Instability Of Free Surface Research Articles

Mitigation of vibration and free-surface instabilities by using anti-vortex baffles in an industrial stirred vessel

Inadequate understanding of mixing dynamics can result in compromised product quality, increased production expenses and revenue loss. This study utilizes vibrometer measurements in both baffled and unbaffled eccentric flat-bottomed stirring vessels, containing a trisodium phosphate–water mixture, to investigate the impact of baffles on reducing vibrations. Through examining the effect of different configurations of baffles, this research effectively eliminates vortices and vibrations generated during agitation, consequently reducing maintenance frequency. Additionally, computational fluid dynamics (CFD) analysis is conducted on baffled and unbaffled stirred tanks, revealing streamlined patterns and mean velocities. By solving the Reynolds-averaged Navier–Stokes (RANS) equations with the k–[Formula: see text] turbulence model, obtained results reveal the importance of baffles in reducing vibrations and vortices while stabilizing the free-surface. However, balancing the number of baffles with impeller blades is crucial to prevent vibration regeneration.

The Disintegration of Magnetic Fluid Layer on Liquid and Solid Substrates in Vertical Magnetic Field

The purpose of this work is to experimentally study the process of decomposition of a horizontal layer of a magnetic fluid lying on a liquid non-wettable substrate into an ordered system of drops under the influence of a vertical magnetic field.Methods. In the experiment, the studied liquids in the form of a two-layer system filling a cylindrical glass cuvette were placed on a horizontal platform in the center of the Helmholtz coils connected to a direct current source. The development of magnetic fluid layer free surface instability was monitored using a high-speed digital video camera installed at the top of the coil system.Results. The dependence of the critical strength of the magnetic field on the thickness of the torn layer and the magnetic susceptibility of the magnetic fluid is determined. The obtained experimental data are compared with the results of existing theoretical studies of the instability of the magnetic fluid layer. Compared to the case of a solid substrate, the rupture of the magnetic fluid layer on a liquid substrate occurred at lower values of the critical field strength. In this case, the development of instability of the free surface magnetic fluid layer on a solid substrate occurs at field strengths twice as high as the instability strength of the interface of the layer on a liquid substrate.Conclusion. The use of a liquid substrate makes it possible to break such layers of the magnetic fluid, the deformation of which on a solid substrate is limited only by a periodic perturbation of the surface. The magnitude of the critical strength leading to the disintegration of a continuous magnetic fluid layer on a liquid substrate increases with increasing thickness of this layer. An increase in the magnetic susceptibility of the magnetic fluid leads to a decrease in the critical field strength.

Influence of cylinder rotation in close to a free surface on flow characteristics

This study investigates the numerical simulation of unsteady flow over a rotating circular cylinder near a free surface using the Volume of Fluid (VOF) method. The hydrodynamic characteristics and flow pattern are analyzed for various Froude numbers (0.3, 0.55, and 1), submergence depths (0.4, 0.6, and 1), and rotation rates (±0.5, ±1, and ±2). The results indicate that both the rotation direction and rate significantly influence flow instability. Counterclockwise rotation reduces free surface instability and drag coefficient. At a rotation rate of 2, the average drag coefficient decreases by 93% compared to a stationary cylinder at Froude number 0.3. Clockwise rotation induces free surface instability and increases the lift and drag coefficients. Higher Froude numbers result in more significant free surface variations, and increased submergence depth leads to stretched vortices.

