Studies on the Motion of Viscous Flows— I 



uBu/^x + v^u/By + w^u/Bz 



uBv/Bx + vBv/By + wBv/'Bz 



wBw/bz. 



Thus the inertia terms introduce a stronger coupling between the equations 

 in the presence of eddies. This coupling offers an explanation for larger trans- 

 fer of momentum and energy in the presence of eddies. 



Consider a surface on which fluid velocity vanishes everywhere. Clearly, 

 the angular velocity of an element of such a surface is zero, and so also is the 

 angular velocity of a tangential line element. The arguments used in Theorem 

 TTT lead to the conclusion that particles on such a surface cannot be in an eddy. 

 The significance of the conclusion stems from the no-slip condition. If a sta- 

 tionary, impervious solid is introduced in an eddy, the fluid particles at the 

 solid boundary would cease to be in the eddy and the value of whirlicity would 

 decline in the neighborhood of the solid boundary. Notice that the viscosity of 

 the fluid enters in the argument only through the no-sHp condition and that the 

 geometry of the surface does not enter the picture at all. Hence, it can be con- 

 cluded that if a stationary, impervious, thin bar is introduced in an eddy of a 

 fluid of low viscosity such as water, there will be a noticeable change in the 

 flow, although the diameter of the rod may be very small in comparison with the 

 diameter of the eddy. A crucial experiment to test this conclusion was made at 

 the University of California, Berkeley. A steady draining vortex was generated 

 in a vertical circular cylinder of 11-1/2 inches diameter, turning at 20 rpm and 

 having an axial hole of one -inch diameter. The apparatus is sketched in Fig. 2 

 and was described by Einstein and Li in 1955 (19). The depression of the water 

 surface indicates roughly the pressure distribution in a horizontal plane, as 

 vertical accelerations are low. Low-pressure gradients in the horizontal plane 

 are associated with small slopes of the free surface. If the introduction of a 

 bar reduces the whirlicity and thereby constrains the whirling motion, its con- 

 sequences on the free surface would be apparent. When a bar of 1/8-inch diam- 

 eter was kept near the boundary of the drain hole, the depression in the water 

 surface reduced by 70%, although the cross-sectional area of the bar was less 

 than 3% of the area of the hole (Fig. 3). When it was placed at the axis of the 

 hole, the depression was reduced by 30%. Comparable reductions were obtained 

 with rectangular and square bars. Such noticeable effects of thin, solid mem- 

 bers on eddies have also been observed in the wake of a bluff body (Roshko, 

 1954) (20) and in the leading-edge vortex from a delta wing (Harvey, 1962). 



This constraining effect has to be taken into account in the interpretation of 

 data obtained by small probes in eddies, and the interference of the flow may be 

 considerably larger than what one would expect from the probe size. The con- 

 straining effect can also be used as an inexpensive and highly effective method 

 of control of vortices near the outlets of reservoirs, or the intake chambers of 

 pumps (J. P. Berge, 1966). 



It is to be expected that vorticity cannot vanish in an eddy; for no line ele- 

 ment has zero angular velocity in a particular plane, and hence the normal 

 component of vorticity, being the average of the angular velocities of all such 



451 



