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Vortex model for the nonlinear evolution of the multimode Richtmyer-Meshkov instability at low Atwood numbers
Abstract
The nonlinear growth of the multimode Richtmyer-Meshkov instability in the limit of two fluids of similar densities (Atwood number A⃗0) is treated by the motion of point potential vortices. The dynamics of a periodic bubble array and the competition between bubbles of different sizes is analyzed. A statistical mechanics model for the multimode front mixing evolution, similar to the single-bubble growth and two-bubble interaction based model used by Alon et al. [Phys. Rev. Lett. 72, 2867 (1994)] for A=1, is presented. Using the statistical bubble merger model, a power law of t0.4 for the mixing zone growth is obtained, similar to that of the bubble front growth for the A=1 case and in good agreement with experiments and full numerical simulations.
... However, the initial perturbation in reality, such as that seeded on the surface between diverse materials in an ICF target, is essentially multi-mode with wavenumbers spanning many orders of magnitude (Miles et al. 2004). Past studies have shown that when the perturbation amplitude on a multi-mode RM unstable interface becomes comparable to its wavelength at the nonlinear regime, the competition between these multiple modes has a vital influence on instability development (Haan 1991;Ofer et al. 1996;Rikanati, Alon & Shvarts 1998;Sadot et al. 1998;Vandenboomgaerde et al. 2002;Niederhaus & Jacobs 2003;Balasubramanian, Orlicz & Prestridge 2013;Di Stefano et al. 2015a,b;Pandian, Stellingwerf & Abarzhi 2017;Elbaz & Shvarts 2018;Mohaghar et al. 2019). Our recent work (Luo et al. 2020;Liang et al. 2021a) further revealed that mode coupling is closely related to the initial spectra (including the wavenumber, phase and initial amplitude of constituent modes) and plays an essential role in RM flows from the very beginning if the initial amplitudes of the modes are large. ...
... A = 1 ) between incompressible fluids. Subsequently, Rikanati et al. (1998) considered the bubble-competition effect on the multi-mode RM instability in the limit of two fluids of similar densities (i.e. A → 0) and proposed a statistical merger model based on the point vortices assumption (Jacobs & Sheeley 1996) to quantify the time-varying amplitude growths of the two bubbles. ...
... A → 0) and proposed a statistical merger model based on the point vortices assumption (Jacobs & Sheeley 1996) to quantify the time-varying amplitude growths of the two bubbles. Rikanati et al. (1998) also found that the asymptotic bubble amplitude growth rate at A = 0 is different from that derived by Hecht et al. (1994) at A = 1 due to the added mass in the A = 0 case. Experimentally, Sadot et al. (1998) performed the first shock-tube experiment to investigate two-bubble competition and confirmed that the growth of the long-wavelength bubble is promoted but the short-wavelength bubble is suppressed in comparison with the single-mode RM instability. ...
Shock-tube experiments on various two-bubble and two-spike interfaces are performed to examine the dependence of bubble competition and spike competition on the initial spectra and density ratio of the interface. The differences in the influences of bubble competition and spike competition on the Richtmyer-Meshkov instability are highlighted for the first time. The bubble-competition effect is mainly dependent on the initial spectra of the two-bubble configuration. In contrast, the spike-competition effect is determined by both the initial spectra and density ratio. The extended buoyancy-drag model is adopted to explain the variation of the drag force imposed on the long-wavelength and short-wavelength structures as the initial conditions change. Based on the spectrum analysis, it is found that the constituent modes of two-bubble and two-spike interfaces have different contributions to the long-wavelength and short-wavelength perturbation growths. A generalised, nonlinear, analytical model is then established to quantify the bubble-competition effect and spike-competition effect considering arbitrary initial spectra and density ratio. The bubble-competition effect is believed to be stronger than the spike-competition effect at a high density ratio because it suppresses the high-frequency perturbation growth more evidently.
... The present study provided a novel viscous single-mode bubble evolution model of RTI that is an extension of the single-mode potential models of Jacobs and Sheeley [17] and Rikanati et al. [19]. Furthermore, a discontinuous Galerkin (DG) spectral element method suitable for miscible fluids is formulated, implemented and applied to RTI turbulent mixing simulation. ...
... This observation implies that the flow is rotational and can be described by vortex dynamics. Therefore, a viscous singlemode bubble evolution model is developed and described, which is an extension of the inviscid model developed by Jacobs and Sheeley [17] and Rikanati et al. [19]. ...
... where a and P 0 are constants. Multiplying Eq. (19) with u 0 and integrating in [À1, 1], then obtain 1 2 ...
Based on the single-mode potential model of Jacobs and Rikanati, Firstly, a viscous single-mode bubble evolution model of Rayleigh Taylor instability (RTI) is developed in this study. Viscous effects of RTI's early stage growth for low Atwood number have been explained. In addition, direct numerical simulations of single mode RTI are studied with Navier-Stokes equations and a transport-diffusive equation for miscible fluids, in which these equations are discritized with discontinuous Galerkin (DG) spectral element method. The turbulent mixing of RTI has kinetic energy dissipation, and the dissipation rate is determined by the inertial and viscous effects. Therefore, the numerical techniques must include a dissipation mechanism for kinetic energy. For this reason, the high accurate spectral element method is employed in this study. Agreement between the theoretical model and the numerical results shows that simulations of RTI is feasible using the mathematical miscible fluid model. The results also suggest that a high order numerical method may provide the capability of simulating small scale fluctuations in turbulent flows with RTI.
... Detailed information is found in (Alon et al. 1993. We mainly elaborate on a recently developed vortex-merger (sometimes referred to as vortex pairing) model for the RM and KH instabilities at low Atwood numbers (Ñuids of similar densities) (Rikanati et al. 1998). ...
... cascade process arises, in which larger and larger structures are continually generated (Brown & Roshko 1974 ;Alon et al. 1993Oron, Alon, & Shvarts 1998 ;Rikanati et al. 1998 ;Sharp 1984 ;Glimm & Sharp 1990 ;Bernal 1988). In the RT and RM instabilities, the fundamental cause for the inverse cascade is competition between structures caused by the reduced drag of large structures. ...
... The model is based on modeling the front by an array of 2D particles (bubbles and spikes or eddies), each evolving according to its single-particle asymptotic behavior obtained from LayzerÏs potential Ñow model (Layzer 1955) for the RT and RM instabilities at A \ 1, and from a vortex model for the RM low Atwood numbers (Jacobs & Sheeley 1996 ;Zabusky & Samtaney 1995) and for the KH instability. In the model, particles overtake smaller neighboring ones to form larger particles ("" particle merger ÏÏ ; Alon et al. 1995) at a rate calculated using the potential Ñow model ) or the vortex model (Rikanati et al. 1998) to describe the twoparticle competition. The results from the model are that asymptotically the TMZ front evolution is dominated by a self-similar growth and the scaling law is obtained. ...
The nonlinear growth of the multimode Rayleigh-Taylor (RT), Richtmyer-Meshkov (RM), and Kelvin-Helmholtz (KH) instabilities is treated by a similar statistical mechanics merger model, using bubbles as the elementary particle in the RM and RT instabilities and eddies in the KH instability. Two particle interaction is demonstrated and merger rates are calculated. Using a statistical merger model, the mixing front evolution scaling law is derived. For the RT bubble front height a scaling law of αAgt2, with α 0.05, is derived. For the RM bubble front, a power law of t0.4 is obtained for all Atwood numbers. For the KH case the mixing zone grows linearly with time through a mechanism of eddy merger. Good agreement with simulations and experiments is achieved.
... Shvarts et al. (1995) compared and validated the performance of the merger models with the model of Haan (1989). Rikanati et al. (1998) constructed a vortex model to account both the single-bubble evolution and two-bubble interaction. The model was an extension of the single-mode vortex models of Jacobs and Sheeley (1996) and Zabusky et al. (1995), and did not rely on the potential flow models where the A = 1 limit must be assumed. ...
... Extending the inviscid vortex models of Jacobs and Sheeley (1996) and Rikanati et al. (1998), Zhang (2013) derived a viscous single-mode bubble evolution model of RTI. The viscous effect was shown to reduce the growth in the initial small amplitude phase of development. ...
... The PM model indicates that the mode amplitude develops as a 1/t decay after saturation, which is consistent with the late bubble growth rate of multi-mode RM instability given by both potential flow models (Alon et al. 1994Oron et al. 2001) and vortex models (Rikanati, Alon & Shvarts 1998). The results in figure 9 show that the PM model generally gives good predictions for the amplitude growths of the fundamental modes with both lower and higher mode numbers. ...
We report the first shock-tube experiments on two-dimensional dual-mode air–SF interfaces with different initial spectra subjected to a convergent shock wave. The convergent shock tube is specially designed with a tail opening to highlight the Bell–Plesset (BP) and mode-coupling effects on amplitude development of fundamental mode (FM). The results show that the BP effect promotes the occurrence of mode coupling, and the feedback of high-order modes to the FM also arises earlier in convergent geometry than that in its planar counterpart. Relatively, the amplitude growth of the FM with a higher mode number is inhibited by the feedback, and saturates earlier. The FM with a lower mode number is affected more heavily by the BP effect, and finally dominates the flow. A new model is proposed to well predict the amplitude growths of the FM and high-order modes in convergent geometry. In particular, for FM that reaches its saturation amplitude, the post-saturation relation is introduced in the model to achieve a better prediction.
... Similarly, Rikanati et al. [179] expanded this model and used multi-mode RM instabilities to validate a model of the mixing zone growth using vortex dynamics. Vortex deposition models have had much success modeling RM instabilities [143,180,181,182,168]. ...
