Instability of monochromatic waves on the surface of a liquid of arbitrary depth

A study is made of the stability, with respect to the spontaneous appearance of modulation, of steady periodic waves of small amplitude on the surface of an ideal liquid depth. The well-known instability of waves on the surface of a liquid of infinite depth disappears on making the transition to small depths.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 45–49, September–October, 1970.     

Instability of a viscoelastic liquid jet with axisymmetric and asymmetric disturbances

The temporal instability behavior of a viscoelastic liquid jet in the wind-induced regime with axisymmetric and asymmetric disturbances moving in an inviscid gaseous environment is investigated theoretically. The corresponding dispersion relation between the wave growth rate and the wavenumber is derived. The linear instability analysis shows that viscoelastic liquid jets are more unstable than their Newtonian counterparts, and less unstable than their inviscid counterparts, for both axisymmetric and asymmetric disturbances, respectively. The instability behavior of viscoelastic jets is influenced by the interaction of liquid viscosity and elasticity, in which the viscosity tends to dampen the instability, whereas the elasticity results in an enhancement of instability. Relatively, the effect of the ratio of deformation retardation to stress relaxation time on the instability of viscoelastic jets is weak. It is found that the liquid Weber number is a key measure that controls the viscoelastic jet instability behavior. At small Weber number, the axisymmetric disturbance dominates the instability of viscoelastic jets, i.e., the growth rate of an axisymmetric disturbance exceeds that of asymmetric disturbances. When the Weber number increases, both the growth rate and the instability range of disturbances increase drastically. The asymptotic analysis shows that at large Weber number, more asymmetric disturbance modes become unstable, and the growth rate of each asymmetric disturbance mode approaches that of the axisymmetric disturbance. Therefore, the asymmetric disturbances are more dangerous than that of axisymmetric disturbances for a viscoelastic jet at large Weber numbers. Similar to the liquid Weber number, the ratio of gas to liquid density is another key measure that affects the viscoelastic jet instability behavior substantially.     

Fractal evolution of interfaces in physical instabilities

A universal method for the theoretical determination of the evolution of the fractal dimension of non-smooth interfaces in the linear stage of physical instabilities is proposed on the basis of a known law of variance of their growth rate. The instability of a thin liquid film with respect to its disintegration into individual jets, the Rayleigh-Taylor instability, and Weibel-type plasma instabilities are studied. It is shown that the fractal dimension of the interface increases for Rayleigh-Taylor-type instability and decreases for thin liquid film instability.     

Instability of Waveguides in a Liquid with Surface Effects

Long surface capillary-gravity waves and waves beneath an elastic plate simulating an ice sheet are considered for a liquid of finite depth. These waves are described by a generalized Kadomtsev-Petviashvili equation containing higher (as compared with the ordinary Kadomtsev-Petviashvili equation) space derivatives. The generalized Kadomtsev-Petviashvili equation has waveguide solutions (waveguides) corresponding to traveling waves which are periodic in the direction of propagation and localized in the transverse direction. These waves result from the instability of uniform (carrier) periodic waves with respect to transverse perturbations. The stability of the waveguides with respect to longitudinal longwave perturbations is studied. The behavior of these perturbations depends on the wavenumber of the carrier periodic wave. Three intervals of wavenumbers corresponding to all the possible types of governing equations are considered.     

Rayleigh-Marangoni-Benard instability in two-layer fluid system

Rayleigh-Marangoni-Bénard instability in a system of two-layer fluids is studied numerically. The convective instabilities in the system of Silicon Oil (10cSt) and Fluorinert (FC70) liquids have been analyzed. The critical parameters at onset of convection are presented in a large range of two-layer depth ratios from 0.2 to 5.0. Numerical results show that the instability of the two-layer system depends strongly on its depth ratio. When the depth ratio increases, the instability mode changes from mechanical coupling to thermal coupling. Between these two typical coupling modes, a time-dependent oscillation is detected. Nevertheless, traveling wave states are found in the case of oscillatory instability. The oscillation mode results from the competition between Rayleigh instability and Marangoni effect. The project supported by the National Natural Science Foundation of China (10372105) and the Knowledge Innovation Program of Chinese Academy of Sciences (KJCX2-SW-L05)     

