From: parfrk@rs6114.ecs.rpi.edu (Karl Andrew Parfrey)
Subject: aerodynamics
Keywords: aerodynamics
Reply-To: parfrk@rpi.edu
Organization: Rensselaer Polytechnic Institute, Troy, NY.
Date: Tue, 17 Nov 1992 01:15:58 GMT

Minor aerodynamics primer:

drag is usually measured in a drag-per-unit-cross-sectional-area-per-
dynamic-pressure (gasp!) manner.  This allows comparing the drag of
differently sized vehicles travelling at different speeds.  The formula
is 
	Drag = 0.5*Rho*V^2*S*Cd       where

		Rho is the air/fluid density through which the
			object is travelling ( Kg/m^3) 
		V is its velocity  (m/s)
		S is its cross sectional area (m^2)
		Cd is its coefficient of drag, which has no units

		This produces drag in units of Kg m/s^2, which is force (Newtons).

The term 0.5*Rho*V^2 is called the "Dynamic Pressure".  It is the
pressure that you would feel if the flow was brought to a dead stop,
i.e. the pressure at the very leading edge of a F1 car nose.  Such a 
location is known as a "stagnation point".

In a wind tunnel you actually measure the density (a function of temperature
and barometric pressure), the airstream velocity, and the drag force.  
From a knowledge of cross sectional area you calculate Cd.

Cd is actually a (weak) function of a dimensionless number known as 
the "Reynolds Number" which describes the flow. 

	Re = (Rho*l*V)/mu      where

		Rho is density
		l is some kind of representative object length
		V is velocity
		mu is viscosity (air has it too; its just a _whole lot
			less_ than oil.  again, a function of temperature
			and barometric pressure.)

The object of wind tunnel testing is to recreate the same Reynold's Number
as the real-life object experiences.  Thus if you are testing 1/3 scale
models (l_model = 1/3 l_real) at the same viscosity and density, you make the
speed 3 times higher (V_model = 3 V_real) so that Re_model = Re_real.
You can fiddle with the air temperature to change viscosity and density
if you want to change the speed.  You can even test underwater if you
find the right combination of model size and flow speed.

There are actually two kinds of drag: parasitic drag (Cd_0) and induced 
drag (Cd_i).  The total drag is the sum of the two.
	
	Cd = Cd_0 + Cd_i

Parasitic drag is caused by the friction of the airflow over the surfaces
of the vehicle.  Representative values:

	Object				Cd_0

	Sphere ()			0.5 (Re < 10^5) or 0.1 (Re > 10^5)
	Cup  )				1.4
	Disk  |				1.1
	Hemisphere (			0.36
	Bullet 				0.2
	Raindrop			0.07
	Wright Bros. Craft (1903)	0.046
	Spirit of St. Louis (1927)	0.029
	P-51 Mustang (1943)		0.017
	F104 Starfighter (1954)		0.017
	Boeing 727 (1963)		0.018
	A7-D Corsair (1965)		0.015
	Good Car (1990s)		0.3xx
	GM "Impact" electric car (1991)	0.2?
	Truck (1990s)			0.6

					C_d

	Galmer G92 IndyCar		1.397 Short Oval
					1.141 Street Circuit
					0.669 Speedway

	So you see that a modern formula car generates drag as efficiently as
a parachute (which would have a Cd like the Cup).

The reason it does so is because of the second kind of drag, induced drag.
When an airfoil generates lift or downforce the resulting force is never
perfectly vertical but rather contains a vertical component (lift/downforce)
and a horizontal component (drag).  This induced drag is inherent in 
the process of producing lift/downforce and cannot be removed by streamlining
or using a smoother wing skin.  The formula is:

	Cd_i = Cl^2 /(pi*e*AR)    where
		
		Cl is the coefficient of lift/downforce, a function
			of the airfoil type and angle of attack
			(lift,downforce = Dynamic Pressure*Cl*Wing Area)
		e is the wing "Oswald efficiency", a function of airfoil
		AR is the wing aspect ratio, the wing length divided by
			the wing width.

Things to note:
	drag/lift/downforce go with velocity squared
	induced drag goes with lift/downforce squared
	the bigger AR, the less induced drag.  thus long narrow wings
		(sailplane, soaring birds such as albatross or vulture)
		are more efficient than short stubby wings (Formula car,
		maneuverable birds such as swallow).  Thus from an
		aerodynamic point of view Formula car wings suck.
		This is one reason why multi-element wings are better than
		a single large wing: each element has a larger aspect ratio.

A more important measure is the L:D ratio - lift (or downforce) divided by
drag.  Some representative numbers:

	Object				L:D
	
	Baseball			0.0
	Rotating Baseball		!=0.0 (principle of curveball) *
	Housecat			0.0
	Rotating Housecat		Still 0.0 **
	Parafoil (rectangular
	  maneuverable parachute)	>3.0
	General aircraft		~10.0-20.0
	Mediocre Sailplane		~35-45
	Unlimited Class Sailplane	65+
	Gallmer G92, short oval		2.64
		     street circuit	2.84
		     speedway		2.92

Note that again that the IndyCar is outperformed by the parachute.  Humbling.
 
  * were there no pavement below the tires to interfere with the airflow
    around them, open wheels would generate a _tremendous_ amount of downforce.
    This intrigues me.  I'll do some numbers and repost... Jacques Cousteau
    has a sailboat which uses a rotating cylindrical sail like this.
  ** empirical data gathered from my kitten Spaz ;^)

One of the reasons ground effects was so quickly embraced is that the L:D
ratio of ground effect downforce is much better than downforce produced with
short, stubby, inefficient wings.  You can make as much downforce with 
wings; you merely need a bigger engine to pull the car through the air.


	-Kap

-- 
Kap Parfrey   (parfrk@rpi.edu)			The opinions expressed
Center for Composite Materials  		herein are not my own
 and Structures 				but rather those of my
Dept of Mechanical Engineering 			employer. I am merely a
Rensselaer Polytechnic Institute		puppet of the regime.
Troy Ny USA 						- Pinochio	
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