Spike Fronts Research Articles

Competition between merging and bifurcation in the generalized Rayleigh-Taylor instability.

The nonlinear evolution of bubble and spike fronts growing through the generalized Rayleigh-Taylor instability are studied by numerical simulations and by solving an extension of Alon's [Phys. Rev. E 48, 1008 (1993)2470-004510.1103/PhysRevE.48.1008] statistical model based on the asymptotic velocity of a single-mode bubble and the merging bubble process. In this work, the generalized Rayleigh-Taylor instability includes a frictional force due to collision with a secondary fluid. Depending on its strength the behavior during the nonlinear stage leads to two different regimes: the first is the classical inertial case where the bubble front is known to grow as h∝t^{2} and evolves towards large structures, and the second is the collisional case where the front grows as h∝t and maintains structures of relatively constant size. In this new regime, the importance of adding the bifurcation process, the opposite process of merging, is highlighted.

Effect of surface tension on late-time growth of high-Reynolds-number Rayleigh-Taylor instability

In this paper, we numerically investigate the late-time growth of high-Reynolds-number single-mode Rayleigh-Taylor instability in a long pipe by using an advanced phase-field lattice Boltzmann multiphase method. We mainly analyze the influence of surface tension on interfacial dynamic behavior and the development of the bubble front and spike front. The numerical experiments indicate that increasing surface tension can significantly reduce the complexity of formed interfacial structure and also prevents the breakup of phase interfaces. The interface patterns in the instability process cannot always preserve the symmetric property under the extremely small surface tension, but they do maintain the symmetries with respect to the middle line as the surface tension is increased. We also report that the bubble amplitude first increases then decreases with the surface tension. There are no obvious differences between the curves of spike amplitudes for low surface tensions. However, when the surface tension increases to a critical value, it can slow down the spike growth significantly. When the surface tension is lower than the critical value, the development of the high-Reynolds-number Rayleigh-Taylor instability can be divided into four different stages, i.e. the linear growth, saturated velocity growth, reacceleration, and chaotic mixing. The bubble and spike velocities at the second stage show good agreement with those from the modified potential flow theory that takes the surface tension effect into account. After that, the bubble front and spike front are accelerated due to the formation of Kelvin-Helmholtz vortices in the interfacial region. At the late time, the bubble velocity and spike velocity become unstable and slightly fluctuate over time. To determine the nature of the late-time growth, we also measure the bubble and spike normalized accelerations at various interfacial tensions and Atwood numbers. It is found that both the spike and bubble growth rates first increase then decrease with the surface tension in general. Finally, we deduce a theoretical formula for the critical surface tension, below which the Rayleigh-Taylor instability takes place and above which tension it does not occur. It is shown that the critical surface tension increases with the Atwood number and also the numerical predictions by the lattice Boltzmann method are also in accord well with the theoretical results.

Analysis of Rayleigh–Taylor instability at high Atwood numbers using fully implicit, non-dissipative, energy-conserving large eddy simulation algorithm

Large eddy simulation of three-dimensional, multi-mode Rayleigh–Taylor instability at high Atwood numbers is performed using a recently developed, kinetic energy-conserving, non-dissipative, fully implicit, finite volume algorithm. The algorithm was especially designed for simulating low-Mach number, variable density/viscosity, transitional, and turbulent flows. No interface capturing mechanism is required. Buoyancy and heat transfer effects can be handled without relying on the Boussinesq assumption. Because of this feature, unlike the pure incompressible ones, it does not suffer from the loss of physical accuracy at high Atwood and Rayleigh numbers. In this study, the mixing phenomenon in Rayleigh–Taylor instability and the effects of high Atwood numbers on the development of the flow are investigated using various diagnostics such as local mole fractions, bubble and spike penetration lengths and growth rates, mixing efficiencies, Taylor micro-scales, and corresponding Reynolds numbers and energy ratios. Additionally, some important terms of the Reynolds stress transport equation are also introduced, such as Reynolds stresses (and their anisotropies) and turbulent production. Results show that Rayleigh–Taylor instability at high Atwood numbers is characterized by rapid development of instability due to the increasing growth rates and higher velocities of spike fronts, larger asymmetry in the mixing region, denser interactions in the non-linear phase, and changes in bubble and spike morphologies. It is also found that interactions of spike-fronts with their surroundings are the primary mechanisms of turbulent production and transition to turbulence. However, late time mean flow measures such as energy ratio and mixedness are not significantly affected. A scaling relation between the spike to bubble penetration ratio and the heavy to light density ratio is also provided.

