Sloshing in partially filled liquid containers—Numerical and experimental study for 2-D problems

This paper deals with the numerical and experimental studies of sloshing of liquid in partially filled prismatic containers subjected to external excitation. Meshless local Petrov-Galerkin (MLPG) method is used for computing the nonlinear sloshing response of liquid in a two-dimensional rigid prismatic tank. At every instant of time, velocity potential is computed at each node and the nodal positions are updated. A local symmetric weak form (LSWF) for nonlinear sloshing of liquid is developed, and a truly meshless method, based on LSWF and moving least squares (MLS) approximation, is presented for the solution of Laplace equation with the requisite boundary conditions. An experimental set-up is also designed to study the behavior of liquid sloshing in partially filled prismatic tank. The resulting slosh heights for various excitation frequencies and amplitudes are compared with the data obtained from the numerical studies. The numerical results are close to that obtained experimentally and little variations in the data are due to ineptness of the experimental set-up and the input parameters.     

FREE VIBRATION ANALYSIS OF BAFFLED LIQUID-STORAGE TANKS BY THE STRUCTURAL-ACOUSTIC FINITE ELEMENT FORMULATION

In fluid-structure interaction systems, baffles as a dynamic damping device are being widely used to suppress the fluid sloshing motion. Meanwhile, natural and dynamic behaviors of such systems are significantly influenced by the baffle parameters, such as the baffle number, the installation location, the inner-hole diameter and the liquid fill height. This paper intends to numerically investigate the parametric eigen characteristics, by the coupled structural-acoustic finite element method (FEM), of baffled cylindrical liquid-storage tank. According to the symmetric two-field formulation, a test FEM program is developed, in which the reduced integration for avoiding the shear and membrane locking and the modified shear correction factor as well as degenerated 8- and 6-node shell elements and 3-D trilinear acoustic elements are used. Through the comparison with the available analytic solutions of no-baffled axisymmetric tank, the validity of the test program and theoretical work is verified. Next, with the verified test FEM program, various combinations of major baffle parameters are intensively examined, in order for the parametric baffle effects on the natural frequency of baffled tanks.     

Transient sloshing in half-full horizontal elliptical tanks under lateral excitation

A semi-analytical mathematical model is developed to study the transient liquid sloshing characteristics in half-full horizontal cylindrical containers of elliptical cross section subjected to arbitrary lateral external acceleration. The problem solution is achieved by employing the linear potential theory in conjunction with conformal mapping, resulting in linear systems of ordinary differential equations which are truncated and then solved numerically by implementing Laplace transform technique followed by Durbin's numerical inversion scheme. A ramp-step function is used to simulate the lateral acceleration excitation during an idealized turning maneuver. The effects of tank aspect ratio, excitation input time, and baffle configuration on the resultant sloshing characteristics are examined. Limiting cases are considered and good agreements with available analytic and numerical solutions as well as experimental data are obtained.     

FREE-VIBRATION ANALYSIS OF LIQUID-FILLED TANK WITH BAFFLES

The natural frequencies of liquid in a liquid-filled cylindrical rigid tank without and with baffles are evaluated. An annular plate is used as a baffle, which is fitted to the inner periphery of a cylindrical tank. Both rigid and flexible baffles are considered. Finite elements are used to discretize both the liquid and the structural domain. The slosh frequencies of liquid are computed for different dimensions, thicknesses and positions of baffles, both rigid and flexible considering the circumferential wave number as one. The axisymmetric and other asymmetric modes are not studied. The results obtained for rigid baffle case are comparable with the existing results. The coupled vibration frequencies of the tank-flexible-baffle system are computed considering the effect of sloshing of liquid.     

A Legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids

A new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids. The formulation makes use of the Legendre polynomials in deriving the finite element interpolation shape functions in the Lagrangian frame of reference. The formulated element uses Gauss–Lobatto–Legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional Gauss–Legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes, which are normally associated with the Lagrangian FE formulation. The numerical performance of the spectral element formulated here is examined by doing the inf–sup test on a standard rectangular rigid tank partially filled with liquid. The eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h-type Lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence. The efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements, and the obtained results are compared with available results in the published literature.     

