Initial Perturbation Amplitude Research Articles

Transition to plastic regime for Rayleigh-Taylor instability in soft solids.

Rayleigh-Taylor instability (RTI) is observed in soft materials that have significant resistance to yield. Estimating the instability threshold is critical to several engineering applications and has been the topic of several studies in past decades. However, limited attention has been given to the elastic-to-plastic (EP) transition threshold, where material properties vary significantly. This study explores the phase transition between the pure elastic and stable plastic regimes in RTI in a soft solid (mayonnaise) using a rotating wheel experimental setup with a time-varying acceleration profile. The material properties of the soft solid are characterized using rheological techniques. Different initial perturbation wavelength and amplitude combinations are used to analyze their role in the EP transition threshold and the subsequent maximum fully recoverable elastic strain. The effects of the initial perturbation dimensions, the steepness of the perturbation, and the mass of the perturbation on the stability of the samples, the EP transition thresholds, and the maximum fully recoverable elastic strains are analyzed. The samples with the largest full elastic recovery potentials and the parameters governing this phenomenon are identified, and the physics behind it is discussed. It is observed that increasing the initial perturbation wavelength decreases the required phase transition acceleration while increasing the maximum fully recoverable elastic strain. Finally, nondimensional parameters that involve the perturbation dimensions, as well as the mechanical properties of the material, are introduced to provide a generalized approach to the problem.

Numerical Diffusion and Turbulent Mixing in Convective Self‐Aggregation

AbstractConvective Self‐Aggregation (CSA) is a common feature of idealized numerical simulations of the tropical atmosphere in Radiative‐Convective Equilibrium (RCE). However, at coarse grid resolution where deep convection is not fully resolved, the occurrence of this phenomenon is extremely sensitive to subgrid‐scale processes. This study examines the role of mixing and entrainment, provided by the turbulence model and the implicit numerical diffusion. The study compares the results of two models, WRF and SAM, by varying turbulence models, initial conditions, and horizontal spatial resolution. At a coarse grid resolution of 3 km, the removal of turbulent mixing prevents CSA in models with low numerical diffusivity but is preserved in models with high numerical diffusivity. When the horizontal grid resolution is refined to 1 km, CSA can only be achieved by increasing explicit turbulent mixing, even with a small amount of shallow clouds. Therefore, the sensitivity of CSA to horizontal grid resolution is not primarily caused by the decrease in shallow clouds. The analysis of the total water path spectrum suggests that the amplitude of initial humidity perturbations introduced by convection in the free troposphere is the key factor. This amplitude is regulated by turbulent mixing and diffusion at small scales. Prior to the onset of CSA, increased mixing makes updrafts more sensitive to the dryness of the free troposphere, which strengthens the moisture‐convection feedback. This leads to an increased distance between convective cores and a stronger humidity perturbation in the free troposphere, which can destabilize the RCE state.

A weakly nonlinear amplitude equation approach to the bypass transition in the two-dimensional Lamb–Oseen vortex

We analytically derive an amplitude equation for the weakly nonlinear evolution of the linearly most amplified response of a non-normal dynamical system. The development generalizes the method proposed in Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), in that the base flow now arbitrarily depends on time, and the operator exponential formalism for the evolution of the perturbation is not used. Applied to the two-dimensional Lamb–Oseen vortex, the amplitude equation successfully predicts the nonlinearities to weaken or reinforce the transient gain in the weakly nonlinear regime. In particular, the minimum amplitude of the linear optimal initial perturbation required for the amplitude equation to lose a solution, interpreted as the flow experiencing a bypass (subcritical) transition, is found to decay as a power law with the Reynolds number. Although with a different exponent, this is recovered in direct numerical simulations, showing a transition towards a tripolar state. The simplicity of the amplitude equation and the link made with the sensitivity formula permits a physical interpretation of nonlinear effects, in light of existing work on Landau damping and on shear instabilities. The amplitude equation also quantifies the respective contributions of the second harmonic and the spatial mean flow distortion in the nonlinear modification of the gain.

