Estimate the terminal speed of a hailstone falling through still air

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluids and Continuum Physics, Ch Dimensional analysis, dynamical similarity and the dimensionless groups

Problem

Estimate the terminal speed of a hailstone falling through still air. Treat it as a smooth sphere of diameter

d = 15\ \text{mm}

made of ice of density

900\ \text{kg m}^{-3}

; air has density

\rho = 1.21\ \text{kg m}^{-3}

and dynamic viscosity

\mu = 1.8\times10^{-5}\ \text{Pa s}

, buoyancy in air is negligible and

g = 9.8\ \text{m s}^{-2}

. The measured drag curve of a smooth sphere, with

C_D = F/(\tfrac{1}{2}\rho U^2 A)

A = \pi d^2/4

the frontal area, and

Re = \rho U d/\mu

both built on the diameter, reads (as pairs

Re

C_D

0.1

240

;

1

27

;

10

4.1

;

10^2

1.1

;

10^3

0.47

;

10^4

0.47

;

10^5

0.47

;

2\times10^5

0.47

;

6\times10^5

0.10

. For

Re \lesssim 1

the curve follows the Stokes line

C_D = 24/Re

; from

10^3

2\times10^5

it is flat at

0.47

; above about

4\times10^5

it has dropped to about

0.10

. ((a)) Estimate the terminal speed, taking the drag coefficient from the flat part of the curve. ((b)) Compute the Reynolds number at that speed and check that your reading of the curve was legitimate.

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