On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations.     

Determination of the hydrodynamic characteristics of a partially filled cavity containing a pendulum

Equations are given for the small oscillations of a viscous liquid-filled cavity containing a pendulum and the dissipative and inertial coefficients are determined on the basis of a systematic application of the finite element method (FEM). In order to improve the accuracy of the values obtained for these coefficients the use of a nonlinear Shanks transformation is proposed. This makes it possible to achieve the required accuracy using much less machine time and memory. The properties of the inertial hydrodynamic characteristics associated with an ideal fluid are studied in relation to cavities lacking a pendulum but fitted with annular ribs. It is shown that as a result of the presence of these rings the estimate given in [6] for the generalized mass of the fundamental mode of the fluid oscillations is inaccurate and must be modified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 91–100, November–December, 1987.     

Diffraction of a plane acoustic wave by a rigid sphere in a cylindrical cavity: An axisymmetric problem

The paper studies the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates. The solution is based on the possibility of transforming partial solutions of the Helmholtz equation between cylindrical and spherical coordinates. Satisfying the interface conditions between the cavity and the acoustic medium and the boundary conditions on the spherical surface yields an infinite system of algebraic equations with indefinite integrals of cylindrical functions as coefficients. This system of equations is solved by reduction. The behavior of the system is studied depending on the frequency of the plane wave     

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.     

Numerical study of mixed convection and conduction in a 2-D square ventilated cavity with an inlet at the vertical glazing wall and outlet at the top surface

A steady state numerical study of combined laminar mixed convection and conduction heat transfer in a ventilated square cavity is presented. The air inlet gap is located at the bottom of a vertical glazing wall and air exits the cavity via a gap located at the top surface. Three locations for the opening at the top surface: left (case a), center (case b) and right side (case c) are considered. All the remaining surfaces are considered adiabatic. The mass, momentum and energy conservation equations were solved using the finite volume method for different Rayleigh numbers in the interval of 10⁴ < Ra < 10⁶ and Reynolds number in the interval of 100 < Re < 700. Temperature, flow field, and heat transfer rates are analyzed. The effect of the interaction between ambient conditions outside the glazing and the air inlet gap at the bottom for different air outlet gap positions at the top surface modifies the flow structure and temperature distribution of the air inside the cavity. The Nusselt number as a function of the Reynolds number was determined for the three cases. It was found that configuration for case (a) removes a higher amount of heat entering the cavity compared to cases (b) and (c). This is due to the short distance between the main stream and the glass wall surface. Thus, the forced airflow entering the cavity is assisted by the buoyancy forces, and most of the cavity remains at the inlet flow temperature, which should be appropriate for warm climates. These results may provide useful information about the heat transfer and fluid flow for future studies.     

Numerical comparison of conjugate and non-conjugate natural convection for internally heated semi-circular pools

In this numerical investigation, a detailed comparison of the conjugate and non-conjugate natural convection within a semi-cylindrical cavity has been presented. The cavity is assumed to be filled with a fluid containing uniformly distributed internal heating sources. The bottom circular wall of the cavity is taken to be thick with finite conductive properties, while the top wall is considered to be isothermal. The Navier–Stokes and energy equations are solved numerically by using the SIMPLER algorithm. A Rayleigh number range from 3.2×10⁶ to 3.2×10¹¹ has been investigated and the effects of solid-to-fluid conductivity ratios of 1.0, 5.0 and 23.0 have been analysed. The present numerical results for a semi-circular cavity with entirely isothermal walls are compared with known results from the open literature. It was found that these results for the non-conjugate problem are in very good agreement. The present results for a conjugate cavity show a remarkable difference from the non-conjugate analysis. The average Nusselt number for the solid–fluid interface shows a decrease while the top wall average Nu number has increased. It has been concluded that these effects increase for a system with a low solid-to-fluid conductivity ratio. It is evident from the present conjugate results that the assumption of isothermal enclosing walls gives somewhat different results when the walls are thick and the solid-to-fluid conductivity ratio is small.     

Natural convection in a cavity with time-dependent flux boundary

Numerical simulations have been carried out to investigate the unsteady natural convection flow in a cavity subjected to a sidewall heat flux varying sinusoidally with time. With all walls non-slip and the upper and lower boundaries and the other sidewall adiabatic, the heating and cooling produces an alternating direction natural convection boundary layer that discharges hot fluid to the top and cold fluid to the bottom of the cavity, generating a time-varying thermal stratification in the cavity interior. Scaling analysis has been conducted for different flow regimes based on the forcing frequency, with the characteristic time scales being the forcing period and the boundary layer development time. The scaling relations are then verified using the simulations, with the results showing overall good agreement with the derived scaling relations.     