Numerical study of Richtmyer–Meshkov instability of light fluid layer with reshock

The Richtmyer–Meshkov instability of a light fluid layer driven by reflected shock is investigated numerically and theoretically to reveal the feature of the interfacial evolution of light fluid layer under reshock. Cases with single- and double-perturbation helium gas layers are investigated to study the evolution and merging of interface, and comparisons with diverse layer thicknesses and initial perturbation amplitude are conducted to explore the coupling effects of the interface. For the single-perturbed case, the amplitude variation of the left interface exhibits a distinct inflection point as impacted by the reflected shock, and the growth rate in the reflected stage is noticeably larger than that in the incident stage. During the merging process of the interfaces, the displacement difference between the two interfaces and the amplitude growth of the interface play a dominant role before and after the reshock, respectively. For the double-perturbed cases, the head-on collision of the two interfaces' spike occurs when the two interfaces are initially anti-phase, and the spike “catching up with” the bubble occurs when the two interfaces are initially in phase. While the initial fluid layer is very thin, the merging of interfaces accelerates and the interface-coupling effect increases. A modified model has been proposed to predict the amplitude growth of the interface after reshock, which agrees well with the numerical results. The distribution and development of vorticity are similar for the studied cases with different initial amplitudes and fluid layer thicknesses.

Self-similar bubble-front evolutions of ablative Rayleigh–Taylor instability seeded by localized perturbations

Two-dimensional numerical simulations are carried out to investigate the nonlinear bubble growth of ablative Rayleigh–Taylor instability (ARTI) seeded by localized perturbations (LPs), where the LPs are described by a Gaussian mode. It is found that the nonlinear bubble-front penetration of LP-seeded ARTI follows the self-similar scaling law αbAT∫gdt2, different from the classical case, where the self-similar behavior is not observed. It is also found that the quadratic growth coefficient αb in the LP-seeded ARTI mainly depends on the initial perturbation amplitude and initial perturbation width. When the perturbation amplitude is small, αb has a value of ∼0.03, which is not sensitive to the perturbation width. As the perturbation amplitude increases, the value of αb increases, and the phenomenon is more significant when the perturbation width is narrower. It is shown that the increase in αb is due to the spike-induced upward jet and the ablation-generated vorticity inside the bubble.

Freeze-out of perturbation growth of single-mode helium–air interface through reflected shock in Richtmyer–Meshkov flows

‘Freeze-out’ of amplitude growth, i.e. the amplitude growth stagnation of a shocked helium–air interface, is realized through a reflected shock, which produces baroclinic vorticity of the opposite sign to that deposited by the first shock. Theoretically, a model is constructed to calculate the relations among the initial parameters for achieving freeze-out. In particular, if the amplitude growth is within the linear regime at the arrival of the reflected shock, the time interval between the impacts of two shock waves is linearly related to the initial perturbation wavelength, and is independent of the initial perturbation amplitude. Experimentally, an air–SF $_6$ (or air–argon) plane interface is adopted to produce a weak reflected shock. Seven experimental runs with specific initial conditions are examined. For all cases, freeze-out is achieved after the reflected shock impact under the designed conditions.

Investigation on the grain size effect on the copper shaped charge jet stretching behavior

The grain size effect on the shaped charge jet (SCJ) stretching process was analytically formulated and experimentally verified by penetration tests. The present analytical model predicts an optimum grain size for the SCJ performance, deduced from the concurrent effect of grain size on flow stress, strain rate sensitivity, and surface roughness. Specifically, reducing the grain size will improve the initial surface roughness and decrease the initial perturbation amplitude, favoring the SCJ stretching. On the other hand, the strain rate sensitivity and flow stress for copper increase with the decrease of grain size, facilitating the perturbation growth and leading to a premature breakup. Thus, the present analytical model predicts that the optimum grain size of the SCJ is about 1–5 μm. The penetration test verified that the shaped charge liner with an average grain size of about 3.6 ± 2.5 μm exhibited the largest penetration depth. The consistent results from the analytical model and the penetration experiments certify the feasibility of the present analytical model on the SCJ performance.

