Natural convection in a square cavity with a heat generating body: Entropy consideration

Conjugate heat transfer in a cavity is an important consideration with regard to cooling of micro-electronic equipment. In the present study, a heat transfer analysis of conditions taking place in a square cavity with a heat source, located in it, is carried out. The natural convection accompanying conduction heat transfer in the heat generating solid body is examined. Air or water are considered as the fluid in the cavity while steel substrate is considered as the heat generating solid body. The location of the solid is changed in the cavity to examine the cooling conditions. The entropy analysis of the system is carried out to determine the irreversibility ratio for each location of the solid body in the cavity. It is found that the heat transfer from the solid body surfaces increases where the surfaces facing the inlet and the exit of the cavity. The entropy generated attains the maximum value for air when the solid body is located at the center of the cavity; in which case, the irreversibility ratio reduces to a minimum value. Received on 26 May 1999     

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.     

Lift Force Acting on a Heavy Solid in a Rotating Liquid-Filled Cavity with a Time-Varying Rotation Rate

The dynamics of a heavy cylindrical body in a liquid-filled horizontal cylindrical cavity with a time-varying rotation rate is experimentally investigated. The body is near the cavity boundary under a centrifugal force and undergoes solid-body rotation together with the liquid and the cavity at a fixed rotation rate. The dependence of the body dynamics on the amplitude and frequency of modulation of the rotation rate is investigated. It is found that at a critical amplitude of modulation (at definite frequency), the heavy body repulses from the cavity boundary and comes into a steady state at some distance from the wall. It is found that the average lift force (repulsive one) is generated by the azimuthal oscillation of the body in the rotating frame of reference and manifests itself at a distance comparable to the thickness of the viscous boundary layer. In the experiments, we observed azimuthal drift of the body due to asymmetric azimuthal oscillations of the body. In the limit of high frequency of the rotation rate modulation, the dependence of the lift force coefficient on the gap between the body and the wall is determined.     

The Thermoelectroelastic State of a Transversally Isotropic Piezoceramic Body with a Spheroidal Cavity in a Uniform Heat Flow

A static thermoelectroelastic problem for an infinite transversally isotropic body containing a spheroidal cavity is explicitly solved. The symmetry axis of the spheroid coincides with the anisotropy axis of the body. It is assumed that at a rather large distance from the cavity the body is in a uniform heat flow directed along the anisotropy axis. Formulas are derived for the stress components and the projections of the electric displacement vector near the cavity, which depend on the heat-flow value, cavity geometry, and the thermoelectroelastic properties of the material. The solution of the problem for a body with a disk-like crack is obtained as a partial case from the solution of the problem for a piezoceramic body with a spheroidal cavity. The stress intensity factors for the force and electric fields are determined near the crack     

Nonstationary interaction of a short blunt body with a cavity in a compressible liquid

The shock-interaction problem for a rigid spherical body and a spherical cavity in a compressible liquid is formulated and solved. Three typical cases of typical dimensions of the body and cavity are examined. An asymptotic solution valid at the earliest stage of interaction is obtained. In the general case, the problem is reduced to an infinite system of integral equations of the second kind. It is numerically solved for the case of a nonsmall air gap __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 40–56, November 2006.     

Stress State of a Soft Ferromagnetic with an Ellipsoidal Cavity in a Homogeneous Magnetic Field

The stress problem is solved for an infinite elastic magnetically soft ferromagnetic containing an ellipsoidal cavity. The body is in a homogeneous magnetic field directed along the shortest axis of the ellipsoid. The main stress-strain and magnetic characteristics of the body are determined. The stress distribution over the cavity surface is analyzed__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 70–78, March 2005.     

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.     

Impact of a long thin body on a cylindrical cavity in liquid: A plane problem

An approach is developed to the investigation of the shock interaction between a long thin cylindrical body and a cylindrical cavity in an infinite compressible perfect liquid. This process accompanies the supercavitation of the body. Three typical cases of cross-sectional dimensions of the body and the cavity are examined. For each case, a mixed nonstationary boundary-value problem with an unknown moving boundary is formulated. The unknown quantities are expanded into Fourier series. An auxiliary problem is solved using the Laplace transform to establish the relationship between the pressure and the velocity on the cavity surface. As a result, the problem is reduced to an infinite system of Volterra equations of the second kind solved simultaneously with the equation of transverse motion and the equation of the contact boundary. An asymptotic solution valid at the initial stage of interaction is obtained for all the three cases, and a numerical solution is found for the most typical case __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 32–53, June 2006.     