Developing a highly accurate numerical framework to study multiphase mixing in high speed flows containing shear layers, shocks, and strong accelerations is critical to many scientific and engineering endeavors. These flows occur across a wide range of scales: from tiny bubbles in human tissue to massive stars collapsing. The lack of understanding of these flows has impeded the success of many engineering applications, our comprehension of astrophysical and planetary formation processes, and the development of biomedical technologies. Controlling mixing between different fluids is central to achieving fusion energy, where mixing is undesirable, and supersonic combustion, where enhanced mixing is important. Iron, found throughout the universe and a necessary component for life, is dispersed through the mixing processes of a dying star. Non-invasive treatments using ultrasound to induce bubble collapse in tissue are being developed to destroy tumors or deliver genes to specific cells. Laboratory experiments of these flows are challenging because the initial conditions and material properties are difficult to control, modern diagnostics are unable to resolve the flow dynamics and conditions, and experiments of these flows are expensive. Numerical simulations can circumvent these difficulties and, therefore, have become a necessary component of any scientific challenge. Advances in the three fields of numerical methods, high performance computing, and multiphase flow modeling are presented: (i) novel numerical methods to capture accurately the multiphase nature of the problem; (ii) modern high performance computing paradigms to resolve the disparate time and length scales of the physical processes; (iii) new insights and models of the dynamics of multiphase flows, including mixing through hydrodynamic instabilities. These studies have direct applications to engineering and biomedical fields such as fuel injection problems, plasma deposition, cancer treatments, and turbomachinery.
... Re can be thought of as the ratio between the largest spatial scale velocity fluctuations allowed in a system (set by the spatial scale of the system) and the length scale at which viscous dissipation will damp out velocity fluctuations (determined by setting Re = 1 and solving for L ν = ν/U). Typically, viscosity will reduce instability growth rates, and the instability growth is more damped for shorter wavelengths [66]. In addition to allowing comparisons between astrophysical systems and the laboratory, similarity 13 scalings are also crucial for understanding how laboratory experiments relate to one another. ...
A shock incident on a contact surface between two materials will deposit vorticity baroclinically given mis-alignments between pressure and density gradients. This vorticity will typically cause any perturbations on the pre-shock interface to grow. The process responsible for that growth will depend on the average tilt between interface and incoming shock, the details of the perturbation on the pre-shock interface, the material properties, and the strength of the incoming shock. If the shock front is parallel to the pre-shock interface the Richtmyer-Meshkov (RM) process will dominate perturbation growth. If the shock front is orthogonal to the pre-shock interface, the shear-driven Kelvin-Helmholtz (KH) instability will typically dominate perturbation growth. Previous theoretical work describing the shock-driven KH instability has assumed that the baroclinic vorticity deposition by the shock along the interface is independent of the post-shock shear flow across that same interface. However, this contradicts Stokes' theorem, which establishes a direct correspondence between the integral vorticity within some region and the shear flow across it through the definition of vorticity (). Here, I generalize a method for estimating integral baroclinic vorticity generation in the KH geometry, first presented in Hurricane, 2008, to both include an arbitrary tilt between the shock and interface, and to furthermore calculate the vorticity distribution along the interface. This vorticity deposition model was previously used in conjunction with a discrete vortex model (DVM), where the vorticity was assumed to be localized to a single point per wavelength. Here, I instead use it in conjunction with a vortex sheet model, which takes into account the full vorticity distribution along the interface. The vortex sheet model can capture both the early and late time behaviors for RM, KH, and oblique geometries so long as secondary processes remain unimportant. Interestingly, for an oblique interface with an appropriate perturbation, vortex sheet models predict that perturbation growth can be dominated by RM at early times while still asymptoting to a KH-like state at late times. I present OMEGA-EP experiments and simulations which confirm this result. I then use the vorticity deposition and vortex sheet models to predict what shock-interface interaction will evolve according to the DVM post-shock. I present simulations and preliminary experimental results which support this prediction.
... However, in real applications, a material interface usually presents multi-mode or random perturbations. In addition to the common flow regimes presented in the single-mode RMI, the multi-mode case includes mode coupling effects such as harmonic generation and bubble merger, and thus possesses much more complex phenomena and mechanisms (Rikanati, Alon & Shvarts 1998;Niederhaus & Jacobs 2003;Sohn 2008;Leinov et al. 2009;Di Stefano et al. 2015;McFarland et al. 2015;Mohaghar et al. 2017). In general, the evolution of the multi-mode RMI can be divided into three stages: the linear stage, the mode competition stage and the bubble merger stage. ...
... The Zufiria-type model was extended by Sohn [22,23] to give a description of the evolution of two or multiple bubbles in both the RT and RM instabilities. A vortex model was constructed by Rikanati et al. [24] to account both the single-bubble evolution and two-bubble interaction, and the development of random perturbations based on the mechanism of bubble merger was described using a statistical model. The two-bubble merger experiments of a single-mode interface were carried out by Sadot et al. [25], and the results indicated that the competition promotes the nonlinear growth of the major bubbles relative to a pure single-mode interface. ...
We report the shock tube experiments on Richtmyer-Meshkov instability to evaluate the bubble merger effect in initial stages. The initial interface is specially designed by alternatively arranging the major and minor inverse-chevron shapes with different amplitudes and wavelengths such that the bubble merger effect is highlighted. The results show that the major and minor shapes develop independently at very early stages, and after a short transient the bubble merger occurs. The bubble merger promotes the width growth of the major shape while it inhibits the width growth of the minor one. However, the bubble merger has a greater effect on the development of the minor shape than the major one. As the initial size difference between two shapes increases, the bubble merger occurs earlier, and the minor shape is affected more heavily. The developments of bubble front difference seem to experience two different linear stages before it enters a nonlinear stage, and collapse well in dimensionless form for different cases. As a result, the initial size difference has a limited effect on the bubble merger in the streamwise direction. The ratio of the major bubble diameter in spanwise direction to the bubble width increases with time, and does not reach a stable phase because the flow is far from the turbulent mixing region.
... In the case of a multimode front, the quantities of interest are not easily reducible to a single measurement. Here, we will examine the data using two methods: the first is to use the evolution of the average bubble height hhi over a section of the interface, a method which has been used in previous studies 41 and which has the advantage of producing a quantity that can be used in analytic analyses of the instability behavior. The second is to operate in Fourier space. ...
We present an experiment using lasers to produce a shock pressure of >10 Mbar, which we then use to drive Richtmyer–Meshkov and Rayleigh–Taylor growth at a 2D multimode perturbed interface. Key features of this platform are that we can precisely reproduce the perturbation from iteration to iteration of the experiment, facilitating analysis, and that the lasers allow us to produce very strong shocks, creating a plasma state in the system. We also implement a Bayesian technique to analyze the multimode spectra. This technique enables us to draw quantitative conclusions about the spectrum, even in the presence of significant noise. For instance, we measure the signal contained in the seeded modes over time, as well as the transition of the initial growth rate of these modes into the overall saturation behavior of the spectrum.
... As can be seen in Equation (2a) or (3a), at the third-order, the linear growth of the fundamental mode is reduced by the nonlinear mode-coupling effect, i.e., third-order negative feedback to the fundamental mode. Weakly nonlinear behaviors in planar RTI have become a field of theoretical, 24-34 experimental [35][36][37] and numerical [38][39][40][41] interest. Strictly speaking, perturbation growth driven only by buoyant force is named as RTI, while the modifications of perturbation behavior by compression and geometrical a) Authors to whom correspondence should be addressed. ...
Harmonic growth in classical Rayleigh-Taylor instability (RTI) on a spherical interface is analytically investigated using the method of the parameter expansion up to the third order. Our results show that the amplitudes of the first four harmonics will recover those in planar RTI as the interface radius tends to infinity compared against the initial perturbation wavelength. The initial radius dramatically influences the harmonic development. The appearance of the second-order feedback to the initial unperturbed interface (i.e., the zeroth harmonic) makes the interface move towards the spherical center. For these four harmonics, the smaller the initial radius is, the faster they grow.
... In the Cartesian geometry, the weakly nonlinear behavior of the RTI has been a field of theoretical, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] experimental, [23][24][25][26] and numerical [27][28][29][30][31][32] interest. In many applications, however, the RTI occurs in a cylindrical or spherical geometry, for which the corresponding investigations are few. ...
The nonlinear saturation amplitude (NSA) of the fundamental mode in the classical Rayleigh—Taylor instability with a cylindrical geometry for an arbitrary Atwood number is analytically investigated by considering the nonlinear corrections up to the third order. The analytic results indicate that the effects of the initial radius of the interface (r 0) and the Atwood number (A) play an important role in the NSA of the fundamental mode. The NSA of the fundamental mode first increases gently and then decreases quickly with increasing A. For a given A, the smaller the r 0/λ (λ is the perturbation wavelength), the larger the NSA of the fundamental mode. When r 0/λ is large enough (r 0 ≫ λ), the NSA of the fundamental mode is reduced to the prediction in the previous literatures within the framework of the third-order perturbation theory.
... A viscous single-mode bubble evolution model of Rayleigh-Taylor Instabilities (RTIs) is developed and described that is an extension of the inviscid model developed by Jacobs and Sheeley (1996) and Rikanati et al. (1998). The basic assumptions for this situation at initial small amplitude phase RTI are divided into two stages, initial vortex formation owing to perturbation, which can be described by potential flow solution, and diffusion process of vortex on the RTI interface. ...
This paper has developed a viscous single-mode bubble evolution model of Rayleigh-Taylor instabilities (RTIs), which is an extension of the single-mode potential models of Jacobs and Rikanati. The viscous vortex model explained the viscous effects of its early stage of RTI development for low Atwood number flow. Furthermore, direct numerical simulations of RTI are studied with Navier-Stokes equations and a transport-diffusive equation. Agreement between the theoretical model and the numerical results shows that simulations of these instabilities is feasible using the mathematical miscible fluid model simulating RTI.
