Converging Richtmyer-Meshkov Instability Research Articles

Convergent Richtmyer–Meshkov instability on two-dimensional dual-mode interfaces

We report the first shock-tube experiments on two-dimensional dual-mode air–SF $_6$ 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.

Quasi-dynamic subgrid-scale kinetic energy equation model for large-eddy simulation of compressible flows

A quasi-dynamic subgrid-scale (SGS) kinetic energy one-equation eddy-viscosity model is introduced in this paper for large-eddy simulation (LES) of compressible flows. With the additional SGS kinetic energy equation, the SGS kinetic energy can be predicted properly. Then, using the dual constraints of SGS kinetic energy and the SGS kinetic energy flux, the eddy-viscosity model can be determined exactly. Taking a similar scheme as the expansion of the SGS stress, other unclosed quantities in the equations to be solved could be well modelled separately. Therefore, with the advance of the equations, all the model coefficients can be determined dynamically. Differing from the classic dynamic procedure, the new methodology needs no test filtering, and thus it could also be called a quasi-dynamic procedure. Using direct numerical simulation of compressible turbulent channel flow, the$a$ $priori$test shows that the key modelled quantities of the suggested model display high correlations with the real values. In LES of compressible turbulent channel flows of the Mach number being$1.5$and 3.0, the proposed model can precisely predict some important quantities, including the mean velocity, Reynolds stress and turbulent flux, and it can also supply more abundant turbulent structures. For the compressible flat-plate boundary layer, the new model can correctly predict the transition process, mean velocity and turbulence intensities in the turbulent region. The results show that the proposed model has the advantage of scale adaptivity. Finally, the new model is applied to LES of turbulent mixing in spherical converging Richtmyer–Meshkov instability, and the accurate results show that the new model has a good ability for LES of complex fluids.

Statistical characteristics of turbulent mixing in spherical and cylindrical converging Richtmyer–Meshkov instabilities

In this paper, the Richtmyer–Meshkov instabilities in spherical and cylindrical converging geometries with a Mach number of approximately 1.5 are investigated by using the high resolution implicit large eddy simulation method, and the influence of the geometric effect on the turbulent mixing is investigated. The heavy fluid is sulphur hexafluoride (SF6), and the light fluid is nitrogen (N2). The shock wave converges from the heavy fluid into the light fluid. The Atwood number is 0.678. The total structured and uniform Cartesian grid node number in the main computational domain is 20483. In addition, to avoid the influence of boundary reflection, a sufficiently long sponge layer with 50 non-uniform coarse grids is added for each non-periodic boundary. Present numerical simulations have high and nonlinear initial perturbation levels, which rapidly lead to turbulent mixing in the mixing layers. Firstly, some physical-variable mean profiles, including mass fraction, Taylor Reynolds number, turbulent kinetic energy, enstrophy and helicity, are provided. Second, the mixing characteristics in the spherical and cylindrical turbulent mixing layers are investigated, such as molecular mixing fraction, efficiency Atwood number, turbulent mass-flux velocity and density self-correlation. Then, Reynolds stress and anisotropy are also investigated. Finally, the radial velocity, velocity divergence and enstrophy in the spherical and cylindrical turbulent mixing layers are studied using the method of conditional statistical analysis. Present numerical results show that the geometric effect has a great influence on the converging Richtmyer–Meshkov instability mixing layers.

Convergent Richtmyer–Meshkov instability on a light gas layer with perturbed inner and outer surfaces

