Re: [Phys-L] raining
Weird. Each Greek [rho] in my previous post got translated into a tilded n
after passing through the Phys-l list. I trust the readers can cope with that.
It's a good thing I didn't use more than one Greek letter.
David Bowman
________________________________________
From: Phys-l <phys-l-bounces@mail.phys-l.org> on behalf of David Bowman via
Phys-l <phys-l@mail.phys-l.org>
Sent: Wednesday, December 20, 2023 8:16 AM
To: Phys-L@Phys-L.org
Cc: David Bowman
Subject: Re: [Phys-L] raining
Elaborating further on my previous post.
Assume the walking/running direction is horizontal/perpendicular to the
vertically falling rain & assuming the falling rain has a uniform distribution
in space and time.
Let ñ = mass density of falling rain (per unit volume of air + rain).
Let L = horizontal distance walked/ran through in the rain.
Let v_t = (mass weighted) mean terminal velocity of falling raindrops
Let v = speed of walking/running
Let A_th = body's projected top facing horizontal cross section area.
Let A_fv = body's projected front facing vertical cross section area.
Let W = mass of water intercepted by moving body.
Then W = ñ*L*(A_fv + (v_t/v)*A_th) .
This is just the projected gross rain interception rate due to simple
kinematics. It obviously neglects the hydrodynamics of tiny deflected air
motions partially carrying drops in the immediate vicinity of body collisions
with drops causing altered localized trajectories of drops that could affect
the collision rate by a tiny amount.
But a much bigger neglected effect is due to wind. Wind causes a number of
complications. Here are 4 of them off the top of my head. First, wind tends
to be gusty & quite time dependent and thus hard to model. Second, wind
changes the direction of the falling rain so the direction of the projected
area perpendicular to it must also change. Third, because of the poly-disperse
distribution of drop sizes there is a distribution of different terminal
velocities of the drops and this causes different sized drops to fall/move in
different directions at different speeds. Fourth, there is a Galilean
transformation involved for the effective walking speed due to any component of
wind which may be parallel the walking direction, and this changes the
effective walking speed for those drops which are carried with that horizontal
component of velocity by the wind.
I suspect there are even more complications from wind, but I'm not in the mood
to try to think of them.
David Bowman