Large-amplitude effects on interface perturbation growth in Richtmyer–Meshkov flows with reshock

Experimental and theoretical studies on the Richtmyer–Meshkov (RM) instability of heavy/light gaseous interfaces with reshock are performed. Both small and large initial perturbation amplitudes of single- and quasi-single-mode interfaces are considered, highlighting the effects of interface amplitude and shape on the linear and nonlinear growths of the RM instability. The results indicate that for small-amplitude interfaces distorted before and after the first reshock arrival, the perturbation growths at linear stages can be well predicted by the impulsive model. For large-amplitude interfaces, however, the reshock acceleration on the evolving interface promotes the mode interaction and enhances the nonlinear effects, making the perturbation growth rates reduced in comparison with those in the singly shocked cases. The complete evolution, especially the bubble evolution, has a strong memory of initial shapes, while for large-amplitude cases, the spike evolution is nearly independent of them owing to the destruction of large-scale vortices and multiple-shock-induced small-scale structures. Compared with that of the single-mode case, the normalized perturbation growths after reshock for the quasi-single-mode cases are more sensitive to initial amplitudes. To better describe the linear growth rates of the RM instability induced by the incident shock and reshock, the reduction factor models for large-amplitude cases are developed, which successfully predict the non-monotonic dependence of linear growth rates on initial perturbation amplitudes. For small-amplitude cases, the nonlinear model proposed for the singly shocked case can predict the reshocked nonlinear growth, while for large-amplitude cases, it is invalid because the perturbation growth shows a linear characteristic after reshock.

Thin-shell effects on nonlinear bubble evolution in the ablative Rayleigh–Taylor instability

The influence of thin-shell effects on the nonlinear evolution of two-dimensional single-mode ablative Rayleigh–Taylor instability (ARTI) is studied in the parameter range of inertial confinement fusion implosions. A new phase of unsaturated nonlinear bubble evolution caused by thin-shell effects is found. This is different from the traditional opinion that the bubble velocity becomes saturated after the ARTI evolution enters a highly nonlinear regime. A modified bubble velocity formula is proposed, based on the Betti–Sanz model [Betti and Sanz, Phys. Rev. Lett. 97, 205002 (2006)], considering the thin-shell effects. It is shown that the bubble velocity becomes saturated in the thick-target case after the ARTI evolution enters a highly nonlinear regime. In this case, the Betti–Sanz bubble dynamics model can predict the evolution of bubble velocity. However, when the thin-shell effects become significant in the case of kD0 < 100, where D0 is the initial thickness of the target and k is the perturbation wavenumber, the difference of the average acceleration between the bubble vertex and the spike tip can be much more significant than that of the thick-target case. In this situation, the nonlinear evolution of the ARTI bubbles will accelerate without saturation until the target breakup, which cannot be depicted by the Betti–Sanz model while the improved theory formula is applicative. The Betti–Sanz model and the improved theory formula are independent of the initial perturbation amplitude.

Bursting on a vortex tube with initial axial core-size perturbations

Axisymmetric rectilinear vortex tubes develop twist waves if their core size varies along the centerline. These twist waves propagate, and when two opposite-signed twist waves meet their interaction leads to vortex bursting. We perform numerical simulations of vortex tubes with a range of initial core-size perturbation amplitudes, and provide a detailed analysis of the bursting dynamics. Our results explain the main mechanisms driving and subsequently arresting the bursting process; quantify the accelerated energy dissipation due to bursting; and show that bursting is susceptible to non-axisymmetric instabilities that further accelerate the decay of the vortex tube.

Receptivity of crossflow instability to discrete roughness amplitude and location

The effect of discrete roughness elements on the development and breakdown of stationary crossflow instability on a swept wing is explored. Receptivity to various element heights and chordwise locations is explored using a combination of experimental and theoretical tools. Forcing configurations, determined based on linear stability predictions, are manufactured and applied on the wing in a low turbulence facility. Measurements are performed using infrared thermography, quantifying the transition front location, and planar particle image velocimetry, providing a reconstruction of stationary crossflow instabilities and their associated growth. Measurements are corroborated with simulations based on nonlinear parabolised stability equations. Results confirm the efficacy of discrete roughness elements in introducing and conditioning stationary crossflow instabilities. Primary instability amplitudes and resulting laminar-turbulent transition location are found to strongly depend on both roughness amplitude and chordwise location. The Reynolds number based on element height is found to satisfactorily approximate the initial forcing amplitude, revealing the importance of local velocity effects in non-zero-pressure gradient flows. Direct estimation of initial perturbation amplitudes from nonlinear simulations suggests the existence of pertinent flow mechanisms in the element vicinity, active in conditioning the onset of modal instabilities. Dedicated velocimetry planes, elucidate the development of a momentum deficit wake which rapidly decays downstream of the element followed by mild growth, representing the first experimental evidence of transient behaviour in swept wing boundary layers. The outcome of this work identifies a strong scalability of the transition dynamics to roughness amplitude and location, warranting the upscaling of roughness elements to more accessible, measurable and spatially resolved configurations in future experiments.

