Two-layer Liquid System Research Articles

An Experimental Study of Three-Dimensional Separation Surface Sloshing in the Wet Storage Tank of a Floating Offshore Platform

In this work, in order to elucidate the three-dimensional (3D) resonant sloshing dynamics of the oil–water interface in an offshore cylindrical wet storage tank, a series of model experiments are conducted in a completely filled cylindrical tank containing two immiscible liquids. To begin with, a series of free damping tests are performed to experimentally determine the viscous damping rate of the system and to examine the corresponding theoretical solutions. Subsequently, the separation surface wave responses at a series of excitation frequencies including the natural frequencies of first five modes are examined. Finally, the rotary sloshing dynamics at the natural frequencies of the first and second natural modes are systematically explored. Interestingly, it is found that the separation surface rotary sloshing in a two-layer liquid system is much more intricate than one-layer liquid rotary sloshing due to the generation of multitudinous short waves in the long wave. As far as we know, this is the first investigation of 3D separation surface rotary wave motion in a two-layer liquid system without a free surface.

Primary and secondary resonance phenomenon for two-layer liquid sloshing in a rectangular container under horizontal excitation

Laboratory experiments were conducted to study primary and secondary resonant sloshing in a laterally excited rectangular container containing two-layer stratified liquids. The findings revealed that primary resonance of the free surface or the separation surface occurs when the forcing frequency closely matches the corresponding natural frequency. Some intriguing phenomena, such as the sudden wave amplitude increase and the downward shift in resonant peaks, can be observed due to the soft-spring effect. Secondary resonance arises when superharmonic or sub-harmonic frequencies associated with liquid sloshing closely align with the natural frequencies of the system. For the free surface, the secondary resonances of the first five modes occur at forcing frequencies closely related to a third of the first mode, half of the second mode, a third of the third mode, a quarter of the fourth mode, and a fifth of the fifth mode natural frequency of the free surface, respectively. Furthermore, secondary resonance can also manifest under other forcing conditions, such as when the harmonic at a fractional multiple is near the second mode natural frequency of the free surface. Regarding the separation surface, it is worth noting that secondary resonance occurs due to the dominant contribution of fractional harmonics of the forcing frequency. To the best of the authors' knowledge, this is the first systematic investigation of primary and secondary resonance behaviors in a two-layer liquid system.

Development and numerical implementation of the mathematical model of a gravity wave at the boundary of separation of a two-layer liquid system

Development and numerical implementation of the mathematical model of a gravity wave at the boundary of separation of a two-layer liquid system

Comparative assessment of liquid sloshing in dry and wet storage tank of floating offshore platform

In order to explore the characteristics of the single-layer liquid sloshing in offshore dry oil storage tank and the two-layer liquid sloshing in offshore wet oil storage tank, two series of experiments were carried out: one was free surface sloshing in a closed rectangular tank partially filled with colored water, and the other was interfacial sloshing in the identical tank but completely filled with white oil and colored water. The tank was mounted on a shake table and was subjected to harmonic horizontal excitation with different excitation amplitudes and a wide range of excitation frequencies, including the first seven natural modes of single-layer or two-layer liquid system. The present study find that the frequency responses of interfacial sloshing wave are analogous to those of the free surface sloshing wave, but smaller in amplitudes. The experiments also produce results that are unique to the two-layer liquid sloshing. For example, when the external excitation frequency is equal to or close to the odd mode natural frequency of two-layer liquid system, a complicated three-dimensional (3D) gravity-capillary wave might be generated at the oil-water interface. Finally, the comparisons of free surface and interfacial sloshing in the viscous damping ratio, higher sloshing modes, impact pressure amplitude and mass center displacement were conducted, which revealed the superiority of wet storage technology in structural safety and dynamic stability.

An experimental study of two-layer liquid sloshing under pitch excitations

The non-resonant and resonant responses of a two-layer liquid system in a tank under pitch excitation were investigated experimentally in this study. The movement of both the free surface and the interface was automatically identified simultaneously by an image processing method, which can rectify the visually tilted frames in a moving system. When the frequency of external excitation was near the natural frequency related to upper layer liquid, free surface resonance can be triggered. On the other hand, when the frequency of the external excitation was close to the natural frequency related to lower layer, resonant response of the interface between two liquids occurred. It is also found that Kelvin-Helmholtz instabilities with different length scales can be generated due to the reverse direction of velocities near the interface under different conditions. Such length scale of the Kelvin–Helmholtz instability can be estimated by using the critical Richardson number. In addition, the bottom of the tank may restrict the development of wave trough on the interface when the depth of lower layer was relatively shallow, while the free surface may limit the wave crest of interface when the thickness of the upper layer was small. Further investigations of the interface displacements for both non-resonant and resonant responses were also conducted in frequency domain.

