F • CONVECTIVE HEAT TRANSFER IN GASES 



TRANSITION 



F,17. Stability of the Laminar Boundary Layer and Relation to 

 Transition. That the heat transfer coefficient for turbulent flow is an 

 order of magnitude (say 10 times) greater than the heat transfer coef- 

 ficient for laminar flow is evident from Fig. F,llk. This difference is due 

 to the fact that the velocity gradient at the wall in turbulent flow is con- 

 siderably greater than the velocity gradient at the wall in laminar flow. 



Although the region of development of the boundary layer between 

 the minimum critical Reynolds number (neutral stability for infinitesimal 

 disturbances) and the Reynolds number of fully turbulent flow is truly 

 the transition region, yet in this discussion the expression "transition" 

 will refer to the beginning of fully turbulent flow. It will be found that 

 the transition Reynolds number so defined will be many times greater 

 (again perhaps 10 times or more) than the minimum critical Reynolds 

 number. 



Since the heat transfer coefficients for turbulent flow are much greater 

 than those for laminar flow, it is desirable to employ ways and means of 

 delaying transition as much as possible. One method of delaying tran- 

 sition is to draw heat out of the laminar boundary layer at the wall. By 

 this means the minimum critical Reynolds number is increased. For two- 

 dimensional infinitesimal disturbances, it was demonstrated by Lees [49] 

 that (1) with subsonic free stream flow, cooling the boundary layer was 

 stabilizing, although the layer would always become unstable for suf- 

 ficiently high Reynolds number, whereas (2) with supersonic free stream 

 flow, cooling was again stabilizing, yet it was possible through sufficient 

 practical cooling to maintain stability for any Reynolds number however 

 large. When the wall is insulated, an increase in free stream Mach num- 

 ber is destabilizing for subsonic or supersonic flow. It was next shown by 

 van Driest [50], through numerical calculation, that the region of complete 

 stability (infinite Reynolds number) extended from Mach number 1 to 9 

 for air when the Prandtl number was taken as 0.75 and the Sutherland 

 viscosity law was used with a free stream temperature of — 67.6°F. The 

 results are given in Fig. F,17a. The minimum critical Reynolds numbers 

 other than infinity were computed using an estimation formula given by 

 Lees in [49]. 



The cooling required for complete stabilization of the laminar bound- 

 ary layer for air under various conditions is plotted in Fig. F,17b. The 

 solid curves (the viscous solution) are the more accurate in that they 

 include the viscous forces in the stability analysis, whereas the dotted 

 curves (the inviscid solution) are stability criteria because they include 

 only the pressure forces and not the viscous forces in the analysis. The 

 condition {Pr = 0.75, p^fx^ = 1) should be used for ordinary wind tunnel 

 work because at low temperatures the Prandtl number is approximately 



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