C,2 • THE MEAN FLOW AND REYNOLDS STRESSES 



C,2. The Mean Flow and the Reynolds Stresses. It is generally 

 assumed that the motion can be separated into a mean flow whose com- 

 ponents are Ui, Ui, and Uz and a superposed turbulent flow whose com- 

 ponents are U\, u^, and u^, the mean values of which are zero. In taking 

 average values, the following principles will be adopted. If A and B are 

 dependent variables which are being averaged, and 8 is any one of the 

 space variables x, y, z, or the time t, then dA/dS = dA/dS and AB = AB, 

 where the bar denotes a mean value. 



When the mean flow is not varying, that is, when the average value 

 defined by 



1 r+' 



A(x, y, z, t) = lim — / A{x, y, z, t')dt' 



is independent of the time t, the time average is the natural mean value 

 to use. Difficulties arise when the flow is variable, and other types of 

 averages have to be introduced. For instance, in the problem of turbu- 

 lent flow near an infinite plate moving with variable velocity, the mean 

 values could be taken over planes parallel to the plate. In more general 

 cases, neither the time nor the space mean values can be conveniently 

 defined to possess all the desired properties. We then consider the sta- 

 tistical average over a large (infinite) number of identical systems (en- 

 semble average). 



The equation of continuity of an incompressible fluid, when averaged, 

 becomes 



f = » (2-1) 



The Navier-Stokes equations of motion are 



P^ = ^— {<Jij - pViVj), Vi = Ui -f Ui (2-2) 



Ot OXj 



where aij is the stress tensor due to pressure and viscous forces. If the 

 mean value is taken, Eq. 2-2 becomes 



P -^ = -^ {^ij - pUiUj — pUiUj) (2-3) 



This equation has the same form as Eq. 2-2, if Vi is replaced by Ui, and 

 the stress <Tij is replaced by aij — pU{Uj. Thus, the equations of mean flow 

 are the same as the ordinary equations of motion except that there are 

 the additional virtual stresses 



Tij = —pUiUj (2-4) 



which represent the mean rate of transfer of momentum across a surface 

 due to the velocity fluctuations. These virtual stresses were first intro- 

 duced by Reynolds [8], and are known by his name. 



< 197 ) 