The instability of an annular helium gas layer surrounded by air with sinusoidal inner and outer interfaces, formed by a novel soap-film technique, impacted by a cylindrically convergent shock is experimentally studied in a semi-annular shock tube. Detailed evolution of the interfaces and wave patterns is captured by a high-speed Schlieren system. The focus is placed on the influences of layer thickness and phase difference between the inner and outer interfaces on the instability development. It is found that the larger the layer thickness, the quicker the early stage development of the outer interface. This is because the layer thickness affects the arrival time of the reflected shock (RS) at the outer interface and further determines the direction of baroclinic vorticity deposited on the outer interface by RS; namely, RS inhibits or promotes the instability growth depending on the layer thickness. It is also found that phase difference between the inner and outer perturbations produces a negligible (an evident) influence on the early stage (late-stage) instability growth at the outer interface, whereas a considerable (weak) influence on the early stage (late-stage) instability growth at the inner interface. This finding suggests that the early stage development of the outer (inner) interface can be modulated by changing the layer thickness (perturbation phase difference). Empirical coefficient in the Charakhch'an model [J. Appl. Mech. 41, 23–31 (2000)] is calculated to be β=0.52 by comparing the prediction with the experimental results. The model with β=0.52 gives a reasonable prediction of the post-reshock growth rate for all the cases considered in this work.

Large eddy simulation of converging Richtmyer–Meshkov instability based on subgrid-scale dissipation similar method

This paper focuses on large eddy simulation of the Richtmyer–Meshkov instability (RMI) in spherical and cylindrical converging geometries with a Mach number [Formula: see text] based on subgrid-scale (SGS) dissipation similar method (SDSM). Based on the converging RMI problem, we obtain from a priori test and theoretical analysis that the suggested method can provide accurate structural correlation while ensuring the computational stability. Comparing the numerical simulation results with direct numerical simulation (DNS) and existing model in converging RMI problem, we could find that the suggested method overcomes some defects of the existing model, such as the Smagorinsky model cannot predict transition accurately and the helicity model can only predicts the quasi-two-dimensional flow precisely. It provides a beneficial tool for the research of converging RMI.

A volume-of-fluid reconstruction based interface sharpening algorithm for a reduced equation model of two-material compressible flow

In this paper we present a numerical method for the simulation of two-material flows governed by the compressible unsteady Euler equations. A system of six equations is used, with a pressure relaxation step which ensures convergence to the equivalent five-equation model. The numerical method employs a volume-of-fluid interface tracking method in order to maintain a sharp interface indefinitely, along with a robust update procedure for interfacial cells which ensures density and internal energy positivity. Numerical results are presented for a range of test cases, including for liquid–gas shock–bubble interaction and cylindrically convergent Richtmyer–Meshkov instability between metal and air. Results are in agreement with theoretical data, and compare well to other approaches designed to sharpen the material interface in the five-equation model.

Effect of Atwood number on convergent Richtmyer–Meshkov instability

Developments of two-dimensional single-mode light/heavy interfaces driven by convergent shock waves are numerically investigated, focusing on the effect of the Atwood number on the Rayleigh–Taylor stabilization, the compressibility and the nonlinearity. Five different test gases, including $$\hbox {CO}_2$$ , Kr, R22, R12 and $$\hbox {SF}_6$$ , are considered with air as the ambient gas. It is clarified for the first time that the unperturbed interface begins to decelerate when the shock focuses at the convergence center, and the acceleration during the deceleration phase is proportional to the Atwood number. During the first reshock, the interface moves outwards with a deceleration until it starts moving inwards. When the initial interface is weakly disturbed, a more obvious amplitude reduction is observed for the case with a larger Atwood number before the reshock, which means that the Rayleigh–Taylor stabilization is stronger. To assess the effect of the Atwood number on the compressibility and the nonlinearity, three models, including a linear incompressible model, a nonlinear incompressible model and a linear compressible model, are adopted to predict the amplitude growth before the reshock. The results show that the nonlinearity is weak, and is almost not influenced by the Atwood number before the reshock. The compressibility, however, greatly changes the amplitude growth. As the Atwood number increases, the compressibility plays a less significant role in the amplitude growth because a heavier gas is harder to be compressed. Although a gas with a larger specific heat ratio is also difficult to be compressed, the specific heat ratio plays a minor role to the compressibility relative to the Atwood number. During the reshock, the amplitude grows linearly until the nonlinearity in the cases with large Atwood numbers is strong enough to reduce the amplitude growth rate.

