A • TRANSITION FROM LAMINAR TO TURBULENT FLOW 



the streamlines were concentric circles in a general way as dependent on 

 the velocity distribution; in particular, on the ratio of du/dr to u/r. If this 

 ratio is —1, the negative sign indicating a velocity decreasing with in- 

 creasing radius, the resulting motion is a potential flow with constant 

 circulation which exhibits neutral stabiUty . If the ratio is less than — 1 , 

 the flow is unstable with respect to three-dimensional disturbances, and 

 the large cellular ring vortices studied by Taylor [78] make their appear- 

 ance. If the ratio is +1, the fluid rotates as a rigid body. Laminar motion 

 is stable for ratios between —1 and +2 as a result of the stabilizing 

 action of the centrifugal field. For ratios greater than +2, turbulent mo- 

 tion is expected. 



Experimental studies of instability and transition in the flow between 

 rotating cylinders have been made by Taylor [78], Lewis [79], Wendt [80], 

 and MacPhail [81]. Let us consider first the case in which the outer cylin- 

 der is at rest and the inner cylinder rotates. Calling the surface speed of 

 the inner cylinder u^ the radius of the inner cylinder r-,, that of the outer r^ 

 Taylor found the two-dimensional laminar flow replaced by steady vortex 

 rings when Wi exceeded the value given by 



7r^v^[l -f- (ro/n)] r (k _ A 



-f 0.00056 



1 - 0.652 ^ - 



fe-)r 



or if (fo/ri) — 1 is small, when 



Wi(ro - O ^ .1 1 r^ + n 



> 41 



1 /^ 



n) 



The experimental values of the critical Reynolds number are in good 

 agreement with this relation. However, the resulting motion is not tur- 

 bulent in the accepted sense, consisting of large ring vortices regularly 

 spaced along the length of the annulus. MacPhail found that transition 

 to turbulence developed gradually. As the cylinder speed is increased the 

 regular vortex rings break up and the fragment vortices travel round and 

 round the inner cylinder. Their axes appear to oscillate about positions 

 of rough alignment with the cylinder axis. The dimension of the eddies in 

 the direction of the cylinder axis remains much greater than those in the 

 other directions, and even at speeds far above that for a laminar break- 

 down, the turbulence spectrum contains a band of preferred frequencies. 

 The whole behavior is reminiscent of that of the Kd,rmdn vortex street 

 behind a circular cylinder at high Reynolds numbers. 



When the inner cylinder is at rest and the outer cylinder is rotating, 

 much higher transition Reynolds numbers are found, and a regular vortex 



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