THERMODYNAMICS OF OPEN SYSTEMS 
Here, A; represents the fraction of kinetic energy of 
mean motion dissipated by turbulence per unit mass 
and time, and A; the work per unit mass and time done 
in the course of eddying motion v”’ by the resultant of 
all the effective forces (excluding Reynolds’ apparent 
force 0Rj/dx’) applied to the unit of mass considered. 
Work A; can be regarded as being energy released by 
instability, per unit mass and time, in the course of the 
eddying motion v”* [11]. When A; < 0, the turbulent 
motion is stable; when A;> 0, it is unstable [11]. In 
the first case A; represents the fraction of turbulent 
kinetic energy transformed into heat per unit mass 
and per unit time; in the second, it represents the 
quantity of heat transformed into turbulent kinetic 
energy [13]. 
Finally we observe that turbulence can maintain 
itself only if «x > O (generalization of the criterion of 
L. F. Richardson). . 
The First Law. The unit mass of the fluid (p, p, v*) 
being a closed system, we can apply to it the principle 
of conservation of energy, from which we obtain [10] 
d dq as 
Pacem ee) — a, (& = 1, 2,3). (22) 
This equation states that, per unit mass and time, the 
merease in the sum of wmternal energy e, kinetic energy k, 
and potential energy v, is equal to the quantity of heat 
recewed dq/dt, plus the surface work done by the sur- 
roundings on the wut of mass under consideration during 
the same time interval. 
Because of the equation of kinetic energy k and the 
equation of continuity in p and v", equation (22) takes 
the form (1), the enthalpy h being defined by ph = 
pe + p. 
By setting 
(23) 
where @ represents the rate of production of heat per 
unit mass and W* the components of heat flux (con- 
ductivity and radiation), equation (22) assumes (taking 
account of the equation of continuity) the form of an 
equation of balance [1, 11]: 
+ ole +k + 9) 
(24) 
+ A ove +k + ot + pk + W* | = po. 
The existence of an equation of continuity for the 
system (jf, p, v") justifies application of the Reynolds 
mean operation to the two sides of (24). By simplifying 
the equation thus obtained, with the help of the equa- 
tion of kinetic energy k,, and of equation (20), we 
finally obtain the equation of balance of mean internal 
energy [11], 
(ie to) ——— 
a (je) + aa [pev® + pho” + W*] 
sa (25) 
Ope — 
= —p — — pA; b 
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535 
This equation shows that the apparent diffusion due to 
turbulence causes a flux of convective heat, described 
by its components [6, 11] 
Wk = ph'v"*, (26) 
Finally, if the equation of continuity in 6 and ve is 
substituted in (25), it follows [11] that 
de yon @ (lb _ qm 
di +99 (1) +4, = Me 
te, @7) 
where d/dt represents the rate of variation of any 
quantity followmg the mean motion and where 
.Wwtw) (8) 
Gi 
represents the heat received per unit volume and per 
unit time. This quantity of heat includes the heat 
produced in a unit volume and the heat added by 
radiation, conduction, and convection (in the sense of 
the physicists). 
Equation (27) states that the quantity of heat re- 
cewed, per unit time and per unit mass of fluid, is equal 
to the sum of the increase in internal energy of the unit 
mass considered during this time interval, the mechanical 
work of expansion performed by this unit mass during 
the same time, and the energy of instability which tt 
releases in this time interval. 
The Second Law. The second law states that the 
degradation of “noble” energy (in this case, kinetic 
energy) into internal energy or into heat is always 
accompanied by an increase in entropy. The evolution 
of the system is determined in this way. 
In the case of a turbulent fluid there occurs simul- 
taneously with this degradation the transformation of a 
fraction of the kinetic energy of mean motion into 
kinetic energy of turbulence, and also, more generally, 
a transformation of the kinetic energy at one scale of 
turbulence into energy at the scale of turbulence im- 
mediately smaller. Thus in a turbulent fluid, kinetic 
energy of motion is degraded not only into heat (the 
case of viscous fluid [11]), but also into kinetic energy 
of “inferior quality.” The second law requires that the 
entropy of the system is always increasing in the course 
of these transformations of energy. 
In order to account for this degradation, let us 
suppose that the mean physical state of a turbulent 
fluid has a corresponding mean specific entropy, which 
is a function of the mean internal energy and of the 
mean specific mass, 
Sm = Sm (e, p). (29) 
This satisfies the Gibbs relation (4), 
dm __1lde__p_ db (30) 
dt I Gh (p)? Tm dt’ 
where 7’, is the “temperature” which characterizes the 
mean thermal state of the fluid and where 
