B,ll • REYNOLDS ANALOGY 



9-5) between the temperature and velocity for the case of Pr = 1. By 

 differentiating with respect to y, we obtain 



(fl = ^^^-')^(f). 



Thus in terms of {dU/dy)^, we can write St and C/ as defined by Eq. 11-2 

 in the following form : 



rpQ rp 



1 ^ 1 (dU\ 



'Jl \dyj. 



so that 



(11-4) 



where 17 = T^/Tl, Me is the Mach number of the stream at ^ = 6, and 

 7 is the ratio of specific heats. Eq. 11-4 can be considered as a generali- 

 zation of the Reynolds analogy to include the effects of the heat transfer, 

 compressibiUty, and the difference in the transport of heat and mo- 

 mentum. As Me > 0, r] 9^ 1, and Z)„ 9^ Dh according to Art. 10, the 

 right-hand side of Eq. 11-4 is in general not unity, and the Reynolds 

 relation (Eq. 11-3) is not obtained. However, if we neglect the effect of 

 compressibility, for example at low speeds, and if the transports of heat 

 and momentum are similar (Dh = Du), the right-hand side of Eq. 11-4 is 

 approximately unity, and the Reynolds analogy (Eq. 11-3) is then found 

 to be valid. In general those restrictive conditions are not present, and 

 the Reynolds analogy will not hold. For example, in the case of a heated 

 plate (77 > 1), the factor between brackets in Eq. 11-4 is smaller than 

 unity, so that in many circumstances we have 



St < icf (ll-5a) 



This inequality is verified by experiments. On the other hand, with in- 

 tense cooling of the plate (77 < 1) the term between brackets in Eq. 11-4 

 may become larger than unity so that we may get 



St > ^Cf (ll-5b) 



In spite of its defects, the Reynolds analogy (Eq. 11-3) is often used in 

 theories of boundary layers with heat transfer because of its simplicity, 

 and sometimes the experiments show that the analogy is a surprisingly 

 good approximation. 



Instead of defining the shear stress and heat transfer on the basis of 



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