Inviscid Liquid Research Articles

Incompressible impulsive sloshing

AbstractThe incompressible impulsive time scale for inviscid liquid sloshing in open rigid containers suddenly put into motion is defined as the intermediate time scale in between the acoustic time scale and the gravitational time scale. Surge and sway boundary-value problems for incompressible impulsive sloshing in some realistic container shapes are solved analytically to the leading order in a small-time expansion. A solution is provided for two types of horizontal cylinders: a triangular cylindrical wedge and a half-filled circular cylinder. The surface velocity and the hydrodynamic force with its corresponding virtual fluid mass are calculated. The cases of constant impulsive velocity and constant impulsive acceleration are linked by transformation equations. Flows with waterline singularities are discussed, being leading-order outer flows in terms of matched asymptotic expansions.

Models for water steam condensing flows

Models for water steam condensing flows The paper presents a description of selected models dedicated to steam condensing flow modelling. The models are implemented into an in-house computational fluid dynamics code that has been successfully applied to wet steam flow calculation for many years now. All models use the same condensation model that has been validated against the majority of available experimental data. The state equations for vapour and liquid water, the physical model as well as the numerical techniques of solution to flow governing equations have been presented. For the single-fluid model, the Reynolds-averaged Navier-Stokes equations for vapour/liquid mixture are solved, whereas the two-fluid model solves separate flow governing equations for the compressible, viscous and turbulent vapour phase and for the compressible and inviscid liquid phase. All described models have been compared with relation to the flow through the Laval nozzle.

Modelling Reflection and Transmission of Acoustic Waves at a Semiconductor: Fluid Interface

The paper concentrates on the study of reflection and transmission characteristics of acoustic waves at the interface of a semiconductor half-space underlying an inviscid liquid. The reflection and transmission coefficients varying with the incident angles are examined. Calculated results are verified by considering the quasilongitudinal () and quasitransverse () waves. The special cases of normal and grazing incidence are also derived and discussed. Finally, the numerical computations of reflection and transmission coefficients are carried out with the help of Gauss elimination method by using MATLAB programming software for silicon (Si) and germanium (Ge) semiconductors. The computer simulated-results have been plotted graphically for Si and presented in tabular form in case of Ge semiconductors. The study may be useful in semiconductors, geology, and seismology in addition to surface acoustic wave (SAW) devices.

Effect of vertical quasiperiodic vibrations on the stability of the free surface of an inviscid liquid layer

The aim of the present paper is to examine the effect of the vertical quasiperiodic oscillations on the stability of the free surface of an ideal horizontal liquid layer. The quasiperiodic motion considered here is characterized by two incommensurate frequencies ω1 and ω2. The governing system of equations is reduced to a quasiperiodic Mathieu equation. In this situation, using the harmonic balance method developed by Rand et al. [10, 11] and Hill’s determinants, we determine the marginal stability curves. We show that the quasiperiodic excitation produces a stabilizing or a estabilizing effect and is strongly depending on the ratio of the frequencies.

Effect of the contact-line dynamics on the oscillations of a compressed droplet

This paper studies the natural and forced oscillations of a deformed droplet of an inviscid liquid surrounded by a different liquid and bounded in the axial direction by solid planes. In equilibrium, the droplet is a figure of revolution and the ratio of its radius to height is significant. The equilibrium contact angle between the side surface of the droplet and the solid surface is different from a right angle. The motion of the contact line is taken into account by setting an effective boundary condition. It is shown that three characteristic ranges of natural frequencies exist.

GENERALIZED THERMOELASTIC WAVES IN MICROSTRETCH PLATES LOADED WITH FLUID OF VARYING TEMPERATURE

In the present paper, the theory of generalized thermo-microstretch elasticity has been employed to study the propagation of straight and circular crested waves in microstretch thermoelastic plates bordered with inviscid liquid layers (or half-spaces), with varying temperature on both sides. The secular equations governing the wave motion in both rectangular and cylindrical plates have been investigated. The results in the case of thin (long wavelength) and thick (short wavelength) plates have also been obtained and discussed as special cases of this work. The secular equation in the case of microstretch coupled with thermoelastic, micropolar thermoelastic and thermoelastic plates can be obtained from the present analysis by an appropriate choice of relevant parameters. The results have been deduced and compared with the relevant publications available in the literature at the appropriate stages of this work. Finally, the analytical developments have been illustrated numerically for aluminum–epoxy-like material sandwiched in the inviscid liquid. The computer simulated results in respect of phase velocity, attenuation coefficient, specific loss factor of energy dissipation and relative frequency shift due to liquid layers on both sides of the plate are presented graphically.

