Diverging Shock Waves Research Articles

Numerical investigation of the interaction between a converging shock wave and an offset cylindrical bubble containing different gases

A numerical study investigating the interaction process between a converging shock wave and a gas bubble placed at an offset location is presented. As a first step, for proofing the reliability of the used numerical scheme, a simulation of relevant available experimental findings of Hosseini and Takayama [“Richtmyer–Meshkov instability induced by cylindrical shock wave loading of cylindrical gaseous inhomogeneities,” AIAA Paper No. 2000-2464, 2000] and Hosseini and Takayama [“Study of a converging shock wave interaction with a gaseous interfaces in an eccentric arrangement,” in Japanese Symposium on Shock Waves, 2000] is conducted; the tested gases were helium (He) and sulfur hexafluoride (SF6). The converging shock wave had a Mach number of 1.18 prior to its impact on the 50 mm diameter gas bubble. Achieving good agreement with the experimental findings ensures the reliability of the applied numerical scheme. After the converging shock wave impacted the gas bubble, different shock waves are created. These shock waves propagate differently than those observed in the case of planar shock wave impacting a cylindrical gas bubble or that of a converging shock wave where the gas bubble is located at the center. Furthermore, once the converging shock wave converged, a diverging shock wave expands and again impacts the remaining gas bubble, thus creating more complex shock wave patterns. The gas contained inside the bubble has an effect on the location of the converging shock wave focus point. In the case of the heavy gas SF6, the focus point is near the center of the converging shock wave, but in the case of light gas He, it is offset from the converging shock wave focus point and outside of the initial location of the He bubble. The new results from the current numerical simulation include more detailed results for both bubbles, which were not reported in Hosseini and Takayama [“Richtmyer–Meshkov instability induced by cylindrical shock wave loading of cylindrical gaseous inhomogeneities,” AIAA Paper No. 2000-2464, 2000] and Hosseini and Takayama [“Study of a converging shock wave interaction with a gaseous interfaces in an eccentric arrangement,” in Japanese Symposium on Shock Waves, 2000]. In addition, a shock wave focusing of the transmitted shock wave inside the SF6 bubble is observed. This later creates a secondary diverging shock wave. Higher pressure is achieved in the SF6 case.

Comparison of efficacy of platelet-rich plasma and extracorporeal shock wave for the treatment of chronic insertional Achilles tendinopathy

To compare clinical efficacy of platelet-rich plasma (PRP) and extracorporeal shock wave in treating chronic insertional Achilles tendinopathy. From February 2019 to August 2021, 42 patients with chronic insertional Achilles tendinopathy were selected and divided into PRP group(20 patients, 28 feet) and shock wave group (22 patients, 29 feet). In PRP group, there were 12 males and 8 females, aged 47.00(28.00, 50.75) years old, and the courses of disease ranged 7.00(6.00, 7.00) months;PRP injection was performed in the Achilles tendon stop area of the affected side. In shock wave group, there were 16 males and 6 females, aged 42.00(35.75, 47.25) years old;and the courses of disease was 7.00(6.00, 8.00) months;shock wave was performed in Achilles tendon stop area of the affected side and triceps surae area. Visual analogue scale (VAS) and Victorian Institute of Sport Assessment-Achilles (VISA-A) were applied to evaluate clnical effect before treatment, 1, 3 and 6 months after treatment, and satisfaction of patients was investigated. VAS and VISA-A score in both groups were significantly improved at 1, 3 and 6 months after treatment than before treatment (P<0.05), VAS and VISA-A score in PRP group at 6 months after treatment were significantly higher than those at 1 and 3 months after treatment, and VAS and VISA-A score in shock wave group were lower than those at 1 and 3 months after treatment (P<0.05). There was no significant difference in VAS and VISA-A score between two groups before treatment, 1 and 3 months after treatment(P>0.05), while VAS and VISA-A score in PRP group were better than those in shock wave group at 6 months after treatment(P<0.05), and the satisfaction survey in PRP group was better than that in shock wave group(P<0.05). PRP injection has a good clinical effect on chronic insertional Achilles tendinopathy with high patient satisfaction, and medium-and long-term effect of PRP injection for the treatment of chronic insertional Achilles tendinopathy is better than that of extracorporeal divergent shock wave.

