(7) 



Theory of Turbulent Flow Between Parallel Plates 



It is convenient to utilize one component X of the vorticity (vorticity may 

 be identified by variables X, Y, z appropriate to each coordinate): 



■■■••.. '■• ';.i->. ■■ :;'1 V aX = aW - SV . ,,i- . /. ;.-v^,-, /: .; ■ -, r 



Eliminating W, 



[D2- \- j0] U + qD [DU + (j^/a) V + jSX] - DP ^ , • ^ .-, -. , 



[D2-X- ji/;] V + jaq [DU+ (jVa) V+ jSX] - j aP = , ' ' . , 



[D2-\-jV^] X = -R„oCp'U , .... 



a-i [D2- \- ja^] J - 2ejSR„„(p'U - [2eR„„q)'D + (y-1) g] X 



- I [2eR„y D + (y- 1) g] V + (7 - 1) j0P = , . ^^, ,..._^ . ...... . .., . 



[DU+ (jVa) V+ jSX] - 7/32gX - (yS/a) /B^gV + 7/32jv/,p - ^2j^y,g- , q . ^ , 



This set may be decomposed ,_ , ,. 



over the core: ™ \ - . 



[D2- \- jMq] U + qD [DU+ (jA./a) V+ jSX] - dp = , 

 [D2- \- JMq] V + jaq [DU+ (jVa) Y+ jSX] - j aP = , 



[D2-\- jMJ X = , (8) 



CT-i [D2- \- jaMp] J + (7- 1) jMgP - (7- 1) gX - (7-1) (Sg/a) V = . 

 [DU+ (jVa) V+ j8X] + 7/3^JMoP - /^^jM^J- 7/52gX - (yhpg/a) V = . 



for the region near the wall (x = ±1): -r-. , - , '\ , '. 



[D2 -\- jo)] U + qD [DU+ (jVa) V+j8X]-DP=0, '. ■. " ; 7 ,,; ^ ^ .'•-,. 



[D2- \- joj] Y + jaq [DU+ (jVa) V + j SX] - j aP = , ■ ,■ ,f. ■ 



[d2-\- j«] X = +gU , ' ' ' (9) 



a-i [d2-\- JCTwUT 2ejSgU- g[+2eD + 7- llX- (Sg/a) [±2eD+ 7-l]V+(7-l)j'^P = 0, 

 [DU+ (jVa) V+ jSX] + 7/S2ja;P - /3^icoJ- y/S^gX - {jh/a) (i^gV = . 



CHARACTERISTIC FUNCTIONS 



These equations may be independently solved in terms of four independent 

 modalities (characteristic functions) within each region. Because of the assump- 

 tion of small compressibility, results will only be sought to first order in fi^. 



715 



