E,2 • BASIC EQUATIONS 



appears in Eq. 2-2 when variable specific heat is considered. Frictional 

 heating is small except at very high velocities and is neglected.^ 



Eq. 2-1 and 2-2 indicate that the molecular and turbulent contribu- 

 tions to the shear stress and heat transfer are additive, and that the 

 turbulent components are both proportional to the density p. However, 

 they contain the unknown quantities u'v' and h'v', so that it appears that 

 additional assumptions must be made before solutions can be obtained. 

 For making these assumptions, it is convenient to introduce Boussinesq- 

 type relations as follows [2] : 



—!—, du ri-i dh 



dy dy 



where e„ and en are known as the eddy diffusivities for momentum and 

 heat transfer, the values for which are dependent upon the amount and 

 kind of turbulent mixing at a point. Inasmuch as the pressure is essenti- 

 ally constant across the passage dh/dy = CpdT/dy. Eq. 2-1 and 2-2 be- 

 come, when these relations are introduced, 



r = (m + peu) -^ (2-3) 



q= -{k^pc,en)~ (2-4) 



The physical significance of e„ and eh lies in the fact that eu/{iJ,/p) is the 

 ratio of turbulent to molecular shear stress [3], and eh/{k/pCp) is the ratio 

 of turbulent to molecular heat transfer. The use of Eq. 2-3 and 2-4 may 

 be preferable to that of Eq. 2-1 and 2-2, inasmuch as the former imply 

 the physical requirement that r and q should be zero when the velocity 

 and temperature gradients are zero. Eq. 2-3 and 2-4 can be written in 

 dimensionless form as follows : 



du* 

 li^/p^J dy* 



(2-5) 



+ ^^.^)^ (2-6) 



Pw Cp jUw/Pw/ dy* 



where 



u* = — -y^=^ y* 



ViVP^ (Mw/Pw) 



T* ^ (^- - ^)gp^- , and a^'JL 

 Qw y/T^/p^ eu 



The subscript w refers to values at ^ = 0, i.e. at the wall. 



^ Terms involving p'v' are sometimes included in Eq. 2-1 and 2-2. These terms 

 can, however, be combined with pv in the equations of momentum, energy, and 

 continuity to give pv. The variable pv, rather than v, then appears in the conservation 

 equations. Consideration of p'v' would be necessary only if it were desired to calcu- 

 late V. 



< 289 ) 



