Spike Velocity Research Articles

Nonlinear, single-mode, two-dimensional Rayleigh-Taylor instability in ideal media.

A model for the single mode, two-dimensional Rayleigh-Taylor instability in ideal, incompressible, immiscible, and inviscid fluids is developed as an extension of a previous linear model based on the Newton's second law [A. R. Piriz et al., Am. J. Phys. 74, 1095 (2006)0002-950510.1119/1.2358158]. It describes the transition from linear to nonlinear regimes and takes into account the mass of fluids participating in the motion during the instability evolution, including the laterally displaced mass. This latter feature naturally leads to the bubble and spike velocity saturation without requiring the usual drag term necessary in the well-known buoyancy-drag model (BDM). In addition, it also provides an explanation to the latter phase of bubble reacceleration without appealing to the vorticity generation due to the Kelvin-Helmholtz instability. The model is in perfect agreement with the BDM buoyancy-drag model, but, apart from extending its range of application, it solves many of its issues of concern and provides a more consistent physical picture.

Effects of the stretching program in male volleyball players with posterior shoulder tightness

BackgroundStretching programs are often used to improve muscle stiffness and thereby posterior shoulder tightness (PST); however, knowledge about their effects on the viscoelastic properties of muscles and performance is limited. ObjectiveThe aim of this study was to investigate the effects of a six-week stretching program on the viscoelastic properties of posterior shoulder muscles, shoulder functional performance and range of motion in volleyball players with PST. Study designA Randomized Controlled Trial. Level of evidenceLevel II. MethodsThe study was conducted in 34 male (aged 19–26) professional volleyball players with PST. They were randomly assigned to stretching and control groups. A stretching routine (combining sleeper stretch and cross body stretch exercises) was performed in the stretching group for six weeks. Measurements were taken during the first assessment and six weeks after the initiation of stretching for the following: shoulder external/internal rotation range of motion, spike velocity, single arm seated shot-put test, and infraspinatus– posterior deltoid muscle stiffness (by shear wave elastography). ResultsWhile the stretching group had statistically better results regarding increased shoulder internal rotation (p < 0.01), spike velocity (p = 0.02) and seated single arm shot put distance (p < 0.01), stretching did not have any effect on tissue stiffness. ConclusionsA 6-week program of sleeper stretch, and cross body stretch exercises improved shoulder internal rotation and functional performance. A clinically meaningful change in muscle mechanical properties cannot be detected after a 6-week stretching program.

Robust implementation of Physical Regime Sensitivity and demonstration on Richtmyer–Meshkov Instability experiments

We present a robust analytic implementation of Physical Regime Sensitivity (PRS), which quantifies sensitivity of an experiment or application to regimes of a material model’s independent variables rather than to model parameters. The new PRS implementation is tested on impact-driven Richtmyer–Meshkov Instability (RMI) strength experiments on tantalum. Sequential PRS hydrocode simulations perturbed strength in local regimes of independent variables like strain rate to quantify the resulting effects on experimentally measured maximum spike velocities in the instabilities. The numeric behavior of the sensitivity calculation is investigated regarding mesh convergence and the parameters controlling the strength perturbation. The sensitivities generally converge with mesh size at least as quickly as the underlying RMI simulations. A wide range of perturbation amplitudes produced similar normalized sensitivities, indicating the perturbation is not changing physical behavior in the simulation. Adjusting the range (width) over which the strength is perturbed allows for resolving fine sensitivity features covering narrow regimes of independent variables or more broad regimes for general interpretation. For interpreting RMI, PRS shows that the maximum spike velocity of the instability is most sensitive to a narrow regime of strain rates near 107/s, which occurs inside the spike as strength effects arrest the instability. The PRS analyses revealed that a modification to the experimental metric used to assess strength would better isolate the strength effects in the regimes of most interest. Using the new metric to generate pressure PRS curves revealed that sensitivities to strength at near-zero pressure were 3.5 times greater than at the high-pressure shock state, supporting the practice of using RMI to isolate strain rate effects when combined with high-pressure experiments. It is also shown that a simple, calibrated constant strength model can reproduce PRS sensitivities of a complex, multivariate Preston–Tonks–Wallace strength model quite well, which supports the robustness of PRS and its utility for experimental design and interpretation prior to the availability of a sophisticated material strength model.