Split radiographic tracer technique to measure the full width of a high energy density mixing layer

We describe a new radiography–based technique for measuring the width of the mixing layer that develops when high–energy–density (HED) fluids of different density mix at an interface. In a traditional radiography target, the denser material includes an embedded, density–matched, high opacity, tracer layer to focus the radiograph on a 2D slice of the 3D flow field. The high opacity layer enhances the contrast between the dense and light materials that permits a clear identification of the spike front, i.e. penetration of the dense material into the light material, but obscures the bubble front, i.e. penetration of the light material into the heavy when the fluids are highly interpenetrated, as in a turbulent mixing layer. On the other hand, removing the tracer layer makes it difficult to identify the interface at all, due to poor contrast and an increased susceptibility to edge effects. Our solution is a split tracer approach where half the target uses the traditional geometry while the other half utilizes a special tracer layer in the light material to produce a radiograph with inverse x-ray opacity characteristics when backlit at a specified x–ray energy. Hydrodynamic equivalency is maintained between the two halves of the target. By making a single target with the traditional and inverse material pairs set side-by-side, we are able to simultaneously measure the penetration of bubbles and spikes in a deeply nonlinear flow, yielding the full mix width. We discuss this new target concept and present our first experiments at the National Ignition Facility (NIF) demonstrating the viability of this technique.

Direct numerical simulations of multi-mode immiscible Rayleigh-Taylor instability with high Reynolds numbers

In this paper, we conduct the high-resolution direct numerical simulations of multimode immiscible Rayleigh-Taylor instability (RTI) with a low Atwood number (At = 0.1) using an improved phase field lattice Boltzmann method. The effect of the Reynolds number on the evolutional interfacial dynamics and bubble/spike amplitudes is first investigated by considering its wide range, from 100 up to a high value of 30 000. The numerical results show that, for sufficiently large Reynolds numbers, a sequence of distinguishing stages in the immiscible RTI can be observed, which includes the linear growth, saturated velocity growth, and chaotic development stages. At the late stage, the RTI induces a complex topology structure of the interface and a mass of dissociative drops can be significantly observed in the system. The accelerations of the bubble and spike front are also measured, and it is reported that their normalized values at the late time are, respectively, approximate to the constant values of around 0.025 and 0.027, exhibiting a terminally quadratic growth. As the Reynolds number is reduced to small ones, the multiple disturbances of the RTI are found to merge into a larger one at the initial stage. Then, the evolutional interfaces display the patterns familiar from the single-mode RTI. The phase interfaces in the whole process become very smooth without the appearance of the breakup phenomenon, and the spike and bubble velocities at the late time approach constant values. Furthermore, we also analyze the effects of the initial conditions in terms of the perturbation wavelength and amplitude, and it is found that the instability undergoes a faster growth at the intermediate stage for a larger wavelength, while the late-time bubble and spike growth rates are insensitive to the changes of the initially perturbed wavelength and amplitude.

Large eddy simulation of high atwood number rayleigh-taylor mixing

Large eddy simulation of Rayleigh-Taylor instability at high Atwood numbers is performed using recently developed, kinetic energy-conserving, non-dissipative, fully-implicit, finite volume algorithm. The algorithm does not rely on the Boussinesq assumption. It also allows density and viscosity to vary. No interface capturing mechanism is requried. Because of its advanced features, unlike the pure incompressible ones, it does not suffer from the loss of physical accuracy at high Atwood numbers. Many diagnostics including local mole fractions, bubble and spike growth rates, mixing efficiencies, Taylor micro-scales, Reynolds stresses and their anisotropies are computed to analyze the high Atwood number effects. The density ratio dependence for the ratio of spike to bubble heights is also studied. Results show that higher Atwood numbers are characterized by increasing ratio of spike to bubble growth rates, higher speeds of bubble and especially spike fronts, faster development in instability, similarity in late time mixing values, and mixing asymmetry.