Energy transport characteristics of converging Richtmyer–Meshkov instability

In this paper, the Richtmyer–Meshkov (RM) instability in spherical and cylindrical converging geometries with a Mach number of about 1.5 is investigated by using the direct numerical simulation method. The heavy fluid is sulfur hexafluoride, and the light fluid is nitrogen. The shock wave converges from the heavy fluid into the light fluid. The main focus is on the energy transport characteristics in the mixing layer during the entire development process from early instability to late-time turbulent mixing. First, the turbulence kinetic energy transport equation is analyzed, and it is found that the production and dissipation mechanisms of the turbulence induced by the spherical and cylindrical converging RM instabilities in the mixing layer are the same. The turbulent diffusion terms are crucial in the whole development processes of the mixing layers. Before the reflected shock waves transit the interfaces, the dissipation terms can be ignored relative to other terms, and after that, the dissipation terms are close to the production terms and play an important role. The compressibility terms are approximate to the production terms and promote the production of turbulence kinetic energy in the later stage. The viscous diffusion terms can be ignored throughout the process. Then, the enstrophy transport equation is researched, and it is found that, in the mixing layers, the baroclinicity terms play a leading role in the early stage, while the vortex stretching terms play a leading role in the later stage, and the vortex stretching term of the spherical converging geometry develops faster than that of the cylindrical converging geometry. The compressibility terms are positive in the early stage, which promote the production of enstrophy. After the reflected shock waves transit the interfaces, the compressibility terms become negative, which inhibit the production of enstrophy. In addition, the results of the present direct numerical simulation also show that the density fluctuation spectra in the centers of the mixing layers of the spherical and cylindrical converging RM instabilities present the obvious −5/3 scaling law.

Convergent Richtmyer–Meshkov instability of heavy gas layer with perturbed inner surface

Abstract

On divergent Richtmyer–Meshkov instability of a light/heavy interface

Abstract

Smoothed particle hydrodynamics simulation of converging Richtmyer–Meshkov instability

The Smoothed Particle Hydrodynamics (SPH) method based on the Harten–Lax–van Leer Riemann solver is improved to study converging Richtmyer–Meshkov instability (RMI). A new density summation algorithm is proposed, which greatly suppresses the pressure oscillation at the material interface. The one-dimensional Sod problem is first simulated for code verification. Then, the SPH program is extended to two dimensions to simulate the converging RMI at a square air/SF6 interface, and the numerical results compare well with the experimental ones [Si et al., “Experimental investigation of cylindrical converging shock waves interacting with a polygonal heavy gas cylinder,” J. Fluid Mech. 784, 225–251 (2015)]. Nonlinear mode coupling and pressure disturbance are found to act evidently, causing a very fast growth spike. Performing a Fourier analysis of the interface profiles, amplitude growths of the first three harmonics are obtained. The first harmonic presents an increasing growth rate at early stages due to geometric convergence. The second harmonic experiences a long period of linear growth due to the counteraction between geometric convergence and nonlinearity, whereas the third harmonic saturates very early for stronger nonlinearity. For all three harmonics, the perturbation growth rate reduces evidently at the late stage due to the Rayleigh–Taylor stabilization caused by interface deceleration. It is found that the instability growth at early stages depends heavily on the incident shock strength, while the late-stage asymptotic growth rate is nearly constant, regardless of shock strength. It is also found that intensifying the incident shock is an effective way to produce extreme thermodynamic state at the geometric center even though it causes a faster instability growth.

Weakly nonlinear multi-mode Bell–Plesset growth in cylindrical geometry**Project supported by the Fundamental Research Funds for the Central Universities, China (Grant No. 2019QS04), the National Natural Science Foundation of China (Grant No. 11574390), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA 25051000).

