Large Initial Amplitude Research Articles

Transmission and reflection of three-dimensional Boussinesq internal gravity wave packets in nonuniform retrograde shear flow

Simulated fully localized internal gravity wave packets with moderately large initial amplitude exhibit finite transmission across a reflection level, due to the combined effects of the wave-induced shear locally cancelling the retrograde background shear, modulational instability, and the nonlinear generation and evolution of secondary wave packets.

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.

FIXATION OF THE RELATION BETWEEN FREQUENCY AND AMPLITUDE FOR NONLINEAR OSCILLATOR HAVING FRACTIONAL TERM APPLYING MODIFIED MICKENS’ EXTENDED ITERATION METHOD

A modified extended iteration procedure is applied to compute the analytical periodic solutions of the nonlinear oscillator having fractional terms. A nonlinear oscillator with force is given to demonstrate the effectiveness and expediency of the iteration scheme. Mickens’ extended iteration method is a well-established method for studying random oscillations. The method is also simple and straightforward to accomplish approximate frequency and the corresponding periodic solution of the strongly nonlinear oscillator. The method gives high validity for both small and large initial amplitudes of oscillations. We have used an appropriate truncation of the obtained Fourier cosine series in each step of iterations to determine the approximate analytic solution of the oscillators. The second, third, and fourth approximate frequencies of the truly nonlinear oscillator with force show a good agreement with their exact values. Also, we have compared the calculated results with some of the existing results. We have shown that the method performs reasonably better.

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.

Nonlinear vibration analysis of the large-amplitude asymmetric response of ship roll motion

Ship roll motion is the single most important factor responsible for ship capsizes. Yet, there is still much to be understood about this subject; especially, how critical parameters influence the asymmetric response during large initial roll amplitudes. Consequently, this paper presents an investigation of the nonlinear vibration of the large-amplitude asymmetric response during ship roll motion. The dynamic model governing the roll motion accounted for quadratic damping and cubic nonlinear restoring moment with softening behaviour. The continuous piecewise linearization method was adopted to solve the governing model. Hence, the effect of four critical parameters (i.e. initial roll amplitude, added mass, quadratic damping and nonlinear restoring moment) on the asymmetric roll response and capsize tendency was investigated. It was observed that capsize tendency was strongly linked to the offset of the equilibrium point and the asymmetric response, which amplified the initial roll amplitude. An increase in the initial roll amplitude or quadratic damping produced a stronger asymmetric response and an increased capsize tendency, while an increase in the added mass or nonlinear restoring moment weakened the asymmetric response and decreased the capsize tendency. Also, an increase in any of the four critical parameters produced a decrease in the frequency of vibration.

Transient nonlinear Rayleigh–Bénard convection with single-mode initiation

Transient nonlinear convection in a finite Prandtl number fluid heated uniformly from below is studied analytically and numerically. We consider the simple geometry of a square cavity with normal-mode compatible boundary conditions. By design, only the marginal state of convection onset contributes to the initial condition for the two-dimensional supercritical convection. The thermal amplitude and the flow amplitude are taken as two independent initial conditions. There are two ways of initiation: (i) soft start with a very small initial amplitude, leading to early exponential growth. (ii) Kick-started transient convection with relatively large initial amplitudes, by which we perform a small-time expansion for benchmarking. Seemingly complicated transient flow occurs with a kick-start where the initial spin and the initial buoyancy torque are in conflict. However, the intricate spiraling flow decays after a couple of reversals of flow directions, and a steady convection settles. This is due to the strict antisymmetry of the temperature perturbation around the mid-point of the cavity, in combination with the symmetry of the flow field.