Numerical Investigation of a Shock Accelerated Heavy Gas Cylinder in the Self-Similar Regime

A detailed numerical simulation of a shock accelerated heavy gas (SF6) cylinder surrounded by air gas is presented. It is a simplified configuration of the more general shock-accelerated inhomogeneous flows which occur in a wide variety of astrophysical systems. From the snapshots of the time evolution of the gas cylinder, we find that the evolution of the shock accelerated gas cylinder is in some ways similar to the roll-ups of a vortex sheet for both roll up into a spiral and fall into a self-similar behavior. The systemic and meaningful analyses of the negative circulation, the center of vorticity and the vortex spacing are in a good agreement with results obtained from the prediction of vorticity dynamics. Unlike the mixing zone width in single-mode or multi-mode Richtmyer-Meshkov instability which doesn’t exist, a single power law of time owing to the bubble and spike fronts follow a power law of tθ with different power exponents, the normalized length of the shock accelerated gas cylinder follows a single power law with θ = 0.43 in its self-similar regime obtained from the numerical results.

On generating initial conditions for turbulence models: the case of Rayleigh–Taylor instability turbulent mixing

A new approach to generate initial conditions for RANS simulations of Rayleigh–Taylor (RT) turbulence is presented. The strategy is to provide profiles of turbulent model variables when it is suitable for the turbulence model to be started, and then use these profiles for the turbulence model initialization. The generation of turbulence model variable profiles is achieved with a two-step process. In the first step, a nonlinear modal model assuming small amplitude initial perturbations, incompressible and inviscid fluids is used to track the growth of modes that exist in a given initial perturbation spectrum, and also modes generated by mode interactions. The amplitude development of each mode represents the penetration distance of the light fluid into the heavy fluid (bubble penetration), for a given mode perturbation. The penetration distance of heavy fluid into the light fluid (spike penetration), for a given mode perturbation, is inferred from the bubble's height by an empirical relation valid for small initial amplitudes, and established by DNS simulations that depend on a nondimensional time, and the density contrast (Atwood number). It is hypothesized that the bubble front position of the RT mixing layer can be approximated by the largest penetration distance among all existing modes. The spike front position is approximated in the same fashion. The nonlinear model is evaluated by comparing the bubble front height evolution predicted by the model against the bubble front height predicted by an incompressible implicit large eddy simulations (ILES) code. Comparisons of results for “top-hat” and two-band initial perturbation spectra at Atwood numbers, AT =0.3 and AT =0.5 for the former, and AT =0.01 and AT =0.5 for the latter, show reasonable agreement. In the second step, the bubble and spike front positions, their derived velocities, and simplified profiles of the mixture fraction distribution of each fluid between the bubble and spike fronts are used with a two-fluid approximation to derive profiles for the turbulence model variables. When initialized with modal model profiles at start time τ0, (i.e., the time when the turbulence model variable profiles are inferred from the modal model results), the RANS simulations provide at all times τ>τ0 profiles that show good agreement with ILES simulations. The procedure for determining the time at which the RANS model should be started is a representative use, other parameters can be used depending on the application. In this paper, for the purpose of demonstration of the full strategy, τ0 is taken as the time at which the mixing layer growth rate parameter α has reached its asymptotic value in the corresponding ILES simulation.

A multiphase buoyancy-drag model for the study of Rayleigh-Taylor and Richtmyer-Meshkov instabilities in dusty gases

AbstractA new multiphase buoyancy-drag model is developed for the study of Rayleigh-Taylor and Richtmyer-Meshkov instabilities in dusty gases, extending on a counterpart single-phase model developed in the past by Srebro et al. (2003). This model is applied to single- and multi-mode perturbations in dusty gases and both Rayleigh-Taylor and Richtmyer-Meshkov instabilities are investigated. The amplitude for Rayleigh-Taylor growth is observed to be contained within a band, which lies within limits identified by a multiphase Atwood number that is a function of the fluid densities, particle size, and a Stokes number. The amplitude growth is subdued with (1) an increase in particle size for a fixed particle number density and with (2) an increase in particle number density for a fixed particle size. The power law index for Richtmyer-Meshkov growth under multi-mode conditions also shows dependence to the multiphase Atwood number, with the index for the bubble front linearly decreasing and then remaining constant, and increasing non-linearly for the spike front. Four new classes of problems are identified and are investigated for Rayleigh-Taylor growth under multi-mode conditions for a hybrid version of the model: (1) bubbles in a pure gas rising into a region of particles; (2) spikes in a pure gas falling into a region of particles; (3) bubbles in a region of particles rising into a pure gas; and (4) spikes in a region of particles falling into a pure gas. Whereas the bubbles accelerate for class (1) and the spikes for class (4), for classes (2) and (3), the spikes and bubbles, respectively, oscillate in a gravity wave-like phenomenon due to the buoyancy term changing sign alternatively. The spike or bubble front, as the case may be, penetrates different amounts into the dusty or pure gas for every subsequent penetration, due to drag effects. Finally, some extensions to the presently developed multiphase buoyancy-drag model are proposed for future research.

