Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

Axisymmetric creeping motion of a slip spherical particle in a nonconcentric spherical cavity

A semianalytical study of the creeping flow caused by a spherical fluid or solid particle with a slip surface translating in a viscous fluid within a spherical cavity along the line connecting their centers is presented in the quasisteady limit of small Reynolds number. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the particle and cavity. The boundary conditions on the particle surface and cavity wall are satisfied by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the ratio of particle-to-cavity radii, the relative distance between the centers of the particle and cavity, the relative viscosity or slip coefficient of the particle, and the slip coefficient of the cavity wall. In the limits of the motions of a spherical particle in a concentric cavity and near a cavity wall with a small curvature, our drag results are in good agreement with the available solutions in the literature. As expected, the boundary-corrected drag force exerted on the particle for all cases is a monotonic increasing function of the ratio of particle-to-cavity radii, and becomes infinite in the touching limit. For a specified ratio of particle-to-cavity radii, the drag force is minimal when the particle is situated at the cavity center and increases monotonically with its relative distance from the cavity center to infinity in the limit as it is located extremely away from the cavity center. The drag force acting on the particle, in general, increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given configuration, but surprisingly, there are exceptions when the ratio of particle-to-cavity radii is large.     

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.     

Transition flow of a fibrous suspension in a pipe as the flow of an anisotropic fluid

The transition flow is considered of a fibrous suspension in a pipe. The flow region consists of two subregions: at the center of the flow a plug formed by interwoven fibers and fluid moves as a rigid body; between the solid wall and the plug is a boundary layer in which the suspension is a mixture of the liquid phase and fibers separated from the plug [1–3]. In the boundary region the suspension is simulated as an anisotropic Ericksen—Leslie fluid [4, 5] which satisfies certain additional conditions. Equations are obtained for the velocity profile and drag coefficient of the pipe, which are both qualitatively and quantitatively in good agreement with the experimental results [6–8]. Within the framework of the model, a mechanism is found for reducing the drag in the flow of a fibrous suspension as compared to the drag of its liquid phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–98, September–October, 1985.     

Oscillations of a body filled with a viscous fluid

The oscillations of a physical pendulum containing a spherical cavity filled with an incompressible viscous liquid were discussed in [1]. In this paper we consider the mote general problem of the motion of an axially symmetric solid with a spherical cavity filled with an incompressible viscous fluid and moving about a fixed point. It is assumed that the center of the cavity and the fixed point lie on the axis of symmetry of the body.     

Influence of the boundary of a column of incompressible liquid in investigating axisymmetric oscillations of a solid sphere in a cavity

Axisymmetric oscillations of a rigid spherical body in a column of ideal incompressible liquid with a plane boundary in the form of a free liquid surface or a rigid wall within a round cylindrical cavity are considered. The potential and pressure fields are plotted; expressions are obtained for the kinetic energy of the system and the hydrodynamic forces acting on the body. The resistance of the liquid to accelerated movement of the body is determined as a function of the distance to the boundary, for various parameter values. For specified oscillations of the body, the results obtained for axisymmetric conditions in a halfspace are compared with those obtained in an infinite cylindrical cavity. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 11–18, December, 1999.     

Wall proximity effects on the effectiveness of upstream control rod

Two dimensional flow over a circular cylinder with an upstream control rod of same diameter is simulated in unbound condition and in wall bounded conditions. The cylinders are placed at various heights from the wall and the inter-distance between cylinders is also varied. The control rod is subjected to different rotation rates. It is found that, in unbound condition, rotating the control rod decreases the critical pitch length (S/D_cr) and increases the drag and Strouhal number of the main cylinder. In presence of plane wall, the shielding provided by the separated shear layers from the control rod in cavity regime is deteriorated due to deflection of shear layers which results in higher drag and large fluctuation of lift coefficient. However, in wake impingement regime, the binary vortices from the control rod are weakened due to diffusion of vorticity and hence, the main cylinder experiences a lower drag and small lift fluctuations than that of unbound condition. The critical height of vortex suppression (H/D_cr) is higher in cavity regime than that of wake impingement regime due to the single extended-bluff body like configuration. The rotation of control rod energizes the wall boundary layer and increases the critical height of vortex suppression. Increasing the rotational rate of control rod decreases the drag force and reduces the amplitude of lift fluctuation. Analysis of the wall shear stress distribution reveals that it suffers a sudden drop at moderate height where the normal Karman vortex shedding changes to irregular shedding consisting of single row of negative vortices. Modal structures obtained from dynamic mode decomposition (DMD) reveal that the flow structures behind the main cylinder are suppressed due to wall and the flow is dominated by the wake of control rod.     

