APPLICATIONS OF ENERGY PRINCIPLES 
kinetic energy of horizontal motions introduces certain 
changes, although these changes are not actually in the 
nature of approximations. 
Study of a Simple System. Let it be supposed that a 
mass of gas is confined in a chamber with a plane bot- 
tom and vertical walls, under the action of gravity 
which we assume to be acting vertically downward. If 
the chamber is of sufficiently great height, it is not 
necessary that it have a top. Likewise, the gas need 
not be an ideal one, since for the time being no use 
will be made of an equation of state. Coriolis forces 
will, for the present, be omitted. Let it be supposed 
further that the gas is in some state of motion induced 
by differential heating and cooling. 
Tf we take x, y, and z to be a Cartesian coordinate 
system with the positive z-axis vertical, we may write 
the equations of motion for the horizontal directions in 
the form 
du _ — 1 dp 
Goo pe 
d la o 
a _ _ lap 
dt OTe? 
Here u and v are the velocity components in the direc- 
tions and y; p is the density; p the pressure; ¢ time; 
and F, and F, are the components of the viscous forces 
in the x and y directions. Generally speaking, the mo- 
tions in the chamber might be turbulent. If we wish to 
regard the dependent variables in equations (1) as rep- 
resenting mean values free of the turbulence compo- 
nents, we shall assume that the only change necessary is 
to include eddy-stress effects in the quantities F,, F, 
after the manner of Reynolds. More will be said con- 
cerning this point later. 
The kinetic-energy equation corresponding to the 
system (1) is 
aVi dV; d Vi 
Beayaou ah P SD Op is 
® Vane dp =) 
We use the symbol d to represent the rate at which 
the turbulence and viscosity are decreasing the kinetic 
energy per unit volume, and V7; = wu? + v®. It is possi- 
ble to rewrite (2) in the following form: 
OH , dHu , 0Ekv , dHw 
a” aa | Gy Mee 
C) 0 tc) 0 © 
eye (ODL Tee CUS AUN br 
(et gitoees) 
where use has been made of the continuity equation 
Op Opu , Opv Opw _ 
a” aeen ay ag (4) 
which in any case must be true, and where H = 14 V; 
is the horizontal kinetic energy per unit volume. The 
quantity represented by the last three terms on the 
569 
left-hand side of (3) is the divergence of the (three- 
dimensional) kinetic energy transport vector EV. The 
quantity in the first parenthesis on the right is the di- 
vergence of the horizontal vector pV,. If equation (3) 
is integrated over an arbitrary volume, both of these 
quantities may be represented as surface integrals with 
the aid of the divergence theorem. Thus, if the limits 
are fixed, we may write 
all 
5 | B ae dy dz 
= | 2v,as — [f rwax — u dy) dz 
+ [fo +) ae ayae 
= fff dar ay ae, 
where V, is taken to be the inward component of veloc- 
ity at the boundary, and dS is a surface element. 
Equation (5) may now be given the following inter- 
pretation. The total horizontal kinetic energy (7) in a 
fixed region may be changing in consequence of: 
1. An advection of new fluid having kinetic energy 
across the boundary. This is represented by the term 
A = {| EV, dS. This is then one mode of redistribution 
of kinetic energy. 
2. The performance of work by pressure forces at 
the boundary in virtue of the displacements due to the 
horizontal velocity components. This is represented by 
the term W = — I/ piv dx — u dy) dz. This is a sec- 
ond mode of redistribution of kinetic energy. 
3. A production of kinetic energy within the volume 
itself. This is represented by the term 
Ss = [[] (+e) ae ava, 
which contains the primary source of kinetic energy. 
4. The action of frictional forces. This effect would 
ordinarily consist of a dissipation and is represented by 
the term D = [[f eax ay az. 
If the limits of integration include all of the fluid in 
the fixed chamber, it is clear that the surface integrals 
must vanish, so that in a mechanically closed system (5} 
reduces to 
(5) 
ene S a) (6) 
Since for such a system the frictional effect would ordi- 
narily lead to dissipation, it follows that S must be 
positive if the total horizontal kinetic energy K is to 
remain constant or increase. If a more or less constant 
amount of kinetic energy is to be present, the dissipa- 
tion must be balanced by a corresponding positive 
average rate of production. 