Reflection and Transmission of Acoustic Waves at Semiconductor - Liquid Interface

The study of reflection and transmission characteristics of acoustic waves at the interface of a semiconductor halfspace underlying an inviscid liquid has been carried out. The reflection and transmission coefficients of reflected and transmitted waves have been obtained for quasi-longitudinal (qP) wave incident at the interface from fluid to semiconductor. The numerical computations of reflection and transmission coefficients have been carried out with the help of Gauss elimination method by using MATLAB programming for silicon (Si), germanium (Ge) and silicon nitride (Si3N4) semiconductors. In order to interpret and compare, the computer simulated results are plotted graphically. The study may be useful in semiconductors, seismology and surface acoustic wave (SAW) devices in addition to engines of the space shuttles.

Bubble generation in quiescent and co-flowing liquids

A numerical simulation has been accomplished to analyze the problem of dynamic bubble formation from a submerged orifice in an immiscible Newtonian liquid under the condition of constant gas inflow. We have considered two cases for the surrounding liquid, namely the liquid in a quiescent condition and the liquid as a co-flowing stream with the gas. The full cycle, from formation to detachment of the bubbles and the corresponding bubble dynamics, was simulated numerically by using a coupled level-set and volume-of-fluid (CLSVOF) method. The role of the liquid to gas mean velocity ratio, the Bond number and the Weber number in the bubble formation process was studied and the order of magnitude of forces involved in bubble dynamics are presented. Our simulation results show that the minimum radius of the neck decreases with a power law behavior and the power law exponent in a co-flowing liquid is less than 1/2 as predicted by the Rayleigh–Plesset theory for quiescent inviscid liquids. Single periodic and double periodic bubbling (with pairing and coalescence) regimes are observed in the present investigation. It is identified that a moderate co-flowing liquid may inhibit the bubble coalescence. The volume of the bubble and the bubble formation time decrease with increasing liquid to gas mean velocity ratios. For small Bond numbers, significant differences pertaining to bubble dynamics are observed between the co-flowing liquid and the quiescent liquid. Furthermore, the generation and breakup of the Worthington jet after bubble pinch-off and formation of tiny drops inside the detached bubbles are observed.

Single- and two-fluid models for steam condensing flow modeling

The paper presents a description of two models dedicated to steam condensing flow modeling. The models are implemented into an in-house computational fluid dynamics (CFD) code that has been successfully applied to a wet steam flow calculation for many years now. Both models use the same condensation model that has been validated against the majority of available experimental data. The state equations for vapor and liquid water, the physical model as well as the numerical techniques of solution to flow governing equations have been presented. For the single-fluid model (SFM), the Reynolds-averaged Navier–Stokes (RANS) equations for vapor/liquid mixture are solved, whereas the two-fluid model solves separate flow governing equations for the compressible, viscous and turbulent vapor phase and for the compressible and inviscid liquid phase. Both described models have been compared with relation to the flow through the Laval nozzle.

Self-destabilizing mechanism of a laminar inviscid liquid jet issuing from a circular nozzle

A laminar inviscid liquid (typically water) jet issuing from a circular nozzle into otherwise quiescent air disintegrates into droplets periodically at a distance from the nozzle. The Plateau-Rayleigh instability theory and others cannot determine this breakup length because they do not have any logic that determines the initial amplitude of the unstable wave responsible for the breakup. In this paper, a closed spatial evolution solution is derived for a uniformly issued liquid jet by applying a theory that identifies the origin of the unstable wave. This solution describes the self-destabilizing mechanism of the liquid jet in the steady breakup state, showing that the initial amplitude of the unstable wave is determined by the capillary wave with upstream propagating speed that is created by the tip contraction at every breakup. Finally, the developed theory is extended to allow for the self-destabilizing mechanism of a liquid jet issuing from a long nozzle, which initially has a parabolic velocity profile and results in a long breakup length.

