Investigation of hot-fluid cavitating flow through an obstacle in a pipeline

The results of investigating hot-fluid cavitating flow in a pipe with a local contraction are presented for a broad temperature interval (water from cold to near-boiling) and various cavitation regimes — from the initial (bubble) to the supercavitation regime. Experimental relations for the amplitudes of the fluctuations and the fundamental frequencies are presented for a Venturi tube with various diffusers and for diaphragms of various dimensions. A flow model which takes into account the fluctuations of the vapor pressure in the cavity and, moreover liquid-cavity mass transfer effects is presented. It is shown that for a given flow geometry there is a limiting Jakob number below which the self-oscillating regime is impossible at any cavitation numbers.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 124–133, May–June, 1993.     

Dependence of the drag of a conical axisymmetric cavitation body on the cone angle and the cavitation number

Numerical investigations of flow past axisymmetric conical captation bodies have shown that the drag coefficient of the cavitation body, calculated from the maximum cross-sectional area of the cavity (midsection), depends on the cavitation number and the cone angle.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 170–173, May–June, 1995.In conclusion, the author is grateful to G. Yu. Stepanov whose critical remarks were helpful in making this paper more conclusive and more illustrative.     

On the formulation of the problem of the motion of a body in water

The selection of solutions describing steady irrotational flow of an ideal incompressible fluid over bodies is considered. The selection is based on restrictions that follow from the physical properties of a real fluid and from the presence of a boundary layer on the body. In particular, for any body one can specify a minimal Euler number below which flow without cavitation becomes physically impossible. In the limiting case of an Euler number equal to zero, only the Kirchhoff scheme is physical admissible, and the cavity section tends to a circle. An equation is derived for the limiting shapes of cavities at small cavitation numbers, and a comparison is made with known results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–26, September–October, 1980.     

Thin axisymmetric cavities in a subsonic compressible flow

The investigation of subsonic compressible flow past thin axisymmetric cavities carried out in [1–3] is continued by the method of asymptotic expansions. The dependence of the elongation of the cavity on the cavitation number and the Mach number is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–177, September–October, 1987.The author is grateful to Yu. L. Yakimov for discussing his results.     

Quasihomogeneous model of cavitation flows in diffuser channels

Starting with the experiments carried out by Reynolds in 1894, the flow in Venturi tubes has traditionally been used to study and demonstrate various forms of cavitation. Numerous authors have carried out experimental research on the various flow regimes in diffuser channels [1–7] or have investigated theoretical models of such flows [6, 8]. The occurrence and development of cavitation is closely associated with the phenomenon of turbulent separation complicated by the presence of two-phase flow in the dissipation zone. For a long time these effects were considered separately, until Gogish and Stepanov [9] proposed a single model of cavitation and separation based on the theory of intense interaction of an incompressible potential flow and a turbulent cavitation layer of variable density and embracing the various stages of cavitation. The object of this study is to demonstrate the possibilities of this model with reference to the simple example of flows accompanied by cavitation and separation in plane and axisymmetric diffuser channels of the Venturi tube type with straight and curved walls. The dissipative flow near the walls is described by a quasihomogeneous model of turbulent two-phase flow, in which the presence of two phases is taken into account only by varying the mean density. The potential core of the flow is considered in the one-dimensional formulation. The displacement thickness serves as the flow interaction parameter. The conditions of ocurrence and development of circulatory flows are determined. Examples of symmetrical and nonsymmetrical flows are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–54, September–October, 1986.     

Slender axisymmetric cavities in a supersonic flow

Slender axisymmetric cavities in a subsonic flow of compressible fluid were investigated in [1–4]. In [5] a finite-difference method was used to calculate the drag coefficient of a circular cone, near which the shape of the cavity was determined for subsonic, transonic, and supersonic water flows; however, in the supersonic case the entire shape of the cavity was not determined. Here, on the basis of slender body theory an integrodifferential equation is obtained for the profile of the cavity in a supersonic flow. The dependence of the cavity elongation on the cavitation number and the Mach number is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–181, January–February, 1989.     

Capillary cavity flow past a circular cylinder

Cavity flow past a circular cylinder is considered accounting for the surface tension on the cavity boundary. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. The solution is based on two derived governing expressions, which are the complex velocity and the derivative of the complex potential defined in an auxiliary parameter region. An integral equation in the velocity magnitude along the free surface is derived from the dynamic boundary condition. The Brillouin–Villat criterion is employed to determine the location of the point of flow separation. The cases of zero surface tension and zero cavitation number are obtained as limiting cases of the solution. Numerical results concerning the effects of surface tension and cavitation development on the cavity detachment, the drag force and the geometry of the free boundaries are presented over a wide range of the Weber and the cavitation numbers.     

