83 



10 

 X(FTI 



FIGURE 7. 



U<v,. 



versus x for several values of 



wall boundary layers with zero pressure gradient. 

 More recently, similar calculations have been 

 performed for boundary layers in favorable pressure 

 gradient flows. Before comparing the flow tube 

 results with such predictions we should estimate 

 the favorable pressure gradient produced by the 

 boundary layer displacement effect in the tube. 

 The most common way to characterize streamwise 

 pressure gradient in a boundary layer is by the 

 similarity parameter S [schlichting (1968)]. For 

 the general class of wedge flows, the external 

 velocity U is given by U = Cx , and the parameter 

 6 is then 2m/ (m -H 1) . Both m and 6 are constants 

 in any wedge flow, and are equal to zero for the 

 zero pressure gradient boundary layer. We have 

 calculated approximate local values of B in the 

 flow tube , using the Blasius growth law for the 

 boundary layer displacement thickness: 



6* = 1.72(vx/Uoo)^ 



(3) 



(The calculation can be iterated to include the 

 effect of pressure gradient upon 6*, but the differ- 

 ence is negligible.) The resulting values of B as 

 a function of x at several values of U are shown 



CO 



in Figure 7. B is proportional to the square root 

 of X, and thus has its largest value at the down- 

 stream end of the tube . 



Figures 3 and 5, which show transition Reynolds 

 numbers versus overheat for the flow tube, also 

 include the theoretical predictions of Wazzan et al. 

 (1970) for a B of 0.07. This represents an approx- 

 imate average of B in the tube for the velocity 

 range of interest. (Calculations using exact B 

 values from the tube will be done in the near future. 

 Note that the experimental results lie near or even 

 above the B = 0.07 prediction for overheats from 

 zero to 13°F (7°C) . At this point the experimental 

 curve quite suddenly levels out, while the predicted 

 curve continues to rise at an increasing slope. 

 The predicted curve reaches its maximum at a Reynolds 

 number of about 250 x lo^ (near 45°C overheat) , 

 while the experiment has never yielded more than 

 42 X 10^ . 



There are several possible reasons for the 

 disagreement between theory and experiment at the 

 higher overheats. (1) The theory does not account 

 for the destabilizing effects of density stratifi- 

 cation, which will become increasingly important as 

 overheat is increased. Buoyancy effects may 



destabilize the flow in three distinct ways: (a) 

 the bottom of the tiabe wall is subject to thermal 

 convection rolls, similar in form to the Goertler 

 instability; (b) the side wall boundary layer will 

 experience a cross-flow due to the rising fluid 

 near the wall; and (c) the top wall boundary layer 

 will grow in thickness faster than normal because 

 of the fluid rising up from the sides. (2) The 

 theory neglects the effects of temperature and 

 viscosity fluctuations upon the growth of the 

 velocity fluctuations . There is evidence that this 

 is a reasonable approximation. (3) The theory relies 

 upon the e transition criterion, which may become 

 increasingly incorrect at higher overheats. This 

 criterion has never before been applied to boundary 

 layers with inhomogeneous physical properties. 

 There is a large distance between the minimum crit- 

 ical point in the boundary layer and the predicted 

 transition point using e . It is questionable 

 whether the region of linear growth can extend over 

 such a large range of Reynolds numbers. (4) Wall 

 roughness is not accounted for in the theory, and 

 the importance of roughness will increase with wall 

 heating (and with increased velocity) due to the 

 thinning of the boundary layer. Roughnesses that 

 are insignificant at zero or low overheat may become 

 important as overheat increases. 



Velocity Profile Measurements 



In view of the differences between experimental 

 results and computed transition Reynolds numbers, 

 measurements have been made of boundary layer 

 ) velocity profiles in the flow tube to try to 

 establish the mechanism of transition. If the 

 buoyancy effects described above are in fact 

 significant, they should produce measurable devi- 

 ations from axisymmetry in the mean velocity profiles. 

 In addition, they might cause transition to occur 

 earlier on the top, side, or bottom wall, depending 

 upon which mechanism is predominant. We therefore, 

 designed the instrumented section (described above) 

 to be installed on the downstream end of the 6.1 m 

 test section. This contains Pitot tubes for mean 

 velocity measurements and flush mounted hot film 

 probes for intermittency measurements. The instru- 

 mented section has been very successful in measuring 

 mean velocity profiles in the flow tube. Figure 8 

 shows a typical measured profile that has been 



