Take now the xz-averages of both members. The quantities
∂u′/∂t, ∂u′/∂x, v, ∂p/∂x have zero averages; so the equation takes the form
,
if the symbol A is used to indicate that the xz-average is to be taken of the quantity following. Moreover, the incompressibility of the fluid is expressed by the equation
whence
.
When this is added to the preceding equation, the first and third pairs of terms of the second member vanish, since the
x-average of any derivate ∂Q/∂x vanishes if Q is finite for infinitely great values of x; and the equation thus becomes
(1)
From this it is seen that if the turbulent motion were to remain continually isotropic as at the beginning, f (y, t) would constantly retain its critical value f(y). In order to examine the deviation from isotropy, we shall determine A∂(u′v)/∂t, which may be done in the following way:—Multiplying the u- and v-equations of motion by v, u′ respectively, and adding, we have