Impulse Moments Influence on Single-Frequency Torsional Oscillations of Nonlinear Flexible Bodies

A method of analytical study of the influence of impulse moments on nonlinear torsional oscillations of a homogeneous constant cross-section of a body under classical boundary conditions of the first, second, and third types have been developed. When the flexible material properties meet the body close to the power law of flexibility, mathematical models of the process have been obtained. They are the boundary value problems for an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of pulse momentum on the oscillatory process. Under the effect of periodic pulse momentum on a flexible body, resonant and non-resonant processes are possible, and resonant processes occur when the amplitude of natural oscillations approaches a fixed value. The peculiarities of resonant oscillations are established. Relative torsional oscillations of a nonlinear flexible body that rotates around the axis with a constant portable angular velocity are considered, taking into account the periodic action of pulse momentum acting in a fixed cross-section. It is shown that: the amplitude of passing through the main resonance is more significant for larger values of the nonlinearity parameter; the case of the action of pulse momentum closer to the middle of the body. If the initial perturbation amplitude is less than the amplitude at which the resonance occurs in the presence of only internal forces of viscous friction, the external periodic impulse moments of resonance processes have occurred. The reliability of the calculation formulas is confirmed by the fact that in the extreme case known from the literature, resources are obtained related to the dynamics of the respective objects under the continuous action of autonomous perturbation.

Evolution mechanism of double-layer heavy gas column interface with sinusoidal disturbance induced by convergent shock wave

Based on Navier-Stokes equations, combining the fifth-order weighted essentially non-oscillatory scheme with the adaptive structured grid refinement technique, the interactions between converging shock and annular SF<sub>6</sub> layers with different initial perturbation amplitudes and thickness are numerically investigated. The evolution mechanism of shock structure and interface are revealed in detail, and the variations of the circulation, mixing rate and turbulent kinetic energy are quantitatively analyzed. The dynamic mode decomposition method is used to analyze the dynamic characteristics of the vorticity. The results show that in the case with large initial perturbation amplitude, the transmitted shock wave forms Mach reflection structures both inside and outside of the inner interface of SF<sub>6</sub> layer, and multiple shock focusing phenomena occur in the center. After the transmitted shock wave penetrates the outer interface, the circulation increases faster, and the “spike” and “bubble” structure on inner interface develop faster, so that the amplitude of the inner and outer interfaces and the gas mixing rate increase. As for the case with larger thickness of the gas layer, the phase of the transmitted shock wave changes inside the layer, which forms “bubble” at the crest of the inner interface and “spike” at the trough. When the thickness of the gas layer decreases, the crest of the inner interface does not move inside after being impacted, and “spike” and “bubble” are generated in the late stage. The dynamic modes show that the main structure of vorticity and the exchange of positive and negative vorticity on the main structure are determined by the modes with weak growth and low frequency, but the modes with weak growth and high frequency determine rapid exchange of positive and negative vorticity at the interface in the cases with weak coupling effect.