Cavitation bubble collapse and rebound in water: Influence of phase transitions

The influence of the mass transfer across the bubble surface due to evaporation and condensation on the cavitation (vapor) bubble collapse and rebound in water under room conditions has been studied. The bubble is spherical with an initial radius of 1.92mm. The dynamics of the vapor in the bubble and the surrounding liquid is governed by the gas dynamics equations. The effects of the liquid viscosity and heat conductivity of both fluids are taken into account. The liquid and vapor states are mainly described by the known wide-range equations by Nigmatulin and Bolotnova. The mass transfer is governed by the accommodation coefficient αac in the Hertz–Knudsen–Langmuir formula. It has been found that at αac in the range 0.001−0.075 the vapor in the bubble is homobaric at collapse. As αac increases in the range αac>0.075, an isentropic compression wave convergent to the bubble center is formed in the bubble. At focusing of this wave at the bubble center, a divergent (reflected from the center) isentropic compression wave occurs. With rising αac the intensities of these convergent and divergent waves increase. The divergent wave partially penetrates to the liquid in the form of an isentropic divergent wave. With growing αac the penetrated wave becomes steeper. At αac≈0.25 the divergent isentropic wave in the bubble transforms into a shock wave during its propagation to the bubble surface. This shock wave partially penetrates to the liquid in the form of a divergent shock wave. At small αac the outgoing pulse in liquid is shockless. Starting with αac≈0.03, it becomes discontinuous. In the range 0.03≤αac≤0.12 the shock pulse results from only large pressure gradients in the liquid in the vicinity of the bubble at the beginning of rebound. In the range 0.12<αac<0.25 the discontinuous outgoing pulse is formed from the divergent isentropic compression wave arisen in liquid as a result of partial escape of the isentropic wave out of the bubble. At αac>0.25 the shock pulse is formed directly on the bubbles surface as a result of the partial penetration of the divergent shock wave from the bubble to liquid. The obtained numerical results are in good agreement with available numerical and experimental data.

Attenuation and inflection of initially planar shock wave generated by femtosecond laser pulse

Evolution of wavefront geometry during propagation and attenuation of initially planar shock waves generated by femtosecond laser pulses in aluminum is studied. We demonstrate that three stages of shock front inflection take place in consistent hydrodynamics and molecular dynamics simulations.During the first stage, the distance traveled by a near-planar wave DSW≲RL is smaller than the radius of heated laser spot RL. Wave attenuation is associated with one-dimensional plane (1D) rarefaction wave coming from the free surface. Such rarefaction wave shapes the shock wave to a 1D triangular pressure profile along direction normal to target surface with a shock front followed by an unloading tail. The second transitional stage starts after propagation of DSW∼RL, at which the unloading lateral waves begin to arrive to a symmetry axis of flow and initiate inflection of the initially planar shock front. Next at the third stage, the wavefront geometry is finally rounded and rapid attenuation of shock pressure begins at DSW≳RL.It is shown that such divergent shock wave cannot generate plastic deformations in aluminum shortly after propagation of DSW∼RL. Thus, we may estimate the maximal peening depth as a radius of focal spot, which sets an upper limit for the laser shock peening.The cessation of plastic deformation is caused by the fall of the shockwave amplitude below the elastic limit. In this case, the elastic-plastic wave transitions to a purely elastic mode of propagation. For large-sized light spots, this transition ends in the 1D mode of propagation.

Shock wave propagation in perfectly conducting rotational axisymmetric two-phase medium with increasing energy under the action of heat conduction and radiation heat flux

The distribution of divergent cylindrical shock waves, under the effect of the variable azimuthal or axial magnetic field in space, in a perfectly conducting rotational axisymmetric dusty gas is studied. The profiles of the fluid variables were determined using numerical estimation. The effects of changing the piston velocity index, rotational parameter, and Shock Cowling number are studied. The result is that pressure and density disappear in the piston, and then the vacuum is produced at the axis of symmetry, which is perfectly in conformity with the conditions for producing a shock wave in the laboratory when the azimuthal magnetic field is present.

