Drag Crisis and Terminal Velocity?

Question

From what I understand, there exists such a concept known as the "drag crisis," at which an object's boundary layer transitions from laminar to turbulent and its drag coefficient decreases dramatically. I'm wondering, what effect does this have on the terminal velocity?

We know the terminal velocity equation as:

$$Vt^2 =(2mg/CdAp).$$ But this assumes that the drag coefficient stays constant, which it does not. What happens to the terminal velocity during the drag crisis? How does the drag crisis apply to this equation? Does it apply at all, or am I misunderstanding the concept?

Answer 1

During the drag crisis the drag coefficient is indeterminate. In this situation, the drag coefficient and everything happening here is spurious. You get a haphazard drag coefficient that is unreliable, inconsistent and unpredictable. What formula do you then apply in such circumstances? It might be worth the trouble going into a detailed study of this, but very few scientists would want to venture into it.

Answer 2

This paper, "Polymer and surface roughness effects on the drag crisis for falling spheres" investigates how adding a polymer to water or adding roughness to the surface of a sphere changes the onset of the drag crisis. You can see from the graph below that adding polymer brings forward the drag crisis, resulting in the spheres falling faster.

As far as the terminal velocity goes, either the falling object will reach terminal velocity before the onset of the drag crisis or after. I think it would be tricky to engineer a situation where an object's terminal velocity was in the middle of the drag crisis, though it could be interesting to try.