Application of Particular Solutions of the Burgers Equation to Describe the Evolution of Shock Waves of Density of Elementary Steps

Particular solutions of the Burgers equations (BE) with zero boundary conditions are investigated in an analytical form. For values of the shape parameter greater than 1, but approximately equal to 1, the amplitude of the initial periodic perturbations depends nonmonotonically on the spatial coordinate, i.e. the initial perturbation can be considered as a shock wave. Particular BE solutions with zero boundary conditions describe a time decrease of the amplitude of initial nonmonotonic perturbations, which indicates the decay of the initial shock wave. At large values of the shape parameter , the amplitude of the initial periodic perturbations depends harmoniously on the spatial coordinate. It is shown that over time, the amplitude and the spatial derivative of the profile of such a perturbation decrease and tend to zero. Emphasis was put on the fact that particular BE solutions can be used to control numerical calculations related to the BE-based description of shock waves in the region of large spatial gradients, that is, under conditions of a manifold increase in spatial derivatives. These solutions are employed to describe the profile of a one-dimensional train of elementary steps with an orientation near <100>, formed during the growth of a NaCl single crystal from the vapor phase at the base of a macroscopic cleavage step. It is shown that the distribution of the step concentration with distance from the initial position of the macrostep adequately reflects the shock wave profile at the decay stage. The dimensionless parameters of the wave are determined, on the basis of which the estimates of the characteristic time of the shock wave decay are made.

Plasma transport simulations of Rayleigh–Taylor instability in near-ICF deceleration regimes

Rayleigh–Taylor (R–T) instability between plasma species is examined in a kinetic test and near-inertial confinement fusion (ICF) regimes. A transport approximation to the plasma species kinetics is used to represent viscosity and species mass transport within a hydrodynamic fluid code (xRage). R–T simulation results are compared in a kinetic test regime with a fully kinetic particle-in-cell approach [vectorized particle-in-cell (VPIC)] and with an analytic model for the growth rate of R–T instability. Single-mode growth rates from both codes and the analytic model are in reasonable agreement over a range of initial wavelengths including the wavenumber of maximum growth rate. Both codes exhibit similar diffusive mixing fronts. Small code-to-code differences arise from the kinetics, while simulation-analytic model differences arise from several sources dominated by the choice of gradients establishing the hydrostatic equilibrium initial conditions. After demonstrating code agreement in the kinetic test regime, which is practically accessible to the VPIC code, then the xRage code, with the fluid plasma transport approximation, is applied to single mode R–T instability under deceleration conditions closer to an ICF implosion, approximated with a carbon (C) shell imploding on a deuterium (D) fuel. The analytic wavelength of maximum instability is limited by the kinetics, primarily in the viscosity, and is found to be ≈10 μm for an ion temperature near 1 keV at this C–D interface, with the most unstable wavelength increasing as temperature increases. The analytic viscous model agrees with simulation results over a range of initial perturbation wavelengths, provided the simulation results are analyzed over a sufficiently short duration (⪅0.2 ns in this case). Details of the fluid structure evolution during this R–T deceleration are compared between the inviscid Euler equations and cases, which include plasma transport over a range in initial wavelengths and initial perturbation amplitudes. The inviscid Euler solutions show a grid-dependent cascade to smaller scale structures often seen in the R–T instability, while simulations with plasma transport in this deceleration regime develop a single vortex roll-up, as the plasma transport smoothes all hydrodynamic fluid structures smaller than several micrometers. This leads to a grid-converged transient solution for the R–T instability when kinetic effects are included in the simulations, and thus represents a direct numerical simulation of the thermal ions during R–T unstable mixing in ICF relevant conditions.

The early-time dynamics of Rayleigh-Taylor mixing with a premixed layer

The early-time dynamics of Rayleigh-Taylor instability with a premixed density gradient layer was investigated by three-dimensional implicit large eddy simulation. Compared with the classic Rayleigh-Taylor instability, it was found that the mixing layer undergoes an inactive stage when the mixing width of the layer is nearly unchanged. The development of hydrodynamic instability with a premixed layer was studied. Energy transfer and vortex generation at the early time show that span-wise vortices predominate at this stage. Afterwards, initial conditions including the parameters of initial perturbations and the mixing width of premixed layer were investigated in detail. In our study, it was found that the amplitude of initial perturbations and the width of premixed layer are important for determining the characteristic time scale of the inactive stage while the wave number has less effect. The influence of initial conditions on the early-time dynamics of mixing layer was also analysed.

What really governs the upper bound of uniform ductility in gradient or layered materials?

