Incompressible Rayleigh-Taylor Instability Research Articles

Rayleigh-Taylor instability in elastic-plastic solid slabs bounded by a rigid wall.

The linear evolution of the incompressible Rayleigh-Taylor instability for the interface between an elastic-plastic slab medium and a lighter semi-infinite ideal fluid beneath the slab is developed for the case in which slab is attached to a rigid wall at the top surface. The theory yields the maps for the stability in the space determined by the initial perturbation amplitude and wavelength, as well as for the transition boundary from the elastic to the plastic regimes for arbitrary thicknesses of the slab and density contrasts between the media. In particular, an approximate but very accurate scaling law is found for the minimum initial perturbation amplitude required for instability and for the corresponding perturbation wavelength at which it occurs. These results allows for an interpretation of the recent experiments by Maimouni etal. [Phys. Rev. Lett. 116, 154502 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.154502].

Stability boundaries for the Rayleigh-Taylor instability in accelerated elastic-plastic solid slabs.

The linear theory of the incompressible Rayleigh-Taylor instability in elastic-plastic solid slabs is developed on the basis of the simplest constitutive model consisting in a linear elastic (Hookean) initial stage followed by a rigid-plastic phase. The slab is under the action of a constant acceleration, and it overlays a very thick ideal fluid. The boundaries of stability and plastic flow are obtained by assuming that the instability is dominated by the average growth of the perturbation amplitude and neglecting the effects of the higher oscillation frequencies during the stable elastic phase. The theory yields complete analytical expressions for such boundaries for arbitrary Atwood numbers and thickness of the solid slabs.

Rayleigh-Taylor-instability experiments with elastic-plastic materials.

A rotating wheel experimental facility was developed to investigate incompressible Rayleigh-Taylor instability in elastic-plastic materials. A soft solid (mayonnaise) was chosen as the elastic-plastic material for experiments; material properties that include shear modulus and yield strength were fully characterized using a vane spindle type rheometer. Initial perturbations of varying amplitudes and wavelengths were generated on the interface of the soft solid using sinusoidal cutting guides. A backlit imaging technique was used in conjunction with a high-speed camera to track the motion of the interface at various phases of the instability. Results for both two- and three-dimensional perturbations were compared to study the acceleration required for instability and the growth after the interface yielded. Exponential growth rates were observed after instability was reached with trends of increasing growth rates for lower initial amplitudes. It was found that the acceleration required for instability increased when initial amplitude and wavelength decreased. Three-dimensional interfaces were found to be more stable. For both cases, a decrease in initial amplitude produced a more stable interface that increased the threshold acceleration required for the instability. Critical amplitude conditions for instability were calculated and compared with various analytical models and other experimental results.

Weakly nonlinear multi-mode Rayleigh-Taylor instability in two-dimensional spherical geometry

A weakly nonlinear model is proposed for the multi-mode incompressible Rayleigh-Taylor instability in two-dimensional spherical geometry. The second-order solutions are derived, which can be applied to arbitrary small initial perturbations. The cosine-type and the Gaussian-type perturbations are discussed in detail. The growth of perturbations at the pole and that at the equator are compared, and the geometry effect is analyzed. It is found that the initial identical perturbation at the pole and the equator in the cross-sectional view will grow asymmetrically. In the linear regime, the perturbation amplitudes at the pole grow faster than those at the equator due to the different topologies. The geometry effect accelerates the ingoing motion and slows down the outgoing motion in the weakly nonlinear regime. This effect is stronger at the pole than that at the equator.

Thin layer model for nonlinear evolution of the Rayleigh-Taylor instability

On the basis of the thin layer approximation [Ott, Phys. Rev. Lett. 29, 1429 (1972)], a revised thin layer model for incompressible Rayleigh-Taylor instability has been developed to describe the deformation and nonlinear evolution of the perturbed interface. The differential equations for motion are obtained by analyzing the forces (the gravity and pressure difference) of fluid elements (i.e., Newton's second law). The positions of the perturbed interface are obtained from the numerical solution of the motion equations. For the case of vacuum on both sides of the layer, the positions of the upper and lower interfaces obtained from the revised thin layer approximation agree with that from the weakly nonlinear (WN) model of a finite-thickness fluid layer [Wang et al., Phys. Plasmas 21, 122710 (2014)]. For the case considering the fluids on both sides of the layer, the bubble-spike amplitude from the revised thin layer model agrees with that from the WN model [Wang et al., Phys. Plasmas 17, 052305 (2010)] and the expanded Layzer's theory [Goncharov, Phys. Rev. Lett. 88, 134502 (2002)] in the early nonlinear growth regime. Note that the revised thin layer model can be applied to investigate the perturbation growth at arbitrary Atwood numbers. In addition, the large deformation (the large perturbed amplitude and the arbitrary perturbed distributions) in the initial stage can also be described by the present model.

Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical and planar geometries

The relationship between the weakly nonlinear (WN) solutions of the Rayleigh-Taylor instability in spherical geometry [Zhang et al., Phys. Plasmas 24, 062703 (2017)] and those in planar geometry [Wang et al., Phys. Plasmas 19, 112706 (2012)] is analyzed. In the high-mode perturbation limit (Pn(cos θ), n≫1), it is found that at the equator, the contributions of mode P2n along with its neighboring modes, mode P3n along with its neighboring modes, and mode Pn at the third order along with its neighboring modes are equal to those of the second harmonic, the third harmonic, and the third-order feedback to the fundamental mode, respectively, in the planar case with a perturbation of the same wave vector and amplitude as those at the equator. The trends of WN results in spherical geometry towards the corresponding planar counterparts are found, and the convergence behaviors of the neighboring modes of Pn, P2n, and P3n are analyzed. Moreover, the spectra generated from the high-mode perturbations in the WN regime are provided. For low-mode perturbations, it is found that the fundamental modes saturate at larger amplitudes than the planar result. The geometry effect makes the bubbles at or near the equator grow faster than the bubbles in planar geometry in the WN regime.

Rayleigh-Taylor instability in accelerated elastic-solid slabs.

We develop the linear theory for the asymptotic growth of the incompressible Rayleigh-Taylor instability of an accelerated solid slab of density ρ_{2}, shear modulus G, and thickness h, placed over a semi-infinite ideal fluid of density ρ_{1}<ρ_{2}. It extends previous results for Atwood number A_{T}=1 [B. J. Plohr and D. H. Sharp, Z. Angew. Math. Phys. 49, 786 (1998)ZAMPA80044-227510.1007/s000330050121] to arbitrary values of A_{T} and unveil the singular feature of an instability threshold below which the slab is stable for any perturbation wavelength. As a consequence, an accelerated elastic-solid slab is stable if ρ_{2}gh/G≤2(1-A_{T})/A_{T}.

A novel two-dimensional coupled lattice Boltzmann model for incompressible flow in application of turbulence Rayleigh–Taylor instability

A novel coupled lattice Boltzmann model is developed for two-dimensional incompressible Rayleigh–Taylor instability. a modified equilibrium distribution function (D2Q13)is proposed in this paper. The present model is stable and reliable up to temperature jumps between top and bottom walls of the order of 50 ϰ the averaged bulk temperature. The regimes of mixing Rayleigh–Taylor instability are discussed using a simple scaling and the scaling relations obtained are validated by the present model. It is demonstrated that excellent agreement between the present results and the other numerical method or analytical solution shows that the present model is an efficient numerical method for Rayleigh–Taylor instability.

Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical geometry

In this research, a weakly nonlinear (WN) model for the incompressible Rayleigh-Taylor instability in cylindrical geometry [Wang et al., Phys. Plasmas 20, 042708 (2013)] is generalized to spherical geometry. The evolution of the interface with an initial small-amplitude single-mode perturbation in the form of Legendre mode (Pn) is analysed with the third-order WN solutions. The transition of the small-amplitude perturbed spherical interface to the bubble-and-spike structure can be observed by our model. For single-mode perturbation Pn, besides the generation of P2n and P3n, which are similar to the second and third harmonics in planar and cylindrical geometries, many other modes in the range of P0–P3n are generated by mode-coupling effects up to the third order. With the same initial amplitude, the bubbles at the pole grow faster than those at the equator in the WN regime. Furthermore, it is found that the behavior of the bubbles at the pole is similar to that of three-dimensional axisymmetric bubbles, while the behavior of the bubbles at the equator is similar to that of two-dimensional bubbles.

Weakly Nonlinear Rayleigh–Taylor Instability in Incompressible Fluids with Surface Tension**Supported by the National Natural Science Foundation of China under Grant Nos 11275031, 11475034, 11575033 and 11274026, and the National Basic Research Program of China under Grant No 2013CB834100.

A weakly nonlinear model is established for incompressible Rayleigh–Taylor instability with surface tension. The temporal evolution of a perturbed interface is explored analytically via the third-order solution. The dependence of the first three harmonics on the surface tension is discussed. The amplitudes of bubble and spike are greatly affected by surface tension. The saturation amplitude of the fundamental mode versus the Atwood number A is investigated with surface tension into consideration. The saturation amplitude decreases with increasing A. Surface tension exhibits a stabilizing phenomenon. It is shown that the asymmetrical development of the perturbed interface occurs much later for large surface tension effect.

