Rotary sloshing

For a rectangular tank (see Fig. 5.2) the angular frequency of the first oscillation mode which is, of course, not rotary sloshing is expressed as ... 

The natural frequency, f(= /T, of liquid surface oscillations in a circular cylindrical tank and a spherical tank are presented in Figs. 5.4 and 5.5, respectively. In these figures, //l is the bath depth, D is the vessel diameter. Hi,/D is the aspect ratio, m in the parentheses denotes the wth mode of surface oscillation in the tangential direction and n is the nth mode in the radial direction. The tangential mode, i.e., rotary sloshing appears first in both tanks. Further information about this study is available elsewhere [13]. 

Similar bath oscillations have also been observed when liquid in a circular cylindrical vessel is drained through a centric single-hole bottom nozzle [20,21] or when a circular cylindrical bath is subjected to external forced oscillation, as described in detail in Sect. 5.1.3 [6,10,13,16,17], The bath oscillation due to external forced oscillation is called sloshing. In particular, a purely tangential mode of sloshing is referred to as rotary sloshing [6],... 

Figure 5.9 shows the relationship between the period of the first kind of swirl motion, Ts, and the bath depth, Hl, for an air-water system. The solid and broken lines denote the periods of the fundamental and second harmonics of the rotary sloshing, respectively. The angular velocity of the /th harmonics of the rotary sloshing, m,, can be derived using the inviscid linear theory mentioned previously [13] thus. 

Fig. 5.9 Comparison of the period of the first kind of swirl motion with the period of rotary sloshing in water bath...

The measured swirl period could be adequately approximated by (5.1) and divided into two categories, the limit being at Hi /D 0.3. The swirl period agrees with the period of fundamental harmonics of the rotary sloshing (t = 1) for H] /D > 0.3 within a scatter of -20 to 0%, whereas agreement with the period of the second harmonics i = 2) for H, /D < 0.3 is within a scatter of 0 to +25%. This limit seems to be associated with the limit between the shallow water wave and the deep water wave in a cylindrical vessel [16,17] given by Hi /D = 0.3. In this case, a wave affected significantly by the bottom wall of the vessel is termed shallow... 

A variety of swirl motions are known to occur in a bath agitated by gas injection when the bath surface is exposed to the atmosphere, as described in Sect. 5.2.1.4 [18,23, 29-37]. In particular, two types of swirl motions typically occur in a circular cylindrical bath agitated by single-nozzle bottom gas injection, as schematically illustrated in Fig. 5.6 [29, 30] One is observed over an aspect ratio, H /D, from approximately 0.2-1.0. The other appears for H /D > 2. No swirl motion occurs when the aspect ratio falls in the range of 1.0-2.0. The former swirl motion is caused by bath surface oscillations due to quasi-periodic generation and subsequent arrival of bubbles at the bath surface. It resembles the rotary sloshing of a water bath contained in a circular cylindrical vessel [16,17,38]. The latter is caused by the Coanda effect [26], which appears when a bubbling jet approaches the side wall of the vessel [29,30,39]. 

Figure 5.29 illustrates the relationship between D/gyl /T and the aspect ratio Hi /D. The solid line indicates the period of rotary sloshing occurring in a cylindrical water bath subjected to externally forced oscillation. The angular frequency of the fundamental wave of the rotary sloshing, u> (= 2ar/ Ts), is given by [18,38] ... 

Hutton RE (1964) Fluid particle motion during rotary sloshing, Trans ASME J Appl Mech 85 123... 

When the gas flow rate Qg was 120 x 10 m /s and the top of the bath was free, there arose a swirl motion similar to the rotary sloshing typical of a bath in a cylindrical vessel subjected to external, horizontal, or vertical forced oscillation. This motion was termed the first type of swirl motion [12],... 

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