A light cylinder under horizontal vibration in a cavity filled with a fluid

The behavior of a light cylindrical body of circular cross-section under horizontal vibration in a rectangular cavity filled with a fluid is experimentally investigated. At critical vibration intensity the body is repelled from the upper side of the cavity and takes up a stable suspended position, in which the gravity field is balanced by the vibrational repulsive force, executing longitudinal oscillations. As the vibrations are intensified, the gap between the cylinder and the wall widens. A new form of instability, namely, the excitation of the tangential motion of the body along the vibration axis, is found to exist on the supercritical range. The cylinder is at a finite distance from the upper side of the cavity and the tangential motion is due to the loss of symmetry of the oscillating motion. The transition of the cylinder to the suspended state and its return to the wall, as well as the excitation of the average longitudinal motion and its cessation, occur thresholdwise and have a hysteresis. The body dynamics are studied as a function of the dimensionless vibration frequency.     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.     

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.     

Uncoupled magnetoelastic problem for a ferromagnetic body with a spherical cavity

An uncoupled stress problem for an unbounded elastic soft ferromagnetic body with a spherical cavity in a magnetic field uniform at infinity is solved. The stresses, displacements, and magnetic quantities in the body are determined. The features of stress distribution over the body and its boundary surface are studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 42–48, October 2007.     

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.     

Experimental Study of Flow in a Cavity on an Axisymmetric Body

The flow past a cylindrical cavity on an axisymmetric body in the range of Mach numbers from 0.6 to 1.18 and the effect of the Mach number in the transition from subsonic to supersonic flow velocities are studied experimentally. In addition, a broad, 5.3—11.3 range of relative elongations of the cavity which permits one to determine the influence of the elongation on flow regimes including flows with closed and open separation zones is studied.     

Vibrational dynamics of a light spherical body in a rotating cylinder filled with a fluid

The behavior of a light spherical body in a rotating horizontal cylindrical cavity filled with a low-viscosity fluid is experimentally investigated both in the absence and in the presence of transverse vibrations. The system is in a centrifuged state. Mean rotation of the sphere relative to the cavity is found to exist. In the absence of the vibrations slow lagging rotation of the body is due to the gravity field. With increase in the cavity rotation velocity the intensity of differential rotation reduces. The vibrations lead to the excitation of differential rotation of the body, either anticipating or lagging, due to the resonance excitation of its inertial oscillations. The differential rotation of the body leads to the formation of the cylindrical Taylor-Proudman column. With increase in the column rotation velocity the instability of its boundary manifests itself as an azimuthal two-dimensional wave.     

空化器出水非定常垂直空泡的研究

对空化器朝水面高速运动产生的非定常垂直空泡进行了理论研究. 建立了受水面和重力影响下的水下垂直空泡长度变化数学方程, 从方程中导出了非定常垂直空泡长度计算公式, 利用公式计算了空化器出水后空泡从脱落、收缩到溃灭的时间. 对带空化器的航行体, 公式还可以计算脱落空泡溃灭高压作用在航行体上的位置, 最后给出了避免溃灭高压作用在航行体上的条件和判据.     

A Magnetoelastic Field in a Ferromagnetic Medium with a Spherical Cavity

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing a spherical cavity is considered. It is assumed that the body is under an external magnetic field. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the cavity are studied. Graphs are presented for the total magnitoelastic and Maxwell stresses as functions of the magnetic induction, the angle of dip, and the mechanical and magnetic properties of the material     

Impact of a spherical rigid body on the surface of a cavity in a compressible liquid: An axisymmetric problem

The shock interaction of a spherical rigid body with a spherical cavity is studied. This nonstationary mixed boundary-value problem with an unknown boundary is reduced to an infinite system of linear Volterra equations of the second kind and the differential equation of motion of the body. The hydrodynamic and kinematic characteristics of the process are obtained __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 11–19, January 2008.     

Investigation of the Flow in an Annular Cavity in a Cylindrical Body in a Supersonic Stream

The results of an experimental investigation of the flow over an annular cavity in a cylindrical body are presented; the cavity-to-body diameter ratio was 0.7 and the incident flow Mach number was 2.84. Using the data on the pressure distribution and optical measurements of the flow pattern, the structure of the flow inside the cavity was studied on the relative cavity length range from 0.5 to 14 including the regimes with both open and closed separation zones.     

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.