Experimental study of the effects of a viscous liquid layer on the cavity dynamics of vertical entry by a sphere into water at low Froude number

The water entry process is relevant to a wide range of engineering applications and has been extensively investigated. Most liquids used in such studies are single-layered, and little attention has been paid to how the structure of a two-layer liquid system affects the splash and cavity formation. In this study, we use high-speed photography to experimentally investigate the water entry of a sphere after it has passed vertically through a layer of highly viscous liquid (dimethicone) at a low Froude number. We investigate the effects of different thicknesses of this dimethicone layer and find that the formation of the splash crown is closely related to both the thickness and the Froude number. In a certain range of dimethicone thickness, the height of the splash interface decreases with the increasing thickness and increases with the decreasing Froude number. The dimensionless interface height at the pinch-off time is found to have a linear relationship with the dimensionless initial velocity of the sphere. Furthermore, the formation of the cavity, including its length and pinch-off depth, depends on the Froude number. However, the pinch-off time is almost independent of the dimethicone thickness and the Froude number, and the cavity length is nearly independent of the dimethicone thickness for all Froude numbers examined.

Experimental study on the cavity dynamics of oblique impact of sphere on a viscous liquid floating on water

Oblique water entry of marine structure involves complex hydrodynamics phenomena including splash formation, asymmetrical cavity and pinch-off that have been observed in previous systematic experiments. However, very few studies focus on such phenomena induced by oblique impact of object in a two-layer liquid system. In this paper, we present an experimental investigation of oblique entry of a sphere into a deep pool of water within a layer of highly viscous dimethicone floating on the water. In the experiment, a high speed digital camera is established to obtain the instantaneous cavity and motion trajectories for both qualitative and quantitative analysis. The results show that the upper-layer viscous liquid introduces a significant change in the splash formation mechanism and the special behaviors are characterized by stratified splash and corresponding asymmetrical shrinkage structures. Meanwhile, it is observed that the difference in splash characteristics caused by different thicknesses of viscous fluid layer is obvious, and the pinch-off depth is independent of the thickness value which studied in our experiments. In addition, it is found that increasing the horizontal impact velocity will cause a delay in the splash closure and the vertical displacement of the sphere during the dimethicone-water enter process is independent of the horizontal impact velocity. Hydrodynamic force model is established to explain the trajectory of the sphere. Furthermore, the splash angle, pinch-off depth, pinch-off time and the water entry depth at pinch-off locations and the corresponding change tendency are obtained qualitatively and quantitatively. These results indicate that the dimethicone layer is critical in determining the cavity dynamics, especially splash formation and pinch-off characteristics.

Convection in a Two-Layer Liquid System in a Finite Cylinder

This paper describes a cylindrical container of finite dimensions, filled with two quiescent immiscible heat-conducting liquids with a common flat interface. The side walls and bases of the vessel are solid, there are no external forces, and the contact angle of the interface with the side wall of the container is π/2. The interface has a surface tension whose strength linearly depends on temperature. When one of the container bases is heated to a critical temperature, there is movement inside the vessel. When modeling takes into account the energy spent on the interface deformation. The emerging spectral problem is solved by the modified Galerkin method. For various liquids, in the case of monotonous vibrations, the dependence of a critical Marangoni number on a container size and a temperature ratio, specified on the cylinder bases, is obtained, and a perturbed motion velocity field is constructed.

Splash formation and cavity dynamics of sphere entry through a viscous liquid resting on the water

The splash formation and cavity dynamics during the entry of an object into a single-phase liquid system have been investigated considerably in the past, but only a few studies focus on such phenomena in a two-layer liquid system. In the current study, we present an experimental investigation of the entry of a falling sphere into a liquid system consisting of a deep pool of water and a thin layer of dimethicone resting on the water surface. To highlight and support the present findings, experiments of the same sphere entering water were performed and direct comparison were made. As anticipated, the upper-layer viscous liquid introduces a significant change in the splash formation and the differences in behavior are characterized by layered films and wave-like instability structures. In the experiments of the two-layer liquid system, observations show that the early sealing of the splash and the generation of a larger-scale splash fingers are strongly associated with large Froude number, Fr = V0(gD)-1/2 (where V0 is the impact velocity at the still free surface, g is the gravitational acceleration and D is the diameter of the sphere). In addition, the dimensionless cavity depth, the pinch-off depth and the pull-away cavity length behind the sphere are found to be in linear relationship with the Froude number for both water-entry cases and water-dimethicone-entry cases. One of the most interesting results is that the pinch-off time is almost independent of the Froude number. The highly viscous liquid in the upper layer is shown to be critical in determining the cavity dynamics.