Temporal analysis of breakup for a power law liquid jet in a swirling gas

The breakup mechanism and instability of a power law liquid jet are investigated in this study. The power law model is used to account for the non-Newtonian behavior of the liquid fluid. A new theoretical model is established to explain the breakup of a power law liquid jet with axisymmetric and asymmetric disturbances, which moves in a swirling gas. The corresponding dispersion relation is derived by a linear approximation, and it is applicable for both shear-thinning and shear-thickening liquid jets. Analysis results are calculated based on the temporal mode. The analysis includes the effects of the generalized Reynolds number, the Weber number, the power law exponent, and the air swirl strength on the breakup of the jet. Results show that the shear-thickening liquid jet is more unstable than its Newtonian and shear-thinning counterparts when the effect of the air swirl is taken into account. The axisymmetric mode can be the dominant mode on the power law jet breakup when the air swirl strength is strong enough, while the non-axisymmetric mode is the domination on the instability of the power liquid jet with a high We and a low Re _n . It is also found that the air swirl is a stabilizing factor on the breakup of the power law liquid jet. Furthermore, the instability characteristics are different for different power law exponents. The amplitude of the power law liquid jet surface on the temporal mode is also discussed under different air swirl strengths.     

The three-dimensional stationary instability in dynamic thermocapillary shallow cavities

In various configurations with thermal convection, three-dimensional stationary patterns occur that consist of pairs of counter-rotating longitudinal rolls. These rolls are investigated in this paper under a variety of experimental conditions. The liquids used are ethanol and the silicone oil hexamethyldisiloxane. The upper surface of the liquid volume is free and very flat because measures against menisci at the side and end walls have been taken. The temperature gradient is applied horizontally via thermally conducting but transparent sapphire end walls, leading to thermocapillary forces at the free surface in addition to the buoyant forces at normal earth's gravity. The geometry of the liquid volume is either rectangular or axisymmetrical (annular). The rectangular set-up is transparent and especially suited for optical observations of tracers in the bulk of the liquid. The annular set-up has the advantages of a large azimuthal (transversal) extent and the absence of side walls. In it a wavelength of λ≈1.3d was observed (where d is the depth of the liquid volume). Temperatures and velocities are measured and used to characterize the instability. Also the region of existence of the instability is studied in layers shallower than in earlier experiments in order to give a larger ratio between thermocapillary and buoyant forces. To find the onset of the instability when increasing the temperature gradient, the amplitude of the instability was derived from measurements and extrapolated. This yields a significantly lower threshold (Ma^c=2300 ± 1000 for d=5 mm) than previous experimental studies. One implementation of the annular gap experiment was performed under microgravity (experiment MAGIA), the other experiments under normal gravity. The results of the experiment under microgravity indicate the absence of the three-dimensional stationary pattern under the absence of gravity. Received: 14 August 2000/Accepted: 17 April 2001     

Characteristics of critical heat flux in two phase thermosyphon – relationship between maximum falling liquid rate and instability on interface of falling liquid film

Critical heat flux in a two-phase thermosyphon is usually dealt with from two different ways in a limitation of liquid flow rate falling along a vertical tube: one is a maximum falling liquid rate due to the envelope method, and the other one due to instability of falling liquid film. The difference between the maximum and instability criteria is first made clear. The CHF in the thermosyphon is shown to be predicted well by the maximum liquid rate due to the maximum criterion better than due to the instability criterion. In addition, the comparison implies that the CHF phenomenon in the thermosyphon is considered to be caused when the falling liquid reaches the maximum value rather than when the instability of the falling liquid on the interface is brought about. Received on 1 December 1997     

Effects of Surface Waves and Marine Soil Parameters on Seabed Stability

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Corona-induced electrohydrodynamic instabilities in low conducting liquids

The rose-window electrohydrodynamic (EHD) instability has been observed when a perpendicular field with an additional unipolar ion injection is applied onto a low conducting liquid surface. This instability has a characteristic pattern with cells five to 10 times greater than those observed in volume instabilities caused by unipolar injection. We have used corona discharge from a metallic point to perform some measurements of the rose-window instability in low conducting liquids. The results are compared to the linear theoretical criterion for an ohmic liquid. They confirmed that the minimum voltage for this instability is much lower than that for the interfacial instability in high conducting liquids. This was predicted theoretically in the dependence of the critical voltage as a function of the non-dimensional conductivity. It is shown that in a non-ohmic liquid the rose window appears as a secondary instability after the volume instability.     