On the friction drag reduction mechanism of streamwise wall fluctuations

Understanding how to decrease the friction drag exerted by a fluid on a solid surface is becoming increasingly important to address key societal challenges, such as decreasing the carbon footprint of transport. Well-established techniques are not yet available for friction drag reduction. Direct numerical simulation results obtained by Józsa et al. (2019) previously indicated that a passive compliant wall can decrease friction drag by sustaining the drag reduction mechanism of an active control strategy. The proposed compliant wall is driven by wall shear stress fluctuations and responds with streamwise wall velocity fluctuations. The present study aims to clarify the underlying physical mechanism enabling the drag reduction of these active and passive control techniques. Analysis of turbulence statistics and flow fields reveals that both compliant wall and active control amplify streamwise velocity streaks in the viscous sublayer. By doing so, these control methods counteract dominant spanwise vorticity fluctuations in the near-wall region. The lowered vorticity fluctuations lead to an overall weakening of vortical structures which then mitigates momentum transfer and results in lower friction drag. These results might underpin the further development and practical implementation of these control strategies.     

Compressible air flow through a collapsing liquid cavity

We present a multiscale approach to simulate the impact of a solid object on a liquid surface: upon impact a thin liquid sheet is thrown upwards all around the rim of the impactor while in its wake a large surface cavity forms. Under the influence of hydrostatic pressure the cavity immediately starts to collapse and eventually closes in a single point from which a thin, needle‐like jet is ejected. The existing numerical treatments of liquid impact either consider the surrounding air as an incompressible fluid or neglect air effects altogether. In contrast, our approach couples a boundary‐integral method for the liquid with a Roe scheme for the gas domain and is thus able to handle the fully compressible gas stream that is pushed out of the collapsing impact cavity. Taking into account that air compressibility is crucial, since, as we show in this work, the impact crater collapses so violently that the air flow through the cavity neck attains supersonic velocities already at cavity diameters larger than 1 mm. Our computational results are validated through corresponding experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of the dynamics of a liquid in a moving axisymmetric cavity

The problem of the motion of an ideal liquid with a free surface in a cavity within a rigid body has been most fully studied in the linear formulation [1, 2]. In the nonlinear formulation, the problem has been solved by the small-parameter method [3] and numerically [4–7]. However, the limitations inherent in these methods make it impossible to take into account simultaneously the large magnitude and the threedimensional nature of the displacements of the liquid in the moving cavity. In the present paper, a numerical method is proposed for calculating such liquid motions. The results of numerical calculations for spherical and cylindrical cavities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–177, March–April, 1984.     

Optimal control of viscous flow at high reynolds numbers

The most promising and most highly developed method for reducing drag in aerodynamics remains control of the flow by blowing and suction. In practice the main control problems remain the reduction of separation and the protracting of the transition of the boundary layer. These problems are solved mainly by experimental methods [1]. Meanwhile the main theoretical question remains unanswered: what is the theoretical minimum drag attainable by control through blowing (or suction)? In the present study an answer is given to this question for the cage of laminar flow round a body by a viscous incompressible fluid at high Reynolds numbers.     

Some problems of axisymmetric cavitation flows

The equations for the shape of a slender axisymmetric cavity [1–3] are used to consider problems relating to pulsations of the cavity shape, the drag of a slender cavity-forming body, and the influence of surface tension on the shape of a steady cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 28–34, January–February, 1982.I thank V. P. Karlikov and Yu. L. Yakimov for helpful discussions of the results.     