Convection in a Two-Layer Liquid System in a Finite Cylinder

Convection in a Two-Layer Liquid System in a Finite Cylinder

Thermocapillary Flow in an Annular Two-Layer Liquid System

By means of a hybrid lattice Boltzmann method, thermocapillary flow, driven by the surface tension owing to a horizontal temperature gradient along the interface in immiscible two-layer liquid system, is simulated numerically. The dynamic behavior of the interface is captured by using phase-field theory. The dependence of flow and interface deformation on the density ratio, Capillary number and aspect ratio, is investigated.

Study of Marangoni-natural convection in a two-layer liquid system with density inversion using a lattice Boltzmann model

The interplay between buoyancy-induced and surface tension-driven convection in a two-layer liquid system is investigated using a hybrid thermal multiphase lattice Boltzmann method. The upper layer and the lower density inversion layer are thermally and mechanically coupled. The global flow patterns are visualized and the local and average heat transfer rates along the wall estimated in terms of the Nusselt number are plotted. The dependence of fluid flow and heat transfer on the density inversion parameter, Marangoni number, and size aspect ratio, etc., is investigated. The present work shows that the flow pattern is qualitatively and quantitatively different from the double-layer case without density inversion.

Oscillatory instability of Rayleigh-Marangoni-Bénard convection in two-layer liquid system

The oscillatory behaviour of the Rayleigh–Marangoni–Bénard convective instability (R-M-B instability) regarding two combinations of two-layer fluid systems has been investigated theoretically and numerically. For the two-layer system of Silicone oil (10cSt) over Fluorinert (FC70), both linear instability analysis and 2D numerical simulation show that the instability of the system depends strongly on the depth ratio Hr = H1/H2 of the two-layer liquid. The oscillatory regime at the onset of R-M-B convection enlarges with reducing Γ = Ra /Ma values. In the two-layer system of Silicone oil (2cSt) over water, it loses its stability and onsets to steady convection at first, then the steady convection bifurcates to oscillatory convection with increasing Rayleigh number Ra. This behaviour was found through numerical simulation above the onset of steady convection in the case of Γ = 2.9, ε = (Ra − Rac ) / Rac = 1.0, and Hr = 0.5. Our findings are different from the previous study of the Rayleigh–Bénard instability and show the strong effects of the thermocapillary force at the interface on the time-dependent oscillations at or after the onset of convection. We propose a secondary oscillatory instability mechanism to explain the experimental observation of Degen et al. [Phys. Rev. E, 57 (1998), 6647–6659].

Buoyant convection of a two-layer liquid system in a cavity: effects of property variations

Buoyant convection of a two-layer liquid system in a cavity: effects of property variations

Transient Interface Shape of a Two-Layer Liquid in an Abruptly Rotating Cylinder

A description is made of the transient shape of interface of a two-layer liquid in an abruptly rotating circular cylinder. The density of the lower layer is higher than that of the upper layer, but the viscosities may assume arbitrary values. The overall Ekman number is much smaller than unity, and the cylinder aspect ratio is 0(1). The classical Wedemeyer model, which deals with the spin-up from rest of a homogeneous fluid, is extended to tackle the two-layer liquid system. If the upper-layer fluid is of higher viscosity, the interface, at small and intermediate times, rises (sinks) in the center (periphery). After reaching a maximum height at the center, the interface tends to the parabolic shape characteristic of the final-state rigid-body rotation. If the lower-layer fluid is of higher viscosity, the interface, at small and intermediate times, sinks (rises) in the center (periphery). The deformation at the center reaches a minimum height, after which the interface approaches the final-state parabola. The gross adjustment process is accomplished over the spin-up time scale, En−1/2Ω−1, where En and Ω denote the lower value of the Ekman numbers of the two layers and the angular velocity of the cylindrical container, respectively. These depictions are consistent with the physical explanations offered earlier. A turntable experiment is performed to portray the transient interface shape. The model predictions of the interface form are in satisfactory agreement with the laboratory measurements.

Steady thermocapillary flows in a two-layer liquid system with flat interfaces

Steady thermocapillary convection is studied in a system of two flat, superposed layers of immiscible liquids with two fluid-fluid interfaces in a configuration similar to that of an encapsulated crystal growth. The layers are bounded on the sides by isothermal vertical walls maintained at different constant temperatures. A simplified analytical solution is used initially to explore different potential flow regimes in a parameter space of large dimensionality. Then the coupled Navier-Stokes and heat transfer equations are solved numerically with a finite element method via FIDAP, in a rectangular cavity filled with two immiscible liquids in the absence of a gravitational field.

Energy transfer between external and internal gravity waves

In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail.

Disturbances due to the motion of a cylinder in a two-layer liquid system

Disturbances due to the motion of a cylinder in a two-layer liquid system