Transient removal of a contaminated fluid from a cavity

A numerical and experimental study of the time-dependent hydrodynamic removal of a contaminated fluid from a cavity on the floor of a duct is presented. The duct flow has a parabolic inlet velocity profile and laminar flows are considered in a Reynolds number range between 50 and 1600 based on the duct height. The properties of the contaminated cavity fluid are assumed to be the same as for the fluid flowing in the duct. Attention is focussed on the convective transport of contaminated fluid out from the cavity and the effect of duct flow acceleration on the cleaning process. Passive markers which are convected with the flow are used in the numerical simulation for the purpose of identifying the contaminated cavity fluid. It is shown that the cleansing of the cavity is more pronounced during the unsteady start-up of the duct flow and the rate of cleaning decreases as the flow reaches a steady state. The cleaning process is enhanced as the cavity aspect ratio is increased and as the duct Reynolds number increases. A ‘volumetric’ approach based on the spread of markers is shown to be useful in determining the fraction of the cavity that remains contaminated after steady conditions have been reached. The distribution of the contaminant in a cavity during the unsteady stage and after steady conditions are reached are identified using passive markers.     

Dynamics of a Spherical Body in a Liquid with Rotational Vibration of the Cavity

The average dynamics of a single solid sphere in a liquid-filled cylindrical cavity in the presence of high-frequency rotational oscillation about the axis of symmetry is studied experimentally. In the cavity there is an impermeable membrane which forces the liquid as a whole to vibrates together with the cavity. Various orientations of the vessel in the gravity force field are considered. The action of an average force of vibrational nature on the sphere and the dependence of this force on the vibration parameters and the body dimensions and density are studied. The force is measured with respect to the floating threshold for the heavy body, when the average vibrational force balances or exceeds the action of the gravity force.     

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.     

Transient solutions for three-dimensional lid-driven cavity flows by a least-squares finite element method

A time-accurate least-squares finite element method is used to simulate three-dimensional flows in a cubic cavity with a uniform moving top. The time- accurate solutions are obtained by the Crank-Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix-free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least-squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time-accurate solutions to large-scale three-dimensional fluid flows, we present results for three-dimensional lid-driven flows in a cubic cavity for Reynolds numbers up to 3200.     

Dynamics of a Fluid in a Rotating Horizontal Cylinder

The behavior of a low-viscosity fluid in a rotating horizontal circular cylinder is investigated experimentally. The stability of the centrifuged layer, the motion of the fluid with respect to the cavity, the excitation of inertial waves on the fluid surface, and the effect of the waves on the stability and flow structure are studied over a wide region of relative occupancy of the cavity. The results are analyzed from the viewpoint of vibrational mechanics in which the motion is generated by the oscillations of the fluid with respect to the cavity and the gravity force plays the role of the force oscillating in the cavity reference system.     

Effect of duct velocity profile and buoyancy‐induced flow on efficiency of transient hydrodynamic removal of a contaminant from a cavity

This paper describes a preliminary numerical analysis of the effect of duct velocity profile and buoyancy‐induced flow generated by the heat source on hydrodynamic removal of contaminants contained in cavities. The process of fluid renewal in a cavity is modelled via a numerical solution of the Navier–Stokes equations coupled with the energy equation for transient flows. The foulant has the same density as the fluid in the duct and the duct velocity profile is considered to be Poiseuille flow and Couette flow, respectively. The results show that the change in Grashof number and duct flow velocity profile causes a dramatic difference in the observed flow patterns and cleaning efficiency. From a cleaning perspective, the results suggest that Couette flow at higher value of Grashof number becomes more effective in further purging of contaminated fluid from a cavity. Copyright © 2004 John Wiley & Sons, Ltd.     

Stability of a spinning liquid-filled spacecraft

Summary The stability of a spinning liquid-filled spacecraft has been investigated in the present paper. Using Galerkin's method, the attitude dynamic equations have been given. The Liapunov direct method was employed to obtain a sufficient condition for stability. Three kinds of characteristic modals were investigated: free motion of inviscid fluid, slosh motion and non-slosh motion. All characteristic problems can be solved numerically by the Finite Element Method or the Boundary Element Method. It has been demonstrated that the viscosity of the fluid has a dissipative effect at large Reynolds number, while the slosh motion plays a destabilizing role. The non-slosh model of fluid does not affect the stability criterion. Accepted for publication 19 October 1996     

Numerical simulation of time‐dependent hydrodynamic removal of a contaminated fluid from a cavity