A new elastic instability in gravity-driven viscoelastic film flow

We examine the linear stability of the gravity-driven flow of a viscoelastic fluid film down an inclined plane. The viscoelastic fluid is modeled using the Oldroyd-B constitutive equation and, therefore, exhibits a constant shear viscosity and a positive first normal stress difference in viscometric shearing flows; the latter class of flows includes the aforesaid film-flow configuration. We show that the film-flow configuration is susceptible to two distinct purely elastic instabilities in the inertialess limit. The first instability owes its origin entirely to the existence of a free surface and has been examined earlier [Shaqfeh et al., “The stability of gravity driven viscoelastic film-flow at low to moderate Reynolds number,” J. Non-Newtonian Fluid Mech. 31, 87–113 (1989)]. The second one is the analog of the centermode instability recently discovered in plane Poiseuille flow [Khalid et al., “Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow,” Phys. Rev. Lett. 127, 134502 (2021)] and owes its origin to the base-state shear; it is an example of a purely elastic instability of shearing flows with rectilinear streamlines. One may draw an analogy of the aforesaid pair of unstable elastic modes with the inertial free-surface and shear-driven instabilities known for the analogous flow configuration of a Newtonian fluid. While surface tension has the expected stabilizing effect on the Newtonian and elastic free-surface modes, its effect on the corresponding shear modes is, surprisingly, more complicated. For both the Newtonian shear mode and the elastic centermode, surface tension plays a dual role, with there being parameter regimes where it acts as a stabilizing and destabilizing influence. While the Newtonian shear mode remains unstable in the limit of vanishing surface tension, the elastic centermode becomes unstable only when the appropriate non-dimensional surface tension parameter exceeds a threshold. In the limit of surface tension being infinitely dominant, the free-surface boundary conditions for the film-flow configuration reduce to the centerline symmetry conditions satisfied by the elastic centermode in plane Poiseuille flow. As a result, the regime of instability of the film-flow centermode becomes identical to that of the original channel-flow centermode. At intermediate values of the surface tension parameter, however, there exist regimes where the film-flow centermode is unstable even when its channel-flow counterpart is stable. We end with a discussion of the added role of inertia on the aforementioned elastic instabilities.

Rayleigh-Taylor instability of viscous liquid film under an inclined substrate with time modulated temperature

The Rayleigh-Taylor instability of viscous liquid film under a temperature-modulated inclined substrate is studied. Long wave approximation and weighted residual model are used respectively to derive the nonlinear evolution equations of liquid film, and it is shown that the two evolution equations by the above two approaches are equivalent. The recurrence relation and damped Mathieu equation are obtained by employing Floquet theory and linear analysis, respectively. When amplitude a and oscillation frequency ω of the temperature exceed critical values, the film surface loses its stability. The stability diagram is obtained by linear analysis based on Floquet theory. Moreover, based upon the line approach, the nonlinear evolution equation is numerically simulated to give the spatiotemporal evolution of the free surface. Heated capillary tension might accelerate the production of suspended droplets due to the appearing of Marangoni number Ma produced by heating inclined wall. The results reveal that the larger Reynolds number Re, Marangoni number Ma and Biot number Bi will result in the instability of free surface. Moreover, the liquid film becomes more unstable at the smaller Froude number Fr and Weber number We.

Beryllium melt instabilities and ejection during unmitigated current quenches in ITER

The dynamics of transient liquid beryllium flows induced on the ITER first wall during the current quench stage of unmitigated vertical displacement events are modelled by means of two-dimensional Navier–Stokes simulations. The study focuses on melt that is driven to the first wall panels’ chamfered edges, where free-surface instabilities are the most likely to be seeded. Beyond their impact on plasma-facing component damage, these instabilities potentially result in material ejection in the form of droplets, which may ultimately solidify into dust and accumulate in the vessel. Based on prior integrated numerical predictions of quenching magnetic equilibria, wall energy deposition and melt-related damage in a concrete worst-case disruption scenario, the simulations suggest that, although the liquid layer is significantly destabilized, only 5% of the total melt mass created on the wall surface is lost through ejection. This result can serve as a basis to refine the estimates of the real transient-induced beryllium dust inventory expected in ITER.

Experimental study of a liquid metal film flow in a streamwise magnetic field

Continuous wetting of a surface with liquid metal is indispensable in many applications, such as in fusion reactors. In the present study, we provide data on the suppression of free-surface instabilities of liquid metal film flows under the action of strong streamwise magnetic fields in analogy to the poloidal fields used in application. We have designed and built up an experimental test setup which allows studying the influence of magnetohydrodynamics on the dynamic behaviour of liquid metal GaInSn film flows in laminar, transient, and turbulent regimes. While the width and the length of the film are adjusted at w = 23 mm and l = 120 mm, respectively, we are able to apply strong uniform magnetic fields up to B = 5 T over the entire fluid-flow volume. Moreover, the setup allows to vary the Reynolds number within the range 200 ≤ Re ≤ 1700. The corresponding Hartmann and Stuart numbers are Ha ≤ 180 and N ≤ 40, respectively. This study shows that a streamwise magnetic field is capable of suppressing free-surface instabilities even in the turbulent regime of the film flow by dampening any motion perpendicular to the applied magnetic field. Plans for future studies include the quantitative investigation of the parameter space. Figs 4, Refs 8.

