Atwood Number Research Articles

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.

Rayleigh-Taylor Instability: A Status Review of Experimental Designs and Measurement Diagnostics

Abstract The focus of experiments and the sophistication of diagnostics employed in Rayleigh-Taylor instability (RTI) induced mixing studies have evolved considerably over the past seven decades. The first theoretical analysis by Taylor and the two-dimensional experimental results by Lewis on RTI in 1950 examined single-mode RTI using conventional imaging techniques. Over the next 70 years, several experimental designs have been used to creating an RTI unstable interface between two materials of different densities. These early experiments though innovative, were arduous to diagnose and provided little information on the internal, turbulent structure and initial conditions of the RT mixing layer. Coupled with the availability of high-fidelity diagnostics, the experiments designed and developed in the last three decades allow detailed measurements of various turbulence statistics that have allowed broadly to validate and verify late-time nonlinear models and mix-models for buoyancy-driven flows. Besides, they have provided valuable insights to solve several long-standing disagreements in the field. This review serves as an opportunity to discuss the understanding of the RTI problem and highlight valuable insights gained into the RTI driven mixing process with a focus on low to high Atwood number (>0.1) experiments.

Experiments and Simulations on the Turbulent, Rarefaction Wave Driven Rayleigh–Taylor Instability

AbstractExperiments were performed to observe the growth of the turbulent, Rayleigh–Taylor unstable mixing layer generated between air and SF6, with an Atwood number of A=(ρ2−ρ1)/(ρ2+ρ1)=0.64, where ρ1 and ρ2 are the densities of air and SF6, respectively. A nonconstant acceleration with an average value of 2300g0, where g0 is the acceleration due to gravity, was generated by interaction of the interface between the two gases with a rarefaction wave. Three-dimensional, multimode perturbations were generated on the diffuse interface, with a diffusion layer thickness of δ=3.6 mm, using a membraneless vertical oscillation technique, and 20 experiments were performed to establish a statistical ensemble. The average perturbation from this ensemble was extracted and used as input for a numerical simulation using the Lawrence Livermore National Laboratory (LLNL) Miranda code. Good qualitative agreement between the experiment and simulation was observed, while quantitative agreement was best at early to intermediate times. Several methods were used to extract the turbulent growth constant α from experiments and simulations while accounting for time varying acceleration. Experimental, average bubble and spike asymptotic self-similar growth rate values range from α=0.022 to α=0.032 depending on the method used, and accounting for variable acceleration. Values found from the simulations range from α=0.024 to α=0.041. Values of α measured in the experiments are lower than what are typically measured in the literature but are more in line with those found in recent simulations.

Lattice Boltzmann simulation of the Rayleigh–Taylor Instability (RTI) during the mixing of the immiscible fluids

The Lattice Boltzmann Method (LBM) was implemented to simulate the two-dimensional (2-D) Rayleigh–Taylor instability (RTI) during the mixing of the immiscible fluids. Two important parameters of the density and the viscosity as represented respectively by the Atwood and the Reynolds numbers were considered. In the calculation the density of two immiscible fluids was calculated by using the equation of state (EOS). In the present simulations, both Atwood and Reynolds numbers were varied in order to investigate the time variations of the interfacial behavior during RTI, the position of bubbles and spikes, and the horizontal average density. The results indicate that the Atwood number has a higher effect than the Reynolds number in the mixing process during RTI. The bubbles and spikes positions go farther in the displacement with the increase of both Reynolds and Atwood numbers. Moreover, the increase of both Reynolds and Atwood numbers accelerate the mixing process during RTI.

The effect of initial conditions on mixing transition of the Richtmyer–Meshkov instability

Abstract

Nonlinear dynamics of double-layer unstable interfaces with non-uniform velocity shear

The interaction of double-layer density stratified interfaces with initial non-uniform velocity shear is investigated theoretically and numerically, taking the incompressible Richtmyer–Meshkov instability as an example. The linear analysis for providing the initial conditions in numerical calculations is performed, and some numerical examples of vortex double layers are presented using the vortex sheet model. We show that the density stratifications (Atwood numbers), the initial distance between two interfaces, and the distribution of the initial velocity shear determine the evolution of vortex double layers. When the Atwood numbers are large, the deformation of interfaces is small, and the distance between the two interfaces is almost unchanged. On the other hand, when the Atwood numbers are small and the initial distance between two interfaces is sufficiently close (less than or equal to the half of the wavelength of the initial disturbance), the two interfaces get closer to each other and merge at the last computed stage due to the incompressibility. We confirm this theoretically expected fact numerically.

