57 



dap 

 dxi 



6 r ^ 



^ H^W^dy / ^ G.W.dy (41) 



i=l 



i=l 







Therefore, to the first approximation 



zj = A C(xi,y)exp[i j {a + eaj)dx - iait] + 0(e) (42) 



where 



g(n) 



2p 



pudn 



(50) 







Note that all fluid properties are made dimension- 

 less by using their freestream values. 



Equations (48) -(50) are supplemented by the fol- 

 lowing boundary conditions: 



where the zj are related to the disturbance variables 

 by Eq. (19) and the constant Ag is determined from 

 the initial conditions. It is clear from Eq. (42) 

 that, in addition to the dependence of the eigen- 

 solutions on x, the eigenvalue ag is modified by 

 eaj. The present solution reduces to those obtained 

 by Nayfeh, Saric, and Mook (1974) and Saric and 

 Nayfeh (1975) for the case of nonheat conducting 

 flows. 



5. THE MEAN FLOW 



For flows whose thermodynamic and transport 

 properties are functions of temperature , the 

 two-dimensional boundary- layer equations for a 

 zero-pressure gradient are 



u = 0, T 



8 , . .. 3 

 — — (p*u*) + T--r 

 dx* ay* 



p*u* 



8u* 

 3x* 



p*v' 



3u* 

 3y* 



(p*v*) 



^ 3_ 

 3y* 



KV* 



3T* 



o*u*c* 



p 3x* 



+ p*v*c 



P 3y 



3y 



(k* 



3u* 

 3y 



3t* 

 3y* 



(43) 



(44) 



(45) 



The temperature dependence of p and p couples the 

 momentum and the energy equations. Note that buoy- 

 ancy and viscous dissipation effects are neglected. 

 Although the stability analysis is applicable to 

 any wall temperature variations, we present stability 

 results for the case of constant wall temperature 

 for which the flow is self similar. Thus, we intro- 

 duce the transformation. 



/i 



n = 



pdy* 



(46) 



T*/T* and g 

 we 



at n 







u •+ 1 and T 



as n ->■ "■ 



(51) 



(52) 



where the subscript w denotes wall values. Equa- 

 tions (48) -(52) are numerically integrated by using 

 Runge-Kutta and Adams-Moulten integration techniques 

 with the liquid thermodynamic and transport prop- 

 erties computed at each integration step. All nu- 

 merical results presented here are for water; the 

 dependence of its thermodynamic and transport prop- 

 erties on the temperature is given in Appendix III. 



6. ANALYTICAL RESULTS AND COMPARISON WITH 

 EXPERIMENTS 



Although the analysis is applicable to both uniform 

 and nonuniform wall heating, results are presented 

 only for the case of uniform wall heating for which 

 the mean flow is self similar. 



The only available experimental results for the 

 stability of uniformly heated boundary-layer flows 

 are those of Strazisar et al. (1975, 1977) . Using 

 a water tunnel, they introduced disturbances by 

 vibrating a ribbon and measured the response by 

 using a temperature compensated hot-film anemometry. 

 They used the r.m.s. of the stream-wise component 

 of the disturbance velocity, u, to calculate the 

 growth rates. They determined the growth rate as a 

 function of frequency at different Reynolds numbers. 



For a parallel mean flow, aj = 0, ag and A are 

 constants, and the l,^ are function of y only. Hence, 

 one can unambiguously define the growth rate a of 

 the distrubance as the imaginary part of a"g ; that 

 is, 



o = - Im(ao) 

 This definition is equivalent to 



(53) 



where R is the freestream x-Reynolds number defined 

 by 



a = Re {—- Unu) = Re(-— Inv) 

 3x 3x 



p*U*x*/y* 

 e e e 



(47) 



Re (-- !lnp) 



oX 



Re(-— {,nT) 

 3x 



(54) 



Introducing this transformation in Eqs. (43) -(45) 

 and solving the continuity equations for v, we trans- 

 form the original set of partial-differential equa- 

 tions into the following set of ordinary-differential 

 equations : 



On the other hand, for a nonparallel mean flow, aj 

 "f , A and ag are functions of x, and the ^j^ are 

 functions of both x and y. Thus, if one generalizes 

 (53) to take into account ea^, one obtains 



3 , 3u, 3u 



T— py ^ + g T— = 



3n 3n 3n 



3 , 3T, 3T 



-^ (P< T— + Pr c g -— = 



3ri 3ri e p 3ri 



(48) 



(49) 



Im(ag 



eaj) 



(55) 



which is not equivalent to (54) . Moreover, the 

 quantity aj and hence o depend on the normalization 

 of the Cn because part of the Cn c^n be absorbed in 



