Three-dimensional Rayleigh-Taylor Instability Research Articles

Experimental investigation of three-dimensional Rayleigh–Taylor instability of a gaseous interface

Validating the theoretical work on Rayleigh–Taylor instability (RTI) through experiments with an exceptionally clean and well-characterized initial condition has been a long-standing challenge. Experiments were conducted to study the three-dimensional RTI of an SF $_6$ –air interface at moderate Atwood numbers. A novel soap film technique was developed to create a discontinuous gaseous interface with controllable initial conditions. Spectrum analysis revealed that the initial perturbation of the soap film interface is half the size of an entire single-mode perturbation. The correlation between the initial interface perturbation and Atwood numbers was determined. Due to the steep and highly curved feature of the initial soap film interface, the early-time evolution of RTI exhibits significant nonlinearity. In the quasi-steady regime, various potential flow models accurately predict the late-time bubble velocities by considering the channel width as the perturbation wavelength. Differently, the late-time spike velocities are described by these potential flow models using the wavelength of the entire single-mode perturbation. These findings indicate that the bubble evolution is influenced primarily by the spatial constraint imposed by walls, while the spike evolution is influenced mainly by the initial curvature of the spike tip. Consequently, a recent potential flow model was adopted to describe the time-varying amplitude growth induced by RTI. Furthermore, the self-similar growth factors for bubbles and spikes were determined from experiments and compared with existing studies, revealing that a large amplitude in the initial soap film interface promotes the spike development.

Non-overlapping High-accuracy Parallel Closure for Compact Schemes: Application in Multiphysics and Complex Geometry

Compact schemes are often preferred in performing scientific computing for their superior spectral resolution. Error-free parallelization of a compact scheme is a challenging task due to the requirement of additional closures at the inter-processor boundaries. Here, sources of the error due to sub-domain boundary closures for the compact schemes are analyzed with global spectral analysis. A high-accuracy parallel computing strategy devised in “ A high-accuracy preserving parallel algorithm for compact schemes for DNS. ACM Trans. Parallel Comput. 7, 4, 1-32 (2020)” systematically eliminates error due to parallelization and does not require overlapping points at the sub-domain boundaries. This closure is applicable for any compact scheme and is termed here as non-overlapping high-accuracy parallel (NOHAP) sub-domain boundary closure. In the present work, the advantages of the NOHAP closure are shown with the model convection equation and by solving the compressible Navier–Stokes equation for three-dimensional Rayleigh–Taylor instability simulations involving multiphysics dynamics and high Reynolds number flow past a natural laminar flow airfoil using a body-conforming curvilinear coordinate system. Linear scalability of the NOHAP closure is shown for the large-scale simulations using up to 19,200 processors.

Interface models for three-dimensional Rayleigh–Taylor instability

We derive interface models for three-dimensional Rayleigh–Taylor instability (RTI), making use of a novel asymptotic expansion in the non-locality of the fluid flow. These interface models are derived for the purpose of studying universal features associated with RTI such as the Froude number in single-mode RTI, the predicted quadratic growth of the interface amplitude under multi-mode random perturbations, the optimal (viscous) mixing rates induced by the RTI and the self-similarity of horizontally averaged density profiles and the remarkable stabilization of the mixing layer growth rate which arises for the three-fluid two-interface heavy–light–heavy configuration, in which the addition of a third fluid bulk slows the growth of the mixing layer to a linear rate. Our interface models can capture the formation of small-scale structures induced by severe interface roll-up, reproduce experimental data in a number of different regimes and study the effects of multiple interface interactions even as the interface separation distance becomes exceedingly small. Compared with traditional numerical schemes used to study such phenomena, our models provide a computational speed-up of at least two orders of magnitude.

