Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/349

there results

${\frac {\partial f(y,t)}{\partial t}}+{\frac {\partial u^{\prime }}{\partial t}}=-f(y,t){\frac {\partial u^{\prime }}{\partial x}}-v{\frac {\partial f(y,t)}{\partial y}}-u^{\prime }{\frac {\partial u^{\prime }}{\partial x}}-v{\frac {\partial u^{\prime }}{\partial y}}$

$-w{\frac {\partial u^{\prime }}{\partial z}}-{\frac {\partial p}{\partial x}}.$

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

${\frac {\partial f(y,t)}{\partial t}}=-A.\left(u^{\prime }{\frac {\partial u^{\prime }}{\partial x}}+v{\frac {\partial u^{\prime }}{\partial y}}+w{\frac {\partial u^{\prime }}{\partial z}}\right)$,

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

${\frac {\partial u^{\prime }}{\partial x}}+{\frac {\partial v}{\partial y}}+{\frac {\partial w}{\partial z}}=0$

whence

$0=-A.\left(u^{\prime }{\frac {\partial u^{\prime }}{\partial x}}+u^{\prime }{\frac {\partial v}{\partial y}}+u^{\prime }{\frac {\partial w}{\partial z}}\right)$.

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

${\frac {\partial f(y,t)}{\partial t}}=-A.{\frac {\partial u^{\prime }v}{\partial y}}.\qquad \qquad$(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

$v{\frac {\partial f(y,t)}{\partial t}}+{\frac {\partial u^{\prime }v}{\partial t}}=-f(y,t){\frac {\partial u^{\prime }v}{\partial x}}-v^{2}{\frac {\partial f(y,t)}{\partial y}}-u^{\prime }{\frac {\partial u^{\prime }v}{\partial x}}$

$-v{\frac {\partial u^{\prime }v}{\partial y}}-w{\frac {\partial u^{\prime }v}{\partial z}}-v{\frac {\partial p}{\partial x}}-u^{\prime }{\frac {\partial p}{\partial x}}$