On the long time simulation of the Rayleigh–Taylor instability

AbstractWe investigate the classical Rayleigh–Taylor instability with a phase‐field method. Despite of the long history of numerical simulations for the Rayleigh–Taylor instability, almost all results were relatively short time experiments. This is partly because of the way of treating the pressure boundary conditions. We implement a time‐dependent pressure boundary condition through a time‐dependent density field at the boundary. Owing to the pressure boundary treatment, we can perform long time evolutions resulting in an equilibrium state. In addition to the bubble and spike fronts, we have found that the width of sides is another important landmark on the interface of the Rayleigh–Taylor instability. Copyright © 2010 John Wiley & Sons, Ltd.

Transition to turbulence and effect of initial conditions on three-dimensional compressible mixing in planar blast-wave-driven systems

Perturbations on an interface driven by a strong blast wave grow in time due to a combination of Rayleigh–Taylor, Richtmyer–Meshkov, and decompression effects. In this paper, results from three-dimensional (3D) numerical simulations of such a system under drive conditions to be attainable on the National Ignition Facility [E. M. Campbell, Laser Part. Beams 9, 209 (1991)] are presented. Using the multiphysics, adaptive mesh refinement, higher order Godunov Eulerian hydrocode, Raptor [L. H. Howell and J. A. Greenough, J. Comput. Phys. 184, 53 (2003)], the late nonlinear instability evolution, including transition to turbulence, is considered for various multimode perturbation spectra. The 3D post-transition state differs from the 2D result, but the process of transition proceeds similarly in both 2D and 3D. The turbulent mixing transition results in a reduction in the growth rate of the mixing layer relative to its pretransition value and, in the case of the bubble front, relative to the 2D result. The post-transition spike front velocity is approximately the same in 2D and 3D. Implications for hydrodynamic mixing in core-collapse supernovae are discussed.

Scaling laws of nonlinear Rayleigh-Taylor and Richtmyer-Meshkov instabilities in two and three dimensions

The late-time nonlinear evolution of the Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities for random initial perturbations is investigated using a statistical mechanics model based on single-mode and bubble-competition physics at all Atwood numbers ( A ) and full numerical simulations in two and three dimensions. It is shown that the RT mixing zone bubble and spike fronts evolve as h∼ α· A· gt 2 with different values of α for the bubble and spike fronts. The RM mixing zone fronts evolve as h∼ t θ with different values of θ for bubbles and spikes. Similar analysis yields a linear growth with time of the Kelvin–Helmholtz mixing zone. The dependence of the RT and RM scaling parameters on A and the dimensionality will be discussed. The 3D predictions are found to be in good agreement with recent Linear Electric Motor (LEM) experiments.

Two-dimensional Navier–Stokes simulations of gaseous mixtures induced by Richtmyer–Meshkov instability

Two-dimensional numerical calculations of the fluid instability of shock-accelerated interfaces between a heavy fluid and a light one are carried out in order to simulate experiments performed by Poggi et al. [Phys. Fluids 10, 2698 (1998)]. In these experiments, the laser Doppler anemometry technique gives measurements of the fluctuating velocity. Experimental data show that a turbulent mixing zone is generated by the incident shock wave. This turbulent regime is reproduced by two-dimensional calculations. Before the first reshock, several quantities in the mixing zone, such as bubble and spike fronts, turbulent kinetic energy, enstrophy, adopt a quasi self-similar behavior versus time. In particular, we can see in numerical simulations the decay of the turbulent kinetic energy before the first reflected shock wave–mixing-zone interaction and its strong enhancement by reshocks. Furthermore, spectral analysis of the numerical results exhibit a k−3 energy spectrum. Experimental measurements also show that the turbulent boundary layers which develop on the shock-tube walls accelerate the fluid flow in the middle of the tube. Numerical simulations clearly reproduce both this acceleration and the lambda-shock structure observed in experiments.

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.