Role of unstable thermal stratifications on the Rayleigh–Taylor instability

Three-dimensional simulations of Rayleigh–Taylor instability (RTI) at the interface of two masses of air are performed by solving the compressible Navier–Stokes equation, following the numerical implementation in Role of non-zero bulk viscosity in three-dimensional Rayleigh–Taylor instability: Beyond Stokes’ hypothesis - Sengupta et al. (2021). The effect of thermal gradients on the RTI is explored by considering two temperature differences between the air masses, ΔT=21.75K and 46.5K, corresponding to Atwood numbers (At) of 0.035 and 0.073. The flow is studied in an isolated box with non-periodic walls along three directions. A non-conducting interface, initially separating the two air masses, is removed at the onset of the numerical experiment. The stages in the evolution of the RTI are explored via the enstrophy transport equation (ETE). The contributions from vortex stretching, viscous diffusion and dissipation are quantified for different thermal gradients, showing completely different dynamics. For both At considered, the viscous dissipation term is found to dominate across the stages of RTI evolution, with vortex stretching becoming prominent beyond the development of spikes and bubbles in the mixing layer. For At=0.035, two interpenetrating rows of bubbles are noted, whereas, for At=0.073, an alternating row of bubbles and spikes are noted with a 19% rise in the growth rate of the mixing layer. The present simulations have been experimentally validated for a combined Kelvin–Helmholtz Rayleigh–Taylor instability (KHRTI). It is found that the vortical structures, representative of the KH instability, are stretched and diffused due to the buoyancy-driven RT mechanism.

Lattice Boltzmann simulation of three-dimensional fluid interfacial instability coupled with surface tension

In this paper, the development of three-dimensional fluid interfacial Rayleigh-Taylor (RT) instability coupled with the surface tension was numerically studied using the mesoscopic lattice Boltzmann method. We mainly analyzed the influence of surface tension on fluid interfacial dynamics and spike/bubble late-time growth. The numerical experiments show that there exists the critical surface tension (<inline-formula><tex-math id="M3">\begin{document}$\sigma_{\rm{c}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20212061_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20212061_M3.png"/></alternatives></inline-formula>) in the three-dimensional RT instability, above which the RT phenomenon does not appear and below which it would take place. It is found that the critical surface tension increases with the fluid Atwood number and the corresponding numerical predictions show good agreements with those of the theoretical analysis <inline-formula><tex-math id="M4">\begin{document}${\sigma_{\rm{c}}}= {{({{\rho_{\rm{h}}}-{\rho_{\rm{l}}}})g}}/{{{k^2}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20212061_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20212061_M4.png"/></alternatives></inline-formula>. In addition, we can find that increasing surface tension reduces the roll-up of the interface and the complexity of interfacial structure, also preventing the breakup of the interface into the individual droplets. The late-time dynamics of phase interface change from the asymmetric development to the symmetry with respect to the middle axis. When the surface tension is sufficiently low, the spike and bubble amplitudes almost no longer change with it, and further increasing the surface tension can slow down the growth of the spike and bubble amplitudes. Furthermore, we can observe that the development of the high-Reynolds-number RT instability under different surface tensions can also be divided into four distinct stages, including the linear growth, saturated velocity growth, reacceleration, and chaotic mixing. The spike and bubble grow with approximately constant velocities at the saturated stage and their asymptotic values are consistent with those of the modified potential flow theory. In the following, the spike and bubble driven by the increasing Kelvin-Helmholtz vortices are accelerated such that their evolutional velocities exceed the solutions of the potential flow model at the reacceleration stage. The reacceleration stage cannot last infinitely and the spike and bubble velocities at the late time fluctuate with time, implying that the growth of the RT instability enters into the chaotic mixing stage. By numerical analysis, we demonstrate that the three-dimensional RT instability at the chaotic mixing stage has a quadratic growth and also report that the spike and bubble growth rates decrease with the surface tension in general.

