Drag CRISIS! The Coefficient of Drag is NOT a Constant!

There’s a mystery out there. Scientists are still working exactly what it is, but it’s real. It’s proven, in theory and in practice.

This article attempts to address an area of aerodynamics and fluid-dynamics which seems to have eluded much of the Internet. I’ve searched long and hard for relevant information. There’s just not a lot out there. Even the all wonderful WikiPedia is begging for someone to update it’s pathetic 100 word explanation.

Aerodynamic drag is a serious business, whether designing ships, cars, aircraft, submarines, or baseballs.

Consider a car at highway speeds…. the majority of power your engine is producing to keep you moving is to overcome two things.

1. Friction of the tires on the pavement. 4000lbs of weight on 4 tires plus gravity = slowing down real fast.
2. Resistance of your vehicle to movement through the air.

One of these two things above will never change, no matter how fast you go. The other does change according to how fast you are going. Air resistance increases with the square of the speed.

At some point, the drag you have to overcome will exceed the amount of power available to you, and you will be pushing a wall of air so thick and hard, that you will never succeed in accelerating beyond it, no matter how much horsepower you throw at it.The penultimate top speed of any design is dictated by the shape and size. The more streamlined, the less drag.

Coefficient of Drag

There is a number that defines how resistive an object is to a flow. It takes a variety of things into consideration, how slippery/shiny/smooth the surface is, how sharp it is (how well air can flow around it), and how big it is relative to the stream of molecules in the flow.

One can even say it’s a number that describes how aerodynamic something is. It’s a number that starts just above 0 and can end WAY WAY up there. We call this the Coefficient of Drag. This can be used in many calculations, and remains a virtual constant, no matter how fast or slow you are going, and no matter what “fluid” you’re in.

A Honda Civic’s Drag Coefficient is 0.36. It doesn’t matter if you’re going 1km/h or 100 km/h, in air, or underwater. The form of the front of that car will always have a drag coeffecient of 0.36

This is a number that we can use to predict the opposing force of motion through a fluid at any given speed.

Knowing the drag coefficient plus the surface area (the frontal area) of the vehicle, we can determine how much opposing force there is “pressing” against the car at different speeds.

Assuming average full-size passenger cars have a drag area of roughly 8.5 ft² (.79 m²), knowing the 0.36 Cd (Coefficient of Drag), and the density and viscosity of air (about 1.2kg/cubic metre), we can use the Drag Equation to find out, in a real world measurements, how much drag a Civic actually produces traveling at different speeds.

Drag Equation

Fd is the force of drag,

ρ is the density of the fluid (Air is about 1.293 kg/m3)

v is the velocity of the object relative to the fluid,

A is the reference area

C_d is the drag coefficient

In order to determine the density of the fluid (in this case air), we can estimate it based on temperature, pressure, and altitude.  In otherwords, the Barometric Formula. For our exercise today, we will assume a cool day down by the ocean, about 1.293kg/m3.

So let’s work it for a few different speeds; 50 km/h, 80 km/h, 100 km/h, 110 km/h, 115 km/h

(1/2)(1.293)(50km/h^2)(0.36)(.79)

=.5(1.293)(13.9m/sec^2)(0.36)(.79)

= .5(1.293)(193.21)(0.36)(.79)

=3.5524479366 kg

(1/2)(1.293)(80km/h^2)(0.36)(.79)
=.5(1.293)(22.2m/sec^2)(0.36)(.79)
= .5(1.293)(492.84)(0.36)(.79)
=9.0615829464 kg

(1/2)(1.293)(100km/h^2)(0.36)(.79)
=.5(1.293)(27.8m/sec^2)(0.36)(.79)
= .5(1.293)(772.84)(0.36)(.79)
= 14.2097917464 kg

(1/2)(1.293)(110km/h^2)(0.36)(.79)
=.5(1.293)(30.6m/sec^2)(0.36)(.79)
= .5(1.293)(936.36)(0.36)(.79)
= 17.2163456856 kg

(1/2)(1.293)(115km/h^2)(0.36)(.79)
=.5(1.293)(31.92m/sec^2)(0.36)(.79)
= .5(1.293)(1018.89)(0.36)(.79)
= 18.7337802294

As you can see, it took from approximately 0-100km/h for the first 10kg of drag force, then only100-120kph for the next 10kg of force! Before we are even beyond 200kph, we have already reached 60kg (130lbs!).

It doesn’t matter how big your engine is, you’ll never make past a certain point (OK, maybe that civic engine can be replaced with a HEMI, but even then, you’d probably only acheive an extra 10kph at best.

Reynolds Number

Here’s another beautiful number we can use. The Reynolds Number is the ratio of inertial forces to viscous forces. In other words, its a number to describe how big and fast something is, compared to the gooeyness and density of the liquid around it. This allows us to build an idea of “equivalency”, and to scale any dataset produced in a test environment (like a wind-tunnel).

Reynolds numbers tells us about “flow regimes”, and allows us to compare the similarity of flows. For example, we can state that dynamically similar aerodynamic realms involving fluid dynamics are only equal if the Reynolds Numbers are equal. NASA explains “It is possible for an experiment with a helium-filled balloon 100 cm in diameter rising in air to be dynamically similar to a 9.60 cm plastic ball falling in water if the Reynolds Numbers are the same.”

Typically, when we find what Reynolds Number we are looking at, we express it as either a number (50,000) or a factorial expression (5x 10^4).

Here’s some examples of Reynolds flow regimes:

Probably one of the most studied elements in fluid-dynamic studies is the sphere. A sphere is easy to scale, and it’s surface is uncomplicated and perfectly smooth. This makes it an ideal candidate to use in our Reynolds number demonstration.

Drag Crisis

The Reynolds number (R), for a baseball is calculated using R=vd/υ. Where the diameter (d) of the baseball is (7.32 cm), v is the velocity relative to air, and υ is the kinematic viscosity of air (about 0.000015 m2/s at 20 C) 6. So the greater the velocity becomes the greater the Reynolds number. A drag crisis occurs when the laminar flow of air in a boundary layer near the ball begins to separate and becomes turbulent. The effect that the turbulence in the boundary layer causes will actually reduce the size of the turbulent wake behind the ball, and reduce the drag force. The drag crisis produces a regime where the aerodynamic drag force actually decreases as the velocity increases.

Links

http://www.brianhetrick.com/casio/tbb1exapumdrg.html -Calculator Programming Tutorial – this was the ONLY broken out equation I saw on the whole internet.

http://www.aerospaceweb.org/question/aerodynamics/q0215.shtml – Golf ball dimples – excellent ideas for reducing drag.

http://www.uam.es/personal_pdi/ciencias/agrait/nico_archivos/docencia/fisica%20de%20fluidos/Life%20at%20Low%20Reynolds%20Number,%20EM%20Purcell%201973.htm – life at Low reynolds Numbers – Excellent read about how things like sperm and cellular life makes motion in the liquid concrete around them

http://en.wikipedia.org/wiki/Density_of_air - Density of Air

http://www.economicexpert.com/a/Drag:equation.htm - A reasonable quick and dirty explanation of the drag equation.