44 



n = 0.5 



(bl 



FIGURE 17. Maximum growth rates for power law wall 

 temperature distributions, T„(x) - T„ = Ax 



Rfi* = 800. 



number is a minimum for n=0, and increases by 

 maximum of 12% for values of n in the range 

 - 1/2 < n < 1. This behavior, which is not con- 

 sistent with the experimental results, may be due 

 to the fact that significant changes in wall tem- 

 perature and therefore in the velocity and temper- 

 ature distributions are taking place over one or 

 two wave-lengths in violation of the parallel-flow 

 assumptions. It is felt that a proper multiple 

 scales formulation of the stability problem, which 

 takes into account the rather rapid variation of 

 wall temperature with x, is required to properly 

 assess the present results for power-law and step 

 function wall temperature variations. The results 

 for non-uniform wall temperature distributions are 

 given in more detail in the paper by Strazisar and 

 Reshotko (1978) . 



to uniform wall heating beginning at the leading 

 edge while the case s = 1 corresponds to a step 

 change in temperature occurring at the measuring 

 station x ^ = 5.5 inches. The peak disturbance 

 growth rate displays a minimum as s increases for 

 each value of AT considered here. The band of 

 amplified disturbance frequencies also moves toward 

 a higher frequency range as s increases. 



Disturbance growth rate behavior as a function 

 of wall temperature distribution is summarized in 

 Figures 17 and 18, where (-ai)inax """^ defined as 

 the maximum disturbance growth rate for a given 

 value of T„(Xj-g£)-Too at fixed values of n is shown 

 in Figure 17b. We see that positive exponents can 

 result in large disturbance growth rates at low 

 wall heating levels. At higher levels of wall 

 heating the relative reduction in ("0!i)j[iax i^^tween 

 any two temperature levels is greatest as n de- 

 creases. 



The variation of (-cij_)r[,ax with s at values of 

 At = 3°F and 5°F is showii in Figure 18. The min- 

 imum in (-ai)niax ^*- ^^^h wall heating level occurs 

 near the minimum critical Reynolds number of the 

 unheated boundary layer. The measured value of 

 (K6*'min.crit ^°^ AT = is 400, which corresponds 

 to s = 0.25 in Figure 18, while the predicted par- 

 allel flow value of (R6*'min.crit = ^^0 for AT = 

 corresponds to s = 0.42. 



An attempt was made to use the program of Lowell 

 and Reshotko (1974) to solve the parallel-flow 

 spatial stability problem for power law, wall 

 temperature distributions since the solution scheme 

 allows the mean flow solution to be read directly 

 into the coefficient matrix of the disturbance 

 growth rate at a fixed frequency and Reynolds 



5. EFFECT OF WALL HEATING ON SEPARATION 



An underwater vehicle is basically a body of rev- 

 olution having generally favorable pressure gra- 

 dients forward of the maximum diameter and adverse 

 pressure gradients downstream of the maximum 

 diameter. If laminar flow can be maintained all 

 the way to the adverse pressure gradient region 

 then the boundary layer will be very easily 

 separated unless measures are taken to delay such 

 separation. 



An obvious way to delay separation is by suction. 

 This however involves the complexities of suction 

 slots, internal ducting and later discharge of 

 the flow removed from the vehicle boundary layer. 

 A "cleaner" possibility for separation delay if it 

 in fact would work is heating. 



Wazzan et al. (1970) showed that heating can 

 cause a separating profile to fill out significantly. 

 Figure 19 indicates that for a Falkner-Skan g = 

 -0.1988, an overheat of 90 °F, converts a separating 

 profile to one having the shape factor of a Blasius 

 boundary layer. This motivated our proposal to 

 investigate experimentally the potential effect of 

 heating on delay of laminar separation. Subsequent 

 calculations by Aroesty and Berger (1975) using an 



■•- S 



FIGURE 18. Maximum growth rates for step change in- 

 creases in wall temperature. 



FIGURE 19. Velocity profiles at various wall tempera- 

 tures for 6 = -0.1988 [wazzan et al. (1970)]. 