Scale-dependent Rayleigh–Taylor dynamics with variable acceleration in a finite-sized domain for three-dimensional flows

Three-dimensional Rayleigh–Taylor instability (RTI) with the time-varying acceleration in a finite domain is investigated in a systematic framework. The acceleration magnitude follows a power law in time with an exponent greater than −2. Applying the group theory, the instabilities are demonstrated considering the irreducible representations for observable periodic structures with a square symmetry in the plane normal to the acceleration. We derive the dynamical system and illustrate the universal form of the solutions in the linear and nonlinear regimes. The scale-dependent dynamics are shown to be single scale and multiscale in the two regimes, respectively. For the nonlinear regime solutions, fundamental scales are derived bridging the solutions in the finite- and infinite-sized domains. Special solutions for bubbles and spikes are identified from a one-parameter family of solutions. The effect of domain confinement is that the velocity and curvature decreases and shear increases as the domain size is reduced. The theory provides predictions for the flow field and demonstrates the interfacial behavior of RTI. Our results are in good agreement with the prior studies and also provide new benchmarks for experiments and simulations.

Role of non-zero bulk viscosity in three-dimensional Rayleigh-Taylor instability: Beyond Stokes’ hypothesis

Three-dimensional direct numerical simulations (DNS) of Rayleigh-Taylor instability (RTI) at the interface of two masses of air with a sharp temperature gradient of 149 K are performed by solving the compressible Navier-Stokes equation (NSE). The flow is studied in an isolated box with non-periodic walls along the three directions. A non-conducting interface separating the two air masses is impulsively removed at the onset of the instability. No external perturbation has been used at the interface to instigate the instability at the onset, corresponding to practical scenarios in experiments. Computations have been carried out for the two configurations reported by Read (Experimental investigation of turbulent mixing by Rayleigh-Taylor instability, Physica D. 12, 45–58 (1984)). The compressible formulation is free from the Boussinesq approximation commonly used for solving the incompressible NSE. The role of non-zero bulk viscosity is quantified by using a model from acoustic attenuation measurements for the bulk viscosity of air. Effects of Stokes’ hypothesis on the onset of RTI and the growth of mixing layer are reported. The incipient stage is shown to have a strong dependence on the constitutive relation used. The small-scale billowing motion is only observed for non-zero bulk viscosities. Following this stage, the growth rates for bubbles and spikes in the mixing layer are found to be underpredicted by 12% with the use of Stokes’ hypothesis. The results imply that the evolution of RTI from the onset to the fully turbulent regime is best captured by using non-zero values of the bulk viscosity.

A three-dimensional phase-field lattice Boltzmann method for incompressible two-components flows

In this paper, a lattice Boltzmann model for the coupled Allen–Cahn–Navier–Stokes equations in three dimensions is presented. Two equations are solved: one for the fluid velocity and one for the order parameter. Both are written within the general multiple-relaxation-time framework, where all the equilibrium and forcing terms are described by using the full set of Hermite polynomials. The resultant practical implementation is compact. The gradient of the order parameter can be computed by the non-local finite differences or the local central moments. The latter suffers from grid-scale oscillations. The very good accuracy properties are demonstrated against nine well-consolidated benchmark tests. Specifically, two groups of tests are tackled. In the former, the velocity field is superimposed. Hence, only the equation for the evolution of the order parameter is solved. These numerical experiments demonstrate the ability of the proposed scheme to capture the correct evolution of the interface. In the latter, two immiscible fluids are considered and the two equations are solved. Simulations of the vertical penetration of a wedge-shaped body, two- and three-dimensional Rayleigh–Taylor instability prove that two-fluids systems can be successfully simulated by our approach.

The three-dimensional weakly nonlinear Rayleigh–Taylor instability in spherical geometry

The fully three-dimensional Rayleigh–Taylor instability in spherical geometry is investigated in the weakly nonlinear regime. A theoretical model is developed for incompressible fluid and ideal Euler equations. Third-order solutions are derived for interface perturbations of spherical harmonic modes, Yn,m. Interface evolution, fundamental mode growth, the generated spectrum, and bubble growth are determined. It is found that the fastest growing modes satisfy the relation m≃(n+1)/2. The generated spectra demonstrate the feedback of mode coupling, which greatly depends on the azimuthal mode numbers. The growth factors are nearly the same for bubbles at different latitudes and bubbles with initially round cross-sectional perturbation shapes grow faster.