Numerical investigations of spike velocity of microjetting from shock-loaded aluminum and tin

Microjetting is induced by the interaction between incident shock wave and material free surface. Moreover, spike velocity is one of the most important parameters to quantify the microjetting processes. The spike velocity over a wide range of shock and surface conditions for aluminum and tin is achieved via systematical Eulerian peridynamic simulations. The four popular spike velocity models are assessed according to the simulation data. The results indicate that the model proposed by Dimonte et al. based on Richtmyer–Meshkov instability (RMI) mechanism is in best agreement with the simulated spike velocities. The further analysis highlights that the dependence of adiabatic index on material states should be taken into account because it can impact the RMI flows. Then an improved spike velocity model is proposed and the comparative investigations indicate that it exhibits an overall better prediction compared to the original velocity model.

Ab. No. 157 To Study the Effect of Posterior Oblique Sling Exercises on Performance Parameter in Volleyball Female Players

Introduction: Review of the literature and on field observations found that there were paucity of evidence showing direct relation of TheraBand and Swiss ball exercise on activation of POS for performance enhancement. Therefore, in present study, we are going to evaluate the effect of POS activation with the help of TheraBand and Swiss ball stability exercises to improve performance parameters. The Purpose of study is to know the effect of posterior oblique sling (POS) activation exercise on Performance Parameters (spike velocity, ROM) in female volleyball players. Methods: 30 female volleyball players were selected by Purposive and convenient sampling and randomly allocated into two groups equally 15 each based on inclusion and exclusion criteria. experimental study on the POS activation exercises at MP volleyball association, Brothers club Indore. The demographic data including age, height, weight, were collected through a data collection sheet. Result: Means of thoracolumbar rotation after Theraband and Swissball exercise are 31.67 and 35.67 serve velocity are 63.60 and 72.40 Showing significant difference between means. The calculated t-value is 2.65 and 8.57 which is significant at the degree of freedom 28 at 0.05 level of significant because the calculated t-value is greater than (2.05) minimum value at 0.05 significant level. Conclusion: In this study statistically Swiss ball exercises is more effective than TheraBand to improving performance parameter through activation of POS in female volleyball players. Implications: There are few studies are available regarding POS. Furthermore, study use a larger sample size, duration etc. to provide strong evidence.

Experimental and Numerical Study of Pulse Detonation Engine with Ethylene as a Fuel

This paper presents an insight on the study of transient, compressible, intermittent pulsed detonation engine with one-step overall reaction model to reduce the computational complexity in detonation simulations. Investigations are done on flow field conditions developing inside the tube with the usage of irreversible one-step chemical reactions for detonations. In the present simulations one and two-dimensional axisymmetric tubes are considered for the investigation. Studies are also performed with different grid sizes which influence the prediction of Von-Neumann spike, CJ Pressure and detonation velocity. Numerical studies are conducted using 66mm diameter tube and compared with the literature values. The simulation result from the single-step reaction model agrees well with the previous published literature of multi-step reaction models. The present studies shows that one-step overall reaction model is sufficient to predict the flow properties with reasonable accuracy. Experiments are conducted in a 25mm dia. tube with Ethylene/Air mixtures. The reactants are premixed and injected into the detonation tube using solenoid valves. The equivalence ratio of the fuel/oxidizer mixture and the length of the detonation tube are varied to understand their effects on detonation parameters. Finally the results from the present numerical study are compared and validated using NASA CEA and experiments.

External focus of attention enhances arm velocities during volleyball spike in young female players.

The aim of this study was to investigate the effect of different volleyball-specific attentional focus instructions on arm velocities of a volleyball spike in young female volleyball players using the Statistical Parametric Mapping method. Twelve young female volleyball players (13.6 ± 0.6 years old, 1.8 ± 0.8 years of experience in volleyball training) were asked to perform a volleyball spike in a standing position in three different attentional focus conditions including internal focus (IF, i.e., pull back your elbow prior to transfer momentum), external focus, (EF, i.e., imagine cracking a whip to transfer momentum), and control (CON, i.e., no-focus instruction). A Qualisys 3D motion capture-system was used to track reflective markers attached to the arm, forearm, and hand. Consequently, four phases of the volleyball spike including wind-up, cocking, acceleration, and follow-through were analyzed. A one-way repeated-measure ANOVA using one-dimensional statistical parametric mapping (SPM1d) showed that players achieved greater velocities in the hand (p < 0.01), forearm (p < 0.01), and arm (p < 0.01) using the EF instructions from the start of the wind-up phase to the acceleration phase. Post-hoc (SPM1d-t-tests-paired) analyses indicated significantly greater arm, forearm, and hand velocities during the EF condition, compared to CON (p < 0.01, p < 0.01, and p < 0.01 respectively) and IF (p < 0.01, p < 0.01, and p < 0.01 respectively) conditions. These findings suggest that EF instructions had an immediate impact on increasing volleyball spike velocity from the start of the wind-up phase to the acceleration phase prior to ball contact.