A scheme of developed cavitation flow past a wedge

A scheme is proposed for studying the symmetric steady flow with developed cavitation of an ideal incompressible and weightless fluid past a wedge in a channel. This scheme generalizes the existing schemes of cavitation flow past bodies and describes flow that is intermediate between the flow past obstacles in the Riabouchinsky and Efros schemes. Detailed calculations have been made for the drag coefficient and the cavity size in the case of a plate in an unbounded flow. The results of the calculations are tabulated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 165–170, January–February, 1983.I thank P. M. Ivanov, Kh. Kh. Kalazhokov, and T. Sokhov for assistance in the computer calculations.     

Modeling and computation of cavitation in vortical flow

The paper presents an investigation of Euler–Lagrangian methods for cavitating two-phase flows. The Euler–Euler methods, widely used for simulations of cavitating flows in ship technology, perform well in regions of moderate flow changes but fail in zones of strong, vortical flow. Reasons are the strong approximations of cavitation models in the Euler concept. Alternatively, Euler–Lagrangian concepts enable more detailed formulations for transport, dynamics and acoustic of discrete vapor bubbles. Test calculations are performed to study the influence of different parameters in the equations of motion and in the Rayleigh–Plesset equation for bubble dynamics. Results confirm that only Lagrangian models are able to describe correctly the bubble behavior in vortices, while Eulerian results deviate strongly. Lagrangian formulations enable additionally the determination of acoustic pressure of cavitation noise. Two-way coupling between the phases is required for large regions of the vapor phase. A new coupling concept between continuous fluid flow and discrete bubble phase is developed and demonstrated for flow through a nozzle. However, the iterative coupling between the phases via volume fractions is computationally expensive and should therefore be applied only in regions where Eulerian treatment fails. A corresponding local concept for combination with an Euler–Euler method is outlined and is in progress.     

Method of approximate account for the wall effect in cavitation flow around bodies in water tunnels

One of the basic questions in the study of advanced cavitation in water tunnels of the closed-circuit type is the establishment of the correspondence between the flow patterns observed in the channel and in an unbounded stream. The objective of the study of the wall effect must be the determination of a connection between the basic characteristics of the phenomenon, i. e., the cavitation numbers, the cavity dimensions, the drag coefficients, etc., for the unbounded flow and the channel flow. A large number of works devoted to this question are known [1–7], but in the majority of them only two-dimensional flows are considered. These studies contain either exact solutions obtained with the aid of the apparatus of functions of a complex variable or solutions in the linearized formulation.At the present time there is urgent need to obtain at least approximate solutions for axisymmetric cavitation flows in a tunnel.In several studies [1, 2, 4] it has been shown that in the case of two-dimensional flows the presence of solid boundaries influences the drag coefficient only through the mechanism of a change of the magnitude of the cavitation number, while the variation of the drag coefficient itself with the cavitation number is not changed in comparison with the unbounded flow. It may be assumed that an analogous situation obtains for the axisymmetric case as well. Then the question of the wall effect may be reduced to establishing the connection between the corresponding cavitation numbers.The present paper makes an attempt to establish the correspondence between the cavitation numbers in the unbounded flow and in the tunnel for which the cavities behind the same body have the same areas of the maximal cross section.     

Calculation of axisymmetric cavities downstream of a disk in subsonic compressible fluid flow

Cavitation subsonic water flow past a disk is calculated in accordance with the Riabouchinsky scheme by a finite-difference method at Mach numbers M0.95 and cavitation number –0.02. The calculated results are compared with the data of slender body theory and the results obtained from some approximate formulas.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–103, March–April, 1996.     

Cavitation flow past an arbitrarily-shaped airfoil

The problem of two-dimensional inviscid incompressible flow past an arbitrarily-shaped airfoil in the presence of developed cavitation is studied in an accurate nonlinear formulation.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 86–90, September–October, 1995.     

Study of cavitation phenomena based on a technique for visualizing bubbles in a liquid pressurized chamber

In this paper, the influence of nozzle geometry on cavitation and near-nozzle spray behavior under liquid pressurized ambient is studied. For this purpose, eight steel drilled plates, with different diameters and degrees of conicity of their holes, are analyzed. A special near-nozzle field visualization technique, using a test rig pressurized with fuel, is used. Due to the difference in refractive index between liquid and vapor phase, bubbles present at the outlet of the orifice are visualized. The pressure conditions at which bubbles start appearing at the orifice outlet are compared with those at which choked flow appears. The results showed that pressure conditions for inception of cavitation obtained in the visualization tests differs from those seen for choked flow (5–8% in terms of cavitation number). In addition to this, the images taken are analyzed to get the angle of the jet formed by fuel bubbles, showing that it increases significantly for those conditions more prone to cavitate. Furthermore, comparison of bubbles generation when increasing or decreasing backpressure indicates the presence of hysteresis in cavitation inception phenomena.     