Evaluation of flow resistance equations for high gradient rivers using geometric standard deviation of bed material

A dataset of 2184 field measurements reported in the literature was used to evaluate the predictive capability of eight conventional flow resistance equations to predict the mean flow velocity in gravel-bed rivers. The results reveal considerable disagreement with the observed flow velocities for relative submergence less than 4 and for the non-uniformity of the bed material greater than 7.5 for all the equations. However, the predictions made using the Smart and Jäggi (1983), Ferguson (2007), and Rickenmann and Recking (2011) equations were closer to the observed values. Furthermore, bedload sediment transport also reduces the predictive capability of the equations considered in this study except for the Recking et al. (2008) equation, which was developed considering active bedload transport. The performance of flow resistance equations improves when corrected by considering the geometric standard deviation of the bed material. Here we present an empirical approach using the whole dataset and its subsets for accounting for the additional energy losses occurring due to the wake vortices, spill losses, and free surface instabilities occurring due to the protrusions from the bed. The results obtained using the validation dataset shows the importance and usefulness of this approach to account for the additional energy losses, especially for the Strickler (1923) and Keulegan (1938) equations.

Fundamental study on chaotic transition of two-phase flow regime and free surface instability in gas deaeration process

Deaeration is a process of eliminating aspirated air from liquid in hydraulic reservoirs to avoid cavitation in the downstream pump blades. The complex fluid dynamics associated with deaeration is investigated. The three-dimensional buoyancy driven chaotic behavior of gas-liquid interfacial two-phase flow is studied. Parametric study is executed to understand change in internal flow physics (bubble coalescence, disintegration, horizontal spread, bubble velocity etc.), strength of accelerating Rayleigh-Taylor instability, turbulent kinetic energy, amplitude of upward velocity near free surface, and rise in free surface level with the variation of parameters like incoming mixture flow rate, incoming volume fraction of air, liquid fill depth, and Atwood number. The computations show increment in cavitation, wavenumber and amplitude of upward velocity towards oscillating free surface with incoming flow rate (Re). Cavitation and free surface instability show incremental trend with volume fraction of incoming air forming a kink (cavitation reduces) due to bubble coalescence in a threshold range of volume fraction of incoming air. With the variation of Atwood number, initially cavitation reduces. But after a critical value (A*) of Atwood number, effect of bubble disintegration, and rise of cavitation become prominent, which is formulated with respect to incoming flow rate (Re). With liquid fill depth, cavitation shows a slight decrement with almost equal deaeration and constant wavelength of free surface oscillation at an increasing buoyancy driven upward velocity. Some glimpse of design solution to reduce the cavitation and enhance the deaeration is also studied and formulated to get better understanding.

Experimental investigation of wave tip variability of impacting waves

We present an experimental study on the variation in wave impact location and present a mechanism for the development of free surface instabilities on the wave crest for repeatable plunging wave impacts on a vertical wall. The existence of free surface instabilities on an impacting wave is well known, but their characteristics and formation mechanism are relatively unknown. The development of the global wave shape is measured using a visualization camera, whereas the local wave shape is measured with an accurate stereo-planar laser-induced fluorescence technique. A repeatable wave is generated with negligible system variability. The global wave behavior resembles that of a plunging breaker, with a gas pocket cross-sectional area defined by an ellipse of constant aspect ratio. The variability of the local wave profile increases significantly as it approaches the wall. The impact location varies by ∼0.5% of the wave height or more than a typical pressure sensor diameter. Additionally, the wave tip accelerates to a velocity of 1.5gh0 compared to the global wave velocity of 1.2gh0. The difference in impact location and velocity can result in a pressure variation of ∼25%. A mechanism for instability development is observed as the wave tip becomes thinner and elongates when it approaches the wall. A flapping liquid sheet develops that accelerates the wave tip locally and this triggers a spanwise Rayleigh–Taylor instability.

