fluid dynamics - Is the wind's force on a stationary object proportional to $v^2$?

I am on a boat docked at Cape Charles, VA, about 30 or 40 miles from the center of Hurricane Irene. This understandably got me thinking about the force of wind on the boat. Since air friction is proportional to the speed squared (except for some friction types I'm sure someone will be kind enough to remind me of), is the wind's force on the boat also proportional to the wind's speed squared? In other words, will a 70 knot wind produce almost twice the force on the stationary boat as a 50 knot wind, all other factors being equal?

Answer

On the most basic level I would say that both your suspicion and your reasoning are correct, although some caveats should follow.

First I should address this on the basis of simple kinematics. Yes, the energy of an air molecule is $m v^2$ but we need to formalize when and if this will translate into pressure. Imagine a pipe that spews a fluid onto a surface, and the fluid comes to a complete rest after impacting the surface. As speed changes the impulse from a molecule in the fluid increases (and impulse is the more true analog to force/pressure than energy), but so does the flow rate, $\dot{m}$. We say that area and density stays the same.

$$\dot{m} = \rho V A$$

$$P=\frac{F}{A}$$

$$F=\dot{m} \Delta V = \rho A V^2$$

$$P = \rho V^2$$

If you study thermal hydraulics, this will not be the last time you see that final expression. Not by a long shot. I should point out that it's not exactly energy per se that leads to the $V^2$ nature, I would say that it's the above math that leads to it.

The example I provide (much like a garden hose spraying someone holding a shield) is a similar system to that which you describe, which is wind in a hurricane hitting the wall of a house. There are several ways that these two systems are not identical, and science has done a lot of work to develop empirical correlations and representative models for fluid pressure losses.

We can look at the system as either a forms loss (on a human-sized scale) or as a boundary layer friction force (on the larger climate level). Equations for the change in pressure of a fluid due to friction or due to a obstacle ("forms" loss) are as follows.

$$\Delta P_{friction} = f \frac{L}{D} \frac{\rho V^2}{2}$$ $$\Delta P_{forms} = K \frac{\rho V^2}{2}$$

In the above equations, both $f$ and $K$ represent constants that we multiply the $V^2$ term by to account for all of the real-life complexity of fluid flow. Yes, these constants will change with the wind speed, so the answer is that the pressure is not perfectly proportional to $V^2$, but we wouldn't be using the above format if this wasn't the closest form that it does follow.