Theory Of Rayleigh-Taylor Instability Research Articles

Comprehensive study on collision patterns of viscous droplets impacting on a heated particle

The collision process involving droplets and heated particles has gained significant attention due to its wide industrial relevance. This study utilizes a high-speed photography to investigate the collision dynamics between viscous droplets and heated particles. The research identifies six distinct collision patterns. In the bubble-breakup mode, the particle experiences the greatest temperature drop, resulting in the most substantial heat transfer. The particle temperature plays a crucial role in determining collision behavior when the Reynolds number exceeds 100 and the Weber number exceeds 55. The maximum spreading area demonstrates a linear relationship with the Weber number, while it reaches a peak and stabilizes with Reynolds numbers in the deposition regime. Contact angle fluctuations are caused by the instability of the contact line. The liquid film thickness exhibits linear and power growth phases, followed by a rapid decrease in the bubble-breakup regime. While the branch-breakup pattern sees smaller fragmented droplet sizes, the atomization-breakup pattern sees flow velocity rise with both Reynolds and Weber numbers. The predicted wavelength of the disturbance in the atomization regime, based on Rayleigh-Taylor instability theory, aligns well with the experimental measurements. The residence time correlates positively with the Weber number.

Effects of initial velocity on the sinking of ilmenite-bearing cumulates in the early lunar mantle: Insight from the theory of Rayleigh-Taylor instability

The early lunar mantle overturn, associated with the sinking of the dense ilmenite-bearing cumulate (IBC) crystallized at the shallow lunar mantle, provides satisfactory explanations for the origination of high-Ti basalt, the abnormally strong magnetic field between ∼ 3.9 and ∼ 3.6 Ga and the low-viscosity zone in the deep lunar mantle, but still poses a debate regarding the initial state of IBC in the early lunar mantle. If the sinking of IBC initiated before the end of lunar magma ocean crystallization, the solidified IBC can acquire a greater thickness and a higher initial velocity at the IBC-mantle boundary. The variation of initial velocity can affect the strain rate of IBC and, correspondingly, the dislocation creep components at the shallow lunar mantle. In this work, we analyze the effects of initial velocity on the dynamics of early lunar mantle by using the theory of Rayleigh-Taylor instability. To couple the effects of diffusion creep and dislocation creep for all major minerals in the lunar mantle, we exploit an improved Minimized Power Geometric (IMPG) model and isostress mixing model to characterize the upper limit and lower limit for the viscosity of the lunar mantle comprising four major minerals, i.e. olivine, orthopyroxene, clinopyroxene and ilmenite. The modeling results suggest that a high initial velocity, in any case, can shorten the onset time, tending to promote the early lunar mantle overturn even in a rheologically-strong lunar mantle. The effect of initial velocity on the overturn wavelength shows a strong dependence on the rheological mixing model. For the isostress mixing model, the increase of initial velocity tends to elongate the overturn wavelength. For the IMPG mixing model, the overturn wavelength is insensitive to the variation of initial velocity. As the actual lunar mantle rheology sandwiches between the rheologies predicted by isostress mixing model and IMPG model, it can be anticipated that the increase of initial velocity tends to elongate the overturn wavelength. In consideration of the importance of the initial velocity on the dynamics of early lunar mantle, future investigations should focus on the dynamics of the solid IBC in the solidifying lunar magma ocean.

Numerical and experimental study of spray impingement and liquid film separation during the spray/wall interaction at expanding corners

The liquid film separation at expanding corners during the spray/wall interaction is usually encountered in internal combustion engines, which has significant influence on the fuel/air mixing and combustion processes, and consequently the engine-out emissions. However, fundamental investigations on the film separation during the spray/wall interaction at expanding corners are very limited, especially under the relevant operating conditions of engines. In the present study, a new film separation and atomization model was proposed, including the sub-models of film separation criterion, film separation mass ratio, and the film atomization model derived from the Rayleigh-Taylor instability theory. By implementing the proposed model into a computational fluid dynamics (CFD) code, the reliability of the new model was validated by the experimental measurements conducted in this study. The results indicate that the effects of the injection pressure, impingement distance, and corner angle of the expanding corner on the evolution of the impinging spray and the wall film dynamics after the corners are satisfactorily reproduced by the present model. Moreover, the predicted size distribution of the detached droplets formed by the film separation is in good agreement with the measurement. It is also found that the expanding corner angle remarkably affects the impinging spray structure and the film separation characteristics, and the predictions of the new film separation model agrees better with the measurements than the previous model.