... Zhang & Sohn (1996 and Velikovich & Dimonte (1996) found a similar result using a nonlinear perturbation series solution to the incompressible flow equations. Jacobs & Sheeley (1996) and Rikanati, Alon & Shvarts (1998) modelled the perturbed fluid interface as a series of point vortices. This appears to give a 1/t asymptotic growth rate if the point vortices are spaced in a uniform array (Likhachev & Jacobs 2005). ...
We derive a growth-rate model for the Richtmyer–Meshkov mixing layer, given arbitrary but known initial conditions. The initial growth rate is determined by the net mass flux through the centre plane of the perturbed interface immediately after shock passage. The net mass flux is determined by the correlation between the post-shock density and streamwise velocity. The post-shock density field is computed from the known initial perturbations and the shock jump conditions. The streamwise velocity is computed via Biot–Savart integration of the vorticity field. The vorticity deposited by the shock is obtained from the baroclinic torque with an impulsive acceleration. Using the initial growth rate and characteristic perturbation wavelength as scaling factors, the model collapses the growth-rate curves and, in most cases, predicts the peak growth rate over a range of Mach numbers (\1. 1\leq {M}_{i} \leq 1. 9\), Atwood numbers (\- 0. 73\leq A\leq - 0. 35\ and \0. 22\leq A\leq 0. 73\), adiabatic indices (\1. 40/ 1. 67\leq {\gamma }_{1} / {\gamma }_{2} \leq 1. 67/ 1. 09\) and narrow-band perturbation spectra. The mixing layer at late times exhibits a power-law growth with an average exponent of \\theta = 0. 24\.
... There is general consensus that the growth of the mixing layer is described by W ∝ t θ . Analytical results suggest either linear growth in time [2], a growth rate of θ = 2/3 − µ [14,15] where µ is a correction dependent on the level of turbulent dissipation, θ = 0.4 [16] or even logarithmic [16][17][18]. ...
This paper presents a numerical study of a reshocked turbulent mixing layer using high-order accurate Implicit Large-Eddy-Simulations (ILES). Existing theoretical approaches are discussed, and the theory of Youngs (detailed in Ref. 1) is extended to predict the behaviour of a reshocked mixing layer formed initially from a shock interacting with a broadband instability. The theory of Mikaelian2 is also extended to account for molecular mixing in the single-shocked layer prior to reshock. Simulations are conducted for broadband and narrowband initial perturbations and results for the growth rate of the reshocked layer and the decay rate of turbulent kinetic energy show excellent agreement with the extended theoretical approach. Reshock causes a marginal decrease in mixing parameters for the narrowband layer, but a significant increase for the broadband initial perturbation. The layer properties are observed to be very similar post-reshock, however, the growth rate exponent for the mixing layer width is higher in the broadband case, indicating that the reshocked layer still has a dependence (although weakened) on the initial conditions. These results have important implications for Unsteady Reynolds Averaged Navier Stokes modelling of such instabilities.
... a * is equal to a 0 ϩ , the postshock amplitude for the light to heavy configuration~see Richtmyer, 1960!, and to a * ϭ a 0 ϩ ϩ a 0 Ϫ !02 for the heavy to light configuration~see Meyer & Blewett, 1972!, where a 0 Ϫ is the preshock amplitude. The linear stage is followed by a nonlinear stage during which the growth velocity reaches an asymptotic C~A!l0t behavior where C~A! is a constant that depends on the Atwood number and the dimensionality~for more details, see Hecht et al., 1994;Rikanati et al., 1998!. Those models~nonlinear classical models! ...
The present article describes an experimental study that is a part of an integrated theoretical (Rikanati et al. 2003) and experiential investigation of the Richtmyer Meshkov (RM) hydrodynamic instability that develops on a perturbed contact surface by a shock wave. The Mach number and the high initial-amplitude effects on the evolution of the single-mode shock-wave-induced instability were studied. To distinguish between the above-mentioned effects, two sets of shock-tube experiments were conducted: high initial amplitudes with a low-Mach incident shock and small amplitude initial conditions with a moderate-Mach incident shock. In the high-amplitude experiments a reduction of the initial velocity with respect to the linear prediction was measured. The results were compared to those predicted by a vorticity deposition model and to previous experiments with moderate and high Mach numbers done by others and good agreement was found. The result suggested that the high initial-amplitude effect is the dominant one rather than the high Mach number effect as suggested by others. In the small amplitude moderate Mach numbers experiments, a reduction from the impulsive theory was noted at late stages. It is concluded that while high Mach number effect can dramatically change the behavior of the flow at all stages, the high initial-amplitude effect is of minor importance at the late stages. That result is supported by a two-dimensional numerical simulation.
... By analogy, spike penetration is well fitted by θ s /θ b ≈ 1 + A , as θ s = θ b for A = 0 (symmetrical spikes and bubbles) and increases to θ s = 1 for A = 1 (linear spike growth rate). Similar results have been obtained by Rikanati et al. (1998) in the case of vortex, rather than bubble, merger. Sadot et al. (1998) obtained good agreement between the potential flow bubble competition model, low Mach-number shock-tube experiments and NS for a two-dimensional bimodal initial configuration. ...
... This predominantly vortex-driven behavior has been more thoroughly investigated using simulations and experiments in the case of single interfaces rather than fluid layers. This is reflected in the correspondingly richer variety of models developed for single interfaces [8][9][10][11][12][13][14]. ...
A preliminary investigation of the impact of initial modal composition on the mixing of a shocked, membraneless fluid layer is performed. The growth patterns that emerge upon the impulsive acceleration of three different initial conditions (varicose, sinuous and large-wavelength sinuous) by a Mach 1.2 shock wave are investigated using planar laser induced fluorescence (PLIF) in an air–SF6–air fluid layer. Time-series images of the flow evolution in each of these cases indicate the presence of concentrated regions of vorticity, with the intensity and stability of the resulting vortex configurations dictating the post-shock evolution. In the sinuous case, self advection of the nonuniformly spaced vortices generates a pattern of two streamwise separated regions of material concentration after first shock. However, upon reshock, substantial mixing occurs and results in a structure where the separated regions merge to create a density distribution with a single, broad plateau. This profile contrasts with the varicose case, in which the streamwise density profile is characterized by a narrow peak.
... By analogy, spike penetration is well fitted by θ s /θ b ≈ 1 + A , as θ s = θ b for A = 0 (symmetrical spikes and bubbles) and increases to θ s = 1 for A = 1 (linear spike growth rate). Similar results have been obtained by Rikanati et al. (1998) in the case of vortex, rather than bubble, merger. Sadot et al. (1998) obtained good agreement between the potential flow bubble competition model, low Mach-number shock-tube experiments and NS for a two-dimensional bimodal initial configuration. ...
Key Words shock-induced turbulence, shock-wave refraction s Abstract The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers. This review focuses on the basic physical processes underlying the onset and development of the Richtmyer-Meshkov instability in simple geometries. It examines the principal theoretical results along with their experimental and numerical validation. It also discusses the different experimental approaches and techniques and how they can be used to resolve outstanding issues in this field.
Shock-tube experiments and theoretical studies have been performed to highlight mode-coupling in an air–SF –air fluid layer. Initially, the two interfaces of the layer are designed as single mode with different basic modes. It is found that as the two perturbed interfaces become closer, interface coupling induces a different mode from the basic mode on each interface. Then mode coupling further generates new modes. Based on the linear model (Jacobs et al. , J. Fluid Mech. , vol. 295, 1995, pp. 23–42), a modified model is established by considering the different accelerations of two interfaces and the waves’ effects in the layer, and provides good predictions to the linear growth rates of the basic modes and the modes generated by interface coupling. It is observed that interface coupling behaves differently to the nonlinear growth of the basic modes, which can be characterized generally by the existing or modified nonlinear model. Moreover, a new modal model is established to quantify the mode-coupling effect in the layer. The mode-coupling effect on the amplitude growth is negligible for the basic modes, but is significant for the interface-coupling modes when the initial wavenumber of one interface is twice the wavenumber of the other interface. Finally, amplitude freeze-out of the second single-mode interface is achieved theoretically and experimentally through interface coupling. These findings may be helpful for designing the target in inertial confinement fusion to suppress the hydrodynamic instabilities.
Shock-tube experiments are performed on the convergent Richtmyer-Meshkov (RM) instability of a multimode interface. The temporal growth of each Fourier mode perturbation is measured. The hydrodynamic instabilities, including the RM instability and the additional Rayleigh-Taylor (RT) effect, imposed by the convergent shock wave on the dual-mode interface, are investigated. The mode-coupling effect on the convergent RM instability coupled with the RT effect is quantified. It is evident that the amplitude growths of all first-order modes and second-order harmonics and their couplings depend on the variance of the interface radius, and are influenced by the mode-coupling from the very beginning. It is confirmed that the mode-coupling mechanism is closely related to the initial spectrum, including azimuthal wavenumbers, relative phases and initial amplitudes of the constituent modes. Different from the conclusion in previous studies on the convergent single-mode RM instability that the additional RT effect always suppresses the perturbation growth, the mode-coupling might result in the additional RT effect promoting the instability of the constituent Fourier mode. By considering the geometry convergence, the mode-coupling effect and other physical mechanisms, second-order nonlinear solutions are established to predict the RM instability and the additional RT effect in the cylindrical geometry, reasonably quantifying the amplitude growths of each mode, harmonic and coupling. The nonlinear solutions are further validated by simulations considering various initial spectra. Last, the temporal evolutions of the mixed mass and normalized mixed mass of a shocked multimode interface are calculated numerically to quantify the mixing of two fluids in the cylindrical geometry.