Motion of a magnetizable liquid in a rotating magnetic field

Moskowitz and Rosensweig [1] describe the drag of a magnetic liquid — a colloidal suspension of ferromagnetic single-domain particles in a liquid carrier — by a rotating magnetic field. Various hydrodynamic models have been proposed [2, 3] to describe the macroscopic behavior of magnetic suspensions. In the model constructed in [2] it was assumed that the intensity of magnetization is always directed along the field so that the body torque is zero. Therefore, this model cannot account for the phenomenon under consideration. We make a number of simplifying assumptions to discuss the steady laminar flow of an incompressible viscous magnetizable liquid with internal rotation of particles moving in an infinitely long cylindrical container in a rotating magnetic field. The physical mechanism setting the liquid in motion is discussed. The importance of unsymmetric stresses and the phenomenon of relaxation of magnetization are emphasized. The solution obtained below is also a solution of the problem of the rotation of a polarizable liquid in a rotating electric field according to the model in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–43, July–August, 1970.     

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.     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Confined creeping flow around an axisymmetric body: increase of the shape effect by a tube wall

Using a specially adapted experimental technique, associated with a visualization method based upon solid tracers, we have obtained the flow pattern induced by the very slow uniform translation of an axisymmetric body along the axis of a vertical tube filled with a viscous liquid, both in a fixed frame (the frame is attached to the tube) and in a “relative” frame (the frame accompanies the body in its translation). The body, whose shape evolves from a sphere to a cylinder frustum, is free from any attachment or interaction with any other body; only the tube wall interaction is relevant. In these conditions, the upstream-downstream symmetry, relative to the creeping regime hypotheses, has been very well verified and quantitative information concerning, in particular, the velocity field has been deduced with sufficient precision (better than 2%) to exercise the control of a numerical process capable of giving all the details of the hydrodynamic field including those not directly available from the experiments. By comparison with the unbounded flows around the same bodies, the strong increase of the shape effect by the presence of the confining tube wall has been pointed out and evaluated, on the drag as well as on the surface vorticity and pressure distributions.     

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.     

Numerical study of the entry of a plate and a disk into a compressible fluid

The dependences of the drag force on the time and the Mach number are found, as also are pressure distribution, and the shape of the free surface. It is shown that with the passage of time the drag force rapidly approaches its asymptotic value, which corresponds to flow around a body by a compressible fluid in accordance with Kirchhoff's scheme. It is also shown that with increasing Mach number the dimensions of the cavity decrease, the unsteady cavity always being narrower than the Kirchhoff cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–107, March–April, 1985.     

Shock-initiated implosion of an elliptical cavity and detonation in a liquid layer

Many papers [1–9] have been devoted to the dynamical analysis of bubble implosion in a liquid layer. Experiments have shown that an initially circular cavity is displaced or transformed into an elliptical cavity during the implosion process due to instability, whereupon its further contraction produces cumulative jets. This problem is important in the study of surface wear in cavitation flow [7] and in the analysis of the impact sensitivity of liquid explosives [1–6]. The onset of accumulation is conveniently investigated by starting with an elliptical cavity or by displacing a circular cavity relative to the impact axis, thereby creating an asymmetrical pressure field about the center of the cavity. In the present article certain theoretical notions are advanced with regard to the onset of the cumulative jet in an elliptical or displaced cavity and its influence on the ignition of liquid explosives due to the formation of minute droplets [4] in the adiabatically heated gas inside the cavity. Experimental data on the jet formation time and the frequency of nitroglycerin detonations qualitatively support the theoretical predictions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 78–85, September–October, 1971.     

细长体水下运动空化流场及弹道特性实验

为了研究细长体水下高速运动时空泡的产生、闭合及脱落特性,及影响细长体空泡形态及弹道特性的复杂因素等,初步开展了细长体模型水下高速运动的实验研究,分析了不同初始空化数下细长体模型在水中高速运动的一系列流动现象,重点研究了空泡的发展、闭合、尾部回射流和尾部脱落特性,以及轴对称细长体模型弹道特性与空泡形态变化之间的关系和转动特性随时间的变化历程等。结果表明:细长体水下高速运动时形成超空泡,空泡头部光滑透明,尾部凝结有汽水混合物且有交替脱落的含气漩涡;初始空化数对细长体的速度衰减有所影响;受初始扰动影响,细长体水下运动伴随有绕头部的转动且初始扰动影响细长体俯仰角随时间的变化历程。