Dynamics of Submerged Towed Cylinders 



More formally, we may consider the work done, AW, on the 

 cylinder over one period of oscillation, t.. In much the same way as 

 was done In [ 24] . We find that 



AW = (l-f,)MuV ' [y^+ U^'l^^o dt - (1 -f2)MUr [y^ + U^']^ dt 



If AW< 0, oscillations will be damped, while If AW > oscillations 

 will be amplified. I.e. the systenn will be unstable. We first note 

 that if f| = f2= 1, then Instability can only arise from viscous effects 

 (cf. [ 24]). We next consider the first two terms of (20). We note 

 that for arbitrarily small U stability will be governed by whether 



X* I • 2 r 2 



y_ dt - (l-fg) J y, dt Is positive or negative; It is clear, 



therefore, that a well streamlined nose (f, ~ 1) and a blunt tall (f g < 1) > 

 both tending to make AW < 0, will promote stability. Fo r higher U, 

 however, the sit uati on becomes more complex, as U|yy'| >y may 

 now obtain, and yy' may be either positive or negative, the bar 

 representing the mean value over one period of oscillation, (It is 

 noted that from Figs. 8 and 9 it may be found that for oscillatory 

 instabilities we generally have (yy')o being strongly negative, and 

 (yy'), also negative but with smaller absolute value.) Stability will 

 depend on the magnitude of f , , fp, yo» Yl ®^^« » ^^^ no simple general 

 rules can be formulated beyond the statement of Eq. (20). 



It was found that the most effective way of stabilizing a towed 

 system Is by making it blunt at the tail, which has the disadvantage 

 of Increasing the towing drag. Clearly, what Is needed Is a blunt 

 tall without separated flow! The present work and that of [ 26] Indi- 

 cate that small £2 and large Co (both associated with a blunt tall) 

 have individually stabilizing effects on the system. Clearly then 

 what we need is a sufficiently small fg for stability, and a small 

 C2 for moderate form drag. From the boundary conditions we note 

 that a small £2 has the effect of reducing the lateral shear exerted 

 by the tail on the cylinder. Accordingly, if the tall is made very 

 flexible with the rest of the body essentially rigid, the full shear 

 force might not be transmitted to the cylinder, simulating the effect 

 of a small fg; yet Insofar as axial flow conditions are concerned, 

 they would be fairly good. Of course, this particular solution might 

 give rise to other problems, e.g. whlplash-type behavior of the tall 

 may be envisaged. 



Another point of possible practiced interest hinges on the 

 fact that a towed flexible body, which is unstable at low towing speeds, 

 may be stable at an Intermediate range of towing speeds. (On the 

 other hand, a rigid towed body of the same shape would be unstable 



1013 



