CAVITATION

IN S W I R L E D F L O W S

A. V. Efimov and T. Yu. Kuznetsova

UDC 532.528.001.57

In connection with the interest of specialists in designs of outlets with swirled flows,* it is urgent to solve the problem of the cavitation conditions of their work. The assertion that such outlets are "reliably protected from cavitation and especially from cavitation erosion" [1] is based only on a calculation of the critical cavitation number for uniform roughness and so far has not been confirmed by experimental studies. In our opinion, a more dangerous exciter of cavitation in such outlets is not the roughness but the ledge at the junction of metal plates, which, in all probability, most often line water conduits at places of potential occurrence of cavitation. The present experimental studies were undertaken for the purpose of offering designers and builders answers to such questions as how do processes of cavitation and cavitation erosion develop in swirled flows, will they always occur here more mildly than in traditional axial flows, and if yes, then why. For different cavitation regimes determined on the basis of the known relative cavitation number k/kcr, we recorded their erosion activity with the use of integral energy indices: magnitude of the shock action of cavitation bubbles Ef and magnitude of the light flashes of these bubbles Eph. The latter quantity, as is known, is directly proportional to the mechanical energy of the bubble [2, 6]. The experiments were conducted on a model of the outlet of the Tel'roam hydrostation dam installed at the Divinogorsk high-head hydraulics laboratory of the Krasnoyarsk hydrostation with a scale 1:12. Ledges with a height of 7 and 12 mm were used as the exciter of cavitation in a 0.8-m-diameter cylindrical conduit. As the device recording the cavitation shocks, we use a DD-2P piezoelectric transducer mounted flush with the wall, designed by the Moscow Special Design Department for Steel Hydraulic Structures (Mosgidrostatl'), with a natural frequency in water of 300 kHz, 2-ramdiameter receiving areas, and sensing elements on a base of piezoelectric ceramic with a specific density of 7.3 g/cm3. An FI~U-39A photomultiplier operating in the range 0.35-0.6 /xm was used for recording electromagnetic radiations of cavitation -- light pulses of luminescent flashes. The need to estimate the corrosiveness of cavitation regimes simultaneously by the DD-2P shock transducer and FI~U-39A photomultiplier is due to the fact that the photomultiplier records the corrosiveness (total light energy) of the jet integrally, as a whole, whereas the DD-2P records only that part of it which arrives at the walls. An AI-1024-95m-17 pulse analyzer was the secondary instrument for recording all pulses, making it possible to obtain the pulse distribution curves according to amplitude during a preset time interval, 50.2 sec, in other words, the amplitude/frequency response, which can be interpreted as the nonnormalized value of the probability density function of pulses in a certain range of amplitudes with their random occurrence during a 50.2 sec time interval. Output of information from the analyzer to the storage was realized by a magnetic tape storage (t~lektronika-302 cassette tape recorder) with subsequent replay in the analyzer with processing on an ES-1840 computer. An $8-12 oscillograph was connected parallel to the analyzer, which enabled checking the stability of the cavitation process in an automatic mode and the character of individual pulses in a storage mode. In calibrating the shock pulses 1 V corresponds to 16.24 Pa and the light flashes (by means of a Yu17M selenium photometer) 1 V corresponds to 2.5 Ix (Fig. 1). The index of the cavitation process was the relative cavitation number k/kcr, in which the circumferential velocity was taken as the characteristic velocity Vchar, Vchar = v cos c~ (where v is the local flow velocity; c~ is the swirl angle) over the height of the cavitation exciter z (Fig. 2).

*Reference is to the design studies of outlets at the Kolyma, Tel'mam, Sarez, and other hydroelectric stations. Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 6, pp. 39-42, June, 1993.

362

0018-8220/93/2706-0362512.50 9

Plenum Publishing Corporation

3 8-- #___,

Oq

EO 0

o

d,-1 O

E

E,-1 Fig. 1. Block diagram of apparatus: 1) intermediate tube; 2) DD-2P piezoelectric transducer; 3) preamplifier; 4) power source; 5) $8-12 oscillograph; 6) Al-1024-95m-17 multichannel pulse-height analyzer; 7) I~lektronika-302 tape recorder; 8) cavitation exciter; 9) cavitation jet; 1) FE~U-39A photomultiplier; 11) power unit of FI~U-39A; 12) capacitor; 13) rectifier; 14) power unit.

1

Fig. 2. Components of velocity in cross section of swirled flow in a cylindrical conduit.

The total energy of the cavitation shock or light pulses, representing the integral sum of the products of the pulse amplitudes and their number and identified with the value of the integral was the energy quantity of corrosiveness of cavitation: F2

Es = f Ff(F)dF,

(1)

F,

where f(F) is the experimental distribution function; P 1 is the lower limit of the pulse effects, determined by the sensitivity of the instrument; F 2 is the upper limit, established on the basis of no realization of an event during 50.2 sec. We will analyze the kinematics of a cavitation bubble and the forces acting on it in a swirled flow.

