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.     

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.     

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.     

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.     

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.     

Three-dimensional separation in the neighborhood of roughness on the surface of an axisymmetric body

Steady-state viscous incompressible fluid flow past an axisymmetric slender body is considered at high Reynolds numbers in the regime with vanishing surface friction in a certain cross-section. In a small neighborhood of this cross-section interaction between the boundary layer flow and the external irrotational stream develops. In order to study the structure of the three-dimensional flow with local separation zones it is assumed that there is three-dimensional roughness on the surface of the body with the scale of the interaction zone. For this zone a numerical solution of the problem is obtained and its nonuniqueness is established. The surface friction line (limiting streamline) patterns with their inherent features are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–79, May–June, 1995.Thus, on the basis of the asymptotic marginal separation theory it is possible to obtain fairly simple solutions describing flows with a complex surface friction line structure.     

Circulation flow at the front surface of a sphere in a supersonic wake

Several theoretical and experimental studies of supersonic flow past a blunt body located in the wake behind another body have been made [1–7]. It has been shown that a reverse-circulation flow can occur in the shock layer at the front surface. The possibility of such a flow forming depends on the nonuniformity of the freestream flow and the Reynolds number. This paper presents new results of the theoretical study of the structure of the shock wave at the front surface of such a sphere, obtained on the basis of numerical solution of Navier-Stokes equations. It is shown that for a fixed nonuniformity of the freestream flow, an increase in the Reynolds number and cooling of the surface of the body lead to the formation of a secondary vortex in the region where the contour of the body intersects the axis of symmetry. A study is made of the variations of the drag and heat transfer parameters over the front surface of a cooled and thermally insulated sphere. The possibility of numerical simulation of the flow on the basis of the Euler equations is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–148, May–June, 1985.     

Cavitation flow calculations for a viscous and capillary fluid

Developed cavitation calculations, where the cavity forms a void directly adjoining and stationary relative to the body, have been carried out almost exclusively within the framework of ideal fluid mechanics [1, 2]. Experiments (for example, [2, 3]), however, indicate that viscosity and capillarity have an undoubted influence on cavitation flows. In the case of developed cavities behind nonlifting bodies this effect has been taken into account [4] in terms of the dependence of the arc abscissa of the beginning of the cavity on the Weber and Reynolds numbers We and Re for a given value of the cavitation number. In calculating a partial cavity (of a length not exceeding that of the body in the flow) it is necessary to take into account the development of the boundary layer on the cavity and the presence of viscous separation zones not only in front of but also behind it. In this paper a method of calculating partial cavitation satisfying these requirements is proposed, and problems relating to the justification of the method are discussed. The cavitation calculations presented employ the flow model described in [5], which takes into account the presence of the boundary layer on the body and the cavity, together with the viscous separation zones. The calculation method is a development of that described in [6] and makes important use of an idea derived from [2, 7]. In this connection, the fact that the characteristics of the boundary layer in cavitation flow past bodies have not been sufficiently studied has made it necessary to resort to a numerical experiment to close the semiempirical relations used in the calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 45–51, November–December, 1985.     

Maximum dimension of ice-rock bodies formed in liquid flow past a linear chain of cold sources

An asymptotic analysis of the limiting situation of liquid flow past a linear chain of cold sources when an ice-rock body with constant cross section grows without limit along the flow is carried out. The dependence of the critical heat flow rate of the cold sources and the transverse dimension of the body on the Péclet number is determined.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 179–181, July–August, 1996.     

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.     

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.     

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.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Subsonic gas-liquid cavitation flow past a disk

The problem of axisymmetric subsonic gas-liquid cavitation flow past a disk in accordance with the Riabouchinsky scheme is solved using the method of [1]. Formulas relating the main flow parameters with the cavitation number, the Mach number on the free boundary and the gas/liquid volume ratio under stagnation conditions are presented.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 202–206, March–April, 1996.     

On the size of cavities in tunnels with continuous and perforated walls

A one-dimensional theory is developed to describe the hydrodynamic interaction of cavities with walls of continuous and perforated working sections, allowance being made for the influence of hydraulic losses and the boundary layers on the walls, and also the ponderability and surface-tension forces. It is shown that if some of the fluid is sucked through the perforated boundaries of the working section one can not only strongly reduce the influence of the walls of the tunnel on the size of a cavity but also appreciably increase the useful loading of the flow by the cavity. The available experiments and some made specially by the authors are compared with the calculations, and this reveals satisfactory agreement in the complete range of cavitation numbers and relative blocking of the flow by the body and cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 163–167, March–April, 1979.     

Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents

The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an explicit geometric construction of approximate eigenfunctions for the linearized Euler operator L in vorticity form acting on Sobolev spaces on two dimensional torus. We show that each nonzero Lyapunov–Oseledets exponent for the flow induced by the steady state contributes a vertical line to the essential spectrum of L. Also, we compute the spectral and growth bounds for the group generated by L via the maximal Lyapunov–Oseledets exponent. When the flow has arbitrarily long orbits, we show that the essential spectrum of L on L₂ is the imaginary axis.     

Dynamics of cavitational demolition in the interaction of a shock wave with a free surface

The problem of cavitation in the tension wave associated with the reflection of a shock wave from a free fluid surface is considered. A method of calculating the cavitation zone dynamics which makes it possible to determine the structure of the cavitation front, including for large space scales, is developed. A procedure for determining the dispersity of the fragment-drops of dispersed fluid, which takes into account the initial size distribution of the cavitation nuclei and the parameters of the incident shock wave, is proposed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 73–80, November–December, 1992.     

Motion of a variable-volume sphere in an ideal fluid near a plane surface

The motion of a spherical cavity in a fluid is investigated. The radius of the sphere varies under the action of a constant pressure at infinity. The problems of the collapse of a cavity moving in an unbounded fluid and of the collapse of a cavity near a plane are solved in the exact formulation. The occurrence of an initial translational velocity or the presence of a solid surface, by contrast with the collapse of a sphere at rest in an unbounded fluid [1], yields a limiting radius at which the process of collapse ceases. A sphere initially at rest near a plane always comes into contact with the plane as a result of collapse. The radius and velocities at which the sphere arrives the plane are calculated for various initial distances from the latter. The possible mechanism of the action of a cavitation bubble on a solid surface is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1971.     

Surface waves and stability of tangential velocity discontinuity on a solid-fluid boundary

Solutions of the Rayleigh-wave type on the boundary of an elastic half-space and a moving layer of ideal fluid are obtained. The limiting cases of zero flow velocity and a tangential velocity discontinuity in the fluid were investigated in [1–3]. In [4] the order of magnitude of the critical flow velocity was estimated. An increase in the velocity scales used in engineering and experimental practice (see [5], for instance) has aroused interest in a more thorough analysis of the effect.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 43–46, May–June, 1981.     

Stokes waves on a cavity surface in a rotating fluid

The axisymmetric flow of an inviscid incompressible fluid rotating about a cavity with constant pressure is considered. Due to the centrifugal force, on the cavity surface waves may exist, in particular, waves with a break in the wave base where the cavity meridional sections form the angle 2/3, i.e. Stokes waves. A method of finding these waves from the boundary-value problem for the fluid velocity potential is described. For an infinite cavity, the dependence of the wave parameters on the cavitation number, calculated using the pressure in the cavity, is given.St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 105–110, November–December, 1996.