Detailed nonlinear dynamics of the liquid spike development in gaseous medium caused by a three-dimensional Rayleigh-Taylor instability

Previous researches on the single-mode Rayleigh-Taylor (RT) instability were mainly focused on the advancing speed of gaseous bubbles, while few of them shed light on the nonlinear dynamics of the liquid spikes (jets) where the droplets are constantly generated in the liquid atomization case. In this paper, a three-dimensional (3D) numerical simulation on a single-mode RT instability (Atwood number is close to unity) with the focus on the development of the liquid spike was conducted by the Coupled Level-Set and Volume-of-Fluid (CLSVOF) method. Nonlinear dynamic characteristics of the RT instability were then analyzed thoroughly based on the detailed information of pressure and velocity fields obtained from the numerical simulation. The numerical results revealed that a local maximum pressure point, where an equilibrium was achieved between the inertial force and pressure gradient, was formed at the root of liquid spike region caused by the horizontal impinging flow. Combined with the theoretical analysis, the relationships among the steady-state characteristic parameters, the initial perturbation wavelength and the inertial acceleration were established. It is found that the velocity and width of the liquid entering the freed liquid spike at the maximum pressure point exhibit the same characteristics as those of a vertical jet emanating downward from an orifice injector under gravity. Thus, theories concerning the low-speed jet can be used to predict the disintegration behavior of a liquid spike caused by RT instability.

Improved three-dimensional color-gradient lattice Boltzmann model for immiscible two-phase flows.

In this paper, an improved three-dimensional color-gradient lattice Boltzmann (LB) model is proposed for simulating immiscible two-phase flows. Compared with the previous three-dimensional color-gradient LB models, which suffer from the lack of Galilean invariance and considerable numerical errors in many cases owing to the error terms in the recovered macroscopic equations, the present model eliminates the error terms and therefore improves the numerical accuracy and enhances the Galilean invariance. To validate the proposed model, numerical simulations are performed. First, the test of a moving droplet in a uniform flow field is employed to verify the Galilean invariance of the improved model. Subsequently, numerical simulations are carried out for the layered two-phase flow and three-dimensional Rayleigh-Taylor instability. It is shown that, using the improved model, the numerical accuracy can be significantly improved in comparison with the color-gradient LB model without the improvements. Finally, the capability of the improved color-gradient LB model for simulating dynamic two-phase flows at a relatively large density ratio is demonstrated via the simulation of droplet impact on a solid surface.

Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate

We investigate the Rayleigh–Taylor instability of a thin liquid film coated on the inside of a cylinder whose axis is orthogonal to gravity. We are interested in the effects of geometry on the instability, and contrast our results with the classical case of a thin film coated under a flat substrate. In our problem, gravity is the destabilizing force at the origin of the instability, but also yields the progressive drainage and stretching of the coating along the cylinder’s wall. We find that this flow stabilizes the film, which is asymptotically stable to infinitesimal perturbations. However, the short-time algebraic growth that these perturbations can achieve promotes the formation of different patterns, whose nature depends on the Bond number that prescribes the relative magnitude of gravity and capillary forces. Our experiments indicate that a transverse instability arises and persists over time for moderate Bond numbers. The liquid accumulates in equally spaced rivulets whose dominant wavelength corresponds to the most amplified mode of the classical Rayleigh–Taylor instability. The formation of rivulets allows for a faster drainage of the liquid from top to bottom when compared to a uniform drainage. For higher Bond numbers, a two-dimensional stretched lattice of droplets is found to form on the top part of the cylinder. Rivulets and the lattice of droplets are inherently three-dimensional phenomena and therefore require a careful three-dimensional analysis. We found that the transition between the two types of pattern may be rationalized by a linear optimal transient growth analysis and nonlinear numerical simulations.

Lattice Boltzmann simulation of three-dimensional Rayleigh-Taylor instability.