Dynamical analysis of screening in scalar-tensor theory

We investigate linear and non-linear dynamics of spherically symmetric perturbations on a static configuration in scalar-tensor theories focusing on the chameleon screening mechanism. We particularly address two questions: how much the perturbations can source the fifth force when the static background is well screened, and whether the resultant fifth force can change the stability and structure of the background configuration. For linear perturbations, we derive a lower bound for the square of the Fourier mode frequency $\omega^2$ using the adiabatic approximation. There may be unstable modes if this lower bound is negative, and we find that the condition of the instability can be changed by the fifth force although this effect is suppressed by the screening parameter. For non-linear perturbations, because we are mainly interested in short wavelength modes for which the fifth force may become stronger, we perform numerical simulations under the planar approximation. For a sufficiently large initial amplitude of the density perturbation, we find that the magnitude of the fifth force can be comparable to that of Newtonian gravity even when the model parameters are chosen so that the static background is well screened. It is also shown that if the screening is effective for the static background, the fluid dynamics is mostly governed by the pressure gradient and is not significantly affected by the fifth force.

A Modified Energy Balance Method to Obtain Higher-order Approximations to the Oscillators with Cubic and Harmonic Restoring Force

This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebraic equations are cumbersome to investigate especially in the case of a large initial oscillation amplitude. This limitation is overcome in the proposed method by using the iterative procedure based on the homotopy perturbation method. The approximated results agree well with the numerically obtained exact solutions. These third-order approximate solutions are found to be almost the same as exact solutions, which cannot be found using the existing energy balance method. Highly accurate result and simple solution procedure are advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.

Estimation of Nonlinear Roll Damping by Analytical Approximation of Experimental Free-Decay Amplitudes

Damping is critical for the roll motion response of a ship in waves. A common method for the assessment of damping in a ship’s rolling motion is to perform a free-decay experiment in calm water. In this paper, we propose an approach for estimating nonlinear damping that involves a linear exponential analytical approximation of the experimental roll free-decay amplitudes, followed by parametric identification based on the asymptotic method. The restoring moment can be strongly nonlinear. To validate this method, we first analyzed numerically simulated roll free-decay data using rolling equations with two alternative parametric forms: linear-plus-quadratic and linear-plus-cubic damping. By doing so, we obtained accurate estimates of nonlinear damping coefficients, even for large initial roll amplitudes. Then, we applied the proposed method to real free-decay data obtained from a scale model of a bulk barrier, and found the simulated results to be in good agreement with the experimental data. Using only free-decay peak data, the proposed method can be used to estimate nonlinear roll-damping coefficients for conditions with a strongly nonlinear restoring moment and large initial roll amplitudes.

Gravitational collapse of a massless scalar field in a periodic box

Gravitational collapse of a massless scalar field with the periodic boundary condition in a cubic box is reported. This system can be regarded as a lattice universe model. We construct the initial data for a Gaussian-like profile of the scalar field taking the integrability condition associated with the periodic boundary condition into account. For a large initial amplitude, a black hole is formed after a certain period of time. While the scalar field spreads out in the whole region for a small initial amplitude. It is shown that the expansion law in a late time approaches that of the radiation dominated universe and the matter dominated universe for the small and large initial amplitude cases, respectively. For the large initial amplitude case, the horizon is initially a past outer trapping horizon, whose area decreases with time, and after a certain period of time, it turns to a future outer trapping horizon with the increasing area.

Voltage–Amplitude Response of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Cantilever Resonators

This paper investigates the voltage–amplitude response of superharmonic resonance of second order (order two) of alternating current (AC) electrostatically actuated microelectromechanical system (MEMS) cantilever resonators. The resonators consist of a cantilever parallel to a ground plate and under voltage that produces hard excitations. AC frequency is near one-fourth of the natural frequency of the cantilever. The electrostatic force includes fringe effect. Two kinds of models, namely reduced-order models (ROMs), and boundary value problem (BVP) model, are developed. Methods used to solve these models are (1) method of multiple scales (MMS) for ROM using one mode of vibration, (2) continuation and bifurcation analysis for ROMs with several modes of vibration, (3) numerical integration for ROM with several modes of vibration, and (4) numerical integration for BVP model. The voltage–amplitude response shows a softening effect and three saddle-node bifurcation points. The first two bifurcation points occur at low voltage and amplitudes of 0.2 and 0.56 of the gap. The third bifurcation point occurs at higher voltage, called pull-in voltage, and amplitude of 0.44 of the gap. Pull-in occurs, (1) for voltage larger than the pull-in voltage regardless of the initial amplitude and (2) for voltage values lower than the pull-in voltage and large initial amplitudes. Pull-in does not occur at relatively small voltages and small initial amplitudes. First two bifurcation points vanish as damping increases. All bifurcation points are shifted to lower voltages as fringe increases. Pull-in voltage is not affected by the damping or detuning frequency.

