E,3 • EXPRESSIONS FOR EDDY DIFFUSIVITY 



that the error in calculated velocities or temperatures should not be large. 

 A Taylor's series expansion for w as a function of transverse distance 

 (changes in the axial direction neglected) then indicates that e^ is a func- 

 tion of du/dy, d^u/dy^, d^u/dy^, etc. If, as a first approximation, we con- 

 sider Cu to be a function only of the first and second derivatives, and 

 apply dimensional analysis, there results 



_ fdu dhA _ 2 Jdu/dyY ,^ ^. 



'" ~ '" \dy dyy ~ " {d'^u/dy'^y ^^^ 



where k is an experimental constant. This expression was obtained by 

 von Kdrman in a somewhat different manner and is generally known as 

 the Karm^n similarity hypothesis [18]. A critical examination of the 

 Kdrmdn hypothesis from the point of view of statistical turbulence theo- 

 ries is given by Lin and Shen [SI]. Eq. 3-1 can be written in dimensionless 

 form, as 



^" K^ ^^I'V^fll (3-1') 



Mw/Pw {d'uVdy^'^Y 



Region close to wall. In the region close to the wall it is assumed 

 that eu is a function only of quantities measured relative to the wall u 

 and y, and of the kinematic viscosity. This assumption includes, to a 

 first approximation, an effect of the derivatives of u with respect to y. 

 Since the flow becomes very nearly laminar as the wall is approached, 

 the first derivative approaches the value u/y, and hence may be omitted 

 since u and y already appear in the functional relation. The second deriva- 

 tive approaches the constant value zero as the wall is approached. The 

 kinematic viscosity is included, inasmuch as the ratio of viscous to inertia 

 effects is high near the wall where the turbulence level is low. The eddies 

 in that region are small, so that the shear stresses between the eddies 

 and the viscous dissipation of the energy in the eddies are large. With 

 these assumptions, and dimensional analysis. 



= ^"(^'^'p)=^'^^^(^) 



The form of the function F cannot be determined by dimensional analysis. 

 On the basis of simplicity, and the fact that F should approach zero at 

 the wall (effect of ju/p large) and should approach one asymptotically as 

 uy/{p./p) becomes large (effect of ju/p small), it is assumed in reference [15] 

 that 



F =^ I - e '^ 

 { 291 ) 