HYDROELASTIC COUPLED VIBRATIONS IN A CYLINDRICAL CONTAINER WITH A MEMBRANE BOTTOM, CONTAINING LIQUID WITH SURFACE TENSION

A linear free hydroelastic vibration analysis of a frictionless liquid with a free surface contained in a cylindrical tank with a flexible bottom has been performed. The side-wall has been treated as rigid and the effect of surface tension taken into consideration. The container bottom was treated as a membrane, while for the free liquid surface the effect of two contact line conditions has been investigated. One edge condition was that of a slipping contact line, while the other one treated the contact line as fixed, ie., an anchored edge. The vibration characteristics of a membrane-liquid coupled system have been investigated for various system parameters, i.e., membrane tension parameter T, liquid surface tension parameter σ, material density parameter ρ, liquid height ratio ?₀ and vibration mode numbers m and n. The degree of coupling between a membrane and a liquid was represented with vibration mode diagrams as well as with frequency diagrams. For axisymmetric coupled vibrations with anchored edge condition, vibration mode exchanging of both a membrane and liquid free surface with membrane bottom tension parameter T has been investigated. An interesting phenomenon which is only observed for a flexible bottom container and an anchored edge free surface condition is presented.     

Low-gravity liquid nonlinear sloshing analysis in a tank under pitching excitation

Under pitch excitation, the sloshing of liquid in circular cylindrical tank includes planar motion, rotary motion and rotary motion inside planar motion. The boundaries between stable motion and unstable motion depend on the radius of the tank, the liquid height, the gravitational intension, the surface tensor and the sloshing damping. In this article, the differential equations of nonlinear sloshing are built first. And by variational principle, the Lagrange function of liquid pressure is constructed in volume integration form. Then the velocity potential function is expanded in series by wave height function at the free surface. The nonlinear equations with kinematics and dynamics free surface boundary conditions through variation are derived. At last, these equations are solved by multiple-scales method. The influence of Bond number on the global stable response of nonlinear liquid sloshing in circular cylinder tank is analyzed in detail. The result indicates that the system's amplitude–frequency response changes from a ‘soft-spring’ to a ‘hard-spring’ in the planar motion with the decreasing of the Bond number, while it changes from a ‘hard-spring’ to a ‘soft-spring’ in the rotary motion. At the same time, jump, lag and other nonlinear phenomena of liquid sloshing are discovered.     

A numerical study of three-dimensional liquid sloshing in tanks

A numerical model NEWTANK (Numerical Wave TANK) has been developed to study three-dimensional (3-D) non-linear liquid sloshing with broken free surfaces. The numerical model solves the spatially averaged Navier–Stokes equations, which are constructed on a non-inertial reference frame having arbitrary six degree-of-freedom (DOF) of motions, for two-phase flows. The large-eddy-simulation (LES) approach is adopted to model the turbulence effect by using the Smagorinsky sub-grid scale (SGS) closure model. The two-step projection method is employed in the numerical solutions, aided by the Bi-CGSTAB technique to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate volume-of-fluid (VOF) method is used to track the distorted and broken free surface. Laboratory experiments are conducted for both 2-D and 3-D non-linear liquid sloshing in a rectangular tank. A linear analytical solution of 3-D liquid sloshing under the coupled surge and sway excitation is also developed in this study. The numerical model is first validated against the available analytical solution and experimental data for 2-D liquid sloshing of both inviscid and viscous fluids. The validation is further extended to 3-D liquid sloshing. The numerical results match with the analytical solution when the excitation amplitude is small. When the excitation amplitude is large where sloshing becomes highly non-linear, large discrepancies are developed between the numerical results and the analytical solutions, the former of which, however, agree well with the experimental data. Finally, as a demonstration, a violent liquid sloshing with broken free surfaces under six DOF excitations is simulated and discussed.     

Constructal Design Applied to the Geometric Evaluation of an Oscillating Water Column Wave Energy Converter Considering Different Real Scale Wave Periods

The present work presents numerical study of the influence of geometry on the performance of an oscillating water column (OWC) wave energy converter by means of a constructal design. The main purpose is to maximize the root mean square hydrodynamic power of device, (P_hyd)_RMS, subject to several real scale waves with different periods. The problem has two constraints: hydropneumatic chamber volume (V _HC ) and total OWC volume (V _T ), and two degrees of freedom: H₁/L (ratio of height to length of the hydropneumatic chamber) and H₃ (OWC submergence). For the numerical solution it was used a computational fluid dynamic (CFD) code, based on the finite volume method (FVM). The multiphasic volume of fluid (VOF) model is applied to tackle with the water–air interaction. The results led to important theoretical recommendations about the design of OWC device. For instance, the best shape for OWC chamber, which maximizes the (P_hyd)_RMS, was achieved when the ratio (H₁/L) was four times higher than the ratio of height to length of incident wave (H/λ), (H₁/L) _o = 4(H/λ). Moreover, the optimal submergence (H₃) was achieved as a function of wave height (H) and water depth (h), more precisely given by the following relation: h ? (3H/4) ≤ (H₃) _o ≤ h.     