Linear Rayleigh-Taylor instability in an accelerated Newtonian fluid with finite width.

The linear theory of Rayleigh-Taylor instability is developed for the case of a viscous fluid layer accelerated by a semi-infinite viscous fluid, considering that the top interface is a free surface. Effects of the surface tensions at both interfaces are taken into account. When viscous effects dominate on surface tensions, an interplay of two mechanisms determines opposite behaviors of the instability growth rate with the thickness of the heavy layer for an Atwood number A_{T}=1 and for sufficiently small values of A_{T}. In the former case, viscosity is a less effective stabilizing mechanism for the thinnest layers. However, the finite thickness of the heavy layer enhances its viscous effects that, in general, prevail on the viscous effects of the semi-infinite medium.

A hybrid Rayleigh-Taylor-current-driven coupled instability in a magnetohydrodynamically collimated cylindrical plasma with lateral gravity

We present an MHD theory of Rayleigh-Taylor instability on the surface of a magnetically confined cylindrical plasma flux rope in a lateral external gravity field. The Rayleigh-Taylor instability is found to couple to the classic current-driven instability, resulting in a new type of hybrid instability that cannot be described by either of the two instabilities alone. The lateral gravity breaks the axisymmetry of the system and couples all azimuthal modes together. The coupled instability, produced by combination of helical magnetic field, curvature of the cylindrical geometry, and lateral gravity, is fundamentally different from the classic magnetic Rayleigh-Taylor instability occurring at a two-dimensional planar interface. The theory successfully explains the lateral Rayleigh-Taylor instability observed in the Caltech plasma jet experiment [Moser and Bellan, Nature 482, 379 (2012)]. Potential applications of the theory include magnetic controlled fusion, solar emerging flux, solar prominences, coronal mass ejections, and other space and astrophysical plasma processes.

Modeling hydrodynamic instabilities of double ablation fronts in inertial confinement fusion

A linear Rayleigh-Taylor instability theory of double ablation (DA) fronts is developed for direct-drive inertial confinement fusion. Two approaches are discussed: an analytical discontinuity model for the radiation dominated regime of very steep DA front structure, and a numerical self-consistent model that covers more general hydrodynamic profiles behaviours. Dispersion relation results are compared to 2D simulations.

Drop formation time from a liquid surface

The surface roughness of the free surface of inviscid liquids excited by thermal energy is proportional to the thermal energy and is inversely proportional to the surface tension of the liquid at a constant frequency. The surface wave of the liquid becomes unstable above a critical acceleration and the critical acceleration to make drops out of the liquid surface is predicted using the Rayleigh–Taylor instability theory. The delayed drop ejection from the surface of the liquid since the applied acceleration, called drop formation time, is inversely proportional to the square root of the difference between the applied acceleration and the critical acceleration.

Experimental investigation on splashing and nonlinear fingerlike instability of large water drops

Abstract The fluid physics of the splashing and spreading of a large-scale water drop is experimentally observed and investigated. New phenomena of drop impact that differ from the conventional Rayleigh–Taylor instability theory are reported. Our experimental data shows good agreement with previous work at low Weber number but the number of fingers or instabilities begins to deviate from the R–T equation of Allen at high Weber numbers. Also observed were multiple waves (or rings) on the spreading liquid surface induced from pressure bouncing (or pulsation) within the impacting liquid. The first ring is transformed into a radially ejecting spray whose initial speed is accelerated to a velocity of 4–5 times that of the impacting drop. This first ring is said to be “splashing,” and its structure is somewhat chaotic and turbulent, similar to a columnar liquid jet surrounded by neighboring gas jets at relatively high impact speed. At lower impact speeds, splashing occurs as a crown-shaped cylindrical sheet. A second spreading ring is observed that transforms into fingers in the circumferential direction during spreading. At higher Weber number, the spreading of a third ring follows that of the second. This third ring, induced by the pressure pulsation, overruns and has fewer fingers than the second, which is still in a transitional spreading stage. Several important relationships between the drop impact speed, the spray ejection speed of the first ring, and the number of fingers of the second and third rings are presented, based on data acquired during a set of drop impact experiments. Issues related to the traditional use of the R–T instability are also addressed.

