SOME MEASUREMENTS OF ATMOSPHERIC TURBULENCE. 
11 
The eddy-viscosity can however be measured rigorously by smoke-puff observations 
made in such a manner as to fit in with Osborne Reynolds’ theory of eddy-stresses. # 
This theory is remarkably free from assumptions which might limit its generality. 
It is to be found in Lamb’s ‘ Hydrodynamics,’ IV. edn., Art. 369. 
The equations of motion are three, such as 
0 ( pV x ) 0 « , 0 / \ , 0 / \ , 0 / \ 
A k =£ + te ( ^ )+ dy + SE (W ’“> 
^-2,„sin <pv Y -c I V s ' 
dx 
dx 
■ ( 13 ) 
where c is the “ molecular ” or “ ordinary ” viscosity. Note that there is no need to 
assume p to be independent of position. Reynolds assumed this, but for a reason that 
does not concern us. It will be necessary however to assume that p , the variation of 
density at a fixed point, is so much smaller in comparison with p than is v' in 
comparison with v, that we may put p = 0. This being so, we find on taking the 
mean that (13) becomes 
j 3 {p v x)\ 
1 dt J 
The left side of (14) is the difference between pv x at the beginning and at the end of 
the period through which the average is taken, divided by the period ; and that is 
what we want. The right side of (14) is of exactly the same form in the mean 
quantities p, v x , v Y , fi H as (13) was in the corresponding instantaneous quantities 
p, v x , v Y , v K ; except that there is added a force per unit volume in the x direction 
equal to minus 
^{pv' x v' x )+ J 1 - {pv' x v f Y ) + | (pv' x v' H ). ...... (15) 
ox on oh 
dp , d\h 0 • 0divv , ™- 
dx +p dx~^ sm + v ^ 
+ ^ {pVx • fix + pv'x ■ v' x ) + — ( pv x . V Y + pv' x . v' Y ) 
+ - ( pv x . v B + pv' x v' K ) . 
(14) 
On working out the corresponding equations for the y and li components, it is seen 
that this additional force per unit volume is just that which would be given by the 
following systems of stresses 
XX — p v x p x ? 
xy — — pv x v' Y ; 
yy = 
-pv'W Y ; 
hh = - P v' n v ' H 
tyh = 
— pv' Y v' n ; 
fox* —— — pV £ 
(16) 
* Major G. I. Taylor tells me that he attempted to measure xh with a balloon on an elastic tether 
in 1914. 
