A. A. Aganin

Evolution of deviations from the spherical shape of a vapor bubble in supercompression

This paper considers the evolution of small deviations of a cavitation bubble from a spherical shape during its single compression under conditions of experiments on acoustic cavitation of deuterated acetone. Vapor motion in the bubble and the surrounding liquid is defined as a superposition of the spherical component and its non-spherical perturbation. The spherical component is described taking into account the nonstationary heat conductivity of the liquid and vapor and the nonequilibrium nature of the vaporization and condensation on the interface. At the beginning of the compression process, the vapor in the bubble is considered an ideal gas with a nearly uniform pressure. In the simulation of the high-rate compression stage, realistic equations of state are used. The non-spherical component of motion is described taking into account the effect of liquid viscosity, surface tension, vapor density in the bubble, and nonuniformity of its pressure. Estimates are obtained for the amplitude of small perturbations (in the form of harmonics of degree n = 2, 3, ... with the wavelength λ = 2πR/n, where R is the bubble radius) of the spherical shape of the bubble during its compression until reaching extreme values of pressure, density, and temperature. These results are of interest in the study of bubble fusion since the non-sphericity of the bubble prevents its strong compression.

Evolution of distortions of the spherical shape of a cavitation bubble in acoustic supercompression

The evolution of small perturbations of the spherical shape of a vapor bubble in the process of its single strong expansion and compression in deuterated acetone is studied. In the mathematical model used the motion of vapor and liquid is broken down into the spherical component and its small nonspherical perturbation. The spherical component is described by the fluid dynamics equations with account for time-dependent heat conduction and evaporation and condensation on the liquid-vapor interface using equations of state constructed from experimental data. In describing the nonspherical component the liquid viscosity and the surface tension are taken into account, while the effect of the bubble content is disregarded. Certain simple analytical formulas are presented which describe the bubble radius at the moment of maximum expansion, its variation in the compression stage, and the evolution of the bubble sphericity distortion in compression.

Numerical simulation of gas dynamics in a bubble during its collapse with the formation of shock waves

The specific features of calculation of a gas in a spherical bubble located in the center of a spherical volume of weakly compressible fluid are considered. The problems of motion of a cold gas to a point and a spherical piston converging to a point are used to evaluate the algorithm. It is shown that significant errors can arise in calculation of spherical waves in the vicinity of the pole. These errors can be substantially reduced by means of artificial viscosity in the Riemann problem.

Modeling strong compression of a gas bubble in fluid

A technique for calculating strong adiabatic compression of a gas bubble in fluid is proposed. The compression results from the pressure applied to the outer surface of the fluid. The motion of the fluid and gas is described by two-dimensional dynamic equations of compressed fluid and gas with realistic equations of its state. The effects of viscosity and thermal conductivity are not allowed for. The bubble surface is defined as a contact interface where there is a surface tension. Coupled Euler-Lagrange coordinates are used, with the bubble surface serving as a coordinate system. A spherical system of coordinates is used as a fixed reference. Equations of gas and fluid dynamics are solved by Godunov’s equations with second-order accuracy in space and time. The economic feasibility of the technique is illustrated by some model problems. The proposed method has been proven to be much more efficient than the classic first-order-approximation Godunov’s schemes traditionally used in solving problems of a highly compressed bubble. One of the scenarios is used to show the influence of slight spherical-shape distortions of the bubble on the evolution of the radially converged shock wave resulting from the strong compression.

Numerical simulation of impact of a jet on a wall

A computing technique for simulating the impact of a high-speed liquid jet on a wet wall is implemented. Such an impact generates shock waves in the jet, in the liquid layer on the wall, and in the gas surrounding the liquid. Also, the interphase boundary is strongly deformed by such an impact. The technique is based on the Constrained Interpolation Profile-Combined Unified Procedure (CIP-CUP) method combined with the dynamically adaptive Soroban grids. The gas-dynamic equations describing the liquid and gas flow are integrated without an explicit separation of the liquid-gas boundary. Such an approach is shown to be efficient for the considered problems. It allows us to obtain solutions without oscillations near the interfaces (including the case where they interact with the shock waves). For illustrative purposes, we provide the computational results for several one-dimensional and twodimensional problems with the typical features of the impact of a high-speed liquid jet on a wall, as well as a comparison with the known analytical and numerical solutions. The computational results for the problem of the impact of a high-speed liquid jet on a wall covered by a thin liquid layer are also presented.