Bell–Plesset (BP) effect caused perturbation growth plays an important role in better understanding of characteristics of the convergence effect. Governing equations for multi-mode perturbation growth on a cylindrically convergent interface are derived. The second-order weakly nonlinear (WN) solutions for two-mode perturbations at the interface which is subject to uniformly radical motion are obtained. Our WN theory is consistent with the numerical result in terms of mode-coupling effect in converging Richtmyer–Meshkov instability. Nonlinear mode-coupling effects will cause irregular deformation of the convergent interface. The mode-coupling behavior in convergent geometry depends on the mode number, Atwood number A and convergence ratio Cr. The A = –1.0 at the interface results in larger perturbation growth than A = 1.0. The growth of generated perturbation modes from two similar modes at the initial stage are smaller than that from two dissimilar modes.

Convergent Richtmyer–Meshkov instability of light gas layer with perturbed outer surface

The convergent Richtmyer–Meshkov instability (RMI) of an layer with a uniform outer surface and a sinusoidal inner surface surrounded by air (generated by a novel soap film technique) is studied in a semiannular convergent shock tube using high-speed schlieren photography. The outer interface initially suffers only a slight deformation over a long period of time, but distorts quickly at late stages when the inner interface is close to it and produces strong coupling effects. The development of the inner interface can be divided into three stages. At stage I, the interface amplitude first drops suddenly to a lower value due to shock compression, then decreases gradually to zero (phase reversal) and later increases sustainedly in the negative direction. After the reshock (stage II), the perturbation amplitude exhibits long-term quasi-linear growth with time. The quasi-linear growth rate depends weakly on the pre-reshock amplitude and wavelength, but strongly on the pre-reshock growth rate. An empirical model for the growth of convergent RMI under reshock is proposed, which reasonably predicts the present results and those in the literature. At stage III, the perturbation growth is promoted by the Rayleigh–Taylor instability caused by a rarefaction wave reflected from the outer interface. It is found that both the Rayleigh–Taylor effect and the interface coupling depend heavily on the layer thickness. Therefore, controlling the layer thickness is an effective way to modulate the late-stage instability growth, which may be useful for the target design.

Mode coupling in converging Richtmyer–Meshkov instability of dual-mode interface

The converging Richtmyer–Meshkov (RM) instability on single- and dual-mode $$\hbox {N}_2$$/$$\hbox {SF}_6$$ interfaces is studied by an upwind conservation element and solution element solver. An unperturbed case is first considered, and it is found that the shocked interface undergoes a long-term deceleration after a period of uniform motion. The evolution of single-mode interface at the early stage exhibits an evident nonlinearity, which can be reasonably predicted by the nonlinear model of Wang et al. (Phys Plasmas 22: 082702, 2015). During the deceleration stage, the perturbation amplitude drops quickly and even becomes a negative (phase inversion) before the reshock due to the Rayleigh–Taylor (RT) stabilization. After the reshock, the interface experiences a phase inversion again or does not, depending on the reshock time. The growth of the second-order harmonic in the deceleration stage clearly reveals the competition between the RT effect and the nonlinearity. For dual-mode interfaces, the growth of the first mode (wavenumber $$k_1$$) relies heavily on the second mode (wavenumber $$k_2$$) due to the mode coupling effect. Specifically, for cases where $$k_2$$ is an even or odd multiple of $$k_1$$, the growth of the first mode is inhibited or promoted depending on its initial amplitude sign and the phase difference between two basic waves, while for cases where $$k_2$$ is a non-integer multiple of $$k_1$$, the second mode has negligible influence on the first mode. Through a systematic study, signs of perturbation amplitudes of the generated $$k_2-k_1$$ and $$k_2+k_1$$ waves are obtained for all possible dual-mode configurations, which are reasonably predicted by a modified Haan model (Phys Fluids B 3: 2349–2355, 1991).

