A,19 • TRANSITION ON BODIES OF REVOLUTION 



Little work has been done in correlating the sphere data with fiat 

 plate data in terms of Re^*. Fage [71] gives Re^* = 945 at transition 

 occurring well ahead of separation on a sphere in an air stream of 0.85 

 per cent turbulence. The equivalent flat plate Reynolds number of 298,000 

 is considerably below the curve of Fig. A, 2b, but it must be noted that 

 transition on the sphere occurred in a region of large adverse pressure 

 gradient (X = — 5 to — 7) in Fage's experiments. 



Tomotika [72] computed the growth of the laminar boundary layer 

 on the surface of a sphere in a uniform stream for an experimental pres- 

 sure distribution obtained at a Reynolds number of 165,000. Separation 

 occurred at an azimuth angle of 81° with Rei*/^'Re = 2.72, hence at an 

 equivalent flat plate Reynolds number of 2.5Re. Thus if the Reynolds 

 number at which transition occurs just ahead of the separation is known, 

 the value of Ret may be computed. The critical Reynolds number of a 

 sphere as usually defined corresponds to a considerably more forward 

 position of transition and to a considerably modified pressure distribu- 

 tion. Examination of sphere drag coefficient and pressure coefficient 

 curves shows a departure from an approximately constant value of the 

 coefficient beginning at about OARe^^. Hence the values of Re^ are proba- 

 bly of the same order as Re„. The maximum value observed for a sphere 

 is about 4 X 10^ as compared with 28 X 10^ for the flat plate in air 

 streams of low turbulence. For a turbulence of 1.0 per cent the value of 

 Re^ from a sphere is about 200,000 as compared with 630,000 for a flat 

 plate. For very high turbulence the values agree; for a turbulence of 

 3 per cent both sphere and plate give a value of Ret of about 100,000. 

 The much lower values derived from the sphere at low turbulence are 

 presumably due to the large adverse pressure gradient at the transition 

 point on the sphere. 



In this discussion the well-known but much smaller effect of the scale 

 of turbulence has been omitted. The most suitable turbulence parameter 

 for the generalized case is {u'/u^){df/Ly^ where df is the displacement 

 thickness of the boundary layer at transition. 



The drag of streamline bodies of revolution as a function of wind 

 tunnel turbulence was studied in 1929 [73] and the observed results were 

 interpreted as due to the effect of turbulence on transition in the bound- 

 ary layer. Computations were made on the crude assumption that the 

 velocity distribution in the boundary layer was linear. It was assumed 

 that transition occurred at values of Res of 1250, 2000, and 2750 (5 being 

 the thickness based on a linear distribution) for turbulence levels of 2.3, 

 1.6, and 1.2 per cent. The corresponding values of Res* are 625, 1000, and 

 1375, and of Re^, 193,000, 333,000, and 630,000. In view of the crude ap- 

 proximations in the theoretical computations, and the experimental errors 

 involved in early hot wire measurements of turbulence, these values are 

 in satisfactory agreement with Fig. A,2b. 



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