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"2

Fig. 3. Path of emergence of a cavitation bubble in a swirled flow: 1) imaginary cone, along the helix of which is located the path of the cavitation bubble; 2) rotating circular layer of water; 3) path of cavitation bubble.

As is known, in axial flows mainly a two-dimensional pattern of flow past a ledge with separation occurs as a consequence of the fact that the inertial forces here are considerably greater than gravitational, the cavitation bubble emerges little along the length of the jet, and therefore the latter for all cavitation regimes "licks" the wall. The values of the energies of the shock actions and light flashes recorded by the DD-2P and Ft~U are proportional to each other in this case for the same cavitation regime [2]. A three-dimensional flow pattern is present in swirled flows. The values of the aforementioned energies here are no longer proportional because, as already stated, the solid angle with respect to which the FEU records radiation enables obtaining the total electromagnetic energy of the entire cavitation jet, and the DD-2P records only the part of the energy of shock actions arriving on the solid boundary of the flow. The obtainment of both values of the energy in such flows is especially important methodologically, since in the case of considerable corrosiveness of the cavitation regime and its mild effect on the wall, we will obtain evidence in behalf of designing structures with swirled flows. The cavitation jet in this case no longer "licks" the wall: it is directed at an angle to the axis of the conduit along the path of a conical helix (Fig. 3)~ The cause of this is that the buoyant force Fbuo becomes substantial here, thanks to which the bubble begins to emerge. We will determine the magnitude of this force and rate of emergence Vemer under the assumption the bubble keeps a spherical shape (it is preserved in bubbles with a size up to 0.1 mm [4]), constant radius R, uniform motion over the length of the jet I, and for cavitation stages within 0.5 < k/kcr < 1 in the presence of still small porosity, i.e., when movement of the bubble in a liquid and not in an emulsion can be taken [4]. Such cavitation stages are the most active erosionally. For a turbulent character of movement of the liquid near a bubble emerging with rate Vetoer, covering a path equal to the level of the cavitation jet l, the bubble transmits to the liquid mass m = 7rR2lp with density p kinetic energy: = --'~~2 2

-- aR~lpv~rner

(2)

2

This energy is spent on the formation and movement of swirls and ultimately is converted to heat. For uniform movement of the bubble the buoyant Fbuo and braking F b forces acting on the bubble will be equal m d~ e m e r _ _ F

-- F -w 4 ,,3 9 -- buo-- b-- . . 1. . 3 at< P l -

aR~pv~mer

2

,

(3)

where j is acceleration due to centrifugal forces j = (Vu)x2/(r - z) in a horizontal conduit of radius r (the longitudinal slope is equal to 0) at the level of the cavitation exciter -- ledge with height z. 364

u emer m/sec

8 7 8 5 4 3 g 1

O

p

10-* 10-a la -~ la-lR

Fig. 4. Dependence of the rate of emergence Vemer of a cavitation bubbles on its radius R w i t h a f l o w velocity v = lOm/sec.

We will determine the rate of emergence of the bubble Vetoer respectively in swirled and axial flows by the formulas: R 8

emer----~/-5- Rg.

(4) (5)

In the formula for the swirled flow the term on g is omitted in view of its smallness. For a cavitation bubble R = 10 -3 m in a conduit of radius r = 1 m with flow velocity vu = 30 m/sec, the rate of emergence in a swirled flow Vemer will be 1.6 and in an axial flow 0.164 m/sec. Thus the value of Vetoer is a swirled flow: a) is an order greater than in an axial flow; b) depends directly proportionally on the flow velocity; c) increases with increase of the size of the bubble moving from the wall toward the central air--vacuum core due to the negative pressure gradient in that direction; d) increases with increase of the size of the cavitation bubble as a result of coagulation with similar bubbles on the path toward the tail of the jet (Fig. 4). Thus, for example, the distance of the bubble from the wall for the earlier selected values of vu, Vetoer, and R at the end of the cavitation jet with length 1 = 0.15 m in a swirled flow will be (clearly underestimated as a consequence of disregard of points "c" and "d" in the calculations) about 0.01 m (the same quantity for the axial flow is an order less). However, it cardinally changes the pattern of erosion on the wall. First, the wall ceases to exert an effect on the bubbles. It collapses now spherically symmetrically. Second, the maximum amplitude of the shock front in the case of such collapse decreases exponentially with distance from the bubble. Thus, for instance, if the maximum anaplitude is observed at a distance of the radius of the collapsed bubble 1.59R, when at distance 4R the amplitude of the shock front is already an order less [5]. Figures 5 and 6 show the characteristic distributions (two of the more than 70) of the pulse amplitudes (on the axis of the abscissas) according to their frequency (the number of pulses in a logarithmic scale on the axis of the ordinates during the time of the experiment). For the most erosion-active stage of cavitation within limits 0.39 < ldkcr < 0.75 at a maximum flow velocity v u = 14.7 m/sec, we note on the graphs the presence of a sharp peak -- the maximum number of shocks in a narrow range of amplitudes (Fig. 5), which has a tendency to smooth out with decrease of flow velocity v or its swirl angle a. Here the circumferential component of the flow velocity v u decreases and the axial component v m increases (Fig. 2), which corresponds to a gradual transition of the swirled flow to an axial one; the amplitude/frequency distributions of the swirled and axial flows become increasingly more similar to one another. The distribution of the light flashes for the same cavitation regimes (Fig. 6) do not have extremes. On converting to the same energy units, Eph is also considerably greater than Ef for the same cavitation regime.