Convergent Richtmyer–Meshkov instability of a heavy gas layer with perturbed outer interface

The evolution of an $\text{SF}_{6}$ layer surrounded by air is experimentally studied in a semi-annular convergent shock tube by high-speed schlieren photography. The gas layer with a sinusoidal outer interface and a circular inner interface is realized by the soap-film technique such that the initial condition is well controlled. Results show that the thicker the gas layer, the weaker the interface–coupling effect and the slower the evolution of the outer interface. Induced by the distorted transmitted shock and the interface coupling, the inner interface exhibits a slow perturbation growth which can be largely suppressed by reducing the layer thickness. After the reshock, the inner perturbation increases linearly at a growth rate independent of the initial layer thickness as well as of the outer perturbation amplitude and wavelength, and the growth rate can be well predicted by the model of Mikaelian (Physica D, vol. 36, 1989, pp. 343–357) with an empirical coefficient of 0.31. After the linear stage, the growth rate decreases continuously, and finally the perturbation freezes at a constant amplitude caused by the successive stagnation of spikes and bubbles. The convergent geometry constraint as well as the very weak compressibility at late stages are responsible for this instability freeze-out.

Numerical study on Rayleigh-Taylor effect on cylindrically converging Richtmyer-Meshkov instability

Evolution of a two-dimensional air/SF6 single-mode interface is numerically investigated by an upwind CE/SE method under a cylindrically converging circumstance. The Rayleigh-Taylor effect caused by the flow deceleration on the phase inversion (RTPI) is highlighted. The RTPI was firstly observed in our previous experiment, but the related mechanism remains unclear. By isolating the three-dimensional effect, it is found here that the initial amplitude (a0), the azimuthal mode number (k0) and the re-shocking moment are the three major parameters which determine the RTPI occurrence. In the variable space of (k0, a0), a critical a0 for the RTPI occurrence is solved for each k0, and there exists a threshold value of k0 below which the RTPI will not occur no matter what a0 is. There exists a special k0 corresponding to the largest critical a0, and the reduction rule of critical a0 with k0 can be well described by an exponential decay function. The results show that the occurrence of the RTPI requires a small a0 which should be less than a critical value, a large k0 which should exceed a threshold, and a right impinging moment of the re-shock which should be later than the RTPI occurrence. Finally, the effects of the incident shock strength, the density ratio and the initial position of the interface on the threshold value of k0 and on the maximum critical a0 are examined. These new findings would facilitate the understanding of the converging Richtmyer-Meshkov instability and would be helpful for designing an optimal structure of the inertia confinement fusion capsule.

Nonlinear behaviour of convergent Richtmyer–Meshkov instability

A novel shock tube is designed to investigate the nonlinear feature of convergent Richtmyer–Meshkov instability on a single-mode interface formed by a soap film technique. The shock tube employs a concave–oblique–convex wall profile which first transforms a planar shock into a cylindrical arc, then gradually strengthens the cylindrical shock along the oblique wall, and finally converts it back into a planar one. Therefore, the new facility can realize analysis on compressibility and nonlinearity of convergent Richtmyer–Meshkov instability by eliminating the interface deceleration and reshock. Five sinusoidal$\text{air}{-}\text{SF}_{6}$interfaces with different amplitudes and wavelengths are considered. For all cases, the perturbation amplitude experiences a linear growth much longer than that in the planar geometry. A compressible linear model is derived by considering a constant uniform fluid compression, which shows a slight difference to the incompressible theory. However, both the linear models overestimate the perturbation growth from a very early stage due to the presence of strong nonlinearity. The nonlinear model of Wanget al.(Phys. Plasmas, vol. 22, 2015, 082702) is demonstrated to predict well the amplitude growth up to a normalized time of 1.0. The prolongation of the linear increment is mainly ascribed to the counteraction between the promotion by geometric convergence and the suppression by nonlinearity. Growths of the first three harmonics, obtained by a Fourier analysis of the interface contour, provide a first thorough validation of the nonlinear theory.

Investigation of convergent Richtmyer–Meshkov instability at tin/xenon interface with pulsed magnetic driven imploding**Project supported by the National Natural Science Foundation of China (Grant Nos. 11605183 and 11502254).

