Taking the xz-average of this, we observe that the first term of the first member disappears, since A . v is zero, and the first term of the second member disappears, since A . ∂(u′v)/∂x is zero. Denoting by 1/3R2 the average value of u2, v2, or ω2, so that R may be called the average velocity of the turbulent motion, the equation becomes
,
where
.
Let p be written (), where p′ denotes the value which p
would have if f were zero. The equations of motion immediately give
;
and on subtracting the forms which this equation takes in the two cases, we have
,
which, when the turbulent motion is fine-grained, so that
f(y, t) is sensibly constant over ranges within which u′, v, 'ω
pass through all their values, may be written
.
Moreover, we have
;
for positive and negative values of u′, v, ω are equally probable; and therefore the value of the second member of this equation is doubled by adding to itself what it becomes when for u′, v, ω
we substitute -u′, -v, -ω; which (as may be seen by inspection of the above equation in Δ2p) does not change the value of p′.