Rain drop equations

Question

Motivated by this question, I am thinking about describing a rain drop beyond the elementary physics. The principal improvement, it seems, would be accounting for the drag force dependence on the size and shape of the drop, which in turn depends on

the pressure (i.e., the hight above the ground)
the presence of wind
the speed of the drop
the dust particles that it might have absorbed while falling

and other factors.

I am wondering what is really essential and what is not... and whether there are established models of rain drops - this seems like a ubiquituous object, appearing not only in meteorology, but in many industrial applications.

Update

E.g., one effect related to the speed of the drop would be whether we use linear or quadratic drag term - see Explanation that air drag is proportional to speed or square speed? (also relevant: terminal velocity, drag equation, Stoke's law - mentioned in the comments by @J.G.).

@J.G. thanks for bringing it up. – Roger V. Oct 13 '21 at 12:21 — Roger V., Oct 13 '21 at 12:21

score 1 · Answer 1 · answered Oct 25 '21 at 09:39

1

Here is one more interesting article from the CFD/VOF perspective inside and outside of a raindrop in liminar and turbulent flow. The oscillations diagram inside the drop is amazing.

answered Oct 25 '21 at 09:39

Samson

535

Thanks for the link! – Roger V. Oct 25 '21 at 09:40

score 0 · Answer 2 · answered Oct 25 '21 at 09:08

This article, The Physics of Falling Raindrops in Diverse Planetary Atmospheres, seems to provide arather good recapitulation of what is essential for description of the rain drops, and provdies references to many relevant texts. To mention the main points:

Rain drops quickly reach terminal velocity with quadratic drag force.
The drag coefficient depends on the shape of the drop (via the Reynolds number), which in turn depends on the drop speed, so the equations for the terminal speed and the drag coefficient need to be solved self-consistently.
As a raindrop falls, it evaporates. The evaporation obviously affects the terminal speed, which thus changes as the drop falls. Importantly, the drop might evaporate before it reaches the ground, that is there is lower limit on the initial size of the drops that will reach the ground as rain.
Raindrop is held together via the surface tension forces, which limit the maximal size of the drops that can form. If this size is smaller than that required for the raindrops to reach the grown, the rain evaporates.

A piece of trivia: raindrops do not have a tear shape, but that of a prolate spheroid (and ellipse rotated about its minor axis, see here).

Rain drop equations

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