Temperature field measurement of an array of laminar premixed slot flame Jets using Mach-Zehnder interferometry

An experimental study was performed to investigate the influence of Reynolds number (Re) and non-dimensional jet-to-jet spacing (S/D_h) on flame shape, structure and temperature field of an array of laminar premixed slot flame jets. Mach-Zehnder interferometry technique is used to obtain an insight to the overall temperature field between single, twin and triple slot flame jets. The slot jets with large aspect ratio (L/W), length of L=60 mm and width of W=6 mm were used to eliminate the three-dimensional effect of temperature field. The effect of jet-to-jet spacing was investigated on flame characteristics under the test conditions of 200≤Re≤400 and equivalence ratio (φ) of unity. The present measurement reveals that the variation of maximum flame temperature with increment of Reynolds number is mainly due to heat transfer effects and is negligible while the flame height is increased. For the cases of twin and triple flame jets by increasing Reynolds number and decreasing non-dimensional jet-to-jet spacing (S/D_h), the interferences between the jets are increased and the jets attracted each other. Strong interference was observed at S/D_h=1.15. For the case of triple jets at this S/D_h, the central jet was suppressed while the side jets deflected towards the inner jet. The interference between jets was found to reduce the heat flux in the jet-to-jet interacting zone due to incomplete combustion. Also the optimum jet-to-jet spacing of triple flame jets is obtained at each Reynolds number to enhance the heat transfer performance of the jets.     

Optical coefficients of Hg1 − x − y Cd x Eu y Se crystals

Optical properties of Hg_{1 ? x ? y} Cd _x Eu _y Se crystals grown by the Bridgman method have been investigated based on the independent reflectance and transmittance measurements, which were performed on a Nicolet 6700 spectrometer at T = 300 K in the wavelength range 0.9 ≤ λ ≤ 26.6 μm. The values of refractive index n, absorption index k, and absorption coefficient α have been determined for the crystals studied. Based on the dependences α = f(hν), the presence of direct allowed interband optical transitions in the crystals is established and the band-gap values are determined. The influence of temperature on the transmittance and band gap are investigated in the range T = 114–300 K.     

Mixed convection and entropy generation in a square cavity using nanofluids

In this work, two-dimensional mixed convection and entropy generation of water-(Cu, Ag, Al₂O₃, and TiO₂) nanofluids in a square lid-driven cavity containing two heat sources, have been numerically investigated. The upper lid and bottom wall of the cavity are maintained at a cold temperature T_C, respectively. The governing equations along with boundary conditions are solved using the finite volume method. Comparisons with the previous results were performed and found to be in excellent agreement. The effects of the solid volume fraction (0≤φ≤0.10), Rayleigh (10³≤Ra≤10⁵) and Reynolds (1≤Re≤500) numbers, and different types of nanofluids on the total entropy generation S_t and on entropy generation due to heat transfer S_h are presented and discussed. Moreover, the heat sources positions have an effect on the total entropy generation and Bejan number. It was found that S_t and S_h decrease with increase of φ, Ra, and Re.     

Stabilization of electron-hole liquid in uniaxially strained germanium in a strong magnetic field

Luminescence spectra of uniaxially and uniformly strained high-purity germanium crystals at liquid-helium temperatures in a magnetic field of up to 14 T have been investigated. In strongly strained Ge crystals, a new line has been detected on the low-energy side of the excitonline in magnetic fields higher than 4 T. Studies of this line’s characteristics as functions of pressure, temperature, and magnetic field have led us to conclude that its presence is due to recombination of electron-hole pairs in an electron-hole liquid. The experimental data suggest that the metallic electron-hole liquid is stabilized in a strong magnetic field. By approximating the shape of the newly detected line using the model of metallic electron-hole liquid, we have obtained the electron-hole liquid density n _EHL(B) and Fermi energies E _Fe,h of electrons and holes. The liquid binding energy ø as a function of magnetic field has been estimated.     

A study of many-body phenomena in metal nanoclusters (Au, Cu) close to their transition to the nonmetallic state

The results of a study of many-body phenomena in gold and copper nanoclusters are presented. The measured conductivity as a function of nanocluster height h was found to have a minimum at h ≈ 0.6 nm. Conductivity was local in character at nanocluster sizes l ≤ l _c ≈ 2.5 nm. Changes in core hole screening and an anomalous increase in the Anderson singularity index α in gold and copper nanoclusters could be caused by changes in permittivity from metallic (? → ∞) to nonmetallic (? ∝ l ²). The many-body phenomenon characteristics observed in the X-ray photoelectron and tunnel spectra of gold and copper nanoclusters as the size of the nanoclusters changed led us to suggest changes in the band structure of the nanoclusters and, therefore, their possible transition from the metallic to nonmetallic state.     