Structures evoking vortex rings can be discerned in shock-accelerated flows ranging from astrophysics to inertial confinement fusion. By constructing an analogy between vortex rings produced in conventional propulsion systems and rings generated by a shock impinging upon a high-aspect-ratio protrusion along a material interface, we extend classical, constant-density vortex-ring theory to compressible multifluid flows. We further demonstrate saturation of such vortex rings as the protrusion aspect ratio is increased, thus explaining morphological differences observed in practice.
The purpose of this article is to analyze the large-scale properties of a Richtmyer–Meshkov turbulent mixing zone with large density contrasts and small shock Mach numbers. The main outcome of the study is the expression of a large-scale invariant of the flow. Its existence is contingent on initial conditions but not on the value of the Atwood number which measures the density contrast. As opposed to the small Atwood case, this invariant is not related to the velocity spectrum. Instead, it is given by the value of the spectrum of the solenoidal component of the momentum at small wave numbers. This result stems from the conservation of angular momentum in variable-density flows. Despite this fundamental difference, this invariant still allows to relate the self-similar growth rate of the mixing zone to the large-scale post-shock initial conditions of the flow. Besides, when the shock is weak, this relation can be extended to the pre-shock deformation of the interface. In particular, when the initial interfacial perturbation is limited to small wavelengths (annular spectrum), the growth rate exponent of the mixing zone is shown to saturate to a minimum value close to 1/4, independently from the Atwood number. The different assumptions and predictions of this work are verified by performing implicit large eddy simulations of a Richtmyer–Meshkov turbulent mixing zone.
This chapter focuses on the evolution of a multi-mode interface induced by a shock wave. To understand the deviation of a multi-mode RM unstable interface from a single-mode one, the development of 2D single-mode interfaces, 2D quasi-single-mode interfaces, 2D multi-mode interfaces, and 3DMS interfaces induced by a shock wave are studied experimentally and theoretically. The soap film technique is first developed to form a 2D gaseous interface free of short-wavelength perturbation, diffusion layer, and three-dimensionality. The dependence of the amplitude growth of a multi-mode interface on its initial spectrum is quantified. A universal nonlinear model is finally established to cover the RM instability of a multi-mode interface from the quasi-linear regime to the late nonlinear regime considering various initial conditions, including different amplitude-wavelength ratios, Atwood numbers and Mach numbers. Finally, the universal nonlinear model is extended to describe the RM instability of a 3DMS interface considering the coupling between 3D modes.
Richtmyer-Meshkov (RM) instability occurs when an initially perturbed interface separating different materials is driven by an impulsive acceleration (e.g., shock wave). Later, the perturbation on the interface grows, and, eventually, the mixing layer might transit to turbulence. RM instability plays an essential role in many applications, such as inertial confinement fusion (ICF), supernova explosion, and scramjet engine. It has been widely used to study gas dynamics, vortex dynamics, and turbulence. Therefore, RM instability has attracted lots of attention worldwide, such as the Lawrence Livermore National Laboratory and the Los Alamos National Laboratory in the U.S.A., the Commissariat a l’Energie Atomique Laboratory in France, and the China Academy of Engineering Physics. The RM instability of a single-mode perturbation interface attracted the most attention because of its simplest mathematic pattern. However, the initial interface perturbation in applications is always random, i.e., multi-mode. Nevertheless, the dependence of the multi-mode RM instability on the initial perturbation spectrum has not been fully understood yet. Moreover, an ICF target consists of multiple layers of different materials in spherical geometry. However, there was limited research on the feedthrough between the interfaces and the reverberating waves involved in a shocked multi-layer system. In addition, an ICF target consists of different phases, including solid, liquid, gas, plasma, etc. Still, the influence of the phase-transition on a shocked multi-phase configuration has been scarcely explored. The present study experimentally and theoretically investigates the RM instability separately coupled with the multi-mode perturbation, the multi-layer system, and the multi-phase configuration.
First, we developed a soap film technique to generate shape-controllable two-dimensional (2D) single-mode interfaces, quasi-single-mode interfaces, multi-mode interfaces, and three-dimensional (3D) interfaces with a minimum-surface feature. Experiments were performed in a horizontal shock-tube with a large-aspect-ratio cross-section. The interfacial morphologies of a shocked single-mode interface captured from the experiments show that the instability evolution involves the slightest experimental uncertainty among all existing results. The performances of the linear model and nonlinear models were thoroughly examined through the temporal variations of the interface amplitude growths. Next, four types of quasi-single-mode interfaces dominated by the fundamental mode were generated with the improved soap film technique. It was found that the mode-competition can be ignored at the early nonlinear regime of the RM instability of a quasi-single-mode interface. A simple nonlinear theory was established to describe the mixing width growths by summing the magnitude growths of a finite number of constituted modes ignoring the mode-competition. More constituted modes needed in the simple nonlinear theory to match the experiments indicate the more pronounced deviation of the quasi-single-mode perturbation from the single-mode one. Later, the RM instability of various multi-mode interfaces consisting of multiple dominated modes was explored. A universal nonlinear model for the magnitude growth of each constituted mode and the mixing width growth of the multi-mode interface was established considering the mode-competition and nonlinearity. The universal nonlinear model was validated with our elaborate shock-tube experiments and the data extracted from the literature considering various initial conditions. Last, the RM instability of a 3D interface with a minimum-surface feature was figured out. The universal nonlinear model was extended to describe the 3D RM instability considering the initial 3D interface spectrum and the coupling between 3D modes.
Second, the hydrodynamic instabilities of a shocked 2D finite-thickness gas layer were experimentally and theoretically examined. Using the extended soap film technique, we generated shape-controllable and thickness-controllable discontinuous heavy gas layers and light gas layers such that the instability of each interface of a layer was concerned. The interface-coupling and the reverberating waves inside a gas layer were found to have a significant influence on the hydrodynamic instabilities. It was the first to quantify the additional Rayleigh-Taylor (RT) instability imposed by rarefaction waves and the additional RT stabilization induced by compression waves on a shocked heavy gas layer. Moreover, the additional RM instability caused by the reverberating shocks inside a light gas layer was well described. In addition, the shock-driven dual layer evolution was theoretically and experimentally investigated for the first time. The coupling between the two layers plays a significant role in influencing the RM instability of the three interfaces of a dual layer. The reverberating waves inside a multi-layer system have non-negligible and diverse influences on the three interfaces. It is evident that the shock-driven hydrodynamic instabilities of a semi-infinite interface, a single layer, and a dual layer are categorically different.
Third, the interaction of a shock wave and a 3D water droplet embedded with a vapour bubble was experimentally studied for the first time. The vapour bubble inside a droplet was generated by depressurizing the ambient gas pressure to the saturation pressure of water in the driven section of a shock-tube. It was proved that the relative size and position of the vapour bubble to the droplet influence the hydrodynamic instabilities. The phase-transition accelerates the breakup of the droplet. The transverse jet inside the droplet induced by RM instability and other mechanisms was clearly observed. A modified Rayleigh-Plesset equation is derived to predict the bubble collapse within a droplet, validated with our shock-tube experiments.
Shock-tube experiments on eight kinds of two-dimensional multi-mode air–SF interface with controllable initial conditions are performed to examine the dependence of perturbation growth on initial spectra. We deduce and demonstrate experimentally that the amplitude development of each mode is influenced by the mode-competition effect from quasi-linear stages. It is confirmed that the mode-competition effect is closely related to initial spectra, including the wavenumber, the phase and the initial amplitude of constituent modes. By considering both the mode-competition effect and the high-order harmonics effect, a nonlinear model is established based on initial spectra to predict the amplitude growth of each individual mode. The nonlinear model is validated by the present experiments and data in the literature by considering diverse initial spectra, shock intensities and density ratios. Moreover, the nonlinear model is successfully extended based on the superposition principle to predict the growths of the total perturbation width and the bubble/spike width from quasi-linear to nonlinear stages.
Inertia-dominated hydrodynamic instabilities at material interfaces are ubiquitous phenomena observed in nature and man-made applications, spanning core collapse supernovae, inertial confinement fusion, supersonic combustion, and cavitation bubble collapse. When subjected to accelerations, perturbations along an interface may grow due to the Rayleigh-Taylor (RT) or Richtmyer-Meshkov (RM) instability, while in the presence of shear, they may grow due to the Kelvin-Helmholtz (KH) instability. The main focus of this thesis is the RM instability. The RM instability occurs when a perturbed interface separating two fluids of different densities is impulsively accelerated, e.g., by the passage of a shock wave. During the interaction of the shock with the interface, baroclinic vorticity is generated along the interface due to the misalignment between the density and pressure gradients, thus leading to perturbation growth. The subsequent interface evolution can be described using vorticity dynamics. Although the early stage of vorticity deposition along the interface is relatively well understood, the late-time vorticity dynamics and their effects on the interface evolution are less well known. Our objective is to understand the role of vorticity dynamics in the late-time evolution of RM-type problems. To examine the vorticity dynamics of the RM instability, we implement a vortex-sheet model allowing us to isolate the different contributions of vorticity production in the evolution of the interface. We first use the vortex-sheet model to understand the relative importance between RM and KH in the evolution of perturbations subjected to an oblique shock under high-energy-density (HED) conditions. At early times, the perturbation growth is dominated by the impulsive acceleration of the shock (RM), as evidenced by our proposed scaling accounting for the normal and tangential components of the shock. At later times, the perturbation growth is modulated by the positive and negative vorticity generated by the shear and the decompression due to the arrival of the rarefaction produced by laser turn off. As the tilt angle is increased, the onset of the shear-dominated dynamics occurs earlier and becomes more pronounced. We further demonstrate the ability of the vortex-sheet model to reproduce roll-up dynamics for non-zero Atwood numbers by comparing to past laser-driven HED experiments. We then explain the mechanisms of vorticity generation in the late-time evolution of the single-mode RM instability. In particular, we explore the generation of secondary opposite-sign vorticity occurring inside the roll-ups as the interface spirals inward. We show that, in the case of a zero Atwood number, opposite-sign vorticity never develops. In this case, the vorticity distribution along the interface is only governed by the rate of change of the sheet surface. Near the vortex core, the rate of change of the sheet surface alternates between positive and negative values, indicating that the interface near the vortex core undergoes a series of contractions and expansions, thus giving rise to oscillations in the corresponding sheet strength. In the case of small Atwood numbers, performing a vorticity budget suggests that opposite-sign vorticity is generated by the nonlinear vorticity advection along the interface. To quantify the amount of opposite-sign vorticity generated along the interface, we consider positive and negative circulations, and their dependence of the strength of the incident shock and the Atwood number. We show that opposite-sign circulation behaves according to a power law in time and that the interface evolution scales in time with respect to the shock Mach number.