Self-Similar Multimode Bubble-Front Evolution of the Ablative Rayleigh-Taylor Instability in Two and Three Dimensions.

The self-similar nonlinear evolution of the multimode ablative Rayleigh-Taylor instability (ARTI) is studied numerically in both two and three dimensions. It is shown that the nonlinear multimode bubble-front penetration follows the α_{b}A_{T}(∫sqrt[g]dt)^{2} scaling law with α_{b} dependent on the initial conditions and ablation velocity. The value of α_{b} is determined by the bubble competition theory, indicating that mass ablation reduces α_{b} with respect to the classical value for the same initial perturbation amplitude. It is also shown that ablation-driven vorticity accelerates the bubble velocity and prevents the transition from the bubble competition to the bubble merger regime at large initial amplitudes leading to higher α_{b} than in the classical case. Because of the dependence of α_{b} on initial perturbation and vorticity generation, ablative stabilization of the nonlinear ARTI is not as effective as previously anticipated for large initial perturbations.

Improved thin layer model of classical Rayleigh-Taylor instability for the deformation of interface

The thin shell (layer) configuration is adopted in inertial-confinement fusion (ICF) implosions. The weakly nonlinear deformation of the thin shell significantly influences the performances of implosion acceleration and fusion ignition, which is an important issue for the study of ICF physics. Based on the thin layer model of Ott (Ott E 1972 Phys. Rev. Lett. 29 1429), an improved thin layer model is proposed to describe the deformation and nonlinear evolution of the perturbed interface induced by the Rayleigh-Taylor instability (RTI). Differential equations describing motion are obtained by analyzing the forces of fluid elements (i.e., Newton's second law), which are then solved by numerical method. Then the position of the perturbed interface with an initial perturbation can be obtained. The linear growth rate obtained from our thin layer approximation agrees with that from the classical RTI. For fixed Atwood number (wave number), the total amplitudes of the bubble and spike obtained from the improved thin layer model agree with those from the three-order weakly nonlinear model. In addition, we compare the deformation and evolution of the layer from our model with results of the numerical simulation. In the linear regime, the amplitudes of the bubble and spike obtained from our model agree with those from the numerical simulation. And the evolution of the perturbed interface obtained from the improved thin layer model is consistent with that from the numerical simulation. In the nonlinear regime, the evolution trends of the total amplitude of the bubble and spike for both the improved thin layer model and numerical results are the same. However, the amplitude of the bubble is obviously greater than that of the spike in the later stage of the perturbation. This is because of some shortcomings in the improved thin layer model. The first shortcoming is that ignoring the dynamical pressure in the pressure difference. In fact, the shear velocity of the fluids plays an important role in the nonlinear regime of the perturbation. The second shortcoming is that the surface area of the upper interface equals the lower interface in the whole perturbation process of the present model. Thus, the present model can be used to describe the nonlinear evolution of the perturbed interface before the mushroom structure. Finally, it is worth noting that the improved thin layer model can be used to describe the deformation and nonlinear evolution of a thin layer for arbitrary Atwood number with a perturbation of large initial amplitude and arbitrary distribution. The initial perturbations of the triangular and rectangular waves are also discussed.

