I originally thought that the terminal velocity of a smooth sphere in free fall through the atmosphere would be an easy question to answer. But for heavy spheres of small diameter, the terminal velocity appears to reach upwards of 200 m/s.

According to this paper the critical mach number for a sphere is about 0.57, which in air corresponds to a speed ~200 m/s. Above this, compressibility effects should start generating wave drag on the sphere among other things, furthur limiting terminal velocity. My question is: How can I model this effect for high speed spheres in free-fall in the atmosphere?

Here is an overview of my current process for determining terminal velocity.

The obvious starting point is to assume quadratic drag and find terminal velocity via

$$v_t = \sqrt{\frac{2mg}{\rho C_dA_c}}$$

The trouble here is the dependancy of $C_d$ on Re, and thus on $v_t$. Since I am trying to generate an emperical model variable over different mass and radius, I opted not to take the iterative approach to solving this dependency. Rather I noted that the value $C_dRe^2$ takes on a constant value dependant only on sphere mass. For a sphere, this is:

$$C_dRe^2=\frac{8mg}{\pi \rho_{air} \nu^2}$$

Using an emperical estimation for the $C_d - Re$ dependency found here which is valid for $Re$ up to $Re=1\times10^6$ (and captures the highly variable turbulence transition):

$$C_D=\frac{24}{Re} + \frac{2.6(\frac{Re}{5.0})}{1+(\frac{Re}{5.0})^{1.52}} + \frac{0.411(\frac{Re}{2.63\times10^5})^{-7.94}}{1+(\frac{Re}{2.63\times10^5})^{-8.00}} + \frac{0.25(\frac{Re}{1\times10^6})}{1+(\frac{Re}{1\times10^6})}$$

A value for $Re$ and thus $C_d$ can be determined. This can then be applied to the terminal velocity equation to find a final value of $v_t$. Here is where I run into problems.

Take the example of a 3cm radius sphere of gold ($m=2.15$). By my calculations this sphere would have a teminal velocity of 295 m/s (or about 660 mph). That is equivalent to $M=.86$, most definitely in the transonic flow regime. So how would this scenario actually play out in real life? Would the sphere actually travel slower than this theroetical 295 m/s? Is the difference in speed signifigant enough that it warrants modelling, or is this current approach a good approximation of terminal velocity? Is there a straightforward way of modelling the impact of high speed wave drag mathematically?


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