In this paper, the three-dimensional (3D) Rayleigh-Taylor instability (RTI) with low Atwood number (A(t)=0.15) in a long square duct (12W × W × W) is studied by using a multiple-relaxation-time lattice Boltzmann (LB) multiphase model. The effect of the Reynolds number on the interfacial dynamics and bubble and spike amplitudes at late time is investigated in detail. The numerical results show that at sufficiently large Reynolds numbers, a sequence of stages in the 3D immiscible RTI can be observed, which includes the linear growth, terminal velocity growth, reacceleration, and chaotic development stages. At late stage, the RTI induces a very complicated topology structure of the interface, and an abundance of dissociative drops are also observed in the system. The bubble and spike velocities at late stage are unstable and their values have exceeded the predictions of the potential flow theory [V. N. Goncharov, Phys. Rev. Lett. 88, 134502 (2002)]. The acceleration of the bubble front is also measured and it is found that the normalized acceleration at late time fluctuates around a constant value of 0.16. When the Reynolds number is reduced to small values, some later stages cannot be reached sequentially. The interface becomes relatively smoothed and the bubble velocity at late time is approximate to a constant value, which coincides with the results of the extended Layzer model [S.-I. Sohn, Phys. Rev. E 80, 055302(R) (2009)] and the modified potential theory [R. Banerjee, L. Mandal, S. Roy, M. Khan, and M. R. Guptae, Phys. Plasmas 18, 022109 (2011)]. In our simulations, the Graphics Processing Unit (GPU) parallel computing is also used to relieve the massive computational cost.

Numerical simulation of the three-dimensional Rayleigh–Taylor instability

The Rayleigh–Taylor instability is a fundamental instability of an interface between two fluids of different densities, which occurs when the light fluid is pushing the heavy fluid. During the nonlinear stages, the growth of the Rayleigh–Taylor instability is greatly affected by three-dimensional effects. To investigate three-dimensional effects on the Rayleigh–Taylor instability, we introduce a new method of computation of the flow of two incompressible and immiscible fluids and implement a time-dependent pressure boundary condition that relates to a time-dependent density field at the domain boundary. Through numerical examples, we observe the two-layer roll-up phenomenon of the heavy fluid, which does not occur in the two-dimensional case. And by studying the positions of the bubble front, spike tip, and saddle point, we show that the three-dimensional Rayleigh–Taylor instability exhibits a stronger dependence on the density ratio than on the Reynolds number. Finally, we perform a long time three-dimensional simulation resulting in an equilibrium state.

Three-dimensional Eulerian method for large deformation of viscoelastic fluid: Toward plate-mantle simulation

Toward the unified simulation of the large deformation of a rigid viscoelastic material (plate) and the convection of a viscous fluid (mantle), an Eulerian scheme with a semi-Lagrangian method is developed. The scheme adopts the CIP-CSLR method for advection terms of staggered grid system in three dimensions. The positive transported profile of a positive quantity is assured by flux corrections in the dimensional splitting method. The Jaumann co-rotational effect of the stress tensor is also integrated into the semi-Lagrangian treatment. This co-rotated semi-Lagrangian method is combined with an exponential time differencing method in the time development of the Maxwell constitutive model. The large time step comparable to, or larger than, the Maxwell relaxation time is successfully realized. Validation tests are performed for the three-dimensional Rayleigh–Taylor instability of a viscoelastic material with jump discontinuity of the mass density and other material properties.