The finite-thickness effect of two superimposed fluids on harmonics in the Richtmyer-Meshkov instability (RMI) for arbitrary Atwood numbers is investigated by using weakly nonlinear analysis up to the third order. When the thickness of the two fluids tends to be infinity, our results can reproduce the classical results where RMI happens at the interface separating two semi-infinity-thickness fluids of different densities. It is found that the thickness has a large influence on the amplitudes of the first three harmonics compared with those in classical RMI. On the one hand, the thickness effect encourages or reduces the amplitudes of the first three harmonics, and on the other hand, it changes the phases of the second and the third harmonics.
This paper investigates the finite-thickness effect of two superimposed fluids on bubbles and spikes in Richtmyer-Meshkov instability (RMI) for arbitrary Atwood numbers by using the method of the small parameter expansion up to the second order. When the thickness of the two fluids tends to be infinity, our results can reproduce the classical results where RMI happens at the interface separating two semi-infinity-thickness fluids of different densities. It is found that the thickness has a large influence on the amplitude evolution of bubbles and spikes compared with those in classical RMI. Based on the thickness relationship of the two fluids, the thickness effect on bubbles and spikes for four cases is discussed. The thickness encourages (or reduces) the growth of bubbles or spikes, depending on not only Atwood number, but also the relationship of the thickness ratio of the heavy and light fluids, which is explicitly determined in this paper. © 2018 Hefei Institutes of Physical Science, Chinese Academy of Sciences and IOP Publishing.
We report the first observations of Kelvin-Helmholtz vortices evolving from well-characterized, dual-mode initial conditions in a steady, supersonic flow. The results provide the first measurements of the instability'svortex merger rate and supplement data on the inhibition of the instability's growth rate in a compressible flow. These experimental data were obtained by sustaining a shockwave over a foam-plastic interface with a precision-machined seed perturbation. This technique produced a strong shear layer between two plasmas at high-energy-density conditions. The system was diagnosed using x-ray radiography and was well-reproduced using hydrodynamic simulations. Experimental measurements imply that we observed the anticipated vortex merger rate and growth inhibition for supersonic shear flow.
In this experimental study, the ablative Richtmyer–Meshkov (RM) and the Rayleigh–Taylor (RT) instabilities were generated by the laser pulse of Gaussian-like power profile. The initial multi-modal perturbation, the inhomogeneous momentum transfer and different Atwood numbers generate different shapes of spikes and bubbles in the central region (CR) and the near-central region (NCR) of the spot. A one-dimensional Gaussian-like power profile causes the formation of the wavy-like rows of aperiodic spikes. The periodic spike segments inside the rows appear due to locally coherent flow. In the NCR, the mushroom-shape spikes tend to the organization on the isotropic square and the anisotropic rhombic lattices. The increase of the lattice periods two, three, or four times indicates formation of superstructures. The growth of sharp asymmetric RM/RT spikes in the CR is fast, uncorrelated and linear, while the growth of the symmetric mushroom-shape ones in the NCR is slow, correlated, and nonlinear.
This paper has developed a viscous single-mode bubble evolution model of Rayleigh Taylor Instabilities (RTIs), which is an extension of the single-mode potential models of Jacobs and Rikanati. The viscous vortex model explained the viscous effects of its early stage of RTI development for low Atwood number flow. Furthermore, direct numerical simulations of RTI are studied with Navier-Stokes equations and a transport-diffusive equation. Agreement between the theoretical model and the numerical results shows that simulations of these instabilities are feasible using the mathematical miscible fluid model simulating RTI.
This paper presents an experimental study of Richtmyer-Meshkov (R-M) instability at an interface between water and air using a rectangular shock tube. When the Atwood number approaches to 1, the R-M asymptotic bubble and spike evolutions are found to obey a power law: hb - t0.55±0.01, hs - t. The power law doesn't change when the Mach number increases from 1.36 to 1.58, but us increases very much. This paper also observes and studies the phenomenon of bubble competition, i.e. lager bubbles overtake their smaller neighbors.
Nonlinear saturation amplitudes (NSAs) of the first two harmonics in classical Rayleigh-Taylor instability (RTI) in cylindrical geometry for arbitrary Atwood numbers have been analytically investigated considering nonlinear corrections up to the fourth-order. The NSA of the fundamental mode is defined as the linear (purely exponential) growth amplitude of the fundamental mode at the saturation time when the growth of the fundamental mode (first harmonic) is reduced by 10% in comparison to its corresponding linear growth, and the NSA of the second harmonic can be obtained in the same way. The analytic results indicate that the effects of the initial radius of the interface (r 0) and the Atwood number (A) play an important role in the NSAs of the first two harmonics in cylindrical RTI. On the one hand, the NSA of the fundamental mode first increases slightly and then decreases quickly with increasing A. For given A, the smaller the r 0 / λ (with λ perturbation wavelength) is, the larger the NSA of the fundamental mode is. When r 0 / λ is large enough ( r 0 ≫ λ ), the NSA of the fundamental mode is reduced to the prediction of previous literatures within the framework of third-order perturbation theory [J. W. Jacobs and I. Catton, J. Fluid Mech. 187, 329 (1988); S. W. Haan, Phys. Fluids B 3, 2349 (1991)]. On the other hand, the NSA of the second harmonic first decreases quickly with increasing A, reaching a minimum, and then increases slowly. Furthermore, the r 0 can reduce the NSA of the second harmonic for arbitrary A at r 0 ≲ 2 λ while increase it for A ≲ 0.6 at r 0 ≳ 2 λ . Thus, it should be included in applications where the NSA has a role, such as inertial confinement fusion ignition target design.
This chapter presents a systematic treatment of shock-wave-induced hydrodynamic mixing instabilities, based on models, simulations, and experiments. The description, based on penetration of the light fluid in the heavy one (bubbles) and the heavy into the light (spikes), provides a comprehensive and understanding of the evolution of the instability for both single-mode and multimode cases. The evolution of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities in the linear and nonlinear stages is also presented. When two fluids of different densities are subjected to an accelerating field, under certain circumstances an instability is created at the contact surface between them. If the acceleration is slowly varied and directed from the heavy fluid to the light one, the Rayleigh-Taylor instability occurs. The theoretical and experimental studies performed regarding the late nonlinear stages of the instability evolution is described. The chapter is devoted to single-mode evolution, but to complete the whole picture, a generalization of the single mode to the multimode case is presented.
▪ Abstract The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers. This review focuses on the basic physical processes underlying the onset and development of the Richtmyer-Meshkov instability in simple geometries. It examines the principal theoretical results along with their experimental and numerical validation. It also discusses the different experimental approaches and techniques and how they can be used to resolve outstanding issues in this field.
A nonlinear theory is developed to describe the cylindrical Richtmyer-Meshkov instability (RMI) of an impulsively accelerated interface between incompressible fluids, which is based on both a technique of Padé approximation and an approach of perturbation expansion directly on the perturbed interface rather than the unperturbed interface. When cylindrical effect vanishes (i.e., in the large initial radius of the interface), our explicit results reproduce those [Q. Zhang and S.-I. Sohn, Phys. Fluids 9, 1106 (1996)] related to the planar RMI. The present prediction in agreement with previous simulations [C. Matsuoka and K. Nishihara, Phys. Rev. E 73, 055304(R) (2006)] leads us to better understand the cylindrical RMI at arbitrary Atwood numbers for the whole nonlinear regime. The asymptotic growth rate of the cylindrical interface finger (bubble or spike) tends to its initial value or zero, depending upon mode number of the initial cylindrical interface and Atwood number. The explicit conditions, directly affecting asymptotic behavior of the cylindrical interface finger, are investigated in this paper. This theory allows a straightforward extension to other nonlinear problems related closely to an instable interface.
The evolution of the three-dimensional planar Richtmyer-Meshkov (RM) instability
during a two shock wave interaction (i.e., reshock) is investigated by means of comparing
numerical simulations and analytical modelling with experimental results of
low Mach numbers (M < 1.5) and fairly high Atwood numbers (A ∼ 0.7). The study
discusses and analyses the differences in the evolution of the mixing zone for two different
types of initial perturbations, namely, multi-mode random initial perturbation
with a narrow or wide bubble size distribution. More specifically, the study is focused
on the agreement between numerical simulations and experiments performed with
an unknown random initial perturbation. Using a large set of experimental results
with different reshock arrival times and Mach numbers, the numerical simulations
results are compared to the experimental results for a variety of different scenarios.
