(1)
Vladimir V. Tarasevich
Associate Prof. of Novosibirsk State University of Architechture and Civil Engineering
NGASU, Leningradskaya st., 113, Novosibirsk, 630008, Russia
Tel.:
(383-2) 669411, Fax (383-2) 161107, E-mail: tvv@iis.nsk.su
Abstract:
The paper is devoted to development of cavity flow
models and deriving on their basis of main qualitative and quantitative
characteristics of a stream, such as a overall flow pattern, criterion of
cavitation initiation, location, initiation time and cavities' lifetime, maximum
pressure estimate etc. The cases both perfect and real liquid were researched.
The comparison with experimental data is represented.
Keywords: pipe, water hammer, cavity, size, duration, collapse, impact, calculation
The water hammer in the pressure pipelines influences on their strength characteristics and is the essential factor for the reliability evaluation of piping system operation [1- 3]. If the saturated vapor pressure is reached in a fluid at a stage of pressure decline, the stream entirety rupture (cavitation) emerges. The additional pressure overshoots emerge under the collapse of cavities (at the stage of recompression), which are dangerous from the point of view of strength parameters of the pipe. The water hammer accompanying by the stream entirety rupture, is observed in pressure conduits of waterpower plans, in pump inlets and in other hydraulic fittings.
THE SINGLE PIPELINE
Let's consider the horizontal pipe of length L, in which the fluid flows with a stationary value of velocity V0 (V0 >0). The constant pressure p0 is specified on the right extremity (x = L):
The non-stationary process arises owing to instantaneous overlapping of pipe cross-section in the left extremity (x=0):
The fluid flow is discribed by the equations of a water hammer [1,2]:
,
(3)
where p=p(x,t) is pressure; V=V(x,t) is velocity; d is pipe diameter; r is liquid density; c is the velocity of water hammer wave [1, 2]; l is hydraulic friction coefficient [2, 3], l= l(|V|). Here the function B(|V|)=l|V|/4d is a strictly increasing function of |V | .
According to aforesaid the initial data will be
,
(4)
A condition of stream entirety saving will be
(5)
where psv is the saturated vapor pressure for the considered liquid (psv << p0).
It is supposed, that the cavity have a vertical walls and fill up all cross-section, and its length is small with comparison to L. The saturated vapor density is neglected for a cavity. Then the left cavity boundary xL and right cavity boundary xR moving is determined by the equations
,
(6)
The conditions are specified for cavity boundaries
(7)
Cavity length is determined by the formula
(8)
where tbc is the instant of cavity beginning. Then the instant of cavity collapse tclp will be defined by equality
(9)
In this case it is easy to find a solution of the mixed problem for a system (3) under the boundary conditions (1) - (2) and initial data (4) . This solution has periodic nature (see figure 1, a). Thus the maximum pressure will be equal according to Zhukowsky formula [1]:
(10)
and the minimum pressure will be equal:
It is follows from (11) and (5) that when
for the left extremity (x=0) there will be p(x,t)< psv at t>0, i.e. condition of stream entirety (5) will be violated. Here
(13)
where N is the entier of K, q is fraction of K.
Let us assume that the cavity described by the equations (6) – (9), will emerge near the left extremity, thus tbc = 0. Using boundary conditions (9) instead of (2), one can obtain the solution of the problem in an interval 0<t<tclp. The obtained solution shows that for 0<t<tclp the condition (5) is valid in all remaining area of current, i.e. the cavitation can be localized by way of one isolated cavity in the beginning of the pipe. It is easy to obtain in this case:
,
when
(14)
where
,
.
Substituting (14) in (8), one can find from (9):
(15)
The boundary condition (2) is recovered after a cavity collapse. It is possible to find a solution of the problem when tcl≤t. The pattern in (x,t) - plane after cavity collapse is represented in figure 2. Thus the maximum pressure will be
(16)
The formula (16) demonstrates, that the function pmax depending on V0 represents a saw-tooth line (see figure 3); i.e., as against a case of solid flow (10), maximum pressure can fall under the growth of an initial velocity V0.