Gradient metallic materials, which can be made by surface mechanical attrition, non-equilibrium growth, or other thermomechanical means, have been deemed as an effective way to overcome the strength-ductility tradeoff as commonly seen in traditional metallic alloys. Although a large number of research works have been devoted to the unique strengthening mechanisms as resulting from the enhanced plastic-strain gradients and back stresses in these novel materials, the disproportionally fewer studies on ductility are rather restricted to the delay of the Considère necking limit due to the elevated work hardening. This view is incomplete, because the onset of necking (i.e., the uniform ductility) depends critically on both the amplitude and wavelength of initial perturbations. Even with a mere simplification of gradient materials by a film-on-substrate or a sandwich structure, this work shows that with random surface perturbations, the upper bound of uniform ductility is dictated by an extremely short-wavelength perturbation at the surface of the outer hard layer (thus denoted as Rayleigh mode). On the other hand, a pre-patterned surface with perturbations of varying periods is practicably unable to evolve into the Rayleigh asymptote and thus can achieve even higher uniform ductility. A revisit of literature experiments is suggested to be undertaken with a particular focus on the spectral growth kinetics of surface morphology and its connection to uniform ductility in these gradient materials.

Investigation of Rayleigh-Taylor instability in copper plate under explosive loading

The Rayleigh–Taylor instability in metal is of great practical interest to a diverse range of fields, and it is significantly different from the traditional fluid interface instability. In this paper, a set of high explosives driven Rayleigh–Taylor instability experiments in copper plate are presented for intensively understanding the perturbation growth behavior of metal interface instability. The perturbation growth history on copper interface at a specific time is recorded by flash radiography. The experimental results show that the evolution of interface perturbation is sensitive to its initial perturbation characters under a given driving pressure, the larger the initial perturbation amplitude, the faster the perturbation grows, but the perturbation wavelength of the interface remains almost unchanged at the explosive loading. The perturbation on the front interface will influence the interface features of the back free interface over time. Namely, the position corresponding to the perturbation trough on the front interface gradually evolves into a peak and vice versa.

Direct numerical simulations of incompressible Rayleigh–Taylor instabilities at low and medium Atwood numbers

Direct numerical simulations of two-dimensional (2D) and three-dimensional (3D), single-mode and multi-mode, incompressible immiscible Rayleigh–Taylor (RT) instabilities are performed using a phase-field approach and high-order finite-difference schemes. Various combinations of Atwood number, Reynolds number, surface tension, and initial perturbation amplitude are investigated. It is found that at high Reynolds numbers, the surface tension, if significant, could prevent the formation of Kelvin–Helmholtz type instabilities within the bubble region. A relationship is proposed for the vertical distance of the bubble and spike vs the Atwood number. The spike and bubble reaccelerate after reaching a temporary plateau due to the reduction of the friction drag as a result of the formation of the spike vortices and also the formation of a momentum jet traveling upward within the bubble region. The interface for a 3D single-mode instability grows exponentially; however, a higher Reynolds number and/or a lower Atwood number could result in a noticeably larger surface area after the initial growth. It is also shown that a 3D multi-mode RT instability initially displays an exponential interface growth rate similar to single-mode RT instabilities. Due to the collapse and merging of individual single-mode instabilities, the interface area for a multi-mode RT instability is strongly dependent to the mesh resolution after the exponential growth rate. However, the ratio of kinetic energy over released potential energy exhibits an almost steady state after the initial exponential growth, with values around 0.4, independently of the mesh resolution.

Effects of the shape of curvature peaks on the size of primordial black holes

We simulate numerically the formation of spherically symmetric primordial black holes (PBHs) seeded by different families of primordial curvature perturbations profiles in a radiation dominated Friedman-Robertson-Walker (FRW) Universe. We have studied the dependency on the curvature profile of the initial mass M BH,i of the PBHs at the time of apparent horizon formation t AH, and the final mass M BH,f after the accretion process, using an excision technique, comparing M BH,i to previous analytical estimations obtained using a compensated PBHs model approach. The analytical estimations are in agreement with numerical results, except for large values of the initial perturbation amplitude, when the compensated model is less accurate. The masses M BH,f and M BH,i do not depend only on the shape around the compaction function peak, but on the full profile of the initial curvature perturbation. We also estimate the accretion effects, and for PBHs with masses relevant for the dark matter abundance, with a final mass equal to the horizon crossing mass, we find M BH,f ≈ 3 M BH,i.