365

20,1"

5

7~3g ,- 21

2,72 "-

1

" "g'1 h

i !

1,00 L

I

o

I

20

i

L_

_

60 P~ MPa

40

Fig. 5. Distribution of the pulse amplitudes of shock cavitation actions according to frequency in a swirled flow.

zzz5- fON 2 g81 -

8

6 3#,6 ,q

7,Jg -

,,00_ 0

'

'

20

40

;og MPa

Fig. 6. Distribution of amplitudes of luminous radiation

according to frequency in a swirled flow.

Let us analyze the probable causes of the presence of peaks on the graphs of the distribution of shock pulses for the most corrosive cavitation regimes. The right short distribution 1 (Fig. 5) is distinguished by a small number of actions, from 1 to 20, with large amplitudes. The other, left distribution 2 (Fig. 6) begins with smaller amplitudes with a considerably greater number of actions, from 7 to 22,000. Taking into account the character of the numerous experimental amplitude/frequency distributions for various cavitation regimes, values of the rates of emergence, exponential law of decay of the shock waves, and kinematics of the bubbles, we can assume that distribution 1 occurs from single bubbles that collapsed directly on the wall, i.e., did not have time to emerge and depart from the wall. On extrapolating distribution 1 to the left (Fig. 5), dashed line, the distribution of the shock pulses of the cavitation bubbles upon their contact with the wall as if in an axial flow is obtained. The left distribution of the shock pulses (Fig. 5, 2) is the amplitude/frequency characteristic of the bulk of the bubbles of the entire cavitation jet. These bubbles emerged to the axis of the flow, i.e., departed from the wall. Upon collapse in the flow, their shock pulse, decaying exponentially, reaches the wall considerably smaller. The energy value of such a distribution Ef (area of the integrand curve 2) is considerably smaller than that for distribution (1). A mild fatigue character of failure of the wall material will be the result of such action of weakened pulses.

CONCLUSIONS

1. With an increase of the flow velocity v from 7.58 to 14.67 m/sec and change of k/kcr in the interval of the most erosion-active values from 0.75 to 0.39, Eph increases proportionally to v 5, although Ef increases proportionally to v 3.

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This indicates that in the presence of general erosion activity of the cavitation regime of the flow, a small erosion action on the wall is observed in swirled flows. 2, The cavitation jet in swirled flows is directed at an angle toward the axis of the conduit as a consequence of the emergence of bubbles in the field of centrifugal forces, This angle and the rate of emergence Vemer increase with increase of the swirl angle c~ and flow velocity v. 3. The milder erosion action of cavitation in a swirled flow compared with an axial flow for the same cavitation regimes is explained by the large values of pressure at the wall (due to centrifugal forces), emergence of bubbles, and direction of the cavitation jet toward the wall.

LITERATURE CITED ,

2. 3. 4.

.

6.

V. V. Volshanik, A. L. Zuikov, and A. P. Mordasov, Swirled Flows in Hydraulic Structures [in Russian], l~nergoatomizdat, Moscow (1990). A. E. Efimov et al., "Microshocks and luminescent flashes in hydrodynamic cavitation," Zh. Fiz. Khim., 62, No. 1 (1988). G. A. Vorob'ev, Protection of Hydraulic Structures from Cavitation [in Russian], l~nergoatomizdat, Moscow (1990). A. V. Efimov, G. A. Vorob'ev, et alo, "Investigation of the structure of a cavitating liquid," Tr. VNIIOFI (Holographic Methods and Equipment Used In Physical Studies and Their Metrological Provision) [in Russian], No. 4 (1977). A. D. Pernik, Problems of Cavitation [in Russian], Sudostroenie, Leningrad (1966). W. P. Mason, Physical Acoustics [Russian translation], Mir, Moscow (1966).

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