Characteristics of high-power radiating imploding discharge with cold start

A qualitative model of the dynamics of a multiterawatt radiating Z-pinch with cold start and high rate of current rise is proposed. The model is used to analyze discharges with currents I ∼ 2–5 MA (with dI/dt > 1013 A/s) through uniform or structured plasma-producing loads, including wire arrays. The most important consequence of cold start is that spatially nonuniform plasma production is prolonged to almost the entire current rise time. Under these conditions, the Ampere force begins to play a dominant role in the plasma dynamics before the plasma-producing load is completely transformed into an accelerated plasma. The results of computations of wire-array vaporization are presented. A formula is proposed for estimating the highest attainable velocity of plasma flow into a heterogeneous liner driven by the Ampere force. It is shown that local imbalance between radial motion of the produced plasma and supply of the plasma-producing substance to be ionized leads to axially nonuniform breakthrough of magnetic flux into the liner, which precedes plasma collapse. The magnetic-flux breakthrough gives rise to a chaotic azimuthal-axial plasma structure consisting of radial plasma jets of relatively small diameter, which is called a radial plasma rainstorm. The breaking-through azimuthal magnetic flux obstructs further current flow in the breakthrough region. Analyses of Z-pinch implosion based on the theory of Rayleigh-Taylor instability or the snowplow model are incorrect under the plasma-rainstorm conditions. The processes taking place in a stagnant Z-pinch include conversion of the energy carried by the current-generated magnetic field into turbulent MHD flow of the ion component of the plasma, its convective mixing with magnetic field, heating, energy transfer from ions to electrons, and emission from the plasma. Under typical experimental conditions, emission plays a key role in the energy balance in an imploding pinch. Z-pinch is modeled by an electric-circuit component that has a time-dependent nonlinear impedance and consumes the magnetic energy supplied by a generator through a magnetically insulated transmission line (MITL). The peak power reached in the circuit is comparable to the peak soft X-ray power output emitted by the pinch in terms of magnitude and timing. Optimum matching conditions are formulated for the generator-MITL-pinch circuit.

Formation of fingers around the edges of a drop hitting a metal plate with high velocity

Water droplets (0.55 or 1.3 mm diameter) were photographed as they impinged on a stainless steel surface. The droplet impact velocity (10–50 m s−1) and the average roughness (0.03 or 0.23 μm) of the test surfaces were varied. The stainless steel substrate was mounted on the end of a rotating arm, giving linear velocities of up to 50 m s−1. Different stages of droplet impact were photographed by synchronizing the ejection of a single droplet with the position of the rotating arm and triggering of a camera. Finger-shape perturbations were observed around the edges of spreading droplets. The maximum diameter to which a droplet spread and the number of fingers formed around it were measured. The size and number of fingers increased with impact velocity and droplet diameter. At sufficiently high velocities, the tips of these fingers detached, producing satellite droplets. By increasing surface roughness, both the number of fingers and the maximum extent of spreading were decreased. At high impact velocities the spreading liquid film became so thin that it ruptured in several places. A mathematical model, based on linear Rayleigh–Taylor instability theory, was used to predict the wavelength of the fastest growing perturbation around a spreading droplet. The corresponding wavenumber agreed reasonably well with the number of fingers around the droplet.