Mechanism of transition to detonation in unconfined volumes

The paper is aimed at numerical study of one of the most hazardous events at a launch place: open space explosion of fuel air mixtures due to accidental loss of containments. A mechanism of transition to detonation in the process of unconfined flame propagation is proposed. It is shown that the detonation onset takes place as a result of local exponential growth of unstable short-wavelength perturbations. Exactly the same mechanism is known to be responsible for the self-similar flame acceleration, however the detonation can arise only in the case of extremely high reaction rate. High reaction rate defines a coupling of the accelerating flamelets with diverging shock waves that leads to the detonation onset.

Isothermal Motion of Hydromagnetic Cylindrical Shock Waves in Self Gravitating Rotating Gas

In this paper the Chisnell2 Chester1 Whitham3 (CCW1-3) method has been used to study the isothermal propagation of cylindrical hydromagnetic diverging shock waves in self gravitating rotating gas in presence of constant axial magnetic field. Assuming density decreasing atmosphere and exploring magnetic field as weak and strong, the analytical expressions have been derived for both weak and strong shocks and finally the effects of overtaking disturbances behind the flow on the motion of shock have also been included

On the special properties of the shear shock wave in an incompressible elastic medium without preliminary deformations

The nonlinear system of hyperbolic equations describing a helical one-dimensional motion of a nonlinear elastic incompressible medium is investigated. Types and velocities of possible divergent shock waves are determined on the basis of a joint analysis of the dynamic compatibility conditions, the characteristics of this system and the Riemann invariants along them. For the single transverse shock wave in previously undeformed medium it is shown that the shearing direction is invariant at the leading shock front determined by the initial action and independent of post-impact loading. The solution of boundary-value problem about shock loading on a circular cylindrical cavity boundary is constructed by the matched asymptotic expansions method and the ray method modified for shock waves.

Interaction between lithotripsy-induced surface acoustic waves and pre-existing cracks

The interaction between pre-existing cracks and surface acoustic waves (SAW) in lithotripsy is investigated. Surface acoustic waves are generated at a water-glass interface by an incident shock wave produced by the spark discharge of a nano pulse lithotripsy (NPL) device or an electromagnetic shock wave lithotripsy (SWL) source. Evidence of SAW, including leaky Rayleigh wave and Scholte wave, will be presented based on photoelastic imaging and numerical stimulations using COMSOL. A clear correlation between SAW and the location of the maximum tensile stress produced on the glass boundary has been identified, which can lead to ring-like fractures on a flat glass surface exposed to NPL-generated spherically divergent shock waves. To simulate cavitation-induced surface pitting in SWL, pre-existing crack will be introduced on the glass surface by microindentation using a Vickers or Knoop indenter. The interaction of SAW with the pre-existing cracks will be examined to characterize crack extension and branching as a function of their location and orientation to the incident shock wave.

Generation of different types of surface acoustic waves by shock wave-Stone interaction

The generation of different types of surface acoustic waves (SAWs) in lithotripsy is investigated by numerical simulations using COMSOL, considering either the case of a weakly focused shock wave as an incident plane wave (in the focal plane) in shock wave lithotripsy (SWL) or a spherically divergent shock wave produced by a spark discharge in Nano Pulse Lithotripsy (NPL). The interaction of these two types of shock waves with an artificial kidney stone immersed in water may generate three different types of SAWs: namely, Scholte wave, acoustic surface evanescent wave (ASEW), and leaky Rayleigh wave (LRW). In particular, we examined the generation of SAWs on a flat, cylindrical, or spherical surface under different incident angles. It was found that the geometry and acoustic properties of the stone could significantly influence the types of SAWs generated and the resultant peak tensile stress produced at the stone-water boundary. Comparison with experimental measurements (i.e., Schlieren and photoelastic imaging) and stone fracture will also be made. [Work supported by NIH through 5R37DK052985-20.]

