C,15 • THE PROCESS OF DECAY 



For large initial Reynolds numbers of turbulence, von Karman and 

 Lin [39] made a tentative proposal to divide the process of decay into 

 three stages: (1) the early stage in which the law (Eq. 14-17) holds, 

 (2) the intermediate stage, in which the law (Eq. 14-15) holds, and (3) 

 the final stage in which the Reynolds number is very low and the law 

 (Eq. 14-14) holds. For estimates of the length of the three periods, we refer 

 to the original article. Here it suffices to say that there is as yet no ex- 

 perimental result available to check the theory for the intermediate stage, 

 and that the recent doubt cast on Loitsiansky's invariant tends to change 

 the basis for such an assumption. 



Detailed discussions will therefore be given only for the early and the 

 final stages. 



Final period of decay. When the Reynolds number of the turbulent 

 motion is very low, as it must eventually happen in the final period of 

 decay of a homogeneous field of turbulence, without external supply of 

 energy, the inertial forces are neghgible and only the viscous forces are 

 effective. Case (a) discussed in Art. 14 then applies. On the other hand, 

 the problem now admits of an explicit solution. In fact, if the quadratic 

 terms are neglected from the equations of Navier-Stokes, we have 



dUi 1 dp nc: -i\ 



-W7 = T^ + v^Ui (15-1) 



By the equation of continuity, this leads to 



Ap = 



Now the only solution of a Laplace equation which is finite throughout the 

 whole space is a constant. Thus the pressure must be independent of po- 

 sition, and the equation for Ui becomes the equation for heat conductions^ 



^ = vAUi (15-2) 



The solution of the initial value problem of this equation is well known 

 to be 



.. /"cO /•» /"« 



Uiix, y, z, t) = j^^^ J-^j- ' ^'^^' ■^' ^' ^^ 



CO / — 00 



exp 



(x - xy + {y- YY + (2 - zy 



dXdYdZ (15-3) 



4:Vt 



From this, the properties of the motion can be exphcitiy calculated. In 



^^ Reissner [60] was the first to attack the problem of turbulence by using the 

 explicit solution of Eq. 15-2. He obtained results analogous to the observed laws of 

 decay. They are, however, more adequate for the discussion of temperature fluctua- 

 tions, and will be taken up again in that connection. The following development is due 

 to Batchelor [51]. 



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