Splashing of molten tin droplets on a rough steel surface

We photographed the impact of molten metal droplets on a flat plate. From these images we measured droplet dimensions during spreading and counted the number of fingers around a splashing drop. Experiments were done using stainless steel substrates with average roughness of 0.06, 0.07, 0.56, and 3.45 μm respectively. The temperature of the substrate was kept at either 25 or 240 °C. Droplet diameter (2.2 mm) and impact velocity (4 m/s) were kept constant, giving a Reynolds number ( Re) of 31 135 and Weber number ( We) of 463. Raising substrate roughness from 0.06 to 0.56 μm enhanced the tendency of droplet to splash, whereas increasing roughness even further to 3.45 μm suppressed splashing. This behaviour was attributed to changes in droplet solidification rate with surface roughness. A simple model of droplet spreading was used to estimate thermal contact resistance between the droplet and surface. Increasing surface roughness was found to raise thermal contact resistance and reduce heat transfer from the droplet to the substrate, delaying the onset of solidification and reducing splashing. The number of fingers formed around a droplet splashing on a smooth surface could be predicted reasonably well by a model based on Rayleigh–Taylor instability theory. Increasing surface roughness reduced the number of fingers while enlarging their size.

Modeling the splash of a droplet impacting a solid surface

A numerical model is used to simulate the fingering and splashing of a droplet impacting a solid surface. A methodology is presented for perturbing the velocity of fluid near the solid surface at a time shortly after impact. Simulation results are presented of the impact of molten tin, water, and heptane droplets, and compared with photographs of corresponding impacts. Agreement between simulation and experiment is good for a wide range of behaviors. An expression for a splashing threshold predicts the behavior of the molten tin. The results of water and especially heptane, however, suggest that the contact angle plays an important role, and that the expression may be applicable only to impacts characterized by a relatively low value of the Ohnesorge number. Various experimental data of the number of fingers about an impacting droplet agree well with predictions of a previously published correlation derived from application of Rayleigh–Taylor instability theory.

On the breakup mechanisms of air-assisted drops in a high speed air

An experimental study was performed to investigate break-up mechanisms of liquid drops injected into a transverse high velocity air jet. The range of conditions included the three drop breakup regimes previously referred to as bag, shear or boundary layer stripping, and ‘catastrophic’ breakup regimes. The results show that the break-up mechanism consists of a series of processes in which dynamic pressure effects deform the drop into a thin liquid sheet. The flattened drop subsequently breaks up into small droplets. At high relative velocity, in the ‘catastrophic’ breakup regime, drops are flattened and fragmented by relatively large wavelength waves whose wavelengths and growth rates are consistent with estimates from Rayleigh-Taylor instability theory. The minute drops that are also produced at this high relative velocity appear to originate from short wave length of Kelvin-Helmholtz waves growing on the larger liquid fragments.

BREAKUP MECHANISMS AND DRAG COEFFICIENTS OF HIGH-SPEED VAPORIZING LIQUID DROPS

An experimental and modeling study was performed to investigate drop drag and breakup mechanisms of liquid drops injected into a transverse high-velocity air jet at room and elevated temperature conditions. The range of conditions included three drop breakup regimes previously referred to as bag, shear or boundary-layer stripping, and "catastrophic" breakup regimes. In the experiments the injected dieselfuel drop's initial diameter was 189 μm, and its velocity was 16 m/s. The transverse cur jet velocity was varied from 70 to 200 m/s, and the air jet's temperature was varied from room temperature up to 450 K. The conditions tested correspond to drop Weber numbers (based on gas density and drop-gas relative velocity) from 56 to 463. Double-pulse, high-magnification photography was used to study the temporal progress of the breakup of the drops. The experiments gave information about the microscopic structure of the liquid breakup process, drop breakup regimes, drop trajectories and drag coefficients, and the acceleration of atomizing drops. The results show that the breakup mechanism consists of a series of processes in which dynamic pressure effects deform the drop into a thin liquid sheet. This drop flattening significantly affects the drop's drag coefficient. It was found that drop trajectories could be modeled adequately using a modified dynamic drag model that accounts for drop distortion. The flattened drop subsequently breaks up into small droplets. At high relative velocities, in the "catastrophic" breakup regime, drops are flattened and fragmented by relatively large-wavelength waves whose wavelengths and growth rates are consistent with estimates from Rayleigh-Taylor instability theory. The minute drops that are also produced at these high relative velocities appear to originate from short-wavelength Kelvin-Helmholtz waves growing on the larger liquid fragments. At high gas temperatures the decreased surface tension and increased vaporization causes the bag to disappear earlier in the bag breakup regime, and causes shortened ligament lengths in the other breakup regimes.

