B,9 • VELOCITY, PRESSURE, AND TEMPERATURE DISTRIBUTIONS 

 If Pr = 1 is assumed, the energy equation (Eq. 8-4) takes the form 



^(^||_pF-»-£7-.) = (9-3a) 



This is similar to the momentum equation (Eq. 8-1), rewritten as follows: 



(9-3b) 



dy 



i' 



— - puv - Upv] = 



/yj 



Further from the equation of continuity, Eq. 8-3, we have p^ = const. 

 Therefore a comparison between Eq. 9-3a and 9-3b leads to the following 

 linear relationship between E and U: 



E = c^T + ^U'-\-^{u' + v' + w') = Cj,Tl 



(1-.)| + . 



(9-4) 



where the constant Tl and 17, as determined by the boundary conditions 

 (Eq. 9-2a and 9-2b), are 



7? = TJTl 



Cr,T1 



Cpl e [ 2 ^ e 



T° is the stagnation temperature a,t y = 8. If we neglect as usual the 

 turbulent intensity in Eq. 9-4, we obtain the approximate relation 



CpT + K f^2 



"^P-^ e 



(1 



''^17: + ^ 



(9-5) 



Eq, 9-5 gives a relation between T and U on the basis that the laminar 

 Prandtl number is unity. Some authors have derived the same relation 

 requiring that the turbulent Prandtl number should also be unity (see 

 e.g. [8]), but the latter condition is superfluous according to the above 

 considerations. 



Eq. 9-5 gives the temperature-velocity relationship including heat 

 transfer. If the waU is insulated, we must have 









but, according to Eq. 9-5, 



dy Cp dy Ux ^ dy 



(9-2c) 



(9-6) 



and since in general (dU/dy)^ 9^ 0, the condition (Eq. 9-2c) imposed upon 

 Eq. 9-6 requires that 



n = l 



hence Eq. 9-5 simpUfies to the following form : 



CpT + ^f/2 = CpT^ (= const) (9-7) 



(91 ) 