This methodology allows a constrained comparison, while requiring good agreement
for all cases. A comprehensive parametric study is conducted, examining the evolution
of the mixing zone (MZ) for different initial amplitudes and wavelengths. It is
found that in order to achieve a good agreement, the numerical simulation must be
performed using a wide enough initial spectrum, which enables a dominant, efficient
bubblemerging process to take place within theMZ. The numerical simulation results
are compared to a model, based on classic single bubble RM evolution formulation,
combined with high amplitude effects consideration and phase reversal treatment in
case of heavy to light reshock passage. The model is also extended for the case of
multi-mode fronts, accounting for a bubblemerging process, determining that theMZ
evolution after the reshock can be classified with high confidence as governed by an
inverse cascade bubble merger, approaching self-similarity.
An experiment, meant to investigate the evolution of Richtmyer–Meshkov (RM) instability in the bubble merger regime and at low Atwood number (A∼0.3), is proposed and theoretically analyzed. This experiment is intended to provide a direct measurement of the two-dimensional bubble-front shape and spectrum evolution in time, along with the power-law coefficient for bubble-front growth (θb). It is unique in its use of a well-characterized two-dimensional initial perturbation, allowing controlled initiation and growth of the instability. The proposed design assures a significant time scale of steady RM conditions, taking advantage of the long drive (∼30 ns) available on the OMEGA-EP laser facility, along with neither a Rayleigh–Taylor (RT) component nor shock-proximity effects, due to the use of a light to heavy configuration. Multimode RM growth for the proposed configuration has been analyzed using two-dimensional, direct numerical simulations, showing significant mode coupling and convergence to power-law growth of the bubble front. The effects of two-dimensional rarefactions were also investigated, and it was found that they introduce no major uncertainties or hazards to the physics. An experiment of this kind has not yet been performed, and therefore would serve to validate numerical results and analytical models presented in literature.
We present simple inviscid and viscous vortex models for the Richtmyer–Meshkov (RM) instability, in the limit of fluids of the same densities. The evolution of the interface is driven by a row of alternating point vortices. A regularization method for the singular vortices is applied to the inviscid model for stable calculations. The regularized vortex model provides fine resolutions for the interface, improving previous results on the evolution of the RM instability. The point vortex model, for the inviscid fluid, is validated by comparing with the results of the full vortex sheet model. The viscous vortex model for the unstable interface is based on the multi-Lamb–Oseen vortices. We show that the viscous vortex model gives the main qualitative features of the evolution of the viscous RM instability.
The nonlinear growth, of the multimode incompressible Kelvin-Helmholtz shear flow instability at all density ratios is treated by a large-scale statistical-mechanics eddy-pairing model that is based on the behavior of a single eddy and on the two eddy pairing process. From the model, a linear time growth of the mixing zone is obtained and the linear growth coefficient is derived for several density ratios. Furthermore, the asymptotic eddy size distribution and the average eddy life time probability are calculated. Very good agreement with experimental results and full numerical simulations is achieved.
Using a statistical mechanics bubble competition model, Alon et al. (1994,1995) have shown that the two-dimensional Rayleigh-Taylor (RT) mixing zone bubble and spike fronts evolves as h=alpha (B/S).A.g.t(2) with alpha (B)similar to0.05 and alpha (S)similar to alpha (s). (1+A). The Richtmyer-Meshkov (RM) mixing zone fronts have been found to evolve as h=a(0).t(theta) with different theta's for bubble and spikes. The model predictions were theta (B)similar to0.4 and theta (S) similar to theta (B) at low A's and rises to 1.0 for A close to 1. Full 2D numerical simulations confirmed these scaling laws. Recent experimental results (Dimonte, 1999,2000) have indicated similar scaling laws of the mixing zone evolution, but there were some discrepancies in the values of the scaling parameters, mainly in the value of theta (B) and the similarity parameter, h/< lambda >. It will be shown, based on full 3D numerical simulations, a Layzer type model acid a 3D statistical-mechanics model that these discrepancies are mainly the effect of dimensionality. Accounting for the 3D nature of the problem results in scaling parameters that are very similar to the experimental values. The 3D single mode evolution, used in this model, was confirmed by shock tube experiments.
In the present study the Mach number and the high-initial amplitudes effects on the evolution of the single-mode shock wave induced instability were investigated. To distinguish between the above-mentioned effects, two sets of shock-tube experiments were conducted: high-initial amplitudes with a low-Mach incident shock; and small amplitude initial conditions with moderate-Mach incident shock. In the high-amplitude experiments a reduction of the initial velocity with respect to the linear prediction was measured. The results were compared to those predicted by a vorticity deposition model and to previous experiments with moderate and high Mach numbers done by others and good agreement was found. The result suggested that the high-initial amplitude effect is the dominant one rather then the high-Mach number effect as suggested by others. In the small amplitude-moderate Mach numbers experiments a reduction from the impulsive theory was noted at late stages.
Experiments conducted on the Omega laser [T. R. Boehly et al., Opt. Commun. 133, 495 (1997)] and simulations show reduced Richtmyer–Meshkov growth rates in a strongly shocked system with initial amplitudes kη0 ⩽ 0.9. The growth rate at early time is less than half the impulsive model prediction, rising at later time to near the impulsive prediction. An analytical model that accounts for shock proximity agrees with the results. © 2003 American Institute of Physics.
The late-time nonlinear evolution of the three-dimensional (3D) Rayleigh–Taylor (RT) and Richtmyer–Meshkov (RM) instabilities for random initial perturbations is investigated. Using full 3D numerical simulations, a statistical mechanics bubble-competition model, and a Layzer-type drag-buoyancy model, it is shown that the RT scaling parameters, αB and αS, are similar in two and three dimensions, but the RM exponents, θB and θS are lower by a factor of 2 in three dimensions. The similarity parameter hB/〈λ〉 is higher by a factor of 3 in the 3D case compared to the 2D case, in very good agreement with recent Linear Electric Motor (LEM) experiments. A simple drag-buoyancy model, similar to that proposed by Youngs [see J. C. V. Hanson et al., Laser Part. Beams 8, 51 (1990)], but using the coefficients from the A = 1 Layzer model, rather than phenomenological ones, is introduced. © 2001 American Institute of Physics.
The late-time development of Richtmyer-Meshkov instability is studied in shock tube experiments. This investigation makes use of the experimental apparatus and visualization methods utilized in the earlier study of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)] but employs stronger shocks and initial perturbations with shorter wavelengths to obtain much later-time (in the dimensionless sense) images of the single-mode instability. These modifications produce a very detailed look at the evolution of the late-time single-mode instability, revealing the transition and development of turbulence in the vortex cores that eventually results in the disintegration of the laminar vortex structures into small scale features. Amplitude measurements taken from these images are shown to be effectively collapsed when plotted in dimensionless variables defined using the wave number and the initial growth rate. The amplitude measurements are compared with several late-time nonlinear models and solutions. The best agreement is obtained with the model of Sadot et al. [Phys. Rev. Lett. 80, 1654 (1998)] which can be slightly improved by modifying the expression for the late-time asymptotic growth rate.
The three-dimensional (3D) turbulent mixing zone (TMZ) evolution
under Rayleigh–Taylor and Richtmyer–Meshkov conditions was
studied using two approaches. First, an extensive numerical study was
made, investigating the growth of a random 3D perturbation in a wide
range of density ratios. Following that, a new 3D statistical model was
developed, similar to the previously developed two-dimensional (2D)
statistical model, assuming binary interactions between bubbles that
are growing at a 3D asymptotic velocity. Confirmation of the
theoretical model was gained by detailed comparison of the bubble size
distribution to the numerical simulations, enabled by a new analysis
scheme that was applied to the 3D simulations. In addition, the results
for the growth rate of the 3D bubble front obtained from the
theoretical model show very good agreement with both the experimental
and the 3D simulation results. A simple 3D drag–buoyancy model is
also presented and compared with the results of the simulations and the
experiments with good agreement. Its extension to the spike-front
evolution, made by assuming the spikes' motion is governed by the
single-mode evolution determined by the dominant bubbles, is in good
agreement with the experiments and the 3D simulations. The good
agreement between the 3D theoretical models, the 3D numerical
simulations, and the experimental results, together with the clear
differences between the 2D and the 3D results, suggest that the
discrepancies between the experiments and the previously developed
models are due to geometrical effects.
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·gt2 with different values of α for the bubble and spike fronts. The RM mixing zone fronts evolve as h∼tθ 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.RésuméL'évolution non linéaire, à long terme, des instabilités Rayleigh–Taylor (RT) et Richtmyer–Meskov (RM) à perturbations initiales aléatoires est étudiée au moyen d'un modèle de mécanique statistique basé sur la compétition de bulles ayant tous les nombres d'Atwood ( A ), et à l'aide de simulations numériques complètes à deux et trois dimensions. Les résultats montrent que les fronts de la zone de mélange RT de bulles et d'aiguilles varient comme h∼αA·gt2 avec différentes valeurs de α pour les fronts de bulles et d'aiguilles. Les fronts de la zone de mélange RM varient comme h∼tθ avec différentes valeurs de θ pour les bulles et pour les aiguilles. Une analyse similaire donne une croissance linéaire en fonction du temps pour la zone de mélange Kelvin–Helmholtz. La dépendance des paramètres d'échelle RT et RM en fonction de A et de la dimension est discutée. Les prédictions des simulations à trois dimensions sont en bon accord avec des expériences récentes.