Weakly nonlinear incompressible Kelvin–Helmholtz instability in plane geometry

A weakly nonlinear (WN) theoretical model for the two-dimensional incompressible Kelvin–Helmholtz instability (KHI) is proposed. Its solution form is the complete expansion in real space. The transition from linear to nonlinear growth is analytically studied via third-order solutions of plane KHI initiated by a single-mode surface perturbation. The difference between the WN growth of the Rayleigh–Taylor instability (RTI) in plane geometry and the WN growth of the KHI in plane geometry is discussed. It is found that there are resonance solutions in the higher harmonics of KHI but not in RTI. The vertex of spikes and bubbles is deflected because of the shearing effect in KHI compared with that in RTI. The spike velocity increases with the Atwood number before a particular time and then inversely after that time. There is no such reversal in RTI. However, the bubble velocity with the Atwood number of KHI has the same evolution rule as that of RTI. In addition to the influence of initial perturbation on the nonlinear saturation amplitude, the influence of Atwood number on the nonlinear saturation amplitude of the fundamental mode is obtained. Finally, compared with the numerical simulations, at the same initial conditions, the analytical solutions up to the third harmonics are well consistent with the simulated datum in the linear and weakly nonlinear stages for a widely varied Atwood number.

Eulerian peridynamic modeling of microjetting from a grooved aluminum sample under shock loading

The micro jetting from a grooved aluminum surface under impact loading is investigated by using Eulerian peridynamics (PD). The simulation results are compared with the published experimental data and the spike velocity model, exhibiting qualitative agreement. The governing mechanism accounting for the formation of micro jetting is elucidated from the perspective of the shock wave interaction with the surface groove. The PD simulation results indicate that the incident shock wave induces progressive groove collapse along the direction of shock wave propagation. The rarefaction waves reflected from the groove edges cause the variation of the velocity vector of PD material points, leading to the material points above and below the symmetric axis of the groove converging toward the symmetric axis and colliding with each other. Then, those collided material points are driven by the incident shock wave propagating along the horizontal symmetric axis and eventually ejected from the groove. The effects of the groove dimensions and the impact velocity on the spike velocity and the ejected mass are discussed. The results show that spike velocity decreases with an increasing groove angle but increases with increasing impact velocity. Furthermore, the ejected mass increases with increasing impact velocity. However, when the depth of the surface groove is fixed and the groove angle increases, the ejected mass first increases and then decreases with the turning point at ∼120°. As the depth of the surface groove increases, the ejected mass increases. The simulation results provide a mechanistic understanding of the micro jetting phenomena and instructive guidance for developing better ejecta models.

Numerical simulation of single- and multi-mode Rayleigh–Taylor instability with surface tension in two dimensions

In this paper, we study the long-time evolutions of the single- and multi-mode Rayleigh–Taylor instability with surface tension in two dimensions by using the vortex sheet model. Applying a spectrally accurate numerical method, we investigate the effects of surface tension and density jump on the instability in various regimes of parameters. Complex phenomena of pinching, capillary waves, elongation, and roll-up appear at the interfaces. For a single-mode interface, surface tension retards the growths of bubble and spike. The effect of surface tension on the bubble and spike velocity is generally small but is large for a spike of an infinite density ratio. For multi-mode interfaces, we focus on an infinite density ratio. We show that bubbles grow with the scaling law h=αAgt2 even in the presence of surface tension, while spikes follow the scaling law weakly. It is found that both the growth rates of bubbles and spikes decrease with surface tension and the growth rate of spikes decreases larger than that of bubbles. The growth rate of the bubble front is in agreements with results of previous numerical simulations and experiments.