The Richtmyer–Meshkov instability at the interface of solid state tin material and xenon gases under cylinder geometry is studied in this paper. The experiments were conducted at FP-1 facility in Institute of Fluid Physics, China Academy of Engineering Physics (CAEP). The FP-1 facility is a pulsed power driver which could generate high amplitude magnetic field to drive metal liner imploding. Convergent shock wave was generated by impacting a magnetic-driven aluminium liner onto a inner mounted tin liner. The convergent evolution of the disturbance pre-machined onto the tin linerʼs inner surface was diagnosed by x-radiography. The spike amplitudes were derived from x-ray frames and were compared with linear theory. An analytical model containing material strength effect was derived and matched well to the experimental results. This sensibility of the disturbance evolution to material strength property shines light to the application of Richtmyer–Meshkov instability to infer material strength.

Long-term effect of Rayleigh–Taylor stabilization on converging Richtmyer–Meshkov instability

The Richtmyer–Meshkov instability on a three-dimensional single-mode light/heavy interface is experimentally studied in a converging shock tube. The converging shock tube has a slender test section so that the non-uniform feature of the shocked flow is amply exhibited in a long testing time. A deceleration phenomenon is evident in the unperturbed interface subjected to a converging shock. The single-mode interface presents three-dimensional characteristics because of its minimum surface feature, which leads to the stratified evolution of the shocked interface. For the symmetry interface, it is quantitatively found that the perturbation amplitude experiences a rapid growth to a maximum value after shock compression and finally drops quickly before the reshock. This quick reduction of the interface amplitude is ascribed to a significant Rayleigh–Taylor stabilization effect caused by the deceleration of the light/heavy interface. The long-term effect of the Rayleigh–Taylor stabilization even leads to a phase inversion on the interface before the reshock when the initial interface has sufficiently small perturbations. It is also found that the amplitude growth is strongly suppressed by the three-dimensional effect, which facilitates the occurrence of the phase inversion.

Progress in experiments of converging Richtmyer-Meshkov instability

When a corrugated interface separating two fluids of different properties is subjected to an impulsive acceleration, such as a shock wave, perturbations initially presented on the interface will grow with time and finally develop into turbulent mixing. This type of hydrodynamic instability is generally referred to as Richtmyer-Meshkov (RM) instability. RM instability is a significant topic in fluid mechanics, and has become increasingly attractive in recent years due to its academic significance in vortex dynamics and compressive turbulence as well as important applications in the fields of inertial confinement fusion, supersonic combustion and supernova explosion. For example in inertial confinement fusion, the implosion of the spherically converging shock wave induces intensive mixing between the outer ablator and the inner fuel, and hence greatly decreases the fusion yield. Over the past decades, the RM instability induced by a planar shock wave has been intensively studied through experiments, simulations and theories. Previous studies show that the initial perturbation on the density inhomogeneity subjected to a planar shock wave experiences successively linear growth, nonlinear growth and turbulent mixing. The pressure disturbance and the baroclinic vorticity are found to be the main factors responsible for the perturbation development. However, in most realities, RM instability occurs in a convergent geometry, and the geometry convergence nature would significantly complicate the instability development. So far, the RM instability triggered by a converging shock has seldom been investigated, and the instability development pattern in convergent geometry as well as the influences of the convergence effects, such as the Bell-Plesset (BP) and Rayleigh- Taylor (RT) effects, on the instability development remains unclear. To perform a converging experiment, one has to overcome two crucial challenges in RM instability experiments: producing a stable converging shock wave and generating a well-controlled initial interface in laboratory conditions, and these are very difficult to realize using the current experimental methods. In this paper, we review the converging RM instability experiments in two types of shock tube facilities, i.e., the coaxial-structure converging shock tube and the conventional converging shock tube. Various interface formation techniques developed based on these shock tube test sections of different structures are also discussed. Special attention is paid to the influences of the typical convergence effects, including the BP effect, the RT effect and the compressibility, on the instability development. Finally, based on the existing studies, we propose the following three issues for future experimental study: (1) RM instability induced by high Mach number shock waves; (2) developments of three-dimensional interfaces subjected to an impulsive impact; (3) interaction of a rippled shock with an unperturbed or a perturbed density interface.