Characterization of plunging liquid jets: A combined experimental and numerical investigation

This paper presents a combined experimental and numerical study of the flow characteristics of round vertical liquid jets plunging into a cylindrical liquid bath. The main objective of the experimental work consists in determining the plunging jet flow patterns, entrained air bubble sizes and the influence of the jet velocity and variations of jet falling lengths on the jet penetration depth. The instability of the jet influenced by the jet velocity and falling length is also probed. On the numerical side, two different approaches were used, namely the mixture model approach and interface-tracking approach using the level-set technique with the standard two-equation turbulence model. The numerical results are contrasted with the experimental data. Good agreements were found between experiments and the two modelling approaches on the jet penetration depth and entraining flow characteristics, with interface tracking rendering better predictions. However, visible differences are observed as to the jet instability, free surface deformation and subsequent air bubble entrainment, where interface tracking is seen to be more accurate. The CFD results support the notion that the jet with the higher flow rate thus more susceptible to surface instabilities, entrains more bubbles, reflecting in turn a smaller penetration depth as a result of momentum diffusion due to bubble concentration and generated fluctuations. The liquid average velocity field and air concentration under tank water surface were compared to existing semi-analytical correlations. Noticeable differences were revealed as to the maximum velocity at the jet centreline and associated bubble concentration. The mixture model predicts a higher velocity than the level-set and the theory at the early stage of jet penetration, due to a higher concentration of air that cannot rise to the surface and remain trapped around the jet head. The location of the maximum air content and the peak value of air holdup are also predicted differently.     

粒子入射角度对冷喷涂涂层形成的影响

采用有限元数值计算方法，研究冷喷涂材料改性过程中铜粒子与铜基体非垂直碰撞的变形行为。针对单个粒子相同入射速度不同入射角度的碰撞条件，探讨粒子与基体的结合强度、侵彻深度以及绝热剪切失稳的发生条件。结果表明:随着粒子入射角度的增大，侵彻深度逐渐减少，粒子与基体的结合强度逐渐减弱。发生绝热剪切失稳的条件是入射速度的法向分量大于碰撞过程的临界速度。     

高Bond数下黏性液滴的Rayleigh-Taylor不稳定性

给出了高Bond数下黏性液滴表面Rayleigh-Taylor线性不稳定性的分析解，这种不稳定性对于超音速气流作用下液滴破碎的早期阶段起着至关重要的作用．基于稳定性分析的结果，导出了用于估算稳定液滴的最大直径及液滴无量纲初始破碎时间的计算式，这些计算式与相关文献给出的实验和分析结果比较显示了良好的一致．     

Orientation instability of shear flow of a nematic liquid crystal

The instability of shear flow of a nematic liquid crystal layer is studied. The case when the orientation vector and the flow velocity vector are parallel is considered. It is shown that the orientation instability of this flow is possible if the anchoring boundary condition is weak and if the splay-bend constants in the Frank energy are taken into account. For this type of instability, periodic structures are possible to appear. Their wave vector belongs to the plane of flow and is perpendicular to the velocity vector. The medium parameters are estimated on the basis of the existence condition for this instability. The period of the appearing periodic structures is evaluated.     

Surface instability and primary atomization characteristics of straight liquid jet sprays

Using the detailed numerical simulation data of primary atomization, the liquid surface instability development that leads to atomization is characterized. The numerical results are compared with a theoretical analysis of liquid–gas layer for a parameter range close to high-speed Diesel jet fuel injection. For intermittent and short-duration Diesel injection, the aerodynamic surface interaction and transient head formation play an important role. The present numerical setting excludes nozzle disturbances to primarily investigate this interfacial instability mechanism and the role of jet head. The first disturbed area is the jet head region, and the generated disturbances are fed into the upstream region through the gas phase. This leads to the viscous boundary layer instability development on the liquid jet core. By temporal tracking of surface pattern development including the phase velocity and stability regime and by the visualization of vortex structures near the boundary layer region, it is suggested that the instability mode is the Tollmien–Schlichting (TS) mode similar to the turbulent transition of solid-wall boundary layer. It is also demonstrated that the jet head and the liquid core play an interacting role, thus the jet head cannot be neglected in Diesel injection. In this study, this type of boundary layer instability has been demonstrated as a possible mechanism of primary atomization, especially for high-speed straight liquid jets. The effect of nozzle turbulence is a challenging but important issue, and it should be examined in the future.     

