F,10 • FLAT PLATE SOLUTION 



are, respectively, 



— du , — du d 



pu^- + pv— = — 



ax ay ay 



(m + ^m) 



dp 

 dy 



— dh , — dh 

 '"Tx + 'Ty 



(" + '"'> ^J-y 



du 



dyj 



+ 



dp 

 dx 



dy 



(k + e,) ^ 



dy . 



_dp 



+ u-f 



dx 



(10-1) 

 (10-2a) 

 (10-2b) 



(10-3) 



where e^ and ej are the eddy coefficients of friction and heat transfer 

 defined by 



du 



dh 



(pvYu' = €^Q- and {pvYh' = ek. 



ek 



dT 



dy 



(10-4) 



respectively. The bars indicate temporal mean values. The ratio 

 Cp{n + e^)/{k + ek) is designated the mixed Prandtl number Prm, which 

 reduces to the molecular Prandtl number Pri^^{ = Cpn/k) in the sublayer 

 at the plate and contains the turbulent Prandtl number Prt(= Cpe^/ck) 

 of the outer region of the turbulent boundary layer. Thus Eq. 10-3 can 

 be rewritten as 



— dh , — dh , , , ( du 

 '''Yx^''Yy=^'-"''^A^y 



+ 



dy 



(m + e^) dh 

 Ptu. dy 



+4' ('°-5) 



Since the above equations have the same form as Eq. 3-1, 3-2, and 3-4 

 for laminar flow, the independent variables x and y are Ukewise trans- 

 formed to X and u by the Crocco transformation ii = u{x, y) and x = x. 

 Hence, Eq. 10-1, 10-2a, and 10-5 become, after eUmination of pv, 



d ( — )Lt -f- e^\ d'T 



dp d ( p, + €^ 

 dx dii 



d_ 

 du 



/ 1 dh\ 

 \Pru. du) 



= (10-6) 



+ (M + O (I + u) % = (10-7) 



in which t = {p. -\- e^du/dy. In order to simplify the problem and still 

 arrive at a practical solution, it is assumed that dh/dx = and 

 dp/dx = 0. Eq. 10-6 and 10-7 then reduce to 



d ( I dh 



du \Prjfx du 



+ 1 