RESERCH PROGRESS ON INTERACTION BETWEEN A SINGLE CAVITATION AND FREE SURFACE

The phenomena of cavitation have been observed in various natural and industrial processes like biomedical engineering and marine engineering. The interaction between a single bubble and free surface, involving the non-spherical collapse of cavitation bubble, nonlinear deformation and instability of free surface, etc., is frontier scientific issues in fluid mechanics and bubble dynamics. In this paper, we summarize research progress and achievements of this field in recent years and discuss typical phenomenon from the perspectives of cavitation dynamics and splash behaviors. For the non-spherical evolution of cavitation bubble, we focus on the key processes such as volume oscillation, jet generation, water hammer effect, collapsing shock wave, and centroid migration based on the dimensionless parameter of kelvin impulse. The mechanism of energy distribution during the cavitation collapse is obtained. For the splash dynamics of free surface, four typical phenomena are summarized based on the generation and development of the thin jet and thick jet: formation of transparent water layer, generation of water column, stable and unstable structure of water crown. Furthermore, we review physical mechanisms on the collapse of cavitation, splash and instability of the free surface by using the theories of Kelvin impulse, Singularity at the curved surface, and Rayleigh-Taylor instability. In addition, we present the effect of curved surfaces on the splash dynamics by investigating the behaviors of free surface of spherical and cylindrical shapes. Finally, remaining challenges and development tendency for future research are given.

Explosive Development of the Kelvin–Helmholtz Quantum Instability on the He-II Free Surface

We analyze nonlinear dynamics of the Kelvin–Helmholtz quantum instability of the He-II free surface, which evolves during counterpropagation of the normal and superfluid components of liquid helium. It is shown that in the vicinity of the linear stability threshold, the evolution of the boundary is described by the |ϕ|4 Klein–Gordon equation for the complex amplitude of the excited wave with cubic nonlinearity. It is important that for any ratio of the densities of the helium component, the nonlinearity plays a destabilizing role, accelerating the linear instability evolution of the boundary. The conditions for explosive growth of perturbations of the free surface are formulated using the integral inequality approach. Analogy between the Kelvin–Helmholtz quantum instability and electrohydrodynamic instability of the free surface of liquid helium charged by electrons is considered.

Formation of the Capillary Ridge on the Free Surface Dynamics of Second-Grade Fluid Over an Inclined Locally Heated Plate

Abstract The formation of capillary ridges is the typical features of thin viscous or viscoelastic fluids over a locally heated plate. This ridge leads to the nonuniformity in the thin film coating. In this work, the formation of capillary ridges on the free surface of thin second-grade non-Newtonian fluid flowing over an inclined heated plate is discussed. The flow is modelled by two-dimensional laws of conservation of mass, momentum, and energy with corresponding boundary conditions at the plate and the free surface. An evolution equation for the description of the liquid thin film height is derived from the two-dimensional balance equations using the long-wave approximation. The resulting nonlinear dynamic equation is discretised implicitly on a uniform grid using the finite volume method. The obtained results on the capillary ridge in the free surface are discussed for the different flow parameters. It is noted that the capillary ridge height is higher for the second-grade viscoelastic fluid in comparison to the Newtonian one. This study can be a starting point to investigate the influence of second-grade viscoelastic parameter on the free surface instability and other phenomena of interest.

Effect of wall deformability on the stability of surfactant-laden visco-elastic liquid film falling down an inclined plane

The stability behavior of a visco-elastic (Upper Convected Maxwell) liquid film flowing down an inclined plane is examined using the standard normal mode linear stability analysis in creeping flow limit. The inclined plane is coated with a deformable solid layer, and the gas-liquid interface is covered with a monolayer of an insoluble surfactant. There exist three modes of instabilities for this composite fluid-solid system: (i) the free surface gas-liquid (GL) mode, (ii) the surfactant or Marangoni mode, and (iii) the liquid-solid (LS) mode. Previous studies have shown that the Marangoni mode remains stable for film flowing down a rigid inclined plane. We show that when the rigid wall is replaced by a deformable wall, the Marangoni mode becomes unstable when a non-dimensional deformability parameter, defined as G=μV/EsR, increases above a threshold value (μ, V, Es, and R are fluid viscosity, free-surface velocity, shear modulus of soft solid layer, and fluid thickness, respectively). It is well known that the GL mode remains stable for Newtonian liquid film at zero Reynolds number, but becomes unstable for a visco-elastic film flowing past a rigid inclined plane even in creeping flow limit purely because of fluid elasticity. Previous studies have shown that the presence of soft solid layer can suppress this free surface GL mode instability. The presence of surfactant also has a stabilizing effect on this GL mode instability in rigid limit. Here, we evaluate whether a soft solid coating can be used to obtain a stable flow configuration for a surfactant-covered visco-elastic film as well particularly (i) in view of the above mentioned Marangoni mode destabilization observed solely due to wall deformability, and (ii) when the surfactant stabilizing contribution remains insufficient to suppress the free-surface instability. We demonstrate that there exists a window in terms of parameter G where the free surface GL mode instability is suppressed without exciting any other mode. We also identify the parameter regimes where it is not possible to achieve stable film flow, however, in these cases, we show that the Marangoni mode can be selectively destabilized by appropriately selecting the thickness and shear modulus of solid layer.