Weakly nonlinear incompressible Rayleigh-Taylor instability growth at cylindrically convergent interfaces

A weakly nonlinear (WN) model has been developed for the incompressible Rayleigh-Taylor instability (RTI) in cylindrical geometry. The transition from linear to nonlinear growth is analytically investigated via a third-order solutions for the cylindrical RTI initiated by a single-mode velocity perturbation. The third-order solutions can depict the early stage of the interface asymmetry due to the bubble-spike formation, as well as the saturation of the linear (exponential) growth of the fundamental mode. The WN results in planar RTI [Wang et al., Phys. Plasmas 19, 112706 (2012)] are recovered in the limit of high-mode number perturbations. The difference between the WN growth of the RTI in cylindrical geometry and in planar geometry is discussed. It is found that the interface of the inward (outward) development spike/bubble is extruded (stretched) by the additional inertial force in cylindrical geometry compared with that in planar geometry. For interfaces with small density ratios, the inward growth bubble can grow fast than the outward growth spike in cylindrical RTI. Moreover, a reduced formula is proposed to describe the WN growth of the RTI in cylindrical geometry with an acceptable precision, especially for small-amplitude perturbations. Using the reduced formula, the nonlinear saturation amplitude of the fundamental mode and the phases of the Fourier harmonics are studied. Thus, it should be included in applications where converging geometry effects play an important role, such as the supernova explosions and inertial confinement fusion implosions.

Linear analysis of incompressible Rayleigh-Taylor instability in solids

The study of the linear stage of the incompressible Rayleigh-Taylor instability in elastic-plastic solids is performed by considering thick plates under a constant acceleration that is also uniform except for a small sinusoidal ripple in the horizontal plane. The analysis is carried out by using an analytical model based on the Newton second law and it is complemented with extensive two-dimensional numerical simulations. The conditions for marginal stability that determine the instability threshold are derived. Besides, the boundary for the transition from the elastic to the plastic regime is obtained and it is demonstrated that such a transition is not a sufficient condition for instability. The model yields complete analytical solutions for the perturbation amplitude evolution and reveals the main physical process that governs the instability. The theory is in general agreement with the numerical simulations and provides useful quantitative results. Implications for high-energy-density-physics experiments are also discussed.

Rayleigh–Taylor instability in elastic-plastic solids

The linear analysis of incompressible Rayleigh–Taylor instability is carried out for thick solid plates accelerated uniformly by a constant pressure. The instability threshold is found and the boundary for the elastic to plastic transition is also determined. It is demonstrated that transition from the elastic to the plastic regime is a necessary condition for the onset of instability but not a sufficient one. The theory is in excellent quantitative agreement with the results of two-dimensional numerical simulations and reveals the main physical mechanisms that control the instability.

Surface tension in incompressible Rayleigh–Taylor mixing flow

We study the effect of surface tension on the incompressible Rayleigh–Taylor instability. We modify Goncharov's local analysis [1] to consider the surface tension effect on the Rayleigh–Taylor bubble velocity. The surface tension damps the linear instability and reduces the nonlinear terminal bubble velocity. We summarize the development of a finite-volume, particle-level-set, two-phase flow solver with an adaptive Cartesian mesh, and results from convergence and validation studies of this two-phase flow solver are provided. We use this code to simulate the single-mode, viscous Rayleigh–Taylor instability with surface tension, and good agreement in terminal bubble velocity is found when compared with analytic results. We also simulate the immiscible Rayleigh–Taylor instability with random initial perturbations. The ensuing mixing flow is characterized by the effective mixing rate and the flow anisotropy. Surface tension tends to reduce the effective mixing rate and homogenizes the Rayleigh–Taylor mixing flow. Finally, we provide a scaling argument for detecting the onset of the quadratic, self-similar Rayleigh–Taylor growth.

Large-eddy-simulation of 3-dimensional Rayleigh-Taylor instability in incompressible fluids

The 3-dimensional incompressible Rayleigh-Taylor instability is numerically studied through the large-eddy-simulation (LES) approach based on the passive scalar transport model. Both the instantaneous velocity and the passive scalar fields excited by sinusoidal perturbation and random perturbation are simulated. A full treatment of the whole evolution process of the instability is addressed. To verify the reliability of the LES code, the averaged turbulent energy as well as the flux of passive scalar are calculated at both the resolved scale and the subgrid scale. Our results show good agreement with the experimental and other numerical work. The LES method has proved to be an effective approach to the Rayleigh-Taylor instability.

Ablative stabilization in the incompressible Rayleigh–Taylor instability

Ablative Rayleigh–Taylor instabilities are analyzed in terms of an incompressible fluid model. A generalized surface wave dispersion relation can be derived for arbitrary step-profiles with ablative mass and heat flow. A systematic overview on different regimes of ablative stabilization and growth reduction is given. Convective stabilization by the incoming and outcoming flows are found included as upper and lower limits on the instability growth. Applications for self-consistent steady-state conditions are discussed and a comparison is made with previous numerical work.