to have both mass and volume. Material things resist changes in their velocity (this is what it means to have mass) and no two material things may occupy the same space at the same time (this is what it means to have volume). The portion of the drag force that is due to the inertia of the fluid -- the resistance to change that the fluid has to being pushed aside so that something else can occupy its space -- is called the pressure drag (or form drag or profile drag).

Solve the definition of pressure for force and substitute in Bernoulli's equation for the pressure in a moving fluid …

P =	F	⇒	F = PA =	⎛ ⎝	1	ρv²	⎞ ⎠	A
	A				2

The factors that affect pressure drag are simple to identify and understand. But the devil lies in the details.

Drag increases with the density of the fluid (ρ). More density means more mass, which means more inertia, which means more resistance to getting out of the way. The two quantities are directly proportional.

R ∝ ρ
Drag increases with area (A). Exactly what we mean by this is subject to debate. To me, and in the context of this model, area is the cross sectional area projected in the direction of motion. (I would further simplify this by calling it the projected area.) Take the cross section of the object in the direction of its motion. This is the area of the tube of fluid that must be cast aside to let the object pass. This is the most logical thing to call the area, but not everyone agrees with me. To some, the word "area" refers to the area of contact between the object and the fluid. This also makes sense, but not in the context I've described above. Surface area is not important when one is dealing with pressure drag, but it is important when dealing with viscous drag. More surface area means more of the object is in contact with the fluid. (Houston, we have a problem.) Both notions of area are factors that affect drag. Viscous drag is just as real as pressure drag. Therefore, whenever one mentions area as a factor in a drag problem, it must be specified whether one is referring to projected are or surface area (or even some other type of area).

R ∝ A
Drag increases with speed (v). I hope that this is self-evident. An object that is stationary with respect to the fluid will certainly not experience any drag force. Start moving and a resistive force will arise. Get moving faster and surely the resistive force will be greater. The hard part of this relationship lies in the detailed way speed affects drag. Are the two quantities directly proportional? Does drag increase as the square of speed? The square root of speed? The cube of speed … ? According to our model, it should be the first of these. Drag should be proportional to the square of speed.

R ∝ v²

But for some situations this is not quite correct. As I said before, drag is a complex phenomena. It is cannot always be written with simple mathematical formulas. My first guess would always be that drag is proportional to the square of speed, but I would not be surprised if, over some range of values, it was found to be directly proportional, or proportional to the ³/₂ power, or even that drag and speed were related by some polynomial. Welcome to the world of empirical modeling -- where relationships are determined by actual physical experiments rather than an ideology of pure theory. Which brings us to our last factor …
Drag is influenced by other factors including shape, texture, viscosity (which results in viscous drag or skin friction), compressibility, lift (which causes induced drag), boundary layer separation, and so on. These factors can be dealt with separately in a more complete theory of drag (how tedious in one sense, but how necessary in another) or they can be piled into one monolithic fudge factor (oh yes, please) called the coefficient of drag (C_d).

R ∝ C_d

Combining all these factors together yields a theoretically limited (but empirically very reasonable) equation. I like to use the symbol R to represent drag, but there's certainly nothing wrong with using D if you wish. Here it is …

R =	1	C_dρAv²
	2

Simple, compact, wonderful. A nice equation to work with -- or is it?

Well, yes and no.

Yes, but it works only as long as the range of conditions examined is "small". That is, no large variations in speed, viscosity, or crazy angles of attack. The way around this is to reduce the coefficient of drag to a variable rather than a constant. (I can live with this.) C_d depends on some yet to be specified set of factors. It is totally acceptable to say that C_d varies with this that or the other quantity according to any set of rules determined by experiment.
No, since speed is squared. [Gasp!] Recall that speed is the derivative of distance with respect to time. Have you ever tried to solve a nonlinear differential equation? No? Well, welcome to hell. Wait, let me rephrase that -- Welcome to Hell! [Ca-rack! Boom!] Ah ha ha ha ha haaaa! [Rumble] You fool! No one can manage the square of a derivative. The mathematics will consume you. [Ca-rack! Boom!] Ah ha ha ha ha haaaa! [Rumble]

Whew. What the hell was that all about? I might not know how to solve every kind of differential equation off the top of my head, but so what. I can always look for the solution in a book of standard mathematical tables or an on-line equivalent.

C_d	object or shape
Selected Drag Coefficients
2.1	rectangular box
1.8~2.0	eiffel tower
1.3~1.5	empire state building
1.0~1.4	skydiver
1.0~1.3	person standing
0.9	bicycle
0.7~1.1	formula one race car
0.6	bicycle with faring
0.5	sphere
0.7~0.9	tractor-trailer, heavy truck
0.6~0.7	tractor-trailer with faring
0.35~0.45	suv, light truck
0.25~0.35	typical car
0.05	airplane wing, normal operation
0.15	airplane wing, at stall
0.020~0.025	airship, blimp, dirigible, zeppelin

terminal velocity

It's much more than the name of a bad movie. It's something every student of aerodynamic drag should understand.

Imagine yourself as a parachute jumper; or better yet, imagine yourself as a BASE jumper. (BASE is an acronym for building, antenna, span, escarpment.) Since none of these platforms is moving horizontally, none of these jumpers has any initial horizontal velocity. Not that it really matters, but this reduces the complexity of the situation just enough. Step off from your platform and draw the free body diagram. With no initial velocity, there is no aerodynamic drag, and the jumper is effectively in free fall with an acceleration of 9.8 m/s².