Coupling characteristics between bubble and free surface in a shallow water environment

The bubble motion in the shallow water environment has significant applications in underwater explosion and seabed exploration, where the bubble characteristics are mainly associated with the two types of stand-off distance parameters γf and γs (the parameters are defined in Section 2.3). Based on the assumption that the liquid is inviscid and incompressible, the influences of γf and γs on the coupling characteristics between the bubble and the free surface in the shallow water environment are solved by the Euler equations with the finite volume method, and the bubble surface and the free surface are tracked by the front tracking method. For the case γs = 0.8, (i) the water spike always keeps rising during the bubble expansion and contraction when γf<1.25, (ii) the water spike velocity increases with the decrease of γf, and the maximum bubble jet velocity is generated when γf = 0.8, (iii) the bubble top elongated is no longer obvious when γf>1.5. For the case γf = 0.6, (i) the toroidal bubble is split into two sub-toroidal bubbles when γs≤ 0.6 and γs≥1.1, (ii) the water spike velocity and the step pressure decrease with the increase of γs.

Theoretical and experimental study on the hypervelocity impact induced microjet from the grooved metal surface

The dynamic process of the hypervelocity impact induced microjet from the grooved metal surface is studied theoretically and experimentally. Based on shock wave theory and Bernoulli's equation, a physical model is proposed to theoretically analyze the formation process of the microjet and predict the spike (head) velocity of the microjet. The hypervelocity impact experiments of the cylindrical aluminum projectile on the metal target with grooves are conducted through two-stage light gas gun, and the sequence shadowgraphy images of the microjetting formation are recorded by using Ultra-high-speed camera. Then, the numerical simulations of the hypervelocity impact-induced microjet are performed by a self-developed SPH (smoothed particle hydrodynamics) code to examine the velocity distribution, mass distribution and density distribution in detail. These numerical results are compared with those obtained by experiments and theoretical analysis, which verifies the validity of numerical model. The numerical results reveal that the front of the shock wave generated by hypervelocity impact is not strictly planar in the target due to the shock wave releasing from both sides of the impact interface. Thus, the shape of the multiple jets generated from the grooves resembles a “trident”, and the upper microjet and the lower microjet deflect upward and downward respectively. The jetting velocity and mass of the middle microjet is larger than that of the microjets on both sides in the impact area. Last, the variation of the spike velocity and jetting mass with the impact velocity and the target thickness is obtained.

Effect of surface tension on late-time growth of high-Reynolds-number Rayleigh-Taylor instability

In this paper, we numerically investigate the late-time growth of high-Reynolds-number single-mode Rayleigh-Taylor instability in a long pipe by using an advanced phase-field lattice Boltzmann multiphase method. We mainly analyze the influence of surface tension on interfacial dynamic behavior and the development of the bubble front and spike front. The numerical experiments indicate that increasing surface tension can significantly reduce the complexity of formed interfacial structure and also prevents the breakup of phase interfaces. The interface patterns in the instability process cannot always preserve the symmetric property under the extremely small surface tension, but they do maintain the symmetries with respect to the middle line as the surface tension is increased. We also report that the bubble amplitude first increases then decreases with the surface tension. There are no obvious differences between the curves of spike amplitudes for low surface tensions. However, when the surface tension increases to a critical value, it can slow down the spike growth significantly. When the surface tension is lower than the critical value, the development of the high-Reynolds-number Rayleigh-Taylor instability can be divided into four different stages, i.e. the linear growth, saturated velocity growth, reacceleration, and chaotic mixing. The bubble and spike velocities at the second stage show good agreement with those from the modified potential flow theory that takes the surface tension effect into account. After that, the bubble front and spike front are accelerated due to the formation of Kelvin-Helmholtz vortices in the interfacial region. At the late time, the bubble velocity and spike velocity become unstable and slightly fluctuate over time. To determine the nature of the late-time growth, we also measure the bubble and spike normalized accelerations at various interfacial tensions and Atwood numbers. It is found that both the spike and bubble growth rates first increase then decrease with the surface tension in general. Finally, we deduce a theoretical formula for the critical surface tension, below which the Rayleigh-Taylor instability takes place and above which tension it does not occur. It is shown that the critical surface tension increases with the Atwood number and also the numerical predictions by the lattice Boltzmann method are also in accord well with the theoretical results.

