A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



should be noted that transition in a pipe actually begins near the entrance 

 to the pipe. If the entrance is bell-mouthed or rounded such that sepa- 

 ration does not occur, a thin boundary layer develops on the pipe wall 

 and grows in thickness until it equals the pipe radius. Because of the 

 continuity relation the flow near the axis must accelerate and hence the 

 static pressure falls more rapidly than in the finally developed flow where 

 the drop in pressure is due only to friction. Transition thus occurs in the 

 wall boundary layer which is subjected to a favorable stabilizing pressure 

 gradient and to whatever turbulence is present in the entering flow. For 

 the same turbulence the critical Reynolds number would be expected to 

 be somewhat higher than for the boundary layer on a plate with zero 

 pressure gradient. 



With sharp-edged entrances, separation occurs at the entry with the 

 formation under some conditions of regular vortex patterns [53,54]- These 

 vortices give very large disturbances and hence low values of the critical 

 Reynolds number. 



For very rough pipes the critical Reynolds number appears to be the 

 same as for a smooth pipe with very disturbed entry conditions. How- 

 ever, if the initial turbulence is small, roughness may produce larger dis- 

 turbances than those already present and reduce the critical Reynolds 

 number. Depending on the shape of the roughness elements, the maxi- 

 mum permissible height to avoid disturbance in a smooth pipe has been 

 estimated to be of the order of 4:/Re^ times the radius of the pipe [49, 

 p. 311]. The fact that any roughness is permissible is thought to be associ- 

 ated with the fact that roughness elements also possess a critical Reynolds 

 number below which they set up no disturbance. For example for a flat 

 plate roughness element, the critical Reynolds number is about 30. The 

 above relation corresponds to a critical Reynolds number of 32 and the 

 assumption that the critical height is small compared to the radius. See, 

 however, the discussion in Art. 5. 



The effect on the critical Reynolds number of curving the axis of 

 cylindrical pipes of circular cross section has been studied by several 

 investigators [55,56]. The breakdown of the laminar flow does not in this 

 case lead immediately to turbulent motion but to a regular type of second- 

 ary motion under the influence of the centrifugal forces. Turbulence sets 

 in at a critical Reynolds number which depends on L/d where L is the 

 radius of curvature of the axis of the pipe and d is the diameter of the 

 pipe. The values obtained increase as L/d is reduced, from about 2300 at 

 L/d = 1025 to 7600 at L/d = 7.6. 



A, 15. Transition in Pipes of Noncircular Cross Section. Transi- 

 tion has been studied in pipes of rectangular, square, and annular cross 

 section. Basing the Reynolds number on the hydraulic radius, values of 

 2100 for the square cross section, 1600 for a rectangular section with ratio 



(40) 