We study the interaction of a shock with a density-stratified gaseous interface (Richtmyer–Meshkov instability) with localized
jagged and irregular perturbations, with the aim of developing an analytical model of the vorticity deposition on the interface
immediately after the passage of the shock. The jagged perturbations, meant to simulate machining errors on the surface of
a laser fusion target, are characterized using Haar wavelets. Numerical solutions of the Euler equations show that the vortex
sheet deposited on the jagged interface rolls into multiple mushroom-shaped dipolar structures which begin to merge before
the interface evolves into a bubble-spike structure. The peaks in the distribution of x-integrated vorticity (vorticity integrated in the direction of the shock motion) decay in time as their bases widen, corresponding
to the growth and merger of the mushrooms. However, these peaks were not seen to move significantly along the interface at
early times i.e. t < 10 τ, where τ is the interface traversal time of the shock. We tested our analytical model against inviscid simulations
for two test cases – a Mach 1.5 shock interacting with an interface with a density ratio of 3 and a Mach 10 shock interacting
with a density ratio of 10. We find that this model captures the early time (t/τ ∼ 1) vorticity deposition (as characterized by the first and second moments of vorticity distributions) to within 5% of
the numerical results.
A shock driven material interface between two fluids of different density is unstable. This instability is known as Richtmyer–Meshkov (RM) instability. In this paper, we present a quantitative nonlinear theory of compressible Richtmyer–Meshkov instability in two dimensions. Our nonlinear theory contains no free parameter and provides analytical predictions for the overall growth rate, as well as the growth rates of the bubble and spike, from early to later times for fluids of all density ratios. The theory also includes a general formulation of perturbative nonlinear solutions for incompressible fluids (evaluated explicitly through the fourth order). Our theory shows that the RM unstable system goes through a transition from a compressible and linear one at early times to a nonlinear and incompressible one at later times. Our theoretical predictions are in excellent agreement with the results of full numerical simulations from linear to nonlinear regimes.
Motivated by the necessity of large-scale mixing in modeling the light
curves of Type Ib/Ic supernovae (SN Ib/Ic), the Rayleigh-Taylor (R-T)
instabilities in exploding helium stars of masses M(alpha) = 3.3, 4, and
6 solar masses are calculated for the first time with a two-dimensional
hydrodynamical code. Mixing of the ejected material induced by the
instabilities is found to be more extensive for smaller mass stars. For
M(alpha) less than about 4 solar masses, Ni-56 is mixed to the outermost
helium envelope. For M(alpha) greater than about 6 solar masses, on the
contrary, the growth of the R-T instability is too weak to convey Ni-56
into the outer layers. The extensive mixing in smaller mass stars is
consistent with the early light curve models for SN 1983N (Type Ib) and
SN 1987M (Type Ic) whose declines are as fast as and even faster than
Type Ia supernovae. On the other hand, no significant mixing for larger
M-alpha may be consistent with the slow decline of SN Ib 1984L. Thus the
observed variation of the SN Ib/Ic light curves can naturally be
accounted for by the variation of mixing as a function of the helium
star mass.
The stability of the supernova ejecta is compared with the
Rayleigh-Taylor instability for a realistic model of SN 1987A. A linear
analysis indicates that the layers around the composition interface
between the hydrogen-rich and helium zones, and become Rayleigh-Taylor
unstable between the helium and metal zones. In these layers, the
pressure increases outward because of deceleration due to the reverse
shock which forms when the blast shock hits the massive hydrogen-rich
envelope. On the contrary, the density steeply decreases outward because
of the preexisting nuclear burning shell. Then, these layers undergo the
Raleigh-Taylor instability because of the opposite signs of the pressure
and density gradients. The estimated growth rate is larger than the
expansion rate of the supernova. The Rayleigh-Taylor instability near
the composition interface is likely to induce mixing, which has been
strongly suggested from observations of SN 1987A.
Techniques have been developed to improve the uniformity of the laser focal profile, to reduce the ablative Rayleigh-Taylor instability, and to suppress the various laser-plasma instabilities. There are now three direct-drive ignition target designs that utilize these techniques. An evaluation of these designs is still ongoing. Some of them may achieve the gains above 100 that are necessary for a fusion reactor. Two laser systems have been proposed that may meet all of the requirements for a fusion reactor. (C) 1998 American Institute of Physics.
A self-consistent model is analyzed for the spherical infall of weakly magnetized plasma into the magnetosphere of a slowly rotating, strongly magnetized neutron star. It is shown that spherical infall is probably a good approximation for X-ray sources which accrete from a stellar wind. The location of the standoff shock which halts the hypersonic infall is estimated along with the emission from the shocked layer. The location of the equilibrium magnetopause and the structure of the magnetic field within it are calculated; it is found that the magnetic poles are true cusps and that the entry of gas due to equilibrium flow across a cusp is almost certainly dominated by the interchange instability near the magnetic equator. The energy principle is applied to derive necessary conditions for the occurrence of this instability. The results indicate that the strong magnetic-pressure gradient stabilizes the gas unless moderately strong radiative cooling takes place and that the cooled plasma enters the magnetosphere as long filaments capable of moving between field lines. The rate at which the equilibrium magnetopause can 'absorb' mass and momentum is derived, the validity of the approximations employed is discussed, and the likely evolution of the sinking filaments is outlined to show that the spatial distribution of the plasma is determined mainly by the dynamics and thermodynamics of the filaments rather than the magnetic-field structure.
Cambridge Core - Fluid Dynamics and Solid Mechanics - A First Course in Fluid Dynamics - by A. R. Paterson
The properties of gravitational instabilities formed within two-layer
fluid systems are well known and have been applied to a variety of
geophysical problems. We present theoretical and experimental results
for the gravitational instability developed by a three-layer system,
comprising a thin low-viscosity low-density fluid layer sandwiched
between two thick layers of equal properties. Linearized equations can
be used to solve for the initial growth rates as a function of
perturbation wavelength. As is the case for two-layer systems, the
results yield a fastest growing wavelength, termed the characteristic
wavelength, whose value is much greater than the thickness of the
low-viscosity layer. The experimental results confirm the ability of the
linearized equations to predict the dominant wavelength of the
instability. However, for very thin layers or smaller viscosity ratios a
second instability is also observed at a scale much greater than the
characteristic wavelength. Numerical solutions show that the wavelength
of this instability matches that of a fast growing but short-lived mode
arising from perturbations which predominantly involve thickening rather
than translation of the buoyant layer. The analytical solution also
shows that at the characteristic wavelength, the displacement of the
lower interface will be initially a factor 2 - √3 = 0.268 that of
the upper interface. As the instability develops the characteristic
diapir structures, the experiments show that the relative magnitude of
these displacements increase, with underlying fluid being drawn up into
the head of the diapir.
Several targets are described that in simulations give yields of 1--30 MJ when indirectly driven by 0.9--2 MJ of 0.35 {mu}m laser light. The article describes the targets, the modeling that was used to design them, and the modeling done to set specifications for the laser system in the proposed National Ignition Facility. Capsules with beryllium or polystyrene ablators are enclosed in gold hohlraums. All the designs utilize a cryogenic fuel layer; it is very difficult to achieve ignition at this scale with a noncryogenic capsule. It is necessary to use multiple bands of illumination in the hohlraum to achieve sufficiently uniform x-ray irradiation, and to use a low-{ital Z} gas fill in the hohlraum to reduce filling of the hohlraum with gold plasma. Critical issues are hohlraum design and optimization, Rayleigh--Taylor instability modeling, and laser--plasma interactions.
The evolution of a randomly perturbed interface between unbounded incompressible fluids undergoing Rayleigh-Taylor instability is analyzed numerically and theoretically. Two-dimensional simulation results, obtained with an interface tracking code, are presented and compared with a theoretical model based on Young's two-phase flow description of the mixing process. The simplifications implied by self-similarity and by high drag enable simple analytic results to be obtained for the profiles of the average volume fractions and velocities of the two materials as a function of penetration depth. Agreement of the analytic results with the simulation data is demonstrated for a wide range of density ratios.
The nonlinear regime of electrostatic perturbations of the equatorial ionospheric F-region generated by Rayleigh-Taylor instability has been discussed, taking into account conductivity along magnetic field lines. A closed nonlinear equation has been derived in the stationary limit for the polarization electric field potential. It coincides with the Karman equation of an ideal liquid. To solve the equation, the averaged variational Whitham method has been proposed. Some solutions localized along and across the geomagnetic field, B, as well as quasi-periodic solutions in the transverse direction, have been investigated. Nonlinear longitudinal localization of perturbations has been shown to be due to electron-ion collisions.
A potential flow model of Rayleigh–Taylor and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. In the model, which extends the work of Layzer [Astrophys. J. 122, 1 (1955)], ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two-dimensional single-mode evolution as well as two-bubble competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities, the latter treated in an impulse approximation. The model predicts that the asymptotic velocity of a single-mode RM bubble of wavelength λ decays as λt−1, in contrast with the constant asymptotic velocity attained in the RT case. Bubble competition, which is believed to determine the multimode front evolution, is demonstrated for both the RT and RM instabilities. The capability of the model to predict bubble growth in a finite-thickness fluid layer is shown. Finally, the model is applied to the evolution of three-dimensional modes with an initial rectangular geometry. The model yields the aspect ratio dependence of the early nonlinear stages, in agreement with third-order perturbation theory. However, in the late nonlinear stage, the model predicts that the bubbles forget the initial geometry and attain the fastest growing shape, with a circular tip. The model results are in good agreement with full hydrodynamic simulations and analytic solutions, where available.
The Richtmyer–Meshkov instability of a two-liquid system is investigated experimentally. These experiments utilize a novel technique that circumvents many of the experimental difficulties that have previously limited the study of Richtmyer–Meshkov instability. The instability is generated by vertically accelerating a tank containing two stratified liquids by bouncing it off of a fixed coil spring. A controlled two-dimensional sinusoidal initial shape is given to the interface by oscillating the container in the horizontal direction to produce standing waves. The motion of the interface is recorded during the experiments using standard video photography. Instability growth rates are measured and compared with existing linear theory. Disagreement between measured growth rates and the theory are accredited to the finite bounce length. When the linear stability theory is modified to account for an acceleration pulse of finite duration, much better agreement is attained. Late time growth curves of many different experiments seem to collapse to a single curve when correlated with the circulation deposited by the impulsive acceleration. A theory based on modeling the late time evolution of the instability using a row of vortices is developed. The growth curve given by this model has similar shape to those measured, but underestimates the late-time growth rate.