Shock wave processes in collisional gas particle mixtures

Structures and propagation of shock waves in high density particle suspensions in gas are investigated theoretically and numerically. A physical and mathematical model which takes into account integral collisions between the particles on the basis of molecular-kinetic approaches of theory of granular materials is applied. The possibility of different types of shock waves, including double front structures is revealed. The role of particle collisions in the dynamics of particle dense layer expansion under an influence of divergent shock wave and in processes of shock wave diffraction past a backward-facing step is analyzed.

Compression of matter by hyperspherical shock waves

A novel compression scheme is proposed, in which hollow targets with specifically curved structures initially filled with uniform matter, are driven by converging shock waves. The self-similar dynamics is analyzed for converging and diverging shock waves. The shock-compressed densities and pressures are much higher than those achieved using spherical shocks due to the geometric accumulation. Dynamic behavior is demonstrated using two-dimensional hydrodynamic simulations. The linear stability analysis for the spherical geometry reveals a new dispersion relation with cut-off mode numbers as a function of the specific heat ratio, above which eigenmode perturbations are smeared out in the converging phase.

Initiation of detonation in explosives on an U-70 proton accelerator

Detonation initiation in a composite explosive based on HMX and TATB loaded by a divergent shock wave was studied on a U-70 proton accelerator using proton radiography. Density distributions behind the initiating shock front at various times were obtained. Detonation failure due to collision of shock and detonation waves was studied for a plastic-bonded TATB sample using radiography. Characteristic features of the explosive transformation under shock-wave loading were determined from the images obtained.

Nonplanar converging and diverging shock waves in the presence of thermal ions in electron-positron plasma

The cylindrical and spherical Korteweg-de Vries-Burger equations have been derived to study the ion acoustic converging and diverging shock waves. The considered plasma is comprised of inertialess electrons, positrons, and inertial thermal ions. It is noticed that the ion temperature, positron concentration, and kinematic viscosity have significant influence on the shock structure and propagation in nonplanar geometries. The strength of shock in spherical geometry is found to dominate over shock strength in cylindrical geometry. The shock wave strength and steepness escalate with time as it moves towards the center and shock enervates as it recedes away from center. The graphical view of the numerical results is presented for illustration. The results may have relevance in the inertial confinement fusion plasmas.

Focusing of shock waves induced by optical breakdown in water

The focusing of laser-generated shock waves by a truncated ellipsoidal reflector was experimentally and numerically investigated. Pressure waveform and distribution around the first (F(1)) and second foci (F(2)) of the ellipsoidal reflector were measured. A neodymium doped yttrium aluminum garnet laser of 1046 nm wavelength and 5 ns pulse duration was used to create an optical breakdown at F(1), which generates a spherically diverging shock wave with a peak pressure of 2.1-5.9 MPa at 1.1 mm stand-off distance and a pulse width at half maximum of 36-65 ns. Upon reflection, a converging shock wave is produced which, upon arriving at F(2), has a leading compressive wave with a peak pressure of 26 MPa and a zero-crossing pulse duration of 0.1 mus, followed by a trailing tensile wave of -3.3 MPa peak pressure and 0.2 mus pulse duration. The -6 dB beam size of the focused shock wave field is 1.6 x 0.2 mm(2) along and transverse to the shock wave propagation direction. Formation of elongated plasmas at high laser energy levels limits the increase in the peak pressure at F(2). General features in the waveform profile of the converging shock wave are in qualitative agreement with numerical simulations based on the Hamilton model.

Nonlinear spherically divergent shock waves propagating in a relaxing medium

The propagation of nonlinear spherically diverging N-waves in atmosphere was studied experimentally and theoretically. The relative effects of nonlinear, dissipation, and relaxation phenomena on the N-wave duration and amplitude were investigated based on the numerical solutions of the modified Burgers equation. It is shown that, under the experimental conditions, the duration of a pulse increases mainly due to nonlinear propagation, whereas the amplitude depends on the combined effects of nonlinearity, dissipation, and relaxation. The frequency response of the measuring system is obtained. The calibration of the amplitude and duration of the experimental waveforms is performed based on the nonlinear lengthening of the propagating pulse. The results of numerical modeling show good agreement with experimental data.

