SCIENCE: THEORY, EXPERIMENT, PRACTICE CAVITATION PARAMETER IN SWIRLING FLOWS A. V. Efimov and T. Yu. Kuznetsova

UDC 532.528.001.57

As is known, the cavitation parameter k is the main index of the presence and development of the cavitation process. The relative cavitation coefficient k/kcr (kcr is the coefficient of the start of the process) in a certain interval of average values characterizes its erosive activity. The physical essence of the hydrodynamic cavitation coefficient - the ratio of the potential energy of the flow at the point in question to the kinetic energy, in other words, the ratio of the absolute values of the annihilation pressure Pan to the occurrence or growth Pocc of cavitation bubbles (cavities) at the level of the exciter of cavitation:

k=P~,/Poco.

(1)

However, when designing hydraulic structures it is useful to know the components of the cavitation parameter for reducing them to prevent cavitation, the occurrence of which can be caused both by design features of elements of the water conduit and by construction defects. It is especially useful to know the components of the cavitation parameter when designing conduits with swirling flows, where a complex three-dimensional picture of the flow and cavitation jet is observed. An analysis of these components with a quantitative estimation is given below. A change in Pocc is related mainly to a change in the flow velocity, in other words, Pocc is the velocity head. As a result of strict orientation of the cavitation exciters along or across the axis of the conduit,* the cavitation-forming velocity will not be the local velocity at the level of the height of the ledge v but one of its prevailing mutually orthogonal components: either v m directed along the longitudinal axis of the conduit (axial velocity) or vu directed normal to this axis (peripheral velocity). Therefore 2

Pocc= ~

~'~A ~ 2

(2)

where v m = v cos c~ and v u = v sin c~; c~ is the swirl angle of the flow, i.e., the angle between vector v and the longitudinal axis of the conduit. The values of these hydraulic quantities can be determined either experimentally or by formutas [1]. It is seen from the relation between v and its components that the resultant flow velocity such that cavitation is prevented in swirling flows can be taken greater than in axial flows. The annihilation pressure of a cavitation bubble is composed of three components: the time-average hydrostatic pressure Pst, dynamic pressure APdyn, and so-called "atmosphere" of the bubble Pbub:

P~=Pst-~-aPdyn@ Pbub9

(3)

Frequently Pst and the fluctuation component APdyn are determined experimentally directly on the conduit wall and not in the flow at the level of the cavitation exciter. Such an approach leads to errors in determining the cavitation parameter even in axial flows [2]. In swirling flows in the presence of large pressure gradients along the radius of the conduit, this leads to an increase of the error. Let us examine the physical essence of Pan, i.e., its components, the significance of these components, and the need to take them into account under various conditions. The hydrostatic pressure Pst theoretically consists of the algebraic sum of three components: Pcf, the centrifugal pressure; Pc, the value of the pressure or vacuum in the core; PG, the gravitational component, i.e., *In the case of examining construction defects, rough welds or ledges at the joint of metal sheets of the conduit lining can be cavitation exciters. Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 7, pp. 28-30, July, 1993.

0018-8220/93/2707-0401512.50 9

Plenum Publishing Corporation

401

Fig. 1. Diagram of the gravitational component along the cross section of the conduit: ~) current angle; l?} slope angle of conduit.

PSt d/)cf "~-Dc-~ P6"

(4)

The greatest role in formula (4) is played by Pce, the value of which depends on the swirl angle of the flow ~. The value of Poe depends on centrifugal acceleration j, which for large swirl angles can exceed g by an order of magnitude and more and is determined not only by the law of variation of the peripheral velocity vu along the radius of the section but also by the thickness of the rotating layer of water (from the core to the ledge), determining the limit of integration:

rled ~2 ]gcf= o ,! "2~rdr=QU2uln(rled--rc) ,

(5)

rc

where rled is the distance from the axis of the pipeline to the ledge; rc is the radius of the vapor-air core of the swirling flow. The pressure in the core Pc = pg(9.94 - Vz/900) - Pv(T) represents the value of the critical vacuum in the presence of a free flow surface, consisting of the algebraic sum of the minimum atmospheric pressure and saturation vapor pressure of the water at the corresponding temperature Pv(T). For small swirl angles c~ and considerable diameters of the conduits, the gravitational component PG cannot be neglected (Fig. 1):

P~=rQg sin

~ cos 13,

(6)

where so is the current angle; /3 is the slope angle of the conduit; the maximum negative effect of PG will be felt at the upper point of a horizontal conduit (for r = 90 ~ and/3 = 0) under the condition of flow without separation. Such a condition will occur when the centrifugal or gravitational forces or accelerations are equal: (v sin

c,)2/r=g.

(7)

In addition to this, the peripheral velocity v sin a should be less than the value for which cavitation erosion is still observed, i.e., so-called threshold velocity (for steel St3 with an air content of the cavitating water close to 0%, this velocity is approximately equal to 15 m/sec) [3]. If we take the value of the threshold velocity equal to 15 m/sec and swirl angle c~ = 30 ~ the gravitational component PG should be taken into account for a radius of the pipeline of the outlet not less than 4.5 m. For values of velocities less than the threshold value, so-called gas cavitation, representing, in essence, degassing due to the longer existence of a cavity and diffusion into it, in this case, of air dissolved in water, begins to prevail. This air has a damping action during collapse of the cavity, which substantially reduces cavitation erosion. In addition to components of the time-average pressure Pst, the cavitation parameter k is affected by its dynamic value APdyn, consisting of fluctuation 2xPfluc and vibration (hydroelasticity) AXPhecomponents:

APayn= Apfluc _~_APlae.