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Now it gets complicated. There is an initial acceleration, therefore there is an increase in speed. With an increase in speed comes an increase in drag and a decrease in net force. This decrease in net force reduces acceleration. Speed is still increasing, just not quite as fast as it was initially.

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Speed increases, but so too does drag. As drag increases, acceleration decreases. Eventually one can imagine a state when the drag and weight forces are equal. At this point the object will be in equilibrium. It will not cease to move, but rather it will cease to accelerate. The speed that an object has in this state is a large as it can be given the circumstances of the jump. We have reached terminal velocity. There can be no speed greater than or less than this one. Given the usual posture of a skydiver, his or her position, the type of clothes he or she wears, and the state of the air on the way down; your typical skydiver will not be able to exceed 55 to 60 m/s (200 to 210 km/h or 125 to 135 mph).

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That is until the parachute opens. Opening the chute significantly increases the projected area of the skydiver, which cranks the aerodynamic drag up proportionally. Drag now exceeds weight and the acceleration is directed upward. Note: this does not mean the skydiver is moving upward. The direction of motion of an object and its direction of acceleration need not be the same. In fact, in this case they are directed opposite one another.

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Summary

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Problems

practice

Determine the terminal velocity of a falling body when …
1. drag is directly proportional to speed
2. drag is proportional to the square of speed
Solution …
1. Answer it.
2. Answer it.
Determine the velocity of a falling body as a function of time when …
1. drag is directly porpotional to speed and
2. drag is proportional to the square of speed.
Solutions …
1. Answer it.
  
  Check the limits of this equation at t = 0 and t = ∞. Talk about terminal velocity and characteristic time.
2. Repeat using the more accurate, but more difficult v² relationship.
  
  Wow, hyperbolic tangent! Check the limits of this equation at t = 0 and t = ∞. Terminal velocity and characteristic time once again.
Now, taste and compare.

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Write something different.
- Answer it.
Write something completely different.
- Answer it.

conceptual

A book and a page from that same book are both held horizontally and then dropped. Which one has …
1. the greater projected area (assuming no significant change in shape)?
2. the greater aerodynamic drag?
3. the greater weight?
4. the greater net force?
5. the greater velocity on impact with the floor?
A BASE jumper steps off the roof of a tall building followed shortly thereafter by a second jumper. The building is tall enough that aerodynamic drag should be considered. What happens to the separation between the two jumpers from the time the second jumper steps off the roof to the time when the first jumper lands on the ground?

calculus

Determine the velocity as a function of time for a car starting from rest, propelled forward by a constant force F against …
1. a drag force directly proportional to speed and
2. a drag force proportional to the square of speed.
An object is thrown vertically upward with an initial speed v₀. The aerodynamic drag is directly proportional to the speed of the object.
1. Determine the following quantities as functions of time from t = 0 until the object stops moving upward …
  1. velocity
  2. position
  3. acceleration
2. How long does it take for the object to reach its maximum height?
3. To what maximum height does the object rise?

statistical

Two different bicycles were tested in a wind tunnel at the Massachusetts Institute of Technology (MIT) — an ordinary "stand up" road bike with drop handlebars and a recumbent bike (a bicycle you ride from a seated position). The drag force was measured at three different wind speeds while coasting and while pedaling. The rider on the road bike adopted three different postures. The recumbent was tested with and without a fairing (a plastic aerodynamic shield). Here are the measurements in their original English units.

		recumbent		road bike
Drag Force (lb) on Two Different Bicycles
speed (mph)		without faring	with faring	on top bar	on drops	aero tuck
coast	10	1.92	2.40	2.40		1.44
	20	10.08	8.64	10.56		7.20
	30	24.00	19.68	26.88		15.84
pedal	10	2.88	3.12	2.64	3.12
	20	9.36	9.84	10.56	10.08
	30	24.48	19.44	26.40	20.40

Here are the same measurement in SI units

		recumbent		road bike
Drag Force (N) on Two Different Bicycles
speed (m/s)		without faring	with faring	on top bar	on drops	aero tuck
coast	4.47	8.54	10.68	10.68		6.41
	8.94	44.84	38.43	46.97		32.03
	13.41	106.76	87.54	119.57		70.46
pedal	4.47	12.81	13.88	11.75	13.88
	8.94	41.64	43.77	46.98	44.84
	13.41	108.90	86.47	117.44	90.74

Determine the drag coefficient on the …

recumbent bike without a fairing
recumbent bike with a fairing
road bike with the rider's hands on the top bar of the handlebars
road bike with the rider's hands on the drops of the handlebars
road bike with the rider in the aero tuck posture

Source: Aerodynamic Performance of Vision Recumbents, Grant Bower, 01 November 1999.

Resources

high-rise cat syndrome
- Mehlhaff, Cheryl and Wayne Whitney. "High-Rise Syndrome in Cats." Journal of the American Veterinary Medical Association. Vol. 191 (1987): 1399-1403.
- Diamond, Jared. "Why Cats Have Nine Lives." Nature. Vol. 332 (14 April 1988): 586-587.
- Diamond, Jared. "How Cats Survive Falls from New York Skyscapers." Natural History. (August 1989):20-26.
- "46 Floors Down, Still Purring." New York Times. (10 July 1994): CY6.
general
- Projectile Question (try to get it wrong), Cambridge Physics Outlet

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