Drag under the Indy car

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As you might be able to tell by the lack of activity on the blog, I’ve been pretty busy the past couple weeks. Which makes it kind of hard to write about Mythbusters at my usual level of detail.

Fortunately, Rhett Allain has already done it for me. He analyzed last week’s episode to figure out whether air pressure is enough to pick up a manhole cover. And it is: an air pressure of \(\SI{101325}{Pa}\), normal atmospheric pressure, applied to a manhole’s surface area of \(\SI{0.369}{m^2}\) (seriously, click the link), gives a force of \(\SI{37400}{N}\) — over eight thousand pounds! That’s way more than enough to pick up a 300-pound manhole cover.

Of course, that would only happen if the manhole cover had a near-perfect vacuum above it. Perhaps if it were a manhole on a spaceship. But that’s not the situation on Mythbusters. (How cool would that be? Mythbusters in space… but I digress.) The manhole cover under an Indy car has air on both sides; the pressure of the air above is reduced, though, due to Bernoulli’s principle.

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$$

Given that the difference in height, \(h_1 - h_2\), is probably negligible, and the air underneath the cover isn’t moving (\(v_2 = 0\)), that gives

$$\frac{1}{2}\rho v_\text{above}^2 = P_\text{below} - P_\text{above}$$

The Mythbusters only measured 37 pounds of force when they ran their car over the manhole cover at \(\SI{150}{mph}\), which corresponds to an air speed of \(\SI{27}{m/s}\), about 61 miles per hour, if the pressure drop is in fact entirely due to the Bernoulli effect. That’s Rhett’s blog post in a nutshell.

Of course, one might wonder, is the pressure drop actually due to the Bernoulli effect? I’m not an expert in fluid dynamics, but I suspect there’s a lot more going on. When a fluid (like air) flows adjacent to a surface, it has some tendency to “stick” to the surface, measured by its viscosity. The region in which the fluid’s flow is strongly affected by its viscosity is called a boundary layer. It’s often characterized by reduced speed of the flow near the surface, as in this image from Wikipedia,

Laminar flow in a boundary layer

but more complicated things, like turbulence, can happen as well.

In the Mythbusters’ Indy car experiment, there are actually two surfaces, the car and the road. It stands to reason that the car will drag the air next to itself along at \(\SI{150}{mph}\) or so, while the road will “drag” the air next to itself along at zero speed — that is, it tends to slow the air down. Each of these two surfaces will attempt to create its own boundary layer of air. If the two boundary layers can both fit in the space between the car and the road, then they will probably do so, but if they don’t fit, messy things happen.

Here’s the formula for the thickness of the boundary layer for fluid flowing over a flat plate:

$$\delta = \begin{cases}C_1 x \biggl(\frac{\nu}{u_0 x}\biggr)^{1/2} & \text{laminar (smooth)} \\ C_2 x \biggl(\frac{\nu}{u_0 x}\biggr)^{1/5} & \text{turbulent}\end{cases}$$

where \(\nu\) is the kinematic viscosity, \(x\) is the distance “downstream” from where the surface begins, \(u_0\) is the speed of the fluid far away from the surface, and \(C_{1,2}\) are constants roughly of order 1. Wikipedia gives \(C_1 = 4.91\) and \(C_2 = 0.382\), so I’ll just run with those values. For air, \(\nu = \SI{1.46e-5}{m^2/s}\). If we’re simulating the Indy car in the Mythbusters’ experiment, it’s reasonable to assume \(u_0\) is not more than \(\SI{150}{mph}\), because nothing makes the air move much faster than that. And finally, \(x\) maxes out at the length of the car, around \(\SI{5}{m}\).

Plugging the numbers in gives \(\delta = \SI{5.1}{mm}\) for laminar flow, which fits well within the 7/8 of an inch clearance they mentioned on the show — but for turbulent flow, I get \(\SI{64}{mm}\), which does not. So I would expect that the airflow underneath an Indy car is strongly affected by boundary effects.

Now, why is this important again? Bernoulli’s principle applies best when viscosity can be neglected; essentially, it works for frictionless fluids. But in a boundary layer, viscosity plays a large role in determining the fluid’s behavior. Evidently, viscosity can’t be neglected for the air under the Indy car, and that means the application of Bernoulli’s principle is somewhat suspect.