Evolution of a small sphericity distortion of a vapor bubble during its supercompression

The possibility of using two models to study the evolution and maximum increase in amplitude of small distortions of sphericity of a bubble during its strong compression in a liquid is investigated. The investigation is performed in the conditions of experiments on acoustic cavitation of deuterated acetone. The first (fully hydrodynamic) model is based on the two-dimensional equations of gas dynamics. It is valid in every stage of the bubble compression. But its use takes up a lot of computational time. The second (simplified) model is derived by splitting the liquid and vapor motion into a spherical part and its small nonspherical perturbation. To describe the spherical component, a onedimensional version of the two-dimensional model is used in this model. The advantage of the simplified model over the full one is its much lower consumption of computational time. At the same time, the evolution of the nonspherical perturbation in this model is described by utilizing a number of assumptions, validity of which is justified only at the initial stage of the bubble compression. It is therefore logical to apply the simplified model at the initial low-speed stage of the bubble compression, while the full hydrodynamic one is applied at its final high-speed stage. It has been shown that such a combination allows one to significantly reduce the computational time. It has been found that the simplified model alone can be used to evaluate the maximum increase of the amplitude of small sphericity distortions of a bubble during its compression.

Nonlinear radial oscillations and spatial translations of a nonspherical gas bubble in a liquid

A mathematical model of dynamics of a gas bubble in a liquid with non-small distortions of its spherical shape has been developed, with allowing for the spatial translations of the bubble, as well as the influence of the gravitational force and the liquid velocity. The liquid viscosity and compressibility are taken into account approximately. It has been shown that in some particular cases the derived equations are coincident with those obtained by the other authors. Some results of solving the problem of oscillations of a moving nonspherical bubble under periodic variation of liquid pressure are presented.

Numerical Simulation of the Evolution of a Gas Bubble in a Liquid Near a Wall

A numerical technique based on the application of the boundary element method is proposed for studying the axially symmetric dynamics of a bubble in a liquid near a solid wall. It is assumed that the liquid is ideally incompressible and its flow is potential. The process of expansion and contraction of a spheroidal bubble is considered, including the toroidal phase of its movement. The velocity and pressure fields in the liquid surrounding the bubble are evaluated along with the shape of the bubble surface and the velocity of its displacement. The numerical convergence of the algorithm with an increase in the number of boundary elements and a refinement of the time step is shown, and comparison with the experimental and numerical results of other authors is performed. The capabilities of the technique are illustrated by solving a problem of collapse of a spheroidal bubble in water. The bublle is located a short distance from the wall.

Deformation of a bubble formed by coalescence of cavitation inclusions and shock wave inside it at strong expansion and compression

Dynamics of a cavitation bubble is considered at its strong expansion and subsequent compression. The bubble is formed by merging of two identical spherical cavitation microcavities in the pressure antinode of the intensive ultrasonic standing wave in the half-wave phase with negative pressure. Deformations of bubble and deformations of radially converging shock waves occurring therein at bubble compression are studied depending on the size of microcavities forming the bubble. It is found that compression of the medium in the bubble by the converging shock wave is kept close to the spherical one only in the case, when the radius of merging microcavities is 1800 times smaller than the radius of the bubble formed by merging at the time of its maximal expansion.

Impact of a cavitation bubble on a wall

Impulse action of a cavitation bubble on a rigid wall is studied depending on the distance between them. We determine the distances at which the periphery pressure maximums on a wall are preserved as well as the distances at which these maximums exceed the water hammer pressure.