Instability onset for submerged cylinders

This paper describes how the global stability of a circular cylinder is affected when submerged in a two-phase gravitational flow. The flow behavior is governed by both the Reynolds and the Froude number, while the depth of the cylinder has been varied to create different scenarios for the stability analysis. The baseflow obtained by the numerical solution of the 2D Navier-Stokes equations has been analyzed, and the first bifurcation (i.e., Hopf type) has been explored for different depths, Reynolds numbers, and Froude numbers. In addition to the typical vortex shedding instabilities associated with the isolated cylinders, the presence of an interface between fluids creates new instabilities associated with the free surface which present more complex and deformed structures. According to the region of the parameter space studied here, two main causes of instabilities have been found: the ones provoked by vortex shedding on the cylinder wake (wake instabilities) at low Froude numbers and the ones produced by the free surface deformation (free surface instabilities) at high Froude numbers. When instabilities are related to vortex shedding, the critical Reynolds number and the frequency of the most unstable mode are comparable to the classical solution without free surface and gravity effects. In all cases, the shape of the most unstable mode is deformed and distorted according to the free surface location, while the critical Reynolds numbers and the frequency associated with the perturbation are both affected by the gravity and the free surface presence.

Manipulation and control of instabilities for surfactant-laden liquid film flowing down an inclined plane using a deformable solid layer

We analyzed the linear stability of surfactant-laden liquid film with a free surface flowing down an inclined plane under the action of gravity when the inclined plane is coated with a deformable solid layer. For a flow past a rigid incline and in the presence of inertia, the gas-liquid (GL) interface is prone to the free surface instability and the presence of surfactant is known to stabilize the free surface mode when the Marangoni number increases above a critical value. The rigid surface configuration also admits a surfactant induced Marangoni mode which remains stable for film flows with a free surface. This Marangoni mode was observed to become unstable for a surfactant covered film flow past a flexible inclined plane in a creeping flow limit when the wall is made sufficiently deformable. In view of these observations, we investigate the following two aspects. First, what is the effect of inertia on Marangoni mode instability induced by wall deformability? Second, and more importantly, whether it is possible to use a deformable solid coating to obtain stable flow for the surfactant covered film for cases when the Marangoni number is below the critical value required for stabilization of free surface instability. In order to explore the first question, we continued the growth rates for the Marangoni mode from the creeping flow limit to finite Reynolds numbers (Re) and observed that while the increase in Reynolds number has a small stabilizing effect on growth rates, the Marangoni mode still remains unstable for finite Reynolds numbers as long as the wall is sufficiently deformable. The Marangoni mode remains the dominant mode for zero and small Reynolds numbers until the GL mode also becomes unstable with the increase in Re. Thus, for a given set of parameters and beyond a critical Re, there is an exchange of dominant mode of instability from the Marangoni to free surface GL mode. With respect to the second important aspect, our results clearly demonstrate that for cases when the stabilizing contribution of surfactant is not sufficient for suppressing GL mode instability, a deformable solid coating could be employed to suppress free surface instability without triggering Marangoni or liquid-solid interfacial modes. Specifically, we have shown that for a given solid thickness, as the shear modulus of the solid layer decreases (i.e., the solid becomes more deformable) the GL mode instability is suppressed. With further decrease in shear modulus, the Marangoni and liquid-solid interfacial modes become unstable. Thus, there exists a stability window in terms of shear modulus where the surfactant-laden film flow remains stable even when the Marangoni number is below the critical value required for free surface instability suppression. Further, when the Marangoni number is greater than the critical value so that the GL mode remains stable in the rigid limit or with the deformable wall, the increase in wall deformability or solid thickness triggers Marangoni mode instability and, thus, renders a stable flow configuration into an unstable one. Thus, we show that the soft solid layer can be used to manipulate and control the stability of surfactant-laden film flows.