A numerical study of the metal jet induced by a shock wave

In this work, a metal jet induced by a shock wave is studied numerically. Different from the previous works on metal jets, we apply a cut-cell based sharp interface numerical method for the study. The evolution of jets is simulated by the in house code CCGF [X. Bai and X. Deng, Adv. Appl. Math. Mech. 9(5), 1052–1075 (2017)], and the interfacial growth rate is computed and compared with some theoretical models. Various initial conditions, including disturbance amplitude and shock wave strength, are considered here. Based on the model of Karkhanis et al. [J. Appl. Phys. 123, 025902 (2018)], a modified model of the spike velocity is presented to achieve better consistency between the numerical simulation and the model formula under more wide initial conditions (here, the scaled perturbed amplitudes involved are 0.125 and 4, and the incident shock wave Mach number is from 2.5 to 8) in this paper. In order to extend the applicability of the empirical models, an approximate formula for the initial velocity V0 is also obtained; a direct prediction of the spike velocity will become possible when the initial perturbed amplitude and incident shock intensity are known. Relevant figures show that the modified model can estimate a more consistent result with the numerical simulation than the VK or GD model.

Comparative investigation of microjetting from tin surface subjected to laser and plane impact loadings via molecular dynamics simulations

The microjetting processes of single-crystal tin with sinusoidal surface defects under laser loading and plane impact loading are investigated by using molecular dynamics simulations via the second nearest-neighbor modified embedded atom method potential. Simulation results exhibit that the jetting factors for laser loading and plane impact loading first increase with the increment of shock breakout pressure and then reach their own saturation values, in agreement with previous experimental observations. However, the jetting factor under laser loading saturates at relatively lower shock breakout pressure than its counterpart under plane impact loading. Structure analysis via x-ray diffraction and radial distribution function techniques indicates that the laser loading leads to higher melting degree, which further facilitates the ejection process. The spike velocity linearly depends on the shock breakout pressure for both loading methods, but the bubble velocity varies with the loading method. In addition, void nucleation and growth within the sample are observed for laser loading due to the interaction of release waves. The simulation results reveal the underlying mechanisms of ejection process under plane impact loading and laser loading, helping enhance the understanding of microjetting phenomena.

Ejecta production from metal Sn into inert gases

Ejecta is produced from the shock-loaded perturbed surface of metals and subsequently breaks into small particles that are an important source of micro-particles/gas mixing during ejecta's transport and conversion. In engineering applications, the surrounding gas is often neglected during ejecta's formation, and many source models have been established based on the vacuum condition. However, the formation of the spike is always accompanied by gas, which has an important effect on the ejecta's mass/velocity distribution and the transformation time for a steady-state shock wave. To study the interaction between ejecta and ambient gases, we explore the ejecta production at the sinusoidal interface in the presence of argon gas. Six values of gas pressure and five interfaces were chosen to study the formation of the spike/micro-jet by using multi-component elastic–plastic hydro-dynamic codes. The results show that gas perturbed by the spike generated a precursory bow-shaped shock and gradually transformed into a plane wave. The transformation time was related to the velocity of the spike tip and the transmitted wave. The total mass of ejecta in gas had no distinct difference with that in vacuum, while it was significantly increased at the jet tip, which indicates that gas resistance reduced the spike velocity but did not influence the bubble. The initial velocity of the spike was insensitive to gas pressure but its decaying rate was positively correlated with gas pressure. As kh0 increased, the initial velocity of the spike tip and its decaying range increased, making it difficult to attain a steady state.

Lattice Boltzmann method simulations of the immiscible Rayleigh-Taylor instability with high Reynolds numbers

In this paper, an advanced phase-field lattice Boltzmann method based on the multiple-relaxation-time collision model is used to simulate the immiscible single-mode Rayleigh-Taylor instability with a moderate Atwoods number in a long tube, and we systematically analyze the effect of the Reynolds number on the interfacial dynamics and the late-time development stages of interface disturbance. The highest Reynolds number in the current simulation reaches up to 10000. The numerical results show that the Reynolds number significantly affects the development of the instability. For high Reynolds numbers, the instability undergoes a sequence of different growth stages, which include the linear growth, saturated velocity growth, reacceleration, and chaotic mixing stages. In the linear growth stage, the developments of the bubble and spike conform to the classical linear growth theory, and it is shown that the growth rate increases with the Reynolds number. In the second stage, the bubble and spike evolve with the constant velocities, and the numerical prediction for spike velocity is found to be slightly larger than the solution of the potential flow theory proposed by Goncharov [&lt;i&gt;Phys. Rev. Lett.&lt;/i&gt; 2002 &lt;b&gt;88 &lt;/b&gt; 134502], which can be attributed to the formation of vortices in the proximity of the spike tip. In addition, it is found that increasing the Reynolds number reduces the bubble saturated velocity, which then is smaller than the solution of the potential model. The nonlinear evolutions of the bubble and spike induce the Kelvin–Helmholtz instability, producing many vortex structures with different scales. Due to the interactions among the vortices, the instability eventually enters into the chaotic mixing stage, where the interfaces undergo the roll-up at multiple layers, sharp deformation, and chaotic breakup, forming a very complicated topology structure. Furthermore, we also measured the bubble and spike accelerations and find that the dimensionless values fluctuates around the constants of 0.045 and 0.233, indicating a mean quadratic growth. And for low Reynolds numbers, the heavy fluid will fall down in the form of the spike, and the interface in the whole process becomes very smooth without the appearances of the roll-up and vortices. The late-time evolutional stages such as the reacceleration and chaotic mixing cannot also be observed.

