A,14 • PIPE OF CIRCULAR CROSS SECTION 



retain the framework of the modified Schhchting method. Much careful 

 experimental work needs to be done, however, before the adequacy of this 

 method of predicting transition can be evaluated. 



At the Ninth International Congress of Applied Mechanics held in 

 Brussels in September, 1956, A. M. 0. Smith, in a paper entitled "Tran- 

 sition, Pressure Gradient, and Stability Theory," advanced the hypothe- 

 sis that transition occurs when the amplification ratio of the initial dis- 

 turbances as computed from the Tollmien-Schlichting theory reaches e^ 

 or about 8100, and showed a comparison of experimental data with com- 

 puted results which indicated reasonably good agreement. It is difficult 

 to understand how the magnitude of the initial disturbance can be 

 omitted; certainly the theory cannot deal with the effects of free stream 

 turbulence on a smooth plate in a flow with zero pressure gradient. How- 

 ever, in many cases the amplijfication ratio varies so rapidly with increas- 

 ing distance along the surface because of the effects of pressure gradient 

 that the computed transition position varies slowly with changes in the 

 selected value of the initial disturbance amplitude or amplification ratio. 



A, 14. Transition to Turbulent Flow in a Pipe of Circular Cross 

 Section. The nature of the flow in a pipe depends on the value of the 

 Reynolds number Re = u^d/v where u^a is the mean velocity, d the diam- 

 eter, and V the kinematic viscosity of the fluid. Since the velocity dis- 

 tribution in laminar flow is parabolic, the Reynolds number may be 

 written as u^^jr/v where w^ax is the velocity at the center and r is the 

 radius. For comparison with transition in boundary layers we note that 

 the Reynolds number Re^* based on the displacement thickness, w^hich in 

 this case is the thickness of an annulus bounded by the pipe wall which 

 would pass all of the fluid at the maximum velocity, is equal to 0.303i?e. 



Transition from laminar to turbulent flow depends greatly on the 

 initial disturbances which in turn depend on the shape of the entrance 

 to the pipe and the disturbances in the flow in the tank or reservoir 

 ahead of the pipe entry. The lowest critical Reynolds number for large 

 initial disturbances has been measured by many investigators [4^, p. 319] 

 with results lying between 1900 and 2320 for Re or 576 and 703 for Rei*. 

 At lower values of Re, initial disturbances die out far downstream. 

 Reynolds was able to increase Re to 13,000 by reducing the initial dis- 

 turbances. Other experimenters have had greater success, obtaining values 

 of Re of 20,000 (Barnes and Coker \50\ and Schiller \51\), 32,000 (Taylor 

 [45, p. 321]), and 50,000 (Ekman \52\). Ekman's value corresponds to 

 Re^* of 15,150. The values vary by a factor of 26, dependent on initial 

 disturbances which were not quantitatively measured. It is probable that 

 the initial turbulence in these experiments varied from several per cent 

 to less than one hundredth of one per cent. 



For comparison with transition measurements in boundary layers it 



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