Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/350

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

${\frac {\partial }{\partial t}}\{A.(u^{\prime }v)\}=-{\tfrac {1}{3}}R^{2}{\frac {\partial f(y,t)}{\partial t}}-Q$,

where

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

Let p be written ($r^{\prime }+{\bar {\omega }}$), where p′ denotes the value which p would have if f were zero. The equations of motion immediately give

$-\nabla ^{2}p=\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial v}{\partial y}}\right)^{2}+\left({\frac {\partial w}{\partial z}}\right)^{2}+2{\frac {\partial v}{\partial z}}{\frac {\partial w}{\partial y}}+2{\frac {\partial w}{\partial x}}{\frac {\partial u}{\partial z}}+2{\frac {\partial u}{\partial y}}{\frac {\partial v}{\partial x}}$;

and on subtracting the forms which this equation takes in the two cases, we have

$-\nabla ^{2}{\bar {w}}=2{\frac {\partial f(y,t)}{\partial y}}{\frac {\partial v}{\partial x}}$,

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

${\bar {w}}=-2{\frac {\partial f(y,t)}{\partial y}}\nabla ^{-2}{\frac {\partial v}{\partial x}}$.

Moreover, we have

$O=A\left\{u^{\prime }{\frac {\partial (u^{\prime }v)}{\partial x}}+v{\frac {\partial (u^{\prime }v)}{\partial y}}+w{\frac {\partial (u^{\prime }v)}{\partial z}}+v{\frac {\partial p^{\prime }}{\partial x}}+u^{\prime }{\frac {\partial p^{\prime }}{\partial y}}\right\}$;

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′.