Modeling the amplitude growth of Richtmyer–Meshkov instability in shock–flame interactions

This paper discusses the shock–flame interactions and the aspects associated with it, including the types of interactions, role of interactions in turbulent flames, high pressure generation during interactions, initial pressure effects on interactions, equivalence ratio effects on turbulent interactions, and the Richtmyer–Meshkov instability (RMI). In particular, the theory of RMI and the models associated with its amplitude growth with time have been discussed. Then, a new analytical model, Al-Thehabey model, is presented based on the impulsive acceleration of Richtmyer, not the gravitational acceleration. This model predicts the amplitude growth of the interface perturbation in terms of Atwood number (A), wave number of the perturbation (k), interface velocity (uc), and time (t). This new model’s prediction of the amplitude growth of the RMI is tested on six different combinations of fluids at different interface velocities. The results of the new model are compared with the results of four other existing analytical models and the new model’s performance fared very well. In addition, the new model’s performance has been compared with the experimental results from a shock wave incident on CO2–air, at Mach number, M = 3.08, interface velocity, u = 699.1 m/s, Atwood number, A = 0.206, and wavelength, λ = 990.0 × 10−6 m. The new model showed much closer results with the experimental ones than all other models used in the evaluation. The advantage of this new model is that it is capable of predicting the amplitude growth for both linear, at the early stages of the instability, and non-linear later regimes of the instability. In addition, it covers a larger time-domain than both the Alon et al. and the Sadot et al. models.

Optimization of capsule dopant levels to improve fuel areal density*

Fuel areal density (ρR) of all recent indirectly driven, cryogenically-layered DT implosions at the National Ignition Facility (NIF) show a deficit when compared to simulations. Across all designs, experimental ρR is lower than in 1D simulations without alpha energy or momentum deposition. A series of layered implosions were fielded at NIF to assess the impact of fuel-ablator instability, as caused by M-band preheat, on lower-than-expected fuel areal density. The stability of the fuel-ablator interface is modified by varying the Atwood number through a series of experiments where capsules were fielded with different ablator dopant levels. A key finding of this campaign is that optimization of 1D physics (shock timing) dominates stabilization of the fuel-ablator interface.

Experimental investigation on single-medium stratified thermal energy storage system

The demand for renewable energy sources is limited by their inherent intermittent nature. The thermal energy storage technique overcomes this shortcoming by allowing storage of naturally occurring energy (e.g. solar) during “non-peak hours”. The article presents an experimental investigation of the sparsely studied ‘single medium thermocline’ (SMT) based single tank sensible thermal energy storage (TES). The work aims to reconsider its potential by minimizing the fluid dynamic perturbations during the charging cycle which disrupt the stability of the thermocline leading to undesirable energy losses. The article discusses the effect of Atwood number (density stratification) and mean temperature on TES effectiveness as well as providing the appropriate thermodynamic insights. Implementation of the experimental procedures and strategies discussed in this article demonstrates a stable thermocline which persists for more than 6 h (hours) with minimal energy degradation.

Fundamental study on chaotic transition of two-phase flow regime and free surface instability in gas deaeration process

Deaeration is a process of eliminating aspirated air from liquid in hydraulic reservoirs to avoid cavitation in the downstream pump blades. The complex fluid dynamics associated with deaeration is investigated. The three-dimensional buoyancy driven chaotic behavior of gas-liquid interfacial two-phase flow is studied. Parametric study is executed to understand change in internal flow physics (bubble coalescence, disintegration, horizontal spread, bubble velocity etc.), strength of accelerating Rayleigh-Taylor instability, turbulent kinetic energy, amplitude of upward velocity near free surface, and rise in free surface level with the variation of parameters like incoming mixture flow rate, incoming volume fraction of air, liquid fill depth, and Atwood number. The computations show increment in cavitation, wavenumber and amplitude of upward velocity towards oscillating free surface with incoming flow rate (Re). Cavitation and free surface instability show incremental trend with volume fraction of incoming air forming a kink (cavitation reduces) due to bubble coalescence in a threshold range of volume fraction of incoming air. With the variation of Atwood number, initially cavitation reduces. But after a critical value (A*) of Atwood number, effect of bubble disintegration, and rise of cavitation become prominent, which is formulated with respect to incoming flow rate (Re). With liquid fill depth, cavitation shows a slight decrement with almost equal deaeration and constant wavelength of free surface oscillation at an increasing buoyancy driven upward velocity. Some glimpse of design solution to reduce the cavitation and enhance the deaeration is also studied and formulated to get better understanding.