Mathematical study of the small oscillations of a pendulum containing an almost homogeneous, incompressible, inviscid liquid and a barotropic gas (Oscillations of a pendulum with almost homogeneous liquid and gas)

The authors study the small oscillations of a pendulum containing an almost homogeneous, incompressible, inviscid liquid (i.e. a liquid whose density in equilibrium is practically a linear function of the height, which differs very little from a constant) and a moving gas. Using functional analysis, they prove that the spectrum is comprised of a countable set of real eigenvalues and an essential spectrum, which fills an interval, and they give an existence and uniqueness theorem for the solution of the evolution problem.

Coupled frequencies of a rectangular hydroelastic system with two fluids

This paper deals with the study of behaviour of an idealized 2D hydroelastic system involving two inviscid liquids with an elastic rectangular container. The main objective is to investigate the influence of the physical parameters on eigenfrequencies and eigenmodes of the system. The study extends the previous results obtained for hydroelastic systems with one fluid. The governing equations describing the behaviour of the system are analyzed by using the concept of normal modes and their solutions presented in the form of infinite series. The expansion coefficients for the velocity potentials are calculated by employing a new inner product that allows the orthogonalisation of the normal modes. An eigenfrequency equation is then derived from the existence condition of a nontrivial solution. The numerical calculations are performed by varying only some relevant parameters.

Propagation of Waves at an Interface between a Liquid Half‐Space and an Orthotropic Micropolar Solid Half‐Space

An inviscid liquid half‐space is considered in welded contact with a orthotropic micropolar solid half‐space. Appropriate plane harmonic solutions of equations governing a liquid half‐space and an orthotropic solid half‐space are obtained. These solutions satisfy the required boundary conditions at the interface to obtain a system of four nonhomogeneous equations in amplitude ratios for incident quasi‐longitudinal displacement wave. The amplitude ratios of various reflected and refracted waves are computed numerically for a particular example of the present model. The effect of anisotropy upon these amplitude ratios is shown graphically for a particular range of the angle of incidence.

The sloshing of stratified liquid in a two-dimensional rectangular tank

The sloshing of inviscid liquid of stratified density in a rectangular tank is analyzed. As the flow is no longer irrotional, the governing equation is found to be quite different from the Laplace equation used for the liquid of constant density. In particular it contains terms of mixed temporal and spatial derivatives. The problem is solved based on the variable separation method and Laplace transform for the constant Vaisala-Brunt frequency. It is found that the stratification of density may have small effects on those natural frequencies associated with the constant density, but many new natural frequencies have appeared as a result of its effect.

Impulsive displacement of a liquid in a pipe at high Reynolds numbers

We consider the problem of an impulsive displacement of a liquid, originally at rest in a circular pipe, which is displaced by another liquid. The purpose of this analysis is to show that at a sufficiently high inertia the initial essentially inviscid motion can be extended to cover the entire displacement process, thus creating an inviscid window to which an inviscid analysis can be applied. We simplify the problem first, by considering a 1-liquid problem where the displacing liquid and displaced liquid are the same. We identify two characteristic times in this problem: the time it takes an inviscid liquid to be displaced, and the time it takes a viscous liquid to attain a steady state. Taking the ratio of the two defines the Reynolds number for the problem and we show that the motion becomes essentially inviscid once the Reynolds number is sufficiently high. We obtain the general solution of the 1-liquid problem which determines the nondimensional viscous displacement time as a function of the Reynolds number. We derive from the general solution: a critical Reynolds number above which the motion remains unsteady throughout the entire displacement process, and a formula which determines quantitatively whether applying an inviscid analysis to the 1-liquid viscous problem at a given Reynolds number is admissible within an acceptable error tolerance. We also show that at the limit the Reynolds number approaches infinity the viscous displacement time approaches the inviscid displacement time and that the velocity profile and the shape of the material surface separating the displacing from the displaced liquid approach their counterpart in the inviscid solution. Second, based on these results we propose that an inviscid solution is applicable to the 2-liquid viscous problem once the condition of a high Reynolds number is independently met by the two participating liquids. We obtain the solution to the inviscid 2-liquid displacement problem and calculate various examples. Finally, we present a stability analysis of the flat interface between the two inviscid liquids, which shows which of the examples is stable, neutrally stable, or unstable. The paucity of data for an impulsive displacement in the high Reynolds number range makes quantitative comparisons difficult. However, the excellent agreement obtained between the critical Reynolds number derived in this analysis and the result obtained in a numerical analysis of the viscous 2-liquid problem elsewhere constitutes at least a partial validation of the theory. Additional confirmation is obviously recommended.