A new method of visualizing liquid flows and measuring velocity fields

A method of visualizing and measuring the velocity field of a liquid flow proposed at the Institute of Mechanics at the Moscow State University in 1967 [3] is discussed. It consists of creating and measuring the size of an artificial cavity behind an extended cavity forming body with small transverse dimension (cavitation probe) placed across the flow at a given position. Because the transverse dimension of the probe is small, the flow deformation is slight.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 101–108, March–April, 1982.     

Motion of a liquid-gas bubble mixture in the incipient cavitation stage

This article is concerned with the nonsteady motion of a mixture of an ideal incompressible liquid with gas bubbles in the incipient stage of cavitation. Solutions are obtained in the first and second approximations. The solutions are applied to pipe flow situations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 151–153, September–October, 1976.The author is indebted to L. I. Sedov for valuable comments.     

Flow past a plate lattice with developed cavitation

Problems of streamline cavitation flow past a lattice were studied in [1–8] using the Kirchhoff scheme. In this scheme the magnitude of the velocity at the free surface is equal to the stream velocity behind the lattice, and the cavitation number is zero (for a lattice the relative velocity and the cavitation number are defined from the stream velocity behind the lattice). In [4, 7] a solution is given of the problem of flow past a lattice using a scheme with an Efros-Gilbargreturn streamline, which permits considering arbitrary cavitation numbers; however, a unique solution is not given. Some other streamline schemes are mentioned in [8].In the following we consider the cavitational flow of an ideal incompressible inviscid and weightless fluid past an infinite lattice of flat plates, using the streamline wake model previously utilized by Wu [9] in studying cavitational flow past an isolated obstacle. In accordance with this model, the streamlines which separate from the body and bound the cavity behind it pass into two curvilinear infinitely long walls, along which the pressure increases and approaches the pressure in the undisturbed stream.It is further assumed that in the hodograph plane there corresponds to the curvilinear walls a cut along some line and that the complex potential takes the same values at points lying on opposite sides of the cut. In particular, at the points of contact of the streamlines with the curvilinear walls the complex potential is the same. In the Wu scheme the latter condition leads to vanishing of the velocity circulation along the contour CABC₁ (Fig. 1).In conclusion the author wishes to thank N. V. Yurtaeva for the accurately performed numerical work.     

Cavitation flow around a plate in a channel by a stream of heavy liquid

The nonlinear problem of cavitation flow around a plate by a stream of heavy liquid is investigated in precise formulation; the plate is located on the horizontal floor of a channel when the gravity vector is directed perpendicular to the wall of the channel. Two flow systems are considered-Ryabushinskii's and Kuznetsov's system [1]. This problem was investigated in linear formulation in [2], Similar problems were considered earlier in [3–7] for unrestricted flow. Below, on the basis of a method proposed by Birkhoff [8, 9], all the principal hydrodynamic and geometric characteristics are calculated for the problem being considered.Translated from Ivestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1973.     

Symmetric cavitation flow past a wedge in the presence of a source on the wedge or in the flow

A solution is found to the problem of symmetric cavitation flow over a wedge by an ideal incompressible fluid (in accordance with Efros's scheme [1]) in the presence of a point source in the flow or on the wedge. Expressions are obtained for the forces exerted on the source and the wedge by the fluid, the conditions under which there is a negative resistance (thrust) are indicated, and the profiles of the free streamlines are constructed for different values of the flow parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 6, pp. 137–141, November–December, 1979.We thank L. I. Sedov for his interest in the work.     

Allowance for the unsteadiness of the flow in determining the critical cavitation number for marine propellers

A method of estimating the critical cavitation number for marine propeller blades is proposed. This method is based on the reduction of the three-dimensional unsteady problem to the three-dimensional steady problem and a series of two-dimensional unsteady problems.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 78–85, January–February, 1993.The authors are grateful to S. V. Kaprantsev for assisting with the experiments.     

Determination of the lift of a partially cavitated hydrofoil

New methods for the velocity circulation around partially cavitated hydrofoils are proposed for both short and long cavities. The calculated results are compared with available experimental data. The calculated lift and cavitation number dependences remain bounded as the cavity end approaches the trailing edge of the hydrofoil.St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 71–74, November–December, 1994.