B,10 • TRANSPORT OF PROPERTIES IN A TURBULENT FLUID 



where $(^ — r) is the mean value of the property <l> when the lump of 

 fluid carrying # found itself at the instant t — t.H can be expanded into 

 series as follows: 



m -r)= m) - ^ j'_^ dtW) (10-9b) 



where the integral term is the displacement of the lump of fluid in the 

 interval of time r. The expansion is valid when the lump of fluid makes 

 only small displacements and when it is assumed that $ is stationary, but 

 nonuniform. 



After substitution for ^{t — r), Eq. 10-9a becomes 



$(^) = $ _ |5 ^ / dre-'' I dt'v{t') 



dy 

 The double integral is 



Thus 



K r dre-^' r dt'vit') = K j^ dre-" j^ dt"v{t - t") 

 = K r di"v{t - t") r dre-'^ 



= jj dt"e-^'"v{t - t") 



$(i) = $(^) - M5 ; dt"e-'^'"v{t - t") 

 dy Jo 



Hence the flux for the transport of $ is: 



_ a$(0 



J = -p^v = -p-^ / dt"e-'^'"v{t - t")v{t) (10-lOa) 



dy Jo 



Consequently the turbulent coefficient of transport D is found as follows: 



D = jj dt"e-'^'"v{t - t")v{t) (10-lOb) 



The transport coefficient D in Eq. 10-10 depends on the autocorrelation 

 function of velocities vijt — t")v{t) and on k. In its turn k depends on a 

 number of factors among which is the property to be transferred. This 

 entails that D may differ according to the nature of properties to be 

 transferred, i.e. heat, momentum, particles, etc. 



Now we shall compare this result with the concept of mixing length, 

 so often used in the study of the turbulent motion. By analogy with the 

 kinetic theory of gases one may suppose that there is a length I, which 

 represents the distance traveled by the lump of fluid between the instant 

 when it was freed from its surroundings carrying with it the mean 

 property of these surroundings, and the instant when it arrives in a 



( 101 ) 