Study of Nonlinear Evolution of Single-Mode and Two-Bubble Interaction under Richtmyer-Meshkov Instability

Shock-tube experiments were performed in order to verify recently developed theoretical models for the evolution of the shock-wave induced Richtmyer-Meshkov instability [Phys. Rev. Lett. 74, 534 (1995)]. Single-mode bubble and spike evolution and two-bubble interaction in both early and late nonlinear stages were investigated in a $M\ensuremath{\approx}1.3$ $\mathrm{Air}\ensuremath{-}\mathrm{to}\ensuremath{-}{\mathrm{SF}}_{6}$ shock-tube experiment. Experimental results for the single-mode and two-bubble cases, showing distinct bubble and spike evolution, were found to be in very good agreement with the theoretical model prediction as well as numerical simulations, verifying the key elements of the bubble-merger model used for the prediction of the multimode bubble and spike front evolution.

A Method for the Electromyographic Mapping of the Detrusor Smooth Muscle

Various methods for detrusor EMG in the living mammal have been described in the literature. These methods do insufficiently take into account signal components that are caused by movement between the electrodes and the bladder wall. Reliable detrusor EMG has not been achieved yet. This study investigates the feasibility of a new experimental set-up, in which the electrical activity of the detrusor smooth muscle can be examined. In six rabbits, after cervical dislocation, laparotomy and after excision of the heart, electrical signals of the detrusor muscle are measured with 240 electrodes. The electrodes are positioned on the serosal surface of the filled and isovolumetric bladder. During the recordings, no bladder contractions are deliberately evoked by any stimulus. Consistent results in all six animals show a repetitive spike pattern on multiple electrodes with a repetition frequency of 1.2 Hz. Spikes are triphasic and have a mean duration of 0.47 s (STD = 0.15 s, n = 40) and a mean amplitude of 0.29 mV (STD = 0.07 mV, n = 40). On adjacent electrodes a time shift between the spikes is found, suggesting the propagation of electrical activity across the detrusor surface. The maximum conduction velocity of an arbitrary spike front in the direction of propagation is approximately 30 mm/s. In two animals slow waves are found on the edge of the highpass filter setting. Extensive control experiments are executed to validate the set-up and to interpret the data obtained by the animal experiments. The bladder is still able to contract thirty minutes post mortem. The heart, as a distant signal source, generates a signal that is present on all electrodes and shows no detectable time shift from one electrode to any other. Motion imposed on the electrodes relative to the bladder wall does not reproduce the slow waves and spikes found in the animal experiments. The control experiments support that the results of the animal experiments show electrical activity originating from the detrusor muscle itself. With the experimental set-up described in this paper, nearly artefact free detrusor EMG can be recorded. An electromyographic map of a considerable detrusor smooth muscle area can be obtained.

Nonlinear evolution of multimode Rayleigh–Taylor instability in two and three dimensions

The nonlinear evolution of the Rayleigh–Taylor instability from multimode initial perturbations is studied by two complementary approaches. First, a statistical-mechanics bubble-merger model is presented, that enables determination of the late-time scaling properties based on single-mode and two-bubble interaction physics. The results for Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) bubble and spike front penetrations are given, as well as scaling laws for mixing under a time-dependent driving acceleration. The second approach is a modal model, in which nonlinear mode coupling and saturation are included in an equation for effective modes that describe the mixed region. The importance of mode coupling in the generation of large structure that dominates the late stage evolution, and the relative importance of long-wavelength components in the initial perturbation spectra on the late-stage evolution are studied. Finally, multimode RT instability in three dimensions is studied by both full-scale simulations and the modal model. The effect and late-stage memory loss of different aspect ratios in the initial perturbation are demonstrated.

Power Laws and Similarity of Rayleigh-Taylor and Richtmyer-Meshkov Mixing Fronts at All Density Ratios.

The nonlinear evolution of large structure in Rayleigh-Taylor and Richtmyer-Meshkov bubble and spike fronts is studied numerically and explained theoretically on the basis of single-mode and two-bubble interaction physics at Atwood numbers ($A$). Multimode Rayleigh-Taylor bubble (spike) fronts are found as ${h}_{B}={\ensuremath{\alpha}}_{B}\mathrm{Ag}{t}^{2}$ [${h}_{s}={\ensuremath{\alpha}}_{s}(A)g{t}^{2}$] with ${\ensuremath{\alpha}}_{B}=0.05$, while Richtmyer-Meshkov bubble (spike) fronts are found as ${h}_{B}={a}_{B}{t}^{{\ensuremath{\theta}}_{B}}$ (${h}_{s}={a}_{s}{t}^{{\ensuremath{\theta}}_{s}(A)}$) with ${\ensuremath{\theta}}_{B}=0.4$ at all $A'\mathrm{s}$. The dependence of these scaling laws and parameters on $A$ and on initial conditions is explained.