58 



A and aj. If one generalizes the definition (54) 

 and uses (42) , one obtains 



Im(aQ + eai) 



eRe(- £nc^ 



(56) 



Thus, the growth rate in (56) depends on the choice 

 of ^n because the axial and transverse variations 

 of the l.^ are not the same. Since the Cn ^^^^ func- 

 tions of both y and x, one may term a stable flow 

 unstable or vice versa. 



Since there are many possible definitions of the 

 growth rate in a nonparallel flow, one should be 

 careful in comparing analytical and experimental 

 results. Saric and Nayfeh (1975, 1977) found that 

 the best correlation between the nonparallel theory 

 and available experimental data for the Blasius flow 

 is obtained if one uses the definition (55) . In 

 this paper, we compare the definitions (55) and (56) 

 evaluated at the value p where Ci is a maximum. 



Figure 1 shows the variation of the calculated 

 disturbance growth rates a/R with frequency FR=(ij/R 

 for T„-Tg =0, 3,5, and 8°F and for the displacement 

 thickness Reynolds number R* = 800. This range of 

 Tw-Tg is chosen for comparison with the existing 

 experimental results. The growth rate is calculated 

 by using the definition (55) and by normalizing Cj 

 so that Cl"*exp(-aoy) as y-»<». This figure indicates 

 that the disturbance growth rate decreases with in- 

 creasing T^^-Tg. The maximum growth rate is reduced 

 by approximately 56% by increasing the wall temper- 

 ature by 5°F. The maximum growth rate is very small 

 when the wall temperature is increased by 8°F at 

 R* = 800. Figure 1 shows that the range of unstable 

 frequencies decreases with increasing T -Tg. 



b 



UJ 



< 



< 



< 



100 120 140 160 



FREQUENCY FR « lO' 



180 



tr 

 b 



< 



FIGURE 1. The variation of the spatial growth rate 

 with frequency for varying wall temperatures at R* = 

 800. Nonparallel, Parallel. 



FIGURE 2 . The variation of the maximum growth rate 

 with streamwise position for varying wall temperatures. 

 Nonparallel, Parallel. 



Figure 2 shows the variation of the maximum 

 growth rate obtained from our analysis with T -Tg. 

 It shows that the maximum growth rate decreases with 

 increasing wall temperature at all Reynolds numbers. 



Figures 1 and 2 show a comparison between the 

 growth rates based on the parallel, (53), and non- 

 parallel, (55), stability theories. The nonparallel 

 maximum growth rates are approximately 30% larger 

 than the parallel ones. Moreover, the nonparallel 

 critical Reynolds number is approximately 20% lower 

 than the parallel one for all the values of T„-T 

 considered as shown in Figure 2. 



Figures 3a- 3d show comparisons of the experi- 

 mental growth rates of Strazisar et al. and the 

 nonparallel growth rates defined by (53) , (55) and 

 (56) for different values of T^-Tg and different 

 values of R* . These figures show good agreement 

 between the growth rate defined by (55) and the ex- 

 perimental results, in contrast with the parallel 

 theory which underpredicts the experimental results 

 by large amounts. Moreover, including the distor- 

 tion of the eigenfunction with streamwise position 

 in the definition of the growth rate yields a growth 

 rate that is very close to the parallel one and 

 hence underpredicts the experimental results by 

 large amounts. 



7. CONCLUSION 



The method of multiple scales is used to analyze 

 the linear nonparallel stability of two-dimensional 

 liquid boundary layers on a flat plate for the cases 

 of uniform and nonuniform wall heatings. We include 

 disturbances in the temperature, density, thermo- 

 dynamic, and transport properties of the liquid in 

 addition to disturbances in the velocities and 

 pressure. The growth rates calculated from non- 

 parallel results without including the distortion 

 of the eigenfunction with streamwise position are 

 in good agreement with the experimental results of 

 Strazisar et al. (1975, 1977). The nonparallel 

 results show that wall heating in water has a sta- 

 bilizing effect on the flow; there is a decrease in 

 the disturbance growth rates, a decrease in the 

 range of unstable frequencies and an increase in 

 the critical Reynolds number. 