Experimental study of the single-mode three-dimensional Rayleigh-Taylor instability

The three-dimensional Rayleigh-Taylor instability is studied in a low Atwood number (A≈0.15) miscible fluid system. The two fluids are contained within a Plexiglas tank that is mounted on vertical rails and accelerated downward by a weight and pulley system. A net acceleration between 13 and 23m∕s2 can be maintained, resulting in an effective body force equivalent to 0.33–1.35 times Earth’s gravity. A single-mode, three-dimensional perturbation is produced by oscillating the tank, which has a square cross section, along its diagonal. Early time measured growth rates are shown to have good agreement with linear stability theory. At late time, the instability exhibits a nonconstant vertical interfacial velocity in agreement with the recent numerical computations of Ramaprabhu et al. [Phys. Rev. E 74, 066308 (2006)]. Both the late-time bubble and spike velocities have values greater than those predicted by both the simple buoyancy-drag model developed by Oron et al. [Phys. Plasmas 8, 2883 (2001)] and the potential flow model of Goncharov [Phys. Rev. Lett. 88, 134502 (2002)]. The disagreement with the models can be attributed to the formation of vortices, in this case vortex rings, observed in the experiments but not accounted for by the models.

Vortex generation and deformation of the interface in three-dimensional Rayleigh-Taylor Instability

We have performed 3-d hydrodynamic Rayeligh-Taylor simulations with initial perturbations with two modes to clarify nonlinear dynamics of the bubbles. By imposing two modes with different initial amplitudes, well defined bubbles with three different sizes, small, medium and large, can be formed. We pay attention to the generation of not only transverse vortex, perpendicular to the gravity, but also the longitudinal one, and relation between the bubble dynamics and the vortex. It is pointed that the longitudinal vortex is generated due to pressure gradient, parallel to the gravity, caused by fluid flow stagnated at the saddle and that three different bubble growth can be understood from the point of vortex dynamics view.

Nonlinear mixing behavior of the three-dimensional Rayleigh–Taylor instability at a decelerating interface

Results are reported from the first experiments to explore the evolution of the Rayleigh–Taylor (RT) instability from intentionally three-dimensional (3D) initial conditions at an embedded, decelerating interface in a high-Reynolds-number flow. The experiments used ∼5 kJ of laser energy to produce a blast wave in polyimide and/or brominated plastic having an initial pressure of ∼50 Mbars. This blast wave shocked and then decelerated the perturbed interface between the first material and lower-density C foam. This caused the formation of a decelerating interface with an Atwood number ∼2/3, producing a long-term positive growth rate for the RT instability. The initial perturbations were a 3D perturbation in an “egg-crate” pattern with feature spacings of 71 μm in two orthogonal directions and peak-to-valley amplitudes of 5 μm. The resulting RT spikes appear to overtake the shock waves, moving at a large fraction of the predeceleration, “free-fall” velocity. This result was unanticipated by prior simulations and models.

On the miscible Rayleigh–Taylor instability: two and three dimensions

We investigate the miscible Rayleigh–Taylor (RT) instability in both two and three dimensions using direct numerical simulations, where the working fluid is assumed incompressible under the Boussinesq approximation. We first consider the case of randomly perturbed interfaces. With a variety of diagnostics, we develop a physical picture for the detailed temporal development of the mixed layer: we identify three distinct evolutionary phases in this development, which can be related to detailed variations in the growth of the mixing zone. Our analysis provides an explanation for the observed differences between two- and three-dimensional RT instability; the analysis also leads us to concentrate on the RT models which (i) work equally well for both laminar and turbulent flows, and (ii) do not depend on turbulent scaling within the mixing layer between fluids. These candidate RT models are based on point sources within bubbles (or plumes) and their interaction with each other (or the background flow). With this motivation, we examine the evolution of single plumes, and relate our numerical results (for single plumes) to a simple analytical model for plume evolution.

On the three-dimensional Rayleigh–Taylor instability

The three-dimensional Rayleigh–Taylor instability is studied using a lattice Boltzmann scheme for multiphase flow in the nearly incompressible limit. This study focuses on the evolution of the three-dimensional structure of the interface. In addition to the bubble and spike fronts, a saddle point is found to be another important landmark on the interface. Two layers of heavy-fluid roll-ups, one at the spike tip and the other at the saddle point, were observed. The secondary instability in the horizontal planes entangles the already complicated structure of the interface. Parallel computations are utilized to accommodate the massive computational requirements of the simulations.