Gravitational Collapse of Massless Fields in an Expanding Universe

Gravitational collapse of a massless scalar field with the periodic boundary condition in a cubic box is reported. This system can be regarded as a lattice universe model. The initial data is constructed for a Gaussian like profile of the scalar field taking the integrability condition associated with the periodic boundary condition into account. For a large initial amplitude, a black hole is formed after a certain period of time. While the scalar field spreads out in the whole region for a small initial amplitude. The difference of the late time expansion law of the lattice universe depending on the final fate of the gravitational collapse is discussed.

A comparative study of the single-mode Richtmyer–Meshkov instability

In this work, the single-mode Richtmyer–Meshkov instability is studied numerically to find a reasonable nonlinear theoretical model which can be applied to predict the interface evolution from the linear stage to the early nonlinear stage. The cut-cell-based sharp-interface methods $$\hbox {MuSiC}^{+}$$ (Chang et al. in J Comput Phys 242:946–990, 2013) and CCGF (Bai and Deng in Adv Appl Math Mech 9(5):1052–1075, 2017) are applied to generate numerical results for comparisons. Classical Air–SF $$_{6}$$ and Air–Helium conditions are applied in this study, and initial amplitude and Atwood number are varied for comparison. Comparisons to the simulation results from the literature show the applicability of $$\hbox {MuSiC}^{+}$$ and CCGF. Comparisons to the nonlinear theoretical models show that ZS (Zhang and Sohn in Phys Lett A 212:149–155, 1996; Phys Fluids 9:1106–1124, 1997), SEA (Sadot et al. in Phys Rev Lett 80:1654–1657, 1998), and DR (Dimonte and Ramaprabhu in Phys Fluids 22:014104, 2010) models are valid for both spike and bubble growth rates, and MIK (Mikaelian in Phys Rev E 67:026319, 2003) and ZG (Zhang and Guo in J Fluid Mech 786:47–61, 2016) models are valid for bubble growth rate, when the initial perturbation is small and the Atwood number is low, but only the DR model is applicable for both spike and bubble growth rates when the initial perturbation amplitude and the Atwood number are large. A new term of non-dimensional initial perturbation amplitude is presented and multiplied to the DR model to get a unified fitted DR model, which gives consistent results to the simulation ones for small and large initial amplitudes.

Transformation of mode-2 internal solitary wave over a pseudo slope-shelf

Numerical simulations are performed to investigate the effect of wave amplitude in a numerical wave tank on the evolution of a convex mode-2 internal solitary wave (ISW) propagating over a pseudo slope-shelf. A finite volume method based on a Cartesian grid system is adopted to solve the Navier-Stokes equations using Improved Delayed Detached Eddy Simulation model for the turbulent closure. Numerical results reveal three types of waveform during wave generation on the flat bottom: (1) pseudo vortex shedding in the case of very large initial amplitude; (2) PacMan phenomenon in large amplitude; and (3) smooth mode-2 ISW for small amplitude. During wave propagation on the plateau, the first type of waveform induces a quasi-elevated mode-1 ISW; the second generates chaotic internal waves with significant reduction in amplitude; while the third renders a slightly deformed mode-2 ISW across the plateau. Moreover, the decrease in the magnitude of leading trough is more intense than that in the leading crest due to strong wave-obstacle interaction in the case of very large initial wave amplitude.

Crossing Over from Attractive to Repulsive Interactions in a Tunneling Bosonic Josephson Junction.

We explore the interplay between tunneling and interatomic interactions in the dynamics of a bosonic Josephson junction. We tune the scattering length of an atomic ^{39}K Bose-Einstein condensate confined in a double-well trap to investigate regimes inaccessible to other superconducting or superfluid systems. In the limit of small-amplitude oscillations, we study the transition from Rabi to plasma oscillations by crossing over from attractive to repulsive interatomic interactions. We observe a critical slowing down in the oscillation frequency by increasing the strength of an attractive interaction up to the point of a quantum phase transition. With sufficiently large initial oscillation amplitude and repulsive interactions, the system enters the macroscopic quantum self-trapping regime, where we observe coherent undamped oscillations with a self-sustained average imbalance of the relative well population. The exquisite agreement between theory and experiments enables the observation of a broad range of many body coherent dynamical regimes driven by tunable tunneling energy, interactions and external forces, with applications spanning from atomtronics to quantum metrology.