Effect of spherically converging stress waves on the phase composition, structure, and physicochemical transformations of the mixture of aluminum and quartz powders

The phase composition and structure of a mixture of aluminum and quartz powders taken in a ratio of 1:1 have been studied after loading by spherical converging shock waves. A number of concentric layers (zones) have been observed in a meridian section of the sample after shock-wave loading. The pressures in the converging shock wave and in the diverging shock wave reflected from the center of the sample have been estimated as a function of the radial position of Lagrangian particles on the basis of the calculation of pressure profiles P(R, t). The data on the phase composition of different zones are considered in respect to the pressures induced by the converging and diverging shock waves at their boundaries. It has been established that pressures below ∼45 GPa cause only additional compacting of the material and deformation of aluminum and quartz. In this case, the quartz grain size substantially decreases up to the transition into the X-ray amorphous state. The attainment of a pressure of ∼45 GPa initiates the solid-state reaction of SiO2 decomposition, which leads to the precipitation of pure silicon and the evolution of oxygen. The beginning of the silicon precipitation and the chemical reaction of Al2O3 formation are separated over the pressure scale. The critical pressure, which is necessary for the solid-state chemical reaction of the Al2O3 formation is about 50 GPa.

Experimental study of Richtmyer-Meshkov instability induced by cylindrical shock waves

The paper describes the results of holographic interferometric flow visualization of the Richtmyer-Meshkov instability induced by cylindrical shock waves propagating across cylindrical interfaces. Experiments were conducted in an annular coaxial vertical diaphragmless shock tube, which can produce converging cylindrical shock waves with minimum disturbances. The shock wave converged and interacted with a cylindrical soap bubble filled with He, Ne, air, Ar, Kr, Xe, or SF6. The soap bubble was placed coaxially in the test section. The effects of density variation on the Richtmyer-Meshkov instability for a wide range of Atwood numbers were determined. Pressure histories at different radii during the shock wave implosion and reflection from the center were measured. Double-exposure holographic interferometry was used and the motion of the converging shock wave and its interaction with the gaseous interface were visualized. The variation of the pressure at the center with interface Atwood number for constant incident shock Mach number was studied. It is found that the dominant mechanism limiting the maximum pressure at the center of convergence is related to the instability of the converging shock wave induced by its interaction with the interface. A short time after the impulsive acceleration, the interface started deforming, and the growth of these perturbations is described. The results show that after diverging shock wave interaction, the reshocked cylindrical interfaces have a higher growth rate of the turbulent mixing zone than that of the reshocked interface in a plane geometry reported by previous works.

Implosion of a spherical shock wave reflected from a spherical wall

The paper describes results of experiments of a converging spherical shock wave reflected from a spherical wall. In order to visualize the motion and the flow field behind the shock waves, an aspheric lens-shaped transparent test section made of acrylic PMMA (polymethyl methacrylate) with an inner spherical cavity was designed and constructed. This test section made optical flow visualization with collimated object beams possible. Spherical shock waves were produced at the centre of the spherical cavity by explosion of silver azide pellets ranging from 1.0 to 10.0 mg with corresponding energies of 1.9 to 19 J. The charges were ignited by irradiation of a pulsed Nd:YAG laser beam. Pressures were also measured at two points with pressure transducers mounted flush at the inner wall of the test section. The pellet was simultaneously ignited on two sides or was shaped to produce a uniform diverging spherical shock wave. This spherically diverging shock wave was reflected from the spherical inner wall of the test section to form a converging spherical shock wave. We visualized the shock-wave motion by using double exposure holographic interferometry and time-resolved high-speed video recording. The sequence of diverging and converging spherical shock-wave propagations and their interaction with gaseous explosion products were observed. The convergence, acceleration and stability of the imploding shock wave in the test section were studied.

Radiographic Study of the Evolution Material of the Density Profile behind the Front of a Divergent Shock Wave in an Explosive

The evolution of the material density profile behind the front of a divergent initiating shock wave in HMX and RDX based explosives in the transitional regime of explosive transformation was studied using flash radiography. The explosives were loaded by a divergent shock wave through a Plexiglas layer of varied thickness. Distinctive features were found in the evolution of the density profile behind the front of a divergent initiating shock wave that did not become a detonation wave and behind the front of an initiating shock wave that became a detonation wave.