B,10 • TRANSPORT OF PROPERTIES IN A TURBULENT FLUID 



where Ci = ^i — Ui is the thermal velocity and m is mass. The mean 

 values of high powers of d are 



(Tij = —pCiCj = stress tensor 

 Qi = —^pCiC^ = thermal flux 

 / is the internal energy, which is equal to C»T for an ideal gas, and finally 



1 (dUi _^ dU\ 



2 \dXj dXiJ 



Eq. 10-3 express the conservation of mass, momentum, and energy re- 

 spectively. In order to express the quantities (Xij and qi in terms of the 

 macroscopic quantities p, Ui, and T (or 7) we have to investigate further 

 the Boltzmann equation (Eq. 10-1). Because it is nonhnear in character, 

 it can only be solved by approximations. For the detailed calculations, 

 reference may be made to the textbook of Chapman and Cowhng [21]. 

 As a first approximation it is found that 



(10-4) 

 , dT 



where p. and k are found as functions of T and depend on the collision 

 cross section. With this approximation, the second equation of Eq. 10-3 

 becomes the Navier-Stokes equation of motion. 



In particular, for a transport in the y direction of a property which 

 either is a scalar, or has a component in the x direction, Eq. 10-4 reduce to 



dU 



(10-5) 



q,= -p^L^Ly = -- 



or, in general, the laminar flux of the transport J of a transferable 

 property $, which is the momentum or temperature in Eq. 10-5, can be 

 written in the following form : 



Jl^m = Aam X" , (10-6) 



where Aam is a laminar phenomenological coefiicient equal to the coef- 

 ficient of viscosity in the case of transport of momentum, and to the 

 coefficient of heat conduction in the case of the transport of heat. 



When we deal with turbulent transport, it is necessary to replace the 

 concept of the molecular colUsions by the turbulent exchanges between 

 fluid elements. Similarly the thermal velocity Ci is replaced by the ve- 

 locity Ui of turbulent motions. If the property $ is to be transferred by 



(99) 



