A. S. Iberall 



We suggest these as two related estimates of a limit cycle fluctuation which 

 represents a near "maximum" for low-frequency noise as a relaxation process 

 by which a system of fluctuation is maintained from wall to wall. 



We will attempt a more compelling estimate, namely, we will attempt to 

 estimate a "critical" Reynolds number. The following crude scheme is used. 



From the small-amplitude linear theory [5,7], we found that the transition 

 from an overdamped (Rayleigh damping) wave for the tube to an underdamped 

 (organ pipe) wave for the tube took place over the range w = 1 - 100. 



We will obtain a result in two ways: 



First, from ^o^ = 100, ! . , 



y - 1 



\^ 



1 + 



1 + q + 



7 - 1 



1 + 



V^ 



,,5/4 

 1 



625 



For turbulent flow, approximately 



Fitted with Laufer's point, g/R^o = 45.5 at R„^ = 30,800, the experi- 

 mentally fitted result is 



g = 0.0196 Ri-'5 . 

 For g = 625, R„„ = 370. 



Second, we propose that "all of a sudden," as Reynolds number increases, 

 the underdamped frequency can become entrained. Thus, as before, from the 

 Strouhal number 



f = 0.089 Hz . 



300 . 



These values 370 and 300 (based on the half separation), may be compared 

 with the standard value of 400 - 700 for the critical Reynolds number for parallel 

 plates (on the basis of mean velocity and half-plate separation— see, for example, 

 Sec. 146 [11]). The estimate is not bad. 



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