Critical fluid-flow phenomenon in a gas-stirred ladle

Measurement of the velocities of bubbles and liquid with a two-element electroresistivity probe and laser-Doppler velocimeter, respectively, during bottom injection of air into a water bath, has confirmed the existence of a critical gas-injection rate. Above the critical flow rate, the change of axial bubble velocity in the air jet, and of liquid velocity with increasing volume flow rate, diminishes markedly. The existence of the critical flow rate is explicable from high-speed motion pictures of the vertical gas jets, which reveal four zones of gas dispersion axially distributed above the orifice: primary bubble at the orifice, free bubble, plume consisting of disintegrated bubbles, and spout at the bath surface. With increasing gas-injection rate, the free-bubble zone expands such that the point of bubble disintegration rises closer to the bath surface. Above the critical flow rate, the free bubbles rise with minimal breakup and erupt from the bath surface with maximum energy discharge. The combined Kelvin-Helmholtz, Rayleigh-Taylor instability theory has been applied to analyze the bubble breakup in the bath and the critical gas-injection rate in a gas-stirred ladle. The criterion for the critical diameter of bubble breakup has been found to depend primarily on the surface tension and density of the liquid. In the analysis, the propagation time of a disturbance on a bubble surface at the “most unstable” wave number has been compared with the bubble rising time in the bath in order to determine the critical gas-flow rate. The predicted critical values are in close agreement with the measured results.

Breakup criteria for fluid particles

A simple mechanistic model is developed based on the combination of Kelvin-Helmholtz and Rayleigh-Taylor instability theory to describe the breakup of freely rising or falling fluid particles, i.e. bubbles and drops. The breakup is predicted to occur if the growth rate of interfacial waves on the leading front is faster than the rate at which waves propagate around the interface to the side of the particle. Based on this theoretical model and available experimental data, simple correlations are developed to predict the maximum size a fluid particle can reach. Predicted values of the breakup diameter are compared with experimental data for cases of freely rising bubbles, falling drops in gas and freely falling or rising drops in immiscible liquids. The results are extended to predict the maximum droplet size in a high-velocity gas field which is the most interesting case in terms of practical applications. Good agreement between predicted values and experimental data indicates that the principal mechanisms involved in the fluid particle breakup process are properly accounted for by the proposed model.

Rayleigh–Taylor instability: Modes and nonlinear evolution

Analytical models are presented for describing modifications of the classical Rayleigh–Taylor instability theory in the context of inertial confinement fusion. The effects of stratification, finite layers, compressibility, convection and heat conduction are analysed and their mutual importance is estimated. It is found, that convective stabilization dominates for usual flow parameters and can account for growth reductions by a factor of 2 to 3. It was further possible to calculate the nonlinear evolution with the help of representative flow models and to follow the dynamics of bubble, spike and vortex formation.

Viscous effects in Rayleigh-Taylor instability

A simple, physical approximation is developed for the effect of viscosity for stable interfacial waves and for the unstable interfacial waves which correspond to Rayleigh-Taylor instability. The approximate picture is rigorously justified for the interface between a heavy fluid (e.g., water) and a light fluid (e.g., air) with negligible dynamic effect. The approximate picture may also be rigorously justified for the case of two fluids for which the differences in density and viscosity are small. The treatment of the interfacial waves may easily be extended to the case where one of the fluids has a small thickness; that is, the case in which one of the fluids is bounded by a free surface or by a rigid wall. The theory is used to give an explanation of the bioconvective patterns which have been observed with cultures of microorganisms which have negative geotaxis. Since such organisms tend to collect at the surface of a culture and since they are heavier than water, the conditions for Rayleigh-Taylor instability are met. It is shown that the observed patterns are quite accurately explained by the theory. Similar observations with a viscous liquid loaded with small glass spheres are described. A behavior similar to the bioconvective patterns with microorganisms is found and the results are also explained quantitatively by Rayleigh-Taylor instability theory for a continuous medium with viscosity.