Rayleigh–Taylor instability at spherical interfaces between viscous fluids: The fluid/fluid interface

Through the computation of the most-unstable modes, we perform a systematic analysis of the linear Rayleigh–Taylor instability at a spherical interface separating two different homogeneous regions of incompressible viscous fluids under the action of a radially directed acceleration over the entire parameter space. Using the growth rate as the dependent variable, the parameter space is spanned by the spherical harmonic degree n and three dimensionless variables: the Atwood number A, the viscosity ratio s, and the dimensionless variable B=(aRρ22/μ22)1/3R, where aR, ρ2, and μ2 are the local radial acceleration at the interface and the density and viscosity of the denser overlying fluid, respectively. To understand the effect of the various parameters on the instability behavior and to identify similarities and differences between the planar and spherical configurations, we compare the most-unstable growth rates αP* (planar) and αS* (spherical) under homologous driving conditions. For all A, when s ≪ 1, the planar configuration is more unstable than the spherical (αP*>αS*) within the interval 0 < B < ∞. However, as s increases to O(1), there is a region for small values of B where αS*>αP*, whereas for larger values of B, αP*>αS* once again. When s ∼ 2, the maximum of αS* for the n = 1 mode is greater than αS* for any other mode (n ≥ 2). For s∼O(10), αS*>αP* for all A within 0 < B < ∞. We find that the instability behavior between the planar and spherical systems departs from each other for s ≳ 2 and diverges considerably for s ≫ 1. In the limit when s → ∞, the planar configuration reduces to the trivial solution αP*≡0 for all B and A, whereas αS* has a non-zero limiting value for the n = 1 mode but vanishes for all the other modes (n ≥ 2). We derive an equation for αS* in this limit and obtain closed form solutions for the maximum of αS* and the value of B at which this occurs. Finally, we compare the most-unstable growth rates between the exact dispersion relation and three different approximations to highlight their strengths and weaknesses.

A viscoelastic two-phase solver using a phase-field approach

In this work we discuss the implementation and the performance of an in-house viscoelastic two-phase solver, based on a diffuse interface approach. The Phase-Field method is considered and the Cahn-Hilliard equation is employed for describing the transport of a binary fluid system. The interface between the two fluids utilises a continuum approach, which is responsible for smoothing the inherent discontinuities of sharp interface models, facilitating studies that are related to morphological changes of the interface, such as droplet breakup and coalescence. The two-phase solver manages to predict the expected dynamics for all the cases investigated, and exhibits an overall good performance. The numerical implementation is able to predict the expected physical response of the oscillating drop case, while the performance is also validated by examining the droplet deformation case. The corresponding history of the deformation is predicted for several systems considering Newtonian fluids, viscoelastic fluids and combinations of both. Finally, we demonstrate the ability of the solver to capture the complex interfacial patterns of the Rayleigh-Taylor instability for different Atwood numbers when Newtonian fluids are considered. In the two regimes identified, the system is modified to consider viscoelastic fluids and the influence of elasticity is investigated.

Analytical model of nonlinear evolution of single-mode Rayleigh–Taylor instability in cylindrical geometry

Abstract

Flow transitions in collisions between vortex-rings and density interfaces

Flow transitions and vortical developments during vortex-ring collisions with a sharp water–oil density interface are studied using planar laser-induced fluorescence and time-resolved particle-image velocimetry techniques. Circular vortex-rings at Reynolds numbers of $$\hbox {Re}=1000, 2000$$ and 4000 colliding with a density interface characterized by an Atwood number of approximately $$A=0.045$$ were investigated. Results show that at $$\hbox {Re}=1000$$ , collision with the density interface produces vortical structures and flow transitions that are relatively similar to those for a solid-boundary collision. However, the dynamics underlying the present vortical formations and behaviour are different from those associated with solid-boundary collisions, in that the former are driven by baroclinic vorticity generation. Flow behaviour at $$\hbox {Re}=2000$$ shows more significant deformation of the density interface by the vortex-ring but overall behaviour remains comparable. Last but not least, at $$\hbox {Re}=4000$$ , the largest Reynolds number investigated here, the vortex-ring penetrates the density interface almost completely. However, buoyancy effects eventually limit its penetration and reverse its translational direction, such that it crosses back into the oil layer again with its vortex core rotational senses reversed as well. At the same time, vortex-ring fluid is shed and a significant trailing-jet is left in the former’s wake.

