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I know from fluid dynamics classes that I can calculate the Reynolds number of a fluid in a pipe using:

$Re = \frac{\rho v d}{\mu}$

Where: $\rho$ is the fluid density, $v$ is the fluid speed, $d$ is the diameter of the pipe and $\mu$ is the viscosity of the fluid.

My question is, how would I calculate the Reynolds number for a free falling body? Say for a sky diver falling facing the earth. In that case I do not know what $d$ would represent... Maybe the diver average diameter? I know in case of a sphere that would be the sphere diameter, but getting back to the sky diver things get a little bit more difficult. If I'd used an average diamter and approximate a sky diver to a sphere I think I might be making a too rough approximation there...

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  • $\begingroup$ I'd imagine there are two different scenarios for a sky-diver; 1. he wants to descend slowly so he turns his body perpendicular against the moving air, this can be approximated by a flat disk with a diameter approximately his length; 2. he wants to descend quickly so he turn his body into the moving air making him a cylinder with some diameter approximately the size of his shoulders. Depending on his physical dimensions, his shoulders are generally smaller than his length :) $\endgroup$ – nluigi Sep 13 '16 at 15:36
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You may choose any length scale associated with the diver (for e.g. his height or his waist, etc.) to define $Re$. Then any $\textit{dimensionless}$ physical quantity (for e.g. drag coefficient experienced by the diver,$C_d$) that you shall measure will be a function of $Re$ alone, $C_d=f(Re)$, $\textit{for any geometrically scaled version of this particular diver}$ (assuming all other conditions, such as air flow, orientation of diver's body w.r.t. fall direction etc., remain the same).

But you may want the measured functional dependence, $C_d=f(Re)$, which has been measured for one particular diver, to be applicable to all divers with as low an error as possible, given that divers are only $\textit{approximately}$ geometrically scaled versions of each other. When this is the case (in fact this is usually the case) it helps to choose a length scale which is dominant in determining $C_d$ rather than make an arbitrary choice. Intuitively one would expect that greater drag is experienced by the diver if his projected area normal to his direction of falling is greater. If this area be denoted by $A$, then a sensible choice for length scale would be $\sqrt{A}$. Whether or not this choice is a good one may only be verified by actual measurements on different divers (while other conditions remain the same), and checking whether the same function $C_d=f(Re)$ holds for all of them with an accuracy acceptable to you.

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