Computational Analysis of the Metal Free-Surface Instability, Fragmentation, and Ejecta under Shock

We conducted numerical simulations of the related processes of interface instability, tensile fragmentation, and jetting resulting from four kinds of typical macro defect perturbations (chevron, sine wave, rectangle, and square) on a Cu free surface under a reflected shock wave when Cu impacts a solid wall at a speed of 2.5 km/s and found that, for the chevron and sine wave cases, the ejecta velocities of the head are 6.28 and 5.88 km/s, respectively. Some parts of the inner material are in a tensile state without any fragmentation, which is observed only in the main body of the material owing to the tension effect. Furthermore, for the other two initial perturbations (rectangle and square), the highest ejecta velocities may even reach 9.14 and 9.59 km/s, respectively. Fragmentation caused by multilayer spallation can be observed on a large scale in the Cu main body, and there are granules in the front area of the ejecta but the degree to which fragmentation occurs is much less in the Cu main body and there is a notable high-speed, low-density granule area in the ejecta head. Finally, we present a detailed analysis of the spatial distribution of the granules, ejecta mass, pressure, temperature, and grid convergence.

Unstable characteristics of two-dimensional parametric sloshing in various shape tanks: theoretical and experimental analyses

The equations of two-dimensional parametric sloshing are derived for a fluid in an arbitrary shape tank. The parametric instability of the liquid free surface is reduced to the Mathieu equation. An energy-growth exponent λ (EGE) and the corresponding non-dimensional energy-growth coefficient (EGC) η are defined for measuring the instability intensity and determining the stability boundaries of the parametrically-excited fluid system. A stability criterion is proposed by using the EGE. On the basis of the numerical tests, the analytical expressions of EGE/EGC are developed for the fluid system that is governed by the Mathieu equation. Under the small excitations, the stability boundaries by using EGE/EGC agree well with those by the traditional theory. The unstable properties, including instability intensity, instability boundaries and competition of unstable patterns etc, of parametric sloshing are analyzed and discussed in details with EGE/EGC for the fluid system. The fluid system always selects such an unstable pattern that possesses a larger energy-growth exponent. A two-dimensional parametric sloshing experiment is respectively conducted for the fluid in the rectangular, circular and U-shaped tanks. The theoretical and experimental stability boundaries of the principal parametric sloshing are obtained and compared for the fluid in three different shape tanks. Some theoretical predictions of the unstable properties of parametric sloshing are verified through the experiment.

Passive manipulation of free-surface instability by deformable solid bilayers.

