211 



CJx 



rotational 

 Flow 

 (Velocity vectors on rotor) 



Shear 

 Flow 



FIGURE 3. 



(rotor) 



Principle of rotor-type vortexmeter 



rotational flow. When the rotating shaft of a 

 vortexmeter is parallel with vorticity axis, the 

 rotor turns with angular velocity of W-S , where S 

 is the slip due to rotational friction of the rotor 

 shaft and W is the vorticity in the fluid. At 

 present, the slip S can be estimated using the 

 following technique. 



Using the simple consideration of the elementary 

 wing, the torque Q due to a rotor element having 

 small length dr in a radial direction can be deter- 

 mined from Eq. (1) , where C-^ , I , and U are lifting 

 derivative, cord length of vane, and advance speed 

 respectively. 



where 



Q(r) = 'spHU^Cj.j^ rdr 



3CL ^ 3_ L.F. 

 'I'^ " 3jr "3/ !5P5.RU2 



(1) 



R. and L.F. are the rotor radius and lifting force 

 respectively. In the flow with uniform vorticity 

 distribution, the magnitude of the torque acting 

 on the rotor becomes: 



R 



Q(u) 



Q(r)dr 



(2/3)pi!,R3uC^^u 



(2) 



calibration - moTor 



C^ 



(ol calibrotion mode ) 



\i^ inner- shaft 



<^= 



mimoture ball bearing 



FIGURE 4. Principle of rotor-type vortexmeter calibra- 

 tion . 



Then, the slip of the rotor in a rotational flow 

 can be determined by following equation, where 

 q shows rotational friction of the shaft. 



{2/3)pJlR3uC 



(3) 



Lr 



For the calculation of S, the rotational friction 

 of the ball bearings g should be determined experi- 

 mentally. This problem will be briefly discussed 

 later. 



As previously stated, since the generation of a 

 stable vortex for the calibration is presently not 

 feasible, a mechanical calibration was attempted in 

 which vorticities mechanically act on the rotor 

 through the shaft of the rotor. This principle of 

 the calibration is shown in Figure 4 where a newly 

 designed rotor shaft is composed of duplicate inner- 

 shaft and outer-tubes. The outer-tube is mounted 

 on the outer rings of the ball bearings and the 

 vanes are fitted on the outer-tube. The inner- 

 shaft is connected to a calibration-motor. 



To obtain the slip S, the vortexmeter is cali- 

 brated in an irrotational flow in which it travels 

 along at a constant speed. The inner-shaft is 

 driven by the calibration-motor at an angular 

 velocity, to, and the rotor turns at an angular 

 velocity, S, in response to the condition of the 

 ball bearing's frictional torque and the hydro- 

 dynamic characteristics of the rotor. 



From the measured vorticity, Uq, we can estimate 

 the vorticity in fluid as oi = aiQ + S. According to 

 the authors' experience, if the frictional torque, 

 g, is approximately 10~^ kg-m it is possible to 

 consider S = O except in the case of fairly slow 

 speed (cf . Figure 25) . This means that the 

 calibration of the vortexmeter seems unnecessary 

 for ordinary test conditions. 



Although ball bearings exhibiting frictional 

 torque values less than g = 0.7 10~^kg-m in air were 

 chosen in manufacturing the vortexmeter, there was 

 no direct measurement of the frictional torque of 

 the miniature ball bearings in water. The frictional 

 torque, g, also can be determined by measuring the 

 torque on the outer-tube generated by inner-shaft 

 turning in water. According to the results of these 

 measurements, it can be said that there is hardly 

 any difference between the frictional torque value 

 of the bearings when they are used in water or 

 air. 



An example of a vortexmeter is shown in Figures 

 5 and 6. The diameter and length of the rotor are 

 30mm and 18mm respectively, section of the vane 

 is lenticular shaped with a thickness ratio t/H 

 = 1/q. A transducer for rotating the rotor is 

 used in connection with a photo-transistor which 

 makes 4 pulses-signals in one revolution. Assuming 

 Cj^ ^ = 0.6TI, g = 10"^kg-m and U = l.S m/s , it is 

 possible to make a rough estimate of the vortex- 

 meter's precision from the value of slip obtained 

 by Eq. (3). From these values, the slip value, S, 

 equals 10~'r.p.s. which corresponds to 1% error 

 relative to a normal vorticity of u = 1 r.p.s. 



As will be mentioned later, the vortex cores of 

 the stern vortex near the hull surface have a very 

 steep gradient in vorticity distribution. There- 

 fore, it is useful to consider the vorticity values 

 measured by the rotor with a finite diameter at 

 such boundaries. It is clear from the Eq. (1) that 

 a mean value of a torque during a turn due to a 

 wind element dr (see Figure 3) corresponds to a 