(8)

The kinematics of the so-called shear layer at the boundary of interaction of coaxially oppositely swirling flows is probably the main source of pressure fluctuations in swirling flows. 402

For calculation of APtquc, the standard deviation of the pressure fluctuations o is determined experimentally, and in the presence of a normal probability distribution law of fluctuations the maximum amplitude of 5 % probability (see below for substantiation of this value) equal from 3~ to 5o depending on the observation time interval T is calculated [4]. The vibration load from the conduit wall on the water APhe is calculated by the formula [5]

APh~=2n~cVa,

(9)

where 0 is the density of water; c is the speed of sound in water; v is the amplitude of the driving-force harmonic frequency; a is the half-range of vertical vibration of 5% probability with installation of the detector in the upper part of the pipeline.* It is considered that the amplitudes of fluctuations or vibrations of 5 % probability and higher affect the state of cavitation by analogy to the fact that the cavitation duration coefficient ,---= (ZAL,v)/T,

(10)

composed on the basis of the amplitudes of a vacuum of 5 % probability, determines the start of visible cavitation. Here ETcav is the sum of the time intervals during which the fluctuating vacuum reaches or exceeds the critical pressure, equal to three values of the saturation vapor pressure

Pcr=3P~(T)

(ll)

According to the data of experimental investigations [6], at a flow velocity equal to 20.2 m/sec, for cavitation stages of maximum erosion activity consideration of the fluctuation and vibration components on the model reduces the relative cavitation coefficient k/kcr by 5.9% in the case of flow past a cylinder with separation and by 3.2% in the case of flow past a ledge. However, under actual conditions the hydroelasticity component Phe ~'22 -

can be taken equal to zero due to the small values of the frequencies and amplitudes of vibrations of the structures compared with the fluctuation component. We will estimate the contribution of the last component Pbub

to the cavitation parameter. The pressure in the bubble Pbub consists of the saturation vapor pressure Pv(T), determined from tables and dependent on temperature T ~ and partial pressure of the air in the bubble Pa" As is known, the mechanism of entry of air into cavitation bubble can be dual: first, due to air dissolved in water. However, this diffusion is a rather long process (diffusion coefficient D = 2 . 1 0 -5 cm2/sec at T = 20~ and during the time of existence of the vapor cavitation bubble of thousandths of a second, its action cannot be neglected. Second, due to air present in water in a disperse state. The increase of the cavitation coefficient k due to an increase of pressure in a bubble with the entry there of gas is proportional to Pa, equal to mg RT

P~= VbubP.

(i2)

where mg is the mass of the gas entering a bubble with volume Vbub, R, #, and T are respectively the gas constant, molecular mass, and absolute temperature of the gas. Thus, Pa ~v2

2 is proportional to mg and inversely proportional to Vbub. Taking into account that during formation of a cavitation bubble from a nucleus its radii increase by 3-4 orders and the volume by 9-12 orders with a constant mass of gas in it, as a result we obtain a negligible value of the pressure of this mass of gas. (If the pressure in the central w a t e r - a i r cavity of the

* Reference is to the location of the d e t e c t o r on the outside of the wall of the m o d e l pipeline.

403

swirling flow P < 105 Pa, then the pressure in the bubble Pbub = Pv(T)) 9 The results of an experimental study [7] indicate the same: with a change in the air content by 6.5 times (from 0.26 to 1.66%), kcr increases by 1.25 times. Thus it can be concluded that a small air content of water within 0.2-1.96% does not have a substantial effect on the cavitation coefficient [8]. A further increase of the air content (from 2 to 7%) leads to an increase of kcr, but in that case it loses sense, since the process of hydrodynamic cavitation is transformed into degassing, in which case the erosion activity of cavitation decreases by several orders.

LITERATURE CITED

.

2. 3. 4.

V. V. Volshanik, A. L. Zuikov, and A. P. Mordasov, Swirling Flows in Hydraulic Structures [in Russian], t~nergoatomizdat, Moscow (1990). R. S. Gal'perin, Cavitation on Hydraulic Structures [in Russian], t~nergoatomizdat, Moscow (1977). Recommendations on Consideration of Cavitation when Designing Hydraulic Outlet Works: P-38-75 [in Russian], VNIIG, Leningrad (1976). S. M. Slisskii, Hydraulic Calculations of High-Head Hydraulic Structures [in Russian], t~nergoatomizdat, Moscow

7.

(1989). Ya. I. Frenkel', "Cavitation on a propeller surface," Inzh.-Fiz. Zh., 14, No. 2 (1969). A. V. Eflmov, "Investigations of cavitation erosion as a function of the hydraulic parameters of the flow, stage of cavitation, and shape of the body exposed to the flow," Candidate's Dissertation, Moscow (1972). L. S. Shmuglyakov, "Relation between the cavitation coefficient and air content of the medium," t~nergomashino-

8.

stroenie, No. 5 (1956). S. S. Papaniotti, "Effect of the gas content of a liquid on cavitation characteristics," Tr. VNIIGidromash, No. 38

5. 6.

(1968).

404