The time‐dependent hydrodynamic removal of a contaminated fluid from a rectangular cavity on the floor of a duct is analysed numerically. Laminar duct flows are considered for Reynolds numbers of 50 and 1600 where the characteristic length is the duct height. Two cases are considered where: (1) the fluid density in the cavity is the same as that for the duct fluid and (2) the cavity fluid has a higher density than the duct fluid but the two fluids are miscible. The flow is solved by a numerical solution of the time‐dependent Navier–Stokes equations. Attention is focused on the convective transport of contaminated fluid out from the cavity and the effect of duct flow velocity profile on the cleaning process. Passive markers are used in the numerical simulation for the purpose of identifying the contaminated cavity fluid. The results show that the flow patterns in the cavity are influenced by the type of duct flow. From a cleaning perspective, the results suggest that it is easier for the duct flow to penetrate a cavity and to remove contaminated cavity fluid when the duct flow is of the Poiseuille type and the aspect ratio is large. Copyright © 2003 John Wiley & Sons, Ltd.     

高速破片撞击充液容器拖拽阶段气腔特性研究

为研究液压水锤效应拖拽阶段的气腔特性,利用数值模拟与实验相结合的方法对破片撞击充液容器的过程进行研究,并分析了破片撞击速度和液体介质对液压水锤效应拖拽阶段气腔的影响。结果表明:破片撞击充液容器时,在液体中形成的气腔形状为圆锥形,其最大直径和长度随破片的运动逐渐增大,气腔长径比最终趋于一稳定变化区域,约在3.8~3.9之间;气腔最大直径随着破片撞击速度的增大而增大;柴油介质中形成气腔的最大直径和长径比变化规律与水介质中形成的相同,气腔长径比最终在4.25左右浮动,柴油介质中形成气腔的最大直径和长径比均大于水介质中形成的。     

DENTING AND FAILURE OF LIQUID-FILLED TUBES UNDER LATERAL IMPACT

In this paper, numerical simulation of impact cases of liquid-filled tube impacted by missiles is conducted with a commercial finite element code LS-DYNA, and the results obtained are compared with the experimental data to verify the validity of the numerical simulation model adopted. With the verified numerical method, the processes of dynamic response of a blunt indenter impacting an empty or liquid-filled three-span continuous tubular beam are studied when the parameter such as the indenter’s mass, liquid’s density or impact velocity is varied and the other conditions are kept the same. The simulation results indicate that the critical perforation energy and the deformation of the wall of the pipe are significantly influenced by the presence of the liquid and the pressure. The liquid filling the tube provides a ‘foundation’ pressure to resist and localize the deformation, which may affect the perforation process and lead to a reduction of the ballistic limit. The simulation results also indicate that the increase of the fluid density filled in the tube will decrease the ballistic limit, but the fluid density must be in some scope. The relationship between the ballistic limit velocity of the tube and the mass of the impact missile is nonlinear in the Cartesian coordinate while it becomes linear through logarithmic transformation.     

Mechanical energy balances for a capillary viscometer

The entrance and exit flow processes for a cylindrical geometry are analyzed by writing macroscopic mechanical energy balances for a capillary viscometer. These equations can be used to compute the entrance and exit excess dissipation integrals from measured pressure differences if viscometric normal stress data are available for the material of interest. Upper and lower bounds are derived for these integrals for cases when high shear rate normal stress data are not available. The utilization of macroscopic mechanical energy balances in the interpretation of capillary viscometer results is illustrated using numerical solutions for a Maxwell fluid and experimental pressure drop data for high density polythylene.     

含裂纹损伤充液圆柱壳的振动响应求解方法

薄壁圆柱壳流体冲击振动响应是一个复杂的流固耦合(FSI)动力学问题,对于薄壳状态监测与缺陷识别具有重要意义。基于Flügge壳体应力理论,得到壳体运动的高阶偏微分方程组(PDE),利用波传播方法获得圆柱壳系统振动响应。将壳体周围流体定义为理想声学介质,通过亥姆霍兹方程描述声压场,得到流固耦合条件下的薄壁圆柱壳受迫振动响应演变规律。针对薄壳裂纹损伤识别问题,基于断裂力学理论建立局部柔度矩阵,结合呼吸型线弹簧模型(LSM),构造裂纹附近应力及位移连续条件,获得含裂纹损伤充液圆柱壳的振动响应,进而给出一种基于振动能量流的裂纹损伤识别方法。研究结果表明：充液圆柱壳耦合系统在非线性激励下,位移响应在沿轴向、周向和径向的传播特性差异明显;裂纹的存在会导致结构局部柔度的降低和耦合系统固有频率下降;归一化输入功率流能够有效地对充液圆柱壳耦合系统进行结构裂纹损伤识别。研究结果可为充液薄壳振动响应方面的研究提供有益参考,也可为流固耦合条件下的结构裂纹损伤识别方面提供技术支持。