A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

The appearance of a fundamental long-time asymptotic regime in the two space-one time dimensional hyperbolic nonlinear Schrödinger (HNLS) equation is discussed. Based on analytical and numerical simulations, a wide range of initial conditions corresponding to initial lumps of moderate energy are found to approach a quasi--self-similar solution. Even relatively large initial amplitudes, which imply strong nonlinear effects, eventually lead to local structures resembling those of the self-similar solution, with appropriate small modifications. This solution has aspects that suggest it is a universal attractor emanating from a wide range of initial data.

Oscillatory Marangoni flows with inertia

When the surface of a liquid has a non-uniform distribution of a surfactant that lowers surface tension, the resulting variation in surface tension drives a flow that spreads the surfactant towards a uniform distribution. We study the spreading dynamics of an insoluble and non-diffusing surfactant on an initially motionless liquid. We derive solutions for the evolution over time of sinusoidal variations in surfactant concentration with a small initial amplitude relative to the average concentration. In this limit, the coupled flow and surfactant transport equations are linear. In contrast to exponential decay when the inertia of the flow is negligible, the solution for unsteady Stokes flow exhibits oscillations when inertia is sufficient to spread the surfactant beyond a uniform distribution. This oscillatory behaviour exhibits two properties that distinguish it from that of a simple harmonic oscillator: the amplitude changes sign at most three times, and the decay at late times follows a power law with an exponent of $-3/2$. As the surface oscillates, the structure of the subsurface flow alternates between one and two rows of counter-rotating vortices, starting with one row and ending with two during the late-time monotonic decay. We also examine numerically the evolution of the surfactant distribution when the system is nonlinear due to a large initial amplitude.

The Richtmyer–Meshkov instability of a ‘V’ shaped air/ interface

The Richtmyer–Meshkov instability on a ‘V’ shaped air/SF$_{6}$ gaseous interface is experimentally studied in a shock tube. By the soap film technique, a discontinuous interface without supporting mesh is formed so that the initial conditions of the interface can be accurately controlled. Five ‘V’ shaped air/$\text{SF}_{6}$ interfaces with different vertex angles ($60^{\circ }$, $90^{\circ }$, $120^{\circ }$, $140^{\circ }$ and $160^{\circ }$) are created where the ratio of the initial interface amplitude to the wavelength varies to highlight the effects of initial condition on the flow characteristics. The wave patterns and interface morphologies are clearly identified in the high-speed schlieren sequences, which show that the interface deforms in a less pronounced manner with less vortices generated as the vertex angle increases. A regime change is observed in the interface width growth rate near a vertex angle of $160^{\circ }$, which provides an experimental evidence for the numerical results obtained by McFarland et al. (Phys. Scr. vol. T155, 2013, 014014). The growth rate of interface width in the linear phase is compared with the theoretical predictions from the classical impulsive model and a modified linear model, and the latter is proven to be effective for a moderate to large initial amplitude. It is found that the initial growth rate of the interface width is a non-monotone function of the initial vertex angle (amplitude–wavelength ratio), i.e. the interface width growth rate in the linear stage experiences an increase and then a decrease as the vertex angle increases. A similar conclusion was also reached by Dell et al. (Phys. Plasmas, vol. 22, 2015, 092711) numerically for a sinusoidal interface. Finally, the general behaviour of the interface width growth in the nonlinear stage can be well captured by the nonlinear model proposed by Dimonte & Ramaprabhu (Phys. Fluids, vol. 22, 2010, 014104).