One-dimensional simulation of thin liquid-film-edge retraction

The phenomenon of liquid film retraction, occurring in bursting bubbles, is investigated in the simplest case of a semi-infinite sheet of inviscid liquid. Discretized equations of motion are solved on a staggered grid, with the boundary condition adapted to handle movement and breakup of liquid domain. Two independent approaches show a perfect agreement. The calculation shows a periodic pinchoff of small droplets, in contrast to the published results on viscid fluids where the liquid accumulates at the rim. A quantitative analysis of simulated liquid profiles shows that the characteristic parameters are very close to the theoretical predictions obtained from energy conservation and laws of motion.

Three‐dimensional modal analysis of sloshing under surface tension

AbstractThe space structures such as satellites, probes or space stations generally contain large amounts of liquids, which can be propellants, cooling liquids, etc. The motion of these liquids can influence the vibrational behavior of the main structure and can potentially disturb the trajectory controller or the stabilization procedures. To avoid hazardous coupling between the inner liquid sloshing and the main structure movements, engineers need to know precisely the evolution of the inner liquid eigenmodes and eigenfrequencies during the mission. In this aim, we propose here a numerical approach to solve the three‐dimensional linear sloshing problem of an incompressible inviscid liquid taking into account the effects of surface tension which are predominant in low‐gravity environment. A variational formulation is derived from the linearization of the motion equations of the liquid near its initial equilibrium state (meniscus) and its discretization by the finite element method gives a classical spectral problem, whose solutions are the sloshing eigenmodes. While the majority of authors consider simple geometries of tanks, the advantage of the formulation presented here is to be easily applicable to real three‐dimensional cases as it will be demonstrated in some application examples. Copyright © 2010 John Wiley & Sons, Ltd.

Effect of anti-slosh baffles on free liquid oscillations in partially filled horizontal circular tanks

An exact two dimensional hydrodynamic analysis based on the linear potential theory is introduced to study the free liquid sloshing characteristics of transverse oscillation modes in a non-deformable horizontal circular cylindrical baffled container which is filled to an arbitrary depth with an inviscid incompressible liquid. Three common baffle configurations are considered: a pair of internal rigid horizontal side baffles of arbitrary extension installed at the free liquid surface, and a surface-piercing or a bottom-mounted vertical rigid baffle of arbitrary extension positioned along the tank vertical axis of symmetry. The problem solution is obtained by the method of successive conformal coordinate transformations, leading to standard truncated matrix eigenvalue problems on simple (rectangular) regions which are then solved numerically for the resonance eigen-frequencies. The effects of liquid fill level, baffle arrangement and length upon the three lowest antisymmetric and symmetric sloshing frequencies and the associated hydrodynamic pressure mode shapes are examined. Also, convergence of the adopted approach with respect to the fill condition, and baffle type/extension is discussed. Limiting cases are considered and the validity of results is established in comparison with the data in the existing literature.

Transience to instability in a liquid sheet

Series solutions are found which describe the evolution to absolute and convective instability in an inviscid liquid sheet flowing in a quiescent ambient gas and subject to a localized perturbation. These solutions are used to validate asymptotic stability predictions for sinuous and varicose disturbances. We show how recent disagreements in growth predictions stem from assumptions made when arriving at the Fourier integral response. Certain initial conditions eliminate or reduce the order of singularities in the Fourier integral. If a Gaussian perturbation is applied to both the position and velocity of a sheet when the Weber number is less than one, we observe absolutely unstable sinuous waves which grow liket1/3. If only the position is perturbed, we find that the sheet is stable and decays liket−2/3at the origin. Furthermore, if both the position and velocity of a sheet are perturbed in theabsenceof ambient gas, we observe a new phenomenon in which sinuous waves neither grow nor decay and varicose waves grow liket1/2with a convective instability.

Role of axisymmetric axial velocity gradients in nonlinear disintegration of liquid sheets

The nonlinear growth of sinuous and dilatational disturbances on thin inviscid liquid sheets with a velocity profile across the thickness is determined. Variations of the velocity profile in the liquid sheet, with maximum velocity at the centre linearly decreasing to a minimum at the interfaces, show an optimum value of the velocity gradient, below which disintegration of the liquid sheet is accelerated. The ligaments are formed in local dilatational and global dilatational modes and are asymmetric, unlike the symmetric ligaments formed with a liquid sheet issuing at constant velocity. The presence of velocity gradients is seen to yield smaller ligaments when breakup occurs in the local dilatational mode.