This study deals with the elastohydrodynamic coupling that occurs in the flow of a liquid layer down an inclined plane lined with a deformable solid bilayer and its consequences on the stability of the free surface of the liquid layer. The fluid is Newtonian and incompressible, while the linear elastic constitutive relation has been considered for the deformable solid bilayer, and the densities of the fluid and the two solids are kept equal. A temporal linear stability analysis is carried out for this coupled solid-fluid system. A long-wave asymptotic analysis is employed to obtain an analytical expression for the complex wavespeed in the low wave-number regime, and a numerical shooting method is used to solve the coupled set of governing differential equations in order to obtain the stability criterion for arbitrary values of the wave number. In a previous work on plane Couette flow past an elastic bilayer, Neelmegam et al. [Phys. Rev. E 90, 043004 (2014)PLEEE81539-375510.1103/PhysRevE.90.043004] showed that the instability of the flow can be significantly influenced by the nature of the solid layer, which is adjacent to the liquid layer. In stark contrast, for free-surface flow past a bilayer, our long-wave asymptotic analysis demonstrates that the stability of the free-surface mode is insensitive to the nature of the solid adjacent to the liquid layer. Instead, it is the effective shear modulus of the bilayer G_{eff} (given by H/G_{eff}=H_{1}/G_{1}+H_{2}/G_{2}, where H=H_{1}+H_{2} is the total thickness of the solid bilayer, H_{1} and H_{2} are the thicknesses of the two solid layers, and G_{1} and G_{2} are the shear moduli of the two solid layers) that determines the stability of the free surface in the long-wave limit. We show that for a given Reynolds number, the free-surface instability is stabilized when G_{eff} decreases below a critical value. At finite wave numbers, our numerical solution indicates that additional instabilities at the free surface and the liquid-solid interface can be induced by wall deformability and inertia in the fluid and solid. Interestingly, the onset of these additional instabilities is sensitive to the relative placements of the two solid layers comprising the bilayer. We show that it is possible to delay the onset of these additional instabilities, while still suppressing the free-surface instability, by manipulating the ratio of the shear moduli and the thicknesses of the two solid layers in the bilayer. At moderate Reynolds number and finite wave number, we demonstrate that an exchange of modes occurs between the gas-liquid and liquid-solid interfacial modes as the solid bilayer becomes more deformable. We demonstrate further that dissipative effects in the individual solid layers have an important bearing on the stability of the system, and they could also be exploited in suppressing the instability. This study thus shows that the ability to passively manipulate and control interfacial instabilities increases substantially with the use of solid bilayers.

Effect of ullage gas on sloshing, Part II: Local effects of gas–liquid density ratio

Gas–liquid density ratio (DR) is a key dimensionless number in sloshing assessment methodologies of membrane containment systems for LNG tanks of floating structures. Earlier studies on the effect of DR were mainly statistical and effects of DR were usually mixed with those of gas compressibility and ullage gas pressure but attributed only to DR. In an attempt to separately study such effects, part I of this work studied the effects of DR far from impact zones (global effects of gas–liquid density ratio) which proved to be small in the studied range of DR (0.0002 to 0.0060). The effects of DR near impact zones and in the instants prior to the detection of any compressibility effects are referred to as local effects and are treated in the current paper (part II).The test setup was identical to the one presented in Part I and consisted of two 2D model tanks representing transverse slices of tank 2 (out of 4) of a membrane LNG carrier with total capacity of 152000m3 at scales 1:20 and 1:40. Both model tests were performed at 20% fill level of the tank heights. Water was the main liquid that was used. In some tests at scale 1:20 a solution of sodium polytungstate (SPT) was also used which had a higher density compared to water. Different ullage gases of helium (He), air, two mixtures of sulfur hexafluoride (SF6) and nitrogen (N2), and pure SF6, all at atmospheric pressure with a range of DRs from 0.0002 to 0.0060 were utilized. Synchronized high-speed video cameras (@4000 fps) and arrays of piezo-electric PCB (112A21 and 112M361) pressure sensors (@40 kHz) monitored and measured impacts on the tank walls. In Part II of the study short and more regular tank motions which generated highly repeatable single impact waves (SIW) were used instead of long irregular tank motions which were considered in part I.By comparing the single impact waves (SIW) generated by identical tank motions but with different DRs, it was observed that DR clearly modifies wave shapes prior to the moment of wave breaking. Larger DRs tend to slow down the wave front and delay breaking. It was also observed that larger DRs slightly slow down wave trough runup as well. Those effects would also lead to a mild shift of impact types by changing the DR (for example Flip-through to slosh or large gas-pocket to small gas-pocket impacts). By comparing single impact waves (SIW) generated by identical tank motions and the same DR but with different gas and liquid densities it was shown that keeping the same DR is essentially needed to keep the same impact geometry as recommended by the existing sloshing assessment methodologies. Free surface instabilities were also very similar for those waves generated with the same tank motions and similar DR but with different gases and liquids. Considering the reduction of wave kinetic energy by heavier11Throughout the paper the term heavy gas would refer to gases with higher density and the term light gas would refer to gases with lower density. This was thought to make the text easier to read (and write) compared to the terms dense and less dense although dense and less dense are more scientific. Nonetheless wherever needed, the value of DR is mentioned explicitly. ullage gases as a relevant source of the statistical reduction of impact pressures and having in mind the mild shift of wave impact types caused by the change of DR it is still to be studied further why the heavier gas leads to smaller statistical pressures.