Phase-resolved spectroscopy of Gaia14aae: line emission from near the white dwarf surface

AM CVn binaries are a class of ultracompact, hydrogen-deficient binaries, each consisting of a white dwarf accreting helium-dominated material from a degenerate or semi-degenerate donor star. Of the 56 known systems, only Gaia14aae undergoes complete eclipses of its central white dwarf, allowing the parameters of its stellar components to be tightly constrained. Here, we present phase-resolved optical spectroscopy of Gaia14aae. We use the spectra to test the assumption that the narrow emission feature known as the `central spike' traces the motion of the central white dwarf. We measure a central spike velocity amplitude of $13.8 \pm 3.2$ km/s, which agrees at the 1 $\sigma$ level with the predicted value of $17.6 \pm 1.0$ km/s based on eclipse-derived system parameters. The orbital phase offset of the central spike from its expected position is $4 \pm 15$ $^\circ$, consistent with 0 $^\circ$. Doppler maps of the He I lines in Gaia14aae show two accretion disc bright spots, as seen in many AM CVn systems. The formation mechanism for the second spot remains unclear. We detect no hydrogen in the system, but we estimate a 3 $\sigma$ limit on H$\alpha$ emission with an equivalent width of -1.14 \AA. Our detection of nitrogen and oxygen with no corresponding detection of carbon, in conjunction with evidence from recent studies, mildly favours a formation channel in which Gaia14aae is descended from a cataclysmic variable with a significantly evolved donor.

Molecular dynamics simulations of ejecta production from sinusoidal tin surfaces under supported and unsupported shocks

Micro-ejecta, an instability growth process, occurs at metal/vacuum or metal/gas interface when compressed shock wave releases from the free surface that contains surface defects. We present molecular dynamics (MD) simulations to investigate the ejecta production from tin surface shocked by supported and unsupported waves with pressures ranging from 8.5 to 60.8 GPa. It is found that the loading waveforms have little effect on spike velocity while remarkably affect the bubble velocity. The bubble velocity of unsupported shock loading remains nonzero constant value at late time as observed in experiments. Besides, the time evolution of ejected mass in the simulations is compared with the recently developed ejecta source model, indicating the suppressed ejection of unmelted or partial melted materials. Moreover, different reference positions are chosen to characterize the amount of ejecta under different loading waveforms. Compared with supported shock case, the ejected mass of unsupported shock case saturates at lower pressure. Through the analysis on unloading path, we find that the temperature of tin sample increases quickly from tensile stress state to zero pressure state, resulting in the melting of bulk tin under decaying shock. Thus, the unsupported wave loading exhibits a lower threshold pressure causing the solid-liquid phase transition on shock release than the supported shock loading.

Hydrodynamic instability at the interface between colliding inhomogeneous mediums

This paper deals with the analysis of the perturbation growth at the interface between the colliding mediums, one of them has an initial perturbation field of density. The main difference of this hydrodynamic instability from the classical Richtmyer–Meshkov instability is the nature of the initial disturbance. The analyzed hydrodynamic instability is caused by the initial perturbation of the medium density.The time evolution of interface between two mediums has been studied analytically in nonlinear approximation. The problem has been solved for the third-order corrections to hydrodynamic quantities. Nonlinear interactions cause the appearance of surface sound wave near the contact discontinuity. This wave decreases the growth of the spike velocity. Spike and bubble evolutions have a saturation stage.The numerical calculations confirm the results of the theoretical analysis.