Structural and some electrophysical properties of the solid solutions Si1 − x Sn x (0 ≤ x ≤ 0.04)

Films of the solid solutions Si_{1 ? x} Sn _x (0 ≤ x ≤ 0.04) on Si substrates have been grown by liquid phase epitaxy. The structural features of the films have been investigated using X-ray diffraction. The temperature behavior of current-voltage characteristics and the spectral dependence of the photocurrent for the heterostructures p-Si-n-Si_{1 ? x} Sn _x (0 ≤ x ≤ 0.04) have been analyzed. The grown epitaxial films of the solid solutions Si_{1 ? x} Sn _x (0 ≤ x ≤ 0.04) have a perfect single-crystal structure with a (111) orientation and a subgrain size of 60 nm. In the epitaxial films at the Si-SiO₂ interfaces between silicon subgrains and SiO₂ nanocrystals, where there are many sites with a high potential, the Sn ions with a high probability substitute for the Si ions and encourage the formation of Sn nanocrystals with different orientations and, as follows from the analysis of the X-ray diffraction patterns, with different sizes: 8 nm (for the (101) orientation) and 12 nm (for the (200) orientation). The current-voltage characteristics of the heterostructures p-Si-n-Si_{1 ? x} Sn _x (0 ≤ x ≤ 0.04) are described by the exponential law J = J ₀exp(qV/ckT) at low voltages (V < 0.2 V) and the square law J = (9qμ _p τ _p μ _n N _d /8d ³)V ² at high voltages (V > 1 V). These results have been explained by the drift mechanism of charge carrier transport in the electrical resistance relaxation mode.     

Formation and composition of the clathrate phase in the H<Subscript>2</Subscript>O-H<Subscript>2</Subscript> system at pressures to 1.8 kbar

The transition of the hexagonal ice phase I_h to the clathrate phase sII has been found in the H₂O-H₂ system at a pressure of about 1 kbar under conditions of an excess of gaseous hydrogen. The pressures of the I_h → sII and sII → I_h transitions have been determined over a temperature range from ?36 to ?18°C, and the pressure dependence of the synthesis temperature of the clathrate phase from a liquid at pressures from 1.0 to 1.8 kbar has been constructed. The solubility of hydrogen in the I_h and sII phases and in liquid water has been measured. The concentration of hydrogen in the clathrate phase sII is about 1.2 wt % (10 mol %) near the boundary of the sII → I_h transition, and it increases to 2 wt % (16 mol %) at a pressure of 1.8 kbar.     

Panoramic diagnostics of shear stresses on the channel wall with a step using the liquid crystals

Measurements results on the shear stresses of surface friction by means of thin-film coatings based on cholesteric liquid crystals and specialized software for digital processing of experimental video are presented in the paper. The calibration dependencies of shear stress relative to the hue and azimuth angle as well as shear stress spatial distribution at subsonic turbulent flow (V _∝ = 84 m/s) around a step, trapezoidal in plane (Reynolds number calculated for step height h, Re _h = 2.57?10⁴), with a base angle of 46° were derived for two geometries of experiment. The experiments demonstrated high sensitivity of liquid crystals to rearrangement of the near-wall flow structure and possibility to obtain quantitative data about mean shear stress levels.     

Asymptotic properties of the solutions of a differential equation appearing in QED and QCD

Stimulated by the study described in the preceding paper, we establish the asymptotic behaviour of the ratio h′(0)/h(0) for g → ∞, where h(r) is a solution, vanishing at infinity, of the differential equation h″(r) = igω(r)h(r) on the domain 0 ≤ r ≤ ∞ and ω(r) = (1 − √rK₁(√r))/r. Some results are valid for more general ω's.     

Martensitic transformations and magnetic properties of nonstoichiometric alloys of the Ni-Mn-In system

Martensitic transformations and magnetic properties of Ni_89-x Mn _x In₁₁ (42 ≤ x ≤ 44) alloys have been investigated. Critical temperatures of magnetic and structural phase transitions in the studied alloy system have been determined. It has been shown that the martensitic transformation induced by the magnetic field is observed in all alloys. Temperature dependences of the spontaneous magnetization of austenite and martensite as well as the magnitude of the critical field, in which martensitic transformation occurs, have been determined.     

Excitation and emission of hexavalent uranium in KMgF3

Single crystals of KMgF₃ containing uranium have been grown in an oxidizing atmosphere. Absorption, selective laser excitation and emission have been obtained at liquid helium temperature. There is a resonance line at 511.45 nm and this has been shown to be due to a magnetic dipole transition between singly degenerate levels. The polarized excitation results are consistent with a charge compensated UO₄ centre having D_4h symmetry.