F • CONVECTIVE HEAT TRANSFER IN GASES 



the other hand, Fig. F,llj and F,llk show that Eq. 11-31 and 11-32 yield 

 friction coefficients independent of Mach number for a constant wall tem- 

 perature. Since it can hardly be expected that the wall temperature has 

 complete control over the variation of fluid properties, i.e. that dissipation 

 can be neglected, it appears reasonable to rule out Eq. 11-31 and 11-32. 

 For completeness, the results for both laminar and turbulent flow for 

 near-insulated flat plates are brought together in Fig. F,lll using Eq. 

 11-2 and 11-29 and s (turbulent) = 0.825. 



F,12. Cone Solution. For geometrical reasons, boundary layers are 

 thinner on cones than on flat plates and therefore it is expected that 

 turbulent boundary layers will have greater heat transfer coefficients for 

 cones than for plates. The von Karmdn momentum integral relation for a 

 boundary layer on a cone in a supersonic stream with zero angle of attack 

 and attached shock wave is 



Tw = :^ / pu{u^ - u)dy + - / pw(We - u)dy (12-1) 



ax Jo X Jo 



in which the coordinate distance x is measured from the cone apex along 

 the cone and y is measured normal to the surface. Then, using Eq. 12-1 

 instead of Eq. 11-27, and following the same procedure as carried 

 out fin the derivation of Eq. 11-28, van Driest has shown [36] that a 

 simple rule exists for the transformation of turbulent heat transfer results 

 from a flat plate to a cone in supersonic flight. The rule states that 

 the local heat transfer coefficient on a cone is equal to the fiat plate solu- 

 tion for one half the Reynolds number on the cone, the Mach number 

 and wall-to-free stream temperature ratio remaining the same; thus the 

 turbulent flow rule is similar to that for laminar compressible flow where 

 the cone solution is equal to the flat plate solution for one-third the 

 Reynolds number on the cone. For turbulent flow, the correction amounts 

 to only about 10 to 15 per cent, whereas for laminar flow it amounts to 

 73 per cent. 



F,13. Stagnation Point Solution. Although it is expected that the 

 flow will be laminar in the immediate neighborhood of the stagnation 

 region of spheres and cylinders, it is possible for the flow to become 

 unstable and eventually turbulent with increasing distance from that 

 region, owing to the low Reynolds number of the local flow there. 



A theoretical analysis can be made for a fully turbulent boundary 

 layer near the stagnation point when it is assumed, as in flat plate flow, 

 that the velocity profile remains similar with distance. Assuming a 

 y-power law for velocity distribution, the coefficient of heat transfer 



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