B • TURBULENT FLOW 



new layer where it is supposed to mix with the new surrounding fluid. 

 In this case, the transport coefficient is vl, which can be written in the 

 integral form 



vl 



= jj dt"v{t - t")v{t) (10-11) 



if the correlation of velocities converges. Eq. 10-11, based on the mixing 

 length, does not distinguish between the transport of heat, momentum, 

 and matter, because the same length is intrinsically implied in all cases. 

 As an illustration, Eq. 10-10 may be applied to the special case of 

 transport of momentum and heat along the y direction. We then obtain 



^ ^ _^uv = pDu^ (10-12) 



q = —pCpT'v = pCpDk — 



where r is the turbulent shear stress, q is the rate of turbulent transport 

 of heat, and D„ and Dh are respectively the transport coefficients of 

 momentum and heat defined by Eq. 10-lOb. The coefficients are com- 

 monly termed "turbulent exchange coefficients." 



The results (Eq. 10-12) can be compared with the Boussinesq formulas 

 of turbulent transport of momentum and heat, written usually in the 

 following form : 



dU 



dy 



1 (10-13) 



dT 



dy 



where e^ is the eddy viscosity and e^ is the eddy heat conductivity, intro- 

 duced by formal analogy to the corresponding laminar viscosity and heat 

 conductivity of the Navier-Stokes and Fourier equations. Eq. 10-13 give 

 neither the structure of the exchange coefficients nor the basis of the 

 transfer. However, they can be made completely identical in form with 

 Eq. 10-12, if the following expressions are assigned to e^ and e^ 



e^j = pDu 

 €fc = pCpDh 



The ratio 





is called the turbulent Prandtl number by analogy to the laminar Prandtl 

 number introduced in Eq. 6-4b. We see that the mixing length theory 



< 102 ) 