Two-equation and multi-fluid turbulence models for Richtmyer–Meshkov mixing

This paper concerns an investigation of two different approaches in modeling the turbulent mixing induced by the Richtmyer–Meshkov instability (RMI): A two-equation K-L multi-component Reynolds-averaged Navier–Stokes model and a two-fluid model. We have improved the accuracy of the K-L model by implementing new modifications, including a realizability condition for the Reynolds stress tensor and a threshold in the production of the turbulence kinetic energy. We examine the models in the one-dimensional (1D) form in the (re)-shocked mixing of a double-planar air and sulfur-hexafluoride (SF6) interface of the Atwood number |At| ≃ 0.6853. Furthermore, we investigated the models’ accuracy to RMI-induced mixing of a (re)-shocked planar-inverse chevron air–SF6 interface. Relevant integral quantities in time, as well as instantaneous profiles and contour plots, are used to assess the models’ accuracy against high-resolution implicit large eddy simulations. The proposed modifications improve the efficiency of the K-L model. The model is designed as a simple model capable of capturing the self-similar growth of Rayleigh–Taylor and Richtmyer–Meshkov flows. The two-fluid model remains more accurate but is also computationally more expensive.

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.

Effects of Atwood and Reynolds numbers on the evolution of buoyancy-driven homogeneous variable-density turbulence

The evolution of buoyancy-driven homogeneous variable-density turbulence (HVDT) at Atwood numbers up to 0.75 and large Reynolds numbers is studied by using high-resolution Direct Numerical Simulations. To help understand the highly non-equilibrium nature of buoyancy-driven HVDT, the flow evolution is divided into four different regimes based on the behavior of turbulent kinetic energy derivatives. The results show that each regime has a unique type of dependency on both Atwood and Reynolds numbers. It is found that the local statistics of the flow based on the flow composition are more sensitive to Atwood and Reynolds numbers compared to those based on the entire flow. It is also observed that at higher Atwood numbers, different flow features reach their asymptotic Reynolds number behavior at different times. The energy spectrum defined based on the Favre fluctuations momentum has less large scale contamination from viscous effects for variable density flows with constant properties, compared to other forms used previously. The evolution of the energy spectrum highlights distinct dynamical features of the four flow regimes. Thus, the slope of the energy spectrum at intermediate to large scales evolves from -7/3 to -1, as a function of the production to dissipation ratio. The classical Kolmogorov spectrum emerges at intermediate to high scales at the highest Reynolds numbers examined, after the turbulence starts to decay. Finally, the similarities and differences between buoyancy-driven HVDT and the more conventional stationary turbulence are discussed and new strategies and tools for analysis are proposed.

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

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

Achieving 280 Gbar hot spot pressure in DT-layered CH capsule implosions at the National Ignition Facility

We are reporting on a series of indirect-drive 0.9-scale CH capsule implosions (inner radius = 840 μm) fielded in low gas-fill (0.6 mg/cm3) hohlraums of 6.72 mm diameter at the National Ignition Facility. Thanks to the 11%-reduction of the capsule size at a given hohlraum diameter compared to previously tested full-scale capsules, we achieved good hot spot symmetry control near 33% cone-fraction and without the need to invoke cross beam energy transfer. As a result, we achieved a hot spot pressure of 280 ± 40 Gbar, which is the highest pressure demonstrated in layered DT implosions with CH capsules to date. Pushing this design to higher velocity resulted in a reduction of neutron yield. Highly resolved capsule simulations suggest that higher Au M-shell preheat resulted in an increase in Atwood number at the ablator–ice interface, which leads to increased fuel-ablator instability and mixing. The results reported here provide important scaling information for next-generation CH designs.

Properties of the mean momentum balance in turbulent Taylor–Couette flow

This paper investigates the properties of the mean momentum balance (MMB) equation in the azimuthal direction of a turbulent Taylor–Couette flow (TCF). The MMB- equation is integrated to determine the properties of the Reynolds shear stress. An approximation is developed for the Reynolds shear stress in the core region of a turbulent TCF, and the dependence of the peak Reynolds shear stress location on the Reynolds number and the gap geometry is revealed. The properties of the global integral of the turbulent Coriolis force are also revealed. For a turbulent TCF with a small gap or at high Reynolds numbers, the global integral of the turbulent Coriolis force is found to be only weakly influenced by the rotation ratio of the cylinders. Two controlling non-dimensional numbers are derived directly from the scaling analysis of the MMB- equation. The first is a geometry Atwood number to characterize the gap geometry, where is the gap half-width, and is the mid-gap radial location. The second is the friction Reynolds number defined as , where is the kinematic viscosity and is the friction velocity at the inner cylinder. A new three-layer structure is proposed for the inner half of a turbulent TCF at sufficiently high Reynolds number, based on the force balance in the MMB- equation. Layer I is an inner layer, where the force balance is between the viscous force and the Reynolds shear force: . Layer III occupies the core of the gap, where the force balance is between the turbulent Coriolis force and the Reynolds shear force: . In Layer II, all three forces contribute to the balance. An inner scaling is developed for Layer I, and an outer scaling is developed for Layer III. The inner and outer scalings are verified against direct numerical simulation data. Similarities and differences between the turbulent TCF and a pressure-driven turbulent channel flow are elucidated.