The statistical behavior for chaotic mixing of Rayleigh-Taylor unstable interfaces is characterized by the merging of small bubbles (the portions of a light fluid adjacent to a heavy fluid) into large bubbles. The length scales change dynamically as a consequence of bubble merger. We study such statistical behavior quantitatively using a renormalization group fixed point model. Our study shows good agreement among the results of the renormalization group model, experimental data and direct numerical simulation of the two-fluid Euler equations.
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 approximate to 1.3 Air-to-SF6 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 keg elements of the bubble-merger model used for the prediction of the multimode bubble and spike front evolution.
The aim of this talk is to survey Rayleigh-Taylor instability, describing the phenomenology that occurs at a Taylor unstable interface, and reviewing attempts to understand these phenomena quantitatively.
Inertial confinement fusion (ICF) is an approach to fusion that relies on the inertia of the fuel mass to provide confinement. To achieve conditions under which inertial confinement is sufficient for efficient thermonuclear burn, a capsule (generally a spherical shell) containing thermonuclear fuel is compressed in an implosion process to conditions of high density and temperature. ICF capsules rely on either electron conduction (direct drive) or x rays (indirect drive) for energy transport to drive an implosion. In direct drive, the laser beams (or charged particle beams) are aimed directly at a target. The laser energy is transferred to electrons by means of inverse bremsstrahlung or a variety of plasma collective processes. In indirect drive, the driver energy (from laser beams or ion beams) is first absorbed in a high‐Z enclosure (a hohlraum), which surrounds the capsule. The material heated by the driver emits x rays, which drive the capsule implosion. For optimally designed targets, 70%–80% of the driver energy can be converted to x rays. The optimal hohlraum geometry depends on the driver. Because of relaxed requirements on laser beam uniformity, and reduced sensitivity to hydrodynamic instabilities, the U.S. ICF Program has concentrated most of its effort since 1976 on the x‐ray or indirect‐drive approach to ICF. As a result of years of experiments and modeling, we are building an increasingly strong case for achieving ignition by indirect drive on the proposed National Ignition Facility (NIF).
A two dimensional radiation diffusion coupled to hydrodynamics
calculation of nonspherical instabilities in the collapsed core from a
massive star is performed. The core properties are taken from a
one-dimensional collapse calculated with detailed microphysics and
neutrino transport. The shocked outer core (mass between 0.7 and 1.3
solar masses) is found to contain three subregions. The innermost is
doubly diffusive (neutron fingers) unstable, the center subregion is
dynamically unstable, and the outermost is stable. It is found that the
unstable part of the outer core overturns in approximately 5 ms without
disturbing the inner unshocked core of mass less than 0.7 solar masses.
This overturned outer core expands as a piston, creating an outgoing
shock wave which may help power envelope ejection. This outer core
overturn would seem to be a generic feature of core collapse which has
heretofore been neglected.
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
This paper presents a new analysis of small amplitude Richtmyer--Meshkov instability. The linear theory for the case of reflected rarefaction waves, a problem not treated in previous work, is formulated and numerically solved. This paper also carries out a systematic comparison of Richtmyer's impulsive model to the small amplitude theory, which has identified domains of agreement as well as disagreement between the two. This comparison includes both the reflected shock and reflected rarefaction cases. Additional key results include the formulation of criteria determining the reflected wave type in terms of preshocked quantities, identification of parameter regimes corresponding to total transmission of the incident wave, discussion of an instability associated with a rarefaction wave, investigation of phase inversions and the related phenomenon of freeze-out, and study of the sensitivity of the numerical solutions to initial conditions.
The initial growth of irregularities on an interface between two ; compressible fluids is studied for impulsive (i.e., shock) acceleration. It was ; found that the ultimate rate of growth is roughly the same as that given by the ; incompressible theory, if the initial comnpression of the irregularities and of ; the fluids is taken into account. (auth)
Results are presented of an experimental study of the stability of the interface of two gases traversed by ashockwave. It is found that the interface is unstable both in the case of shock wave passage from the lighter to the heavier gas and for passage in the opposite direction. The interface disturbance grows linearly with time in the first approximation.
Direct two-dimensional numerical simulation and experiments, in which small rocket motors accelerate a tank containing two fluids, have been used to investigate turbulent mixing by Rayleigh-Taylor instability at a wide range of density ratios. The experimental data obtained so far has been used to calibrate an empirical model of the mixing process which is needed to make predictions for complex applications. The model devised, which is a form of turbulence model, is based on the equations of multiphase flow. These equations describe velocity separation arising from the action of a pressure gradient on fluid fragments of different density. The dissipation arising from the drag between the fluid fragments is treated as a source of turbulence kinetic energy which is then used to define turbulent diffusion coefficients. Gradient diffusion processes are thereby included in the model.
Richtmyer-Meshkov instability is a fingering instability which occurs at a material interface accelerated by a shock wave. We present an analytic, explicit prediction for the growth rate of the unstable interface. The theoretical prediction agrees, for the first time, with the experimental data on air-SF6, and is in remarkable agreement with the results of recent full nonlinear numerical simulations from early to late times. Previous theoretical predictions of the growth rate for air-SF6 unstable interfaces were about two times larger than the experimental data.
A novel experimental technique using solid fuel rocket motors, has been developed at AWRE Foulness to study the growth of Rayleigh-Taylor instabilities in fluids.The technique, which achieves near constant accelerations up to 750 m s-2 over distances of 1.25 m, is currently providing results for comparison with the theoretical studies on interfacial mixing by Youngs.
Two-dimensional hydrodynamic codes are used to simulate the growth of perturbations at an interface between two fluids of different density due to Rayleigh-Taylor instability. Problems where the interface is subjected to a constant acceleration and where it is accelerated and decelerated by shock waves are considered. Emphasis is placed on the case when the initial perturbation consists of many different wavelength modes. Results are compared with the experimental data of Andronov et al. (1976) and Read (1983). The use of a simple empirical model of the mixing process based on the equations of two-phase flow is discussed.
Motivated by considerations of the solar toroidal magnetic field, the behavior of a layer of uniform magnetic field embedded in a convectively stable atmosphere is studied. Since the field can support extra mass, such a configuration is top-heavy and thus instabilities of the Rayleigh-Taylor type can occur. For both static and rotating basic states, the evolution of the interchange modes (no bending of the field lines) is followed by integrating numerically the nonlinear compressible MHD equations. The initial Rayleigh-Taylor instability of the magnetic field gives rise to strong shearing motions, thereby exciting secondary Kelvin-Helmholtz instabilities which wrap the gas into regions of intense vorticity. The subsequent motions are determined primarily by the strong interactions between vortices which are responsible for the rapid disruption of the magnetic layer.
A statistical model of Rayleigh-Taylor bubble fronts in two dimensions is introduced. Float and merger of bubbles lead to a scale-invariant regime, with a stable distribution of scaled bubble radii and a constant front acceleration. The model is solved for a simple merger law, showing that a family of such stable distributions exists. The basins of attraction of each of these are mapped. The properties of the scale-invariant distributions for various merger laws, including a merger law derived from the Sharp-Wheeler model, are analyzed. The results are in good agreement with computer simulations. Finally, it is shown that for some merger laws, a runaway bubble regime develops. A criterion for the appearance of runaway growth is presented.
Turbulent mixing due to the Rayleigh-Taylor instability is experimentally found to vary strongly with the temporal acceleration profile {ital g}({ital t}). For constant {ital g}, the bubble amplitude {ital h}{sub {ital b}} increases as {ital gt}{sup 2} consistent with previous results. For sustained acceleration profiles with {ital dg}/{ital dt}{ne}0, {ital h}{sub {ital b}} increases, not with the displacement {ital Z}={integral}{integral}{ital g}{ital dt}{prime}{ital dt}, but with the length {ital S}=0.5 [{integral}{radical}{ital g}{ital dt}]{sup 2}. For an impulsive acceleration, mixing is minimized with {ital h}{sub {ital b}}{approximately}{ital Z}{sup 0.4}. These results are used to test mix models. {copyright} {ital 1996 The American Physical Society.}
A renormalization-group fixed point is found, corresponding to chaotic mixing in the Rayleigh-Taylor instability problem. The outer envelope of the mixing region, adjacent to the heavy fluid, is dominated by a merger of unstable modes ( bubbles of light fluid) and dynamically changing length scales. A statistical model is introduced as an approximation to the full two-fluid Euler equation to describe the mixing envelope. Molecular-chaos and continuous-time approximations to this model define an approximate renormalization-group equation, which is shown to have a nontrivial fixed point.
The late time evolution and structure of 2D Rayleigh-Taylor and Richtmyer-Meshkov bubble fronts is calculated, using a new statistical merger model based on the potential flow equations. The merger model dynamics are shown to reach a scale invariant reigme. It is found that the Rayleigh-Taylor front reaches a constant acceleration, growing as 0.05gt2, while the Richtmyer-Meshkov front grows as at0.4 where a depends on the initial perturbation. The model results are in good agreement with experiments and simulations.
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 hB = alphaBAgt2 [hs = alphasAgt2] with alphaB = 0.05, while Richtmyer-Meshkov bubble (spike) fronts are found as hB = aBtthetaB hs = astthetas\(A\) with thetaB = 0.4 at all A's. The dependence of these scaling laws and parameters on A and on initial conditions is explained.