A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



A,21. Transition in Flow near Rotating Disks. If a smooth 



circular disk is rotated about an axis through its center and perpendicular 

 to its plane in a large body of fluid, the flow near the disk may be laminar 

 or turbulent, dependent on the Reynolds number wr'^/v where w is the 

 angular velocity and r is the radius. From Goldstein's analysis [82] of 

 Schmidt and Kempf's measurements of the resisting moment, the critical 

 Reynolds number is found to be about 80,000. Riabouchinsky [83] ob- 

 served about 230,000. 



Theodorsen and Regier [84] obtained a critical Reynolds number of 

 310,000. Large roughness reduced the value only to 220,000 but the intro- 

 duction of disturbances from a small high pressure air jet near the center 

 of the disk reduced the transition Reynolds number to about 125,000. 

 A hot wire anemometer showed regular fluctuations in the transition 

 region of frequency of 200 cps on a disk of 1-foot radius rotated at 

 525 rpm. Transition in this case occurred at a radial distance of 9.6 inches, 

 the flow being laminar at smaller radii and turbulent at larger radii. 



The detailed mechanism of transition in the flow near rotating disks 

 has been studied by Gregory, Stuart, and Walker [85]. They found that 

 the laminar flow broke down into a regular vortex system whose axes lay 

 along the spiral streamlines of the flow. The critical Reynolds number for 

 the appearance of this instability varied from 180,000 to 200,000; tran- 

 sition to general turbulence occurred at Reynolds numbers of 270,000 to 

 299,000, i.e. close to the values obtained by Theodorsen and Regier [84]- 

 See also [86]. 



Transition on the rotating disk resulted from amplified waves cover- 

 ing a certain band of frequencies. Some of the waves were found to be 

 stationary, relative to the surface of the disk, giving rise to the observed 

 vortex pattern. Stuart computed the solution of the disturbance equa- 

 tion for disturbances of zero wave velocity on the rotating disk. The 

 angle of the spiral was in extremely good agreement with experiment 

 while the wave number, which gives the spacing of the vortices, was 

 approximately four times too large. The discrepancy is believed to be 

 due to the neglect of viscosity in the theoretical computations, only the 

 inertial forces being considered. Transition in this case also is the end 

 result of instabihty of the laminar flow and the Reynolds number for 

 transition is a function of disturbances in the flow field. 



A,22. Transition in Flow at Boundary of a Jet. We have pre- 

 viously discussed in Art. 7 the instabihty of the laminar mixing region at 

 the edge of a jet or wake. Even when the motion within the jet and 

 mixing region are fully turbulent, the secondary motions in the surround- 

 ing fluid are usually laminar in character, certainly so at large distances. 

 Corrsin [87] has described the annular transition region and the laminar 

 "collar" in an axially symmetrical heated jet of air. Hot wire anemometer 



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