Application of maximum entropy method for droplet size distribution prediction using instability analysis of liquid sheet

This paper describes the implementation of the instability analysis of wave growth on liquid jet surface, and maximum entropy principle (MEP) for prediction of droplet diameter distribution in primary breakup region. The early stage of the primary breakup, which contains the growth of wave on liquid–gas interface, is deterministic; whereas the droplet formation stage at the end of primary breakup is random and stochastic. The stage of droplet formation after the liquid bulk breakup can be modeled by statistical means based on the maximum entropy principle. The MEP provides a formulation that predicts the atomization process while satisfying constraint equations based on conservations of mass, momentum and energy. The deterministic aspect considers the instability of wave motion on jet surface before the liquid bulk breakup using the linear instability analysis, which provides information of the maximum growth rate and corresponding wavelength of instabilities in breakup zone. The two sub-models are coupled together using momentum source term and mean diameter of droplets. This model is also capable of considering drag force on droplets through gas–liquid interaction. The predicted results compared favorably with the experimentally measured droplet size distributions for hollow-cone sprays.     

Development and elaboration of numerical method for simulating gas–liquid–solid three-phase flows based on particle method

A numerical method for simulating gas–liquid–solid three-phase flows based on the moving particle semi-implicit (MPS) approach was developed in this study. Computational instability often occurs in multiphase flow simulations if the deformations of the free surfaces between different phases are large, among other reasons. To avoid this instability, this paper proposes an improved coupling procedure between different phases in which the physical quantities of particles in different phases are calculated independently. We performed numerical tests on two illustrative problems: a dam-break problem and a solid-sphere impingement problem. The former problem is a gas–liquid two-phase problem, and the latter is a gas–liquid–solid three-phase problem. The computational results agree reasonably well with the experimental results. Thus, we confirmed that the proposed MPS method reproduces the interaction between different phases without inducing numerical instability.     

Gravitational instability of thin gas layer between two thick liquid layers

We consider the problem of gravitational instability (Rayleigh–Taylor instability) of a horizontal thin gas layer between two liquid half-spaces (or thick layers), where the light liquid overlies the heavy one. This study is motivated by the phenomenon of boiling at the surface of direct contact between two immiscible liquids, where the rate of the “break-away” of the vapor layer growing at the contact interface due to development of the Rayleigh–Taylor instability on the upper liquid–gas interface is of interest. The problem is solved analytically under the assumptions of inviscid liquids and viscous weightless vapor. These assumptions correspond well to the processes in real systems, e.g., they are relevant for the case of interfacial boiling in the system water-n-heptane. In order to verify the results, the limiting cases of infinitely thin and infinitely thick gas layers were considered, for which the results can be obviously deduced from the classical problem of the Rayleigh–Taylor instability. These limiting cases are completely identical to the well-studied cases of gravity waves at the liquidliquid and liquid–gas interfaces. When the horizontal extent of the system is long enough, the wavenumber of perturbations is not limited from below, and the system is always unstable. The wavelength of the most dangerous perturbations and the rate of their exponential growth are derived as a function of the layer thickness. The dependence of the exponential growth rate on the gas layer thickness is cubic.     

圆环旋转粘性液膜射流三维不稳定性研究

利用线性稳定性理论进行了射流液体粘性对圆环旋转液膜射流稳定性影响的研究,推导出了三维扰动下具有固体旋涡型速度分布的圆环旋转粘性液膜射流的色散方程;在此基础上进行了类反对称模式与类对称模式下的圆环旋转粘性液膜射流的三维不稳定性分析。研究结果表明,在类反对称模式下,液体粘性超过一定值后,射流最大扰动增长率随液体粘性的增加而迅速减小;轴对称模态的射流特征频率产生一个突降变化;随液体粘性增加,轴对称模态不稳定波数范围减小,非轴对称模态不稳定波数范围呈现出先减小后增大趋势。在类对称模式下,液体粘性对射流最大扰动增长率的影响主要体现在对非轴对称模态的影响上;液体粘性只在粘性较大时才会对非轴对称模态射流特征频率产生一定影响;液体粘性超过一定值后,轴对称模态与非轴对称模态的不稳定波数范围都会快速下降。     

Parametric convection of a low-conducting liquid in an alternating electric field

The development of parametric electroconvective instability of a low-conducting liquid in a horizontal capacitor under the action of an alternating electric field is studied. The conditions on which the free charge is generated due to autonomous unipolar injection are considered. The charge distributions over the liquid at rest are determined analytically and numerically, respectively, for low and arbitrary amplitudes of the external field. The limits of parametric electroconvective instability are found within the framework of the linear theory. It is shown that the perturbations can manifest themselves in different ways depending on the modulation amplitude and frequency of the electric field and be of the subharmonic, synchronous, or quasiperiodic type of the response. The characteristics of the resonance action, namely, the amplitudes and the frequencies necessary for the efficient generation of electroconvection, are determined.