The minimum values of pressures (see Fig. ) are determined by the formulas:
One can determine from here that p6<psv always, and p5<psv when N>1. Thus, a cavity arises in an instant tbc1 near the left extremity (x = 0) when N > 1, and one more cavity arises in an instant tbc2 in an inner part of a stream at point xbc2. On the contrary, when N=1, a cavity arises in an inner part of a stream at point xbc2 at first, and a cavity arises later on in an instant tbc3 near the left extremity (x = 0). In the presence of these two cavities the condition (1) is fulfilled in a remaining part of a stream everywhere, i.e. it is sufficient to localize the cavitation in the form of two isolated cavities. Here
,
,
,
Let us consider an initial stage of the process. It is obvious, that the perturbation will spread along characteristic curve x = ct. We obtain the equations for pressure p*(t)=p(ct,t) and velocity V*(t)=V(ct,t) behind a wavefront of perturbation:
,
(17)
The solution without taking a cavity into consideration gives
when
(18)
Formulas (18) and (1) imply, that under fulfillment of a condition,
, where
,
(19)
the stream entirety rupture takes place in the beginning of the pipe (x=0) when t>0. The inequality (19) is criterion of a cavitation appearance (in the form of small-bubble tail area) immediately behind a perturbation wavefront, and is more strict requirement than (12). However, if to consider (12) as criterion of a cavitation initiation generally, i.e. not only within the first phase at once behind a rarefaction wave front, but also in the subsequent moments, then the criterion (12) can be considered as criterion of cavitation initiation in this case too as numerical experiments demonstrate. It is interesting to note, that (19) ®(12) when e®0. Here
(20)
Let's try to localize a cavitation just as it was done above, having located a solitary cavity described by conditions (6)-(7), in the beginning of the pipe. One can determine the solution of the system (17) will be such, that the value p*(t)<p0 when t>0. In other words, the condition (5) is violated if there is alone isolated cavity at point x = 0 when 0<t<L/c, i.e. the stream entirety rupture will be existing as before. Analogously, it is possible to show, that the adding of any finite number of cavities does not result in realization of an inequality (1). Thus, in a case (l ¹ 0), i.e. under taking into account the viscous properties of a fluid, it is impossible to localize a cavitation by way of separate isolated cavities. Therefore, apart from birth of large cavities the appearance of small bubbles' tail area is inevitable.
Let us consider dynamics of the "detached" fluid column. On the left extremity of this fluid column the cavity with boundary conditions (6) –(9) is disposed. As shown above, an extensive zone of small-bubble cavitation will arise in the detached fluid column, which will pulse according to transiting compression and rarefaction waves. In view of considerable energy dissipation in bubbles, the undular process will damp fast in the detached fluid column. Therefore without an essential error it is possible to consider the moving of the liquid column as moving of uniform “liquid rod” for estimate of the average velocity of liquid column. A motion equation of such a rod is
(21)
where Vst is the average velocity of detached liquid column, w is pipe cross-section area; r – is liquid density; t»lr|Vst|Vst/8 is liquid friction stress; c is wetted perimeter.
The solution of (21) with (9) gives
(22)
Let us note, that formula (22) becomes transformed to the envelope of (15) when e®0.
As the results of analytical modeling for perfect fluid (see Fig. ) show, the maximum pressure p3 is generated by the part of a liquid column, continuing mechanically (as though a cavity still existed) and dissipating the kinetic energy completely up to the instant tclp+T. Therefore with some store it is possible to determine maximum pressure by the Zhukowsky formula for the liquid column velocity at instant tclp+T:
One can obtain using (23), (22) and solution of (21):
It is interesting to note, that formula (24) turns into the envelope of (16) when e®0. The results of calculation by the formula (24) in comparison to experimental data are represent in fig.3.
The main parameters of the oscillatory process of the water hammer accompanying by a cavitation of a stream have been estimated in the paper. The possibility of secondary cavities appearance is predicted and the conditions of their emerging are obtained. It is proved, that in the real fluid (under taking into account the forces of viscosity) the process of a cavitation cannot be localized by way of alone isolated cavity, i.e. it will be accompanied by appearance of the dispelled small-bubble medium. The evaluation of maximum pressure are obtained for this case. The comparison with the results of computer calculations by more precise technique and experimental data has shown a good degree of the concordance.
References
[1] Zhukowsky N.E. About a water hammer in water pipes. - Journal of Polytechnic Society, 1899, No.5. (in Russian).
[2] Streeter, V.L. and Wylie, E.B., Hydraulic Transients. Mc Graw-Hill, 1968.
[3] Kartvelishvili N.A. Dynamics of pressure pipelines. - Moscow: Energia, 1979. (in Russian)
[4] Smirnov D.N., Zubov L.B. The water hammer in pressure conduits. - Moscow: Stroiizdat, 1975. (in Russian).
Fig. 1 Pressure at the left extremity of pipe (x/L=0,02): a - without taking the stream entirety rupture into consideration; b - taking the stream entirety rupture into account. Here p*=(p–psv)/(p0–psv)
Fig. 2 Secondary cavitaion evolution.
C1 is primary cavity; C2, C3 are the secondary cavities.
Fig. 3 pmax depending on V0 : 1 – experimental data [4]; 2 – by formula (16); 3 – by Zhukowsky formula (10); 4 – computer calculation taking into account the friction (l¹0); 5 – calculation by formula (24).
