† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11605183 and 11502254).
The Richtmyer–Meshkov instability at the interface of solid state tin material and xenon gases under cylinder geometry is studied in this paper. The experiments were conducted at FP-1 facility in Institute of Fluid Physics, China Academy of Engineering Physics (CAEP). The FP-1 facility is a pulsed power driver which could generate high amplitude magnetic field to drive metal liner imploding. Convergent shock wave was generated by impacting a magnetic-driven aluminium liner onto a inner mounted tin liner. The convergent evolution of the disturbance pre-machined onto the tin linerʼs inner surface was diagnosed by x-radiography. The spike amplitudes were derived from x-ray frames and were compared with linear theory. An analytical model containing material strength effect was derived and matched well to the experimental results. This sensibility of the disturbance evolution to material strength property shines light to the application of Richtmyer–Meshkov instability to infer material strength.
When a shock wave passes through an interface of different density materials, disturbances on the interface will experience instability growing which is known as the Richtmyer–Meshkov instability (RMI). Understanding the characteristics of this phenomenon is of great importance in a wide range of scientific and engineering applications among which the instability encountered in inertial confinement fusion (ICF)[1,2] is of great interest. During the past 20 years, considerable amounts of analytical, computational, and experimental works dealt with the planar version of this problem.[3,4] It is benefit to focus on the basic physical process underlying the onset and development of the RMI in simple geometries. Many aspects that affect the RMI behavior have been studied thoroughly under planer configuration, such as the Mach number effects,[5,6] the initial condition effects,[7,8] the Atwood number effects,[9,10] the non-equilibrium characteristics,[11] and so on. However, the imploding of the ICF capsule is in convergent geometry which cannot be investigated by the planner shock wave.[12] In convergent geometry, the RMI differs from the planar case due to the geometrical effects on the instability which has been already recognized and studied by theoretical and numerical methods.[13–19] The experimental study, however, is very limited. To better understand the dynamics of the RMI in convergent geometry and validate models in such configuration, sufficient experimental data are needed. Recently, some researchers are working hard to generate convergent shock wave in shock tubes.[20,21] Holder et al.[22] reported laboratory experiments with a convergent shock tube, using a detonable gas mixture to produce a cylindrically convergent shock. But, such a device is lack of flexibility due to the use of explosive gas. Zhai et al.,[23] Apazidis et al.,[24] and Biamino et al.[25] have experimentally proven the possibility of generating a converging cylindrical shock wave using a conventional shock tube. Most of the experiments studied RMI behavior at gas/gas interface. In this paper, however, we present an alternative way using magnetic driving method to generate convergent shock wave in solid materials and study the RMI of solid/gas interface.
The RM instability phenomena have been thoroughly studied for perfect fluid and gases. Recently, however, the RM instability in solid materials is of growing interest.[18,26–28] In materials with strength, the RMI growth is arrested and stabilized at an amplitude which is in inverse proportion to yield stress Y.[29] Dimonte et al.[30] developed a model for the strength suppression of the RMI that can be used to infer yield strength under shock loading. The RMI has also been adopt to explain the ejecta phenomenon in solid materials. Buttler et al.[31] developed an ejecta source term model that links to the surface perturbations of shocked materials. The RMI evolution was also recommended as an alternative method for evaluating the viscosity of metals. Mikaelian[32] presented an analytical expression for the amplitude of perturbations at the interface of two viscous fluids and found a relation between fluid viscosity and solid strength. Besides the above upcoming applications (all of which are in planner regime), the RMI in solid materials under convergent geometry has also critical effect on some engineering applications.[33–35] With the above motivation, a series of experiments were conducted to investigate the convergent RMIphenomenon at the interface between solid tin material and xenon gas in this paper.
In solid materials shock waves are mostly produced by impacting. A variety of driven methods exist, such as projectile of gun type, explosive systems, electrical, electromagnetic, and combined accelerators, devices on the basis of radiation sources (laser radiation and x-radiation, electron, and ion beams, etc.). The method of impactor driven by magnetic field pressure possesses a number of important advantages:
(i) Absence of action of explosive products on the liner means a possibility of its inertial convergence, excludes additional weakly controlled influence on the target, and eliminates a necessity to take this additional action into account during numerical analysis.
(ii) A possibility to set the current pulse of required shape, i.e., the availability of a broad range of characteristics of the impactor drive, a possibility to obtain different impact velocities.
(iii) High symmetry of loading conditioned by the method of magnetic field generation and absence of additional factors (for example, HE initiation system or non-uniformity of laser radiation).
(iv) “transparency” of the magnetic field allows recording the velocity not only of the inner but also the outer surface of the impactor.
With so much advantages, we choose magnetic-driven liner-on-target method to generate shock waves in the tin material. A general magnetically driven cylinder liner-on-target shock experiment is shown in Fig.
The FP-1 facility (Fig.
The impacting condition on the tin target was derived from velocity history of the inner surface which were measured with a smooth tin liner before the RMI experiment. A DISAR velocimetry system[39] was used to obtain the velocity history. Rogowski coil was mounted in the load section of the FP-1 facility to collect the current data passing through the liner. A 1D MHD code[40] in cylindrical coordinate regime was used to analysis the liner imploding procedure with the current data as input condition. Inner surface velocity history was compared between the codeʼs prediction and experimentʼs measurement, which will validate the code. Then the impact condition was inferred by the validated code.
Radiography is the main diagnostic for this RMI experiment. It allows the visualization of the disturbance evolution in the otherwise inaccessible target interior during the course of the experiment. The x-ray source size is 1 mm to 1.5 mm with about a 50-ns pulse duration. A Marx bank with charge voltage of 46 kV delivered about 450-kV peak voltage to the x-ray diodes which could generate about 5.2×10−6 C/kg x-ray at 1-m distance. X-ray film was collected by a high-resolution CCD camera after a fluorescent screen. Both the x-ray diode and camera were carefully protected from high speed fragments along the axial orientation. Figure
Before the RMI experiment, a velocity measurement experiment was conducted to provide free surface velocity data for 1D MHD code validation and impact condition determination. Figure
MHD code calculation and experimental data which shows excellent agreement. Then, with the validated code we can derive that the impact pressure is 9.75 GPa and shock velocity is 3.22 km/s in the tin target. Under this condition the tin target is still in solid state.
The radiography Marx bank generate only one pulse in one shot which means only one figure could be obtain in one shot. However, to study RMI problem we need a time series pictures. So, sequent shots should be conducted with the same driving condition which demand a perfect discharging repeatability of the facility. The repeatability property of the FP-1 facility were examined and verified before the RMI experiment. Figure
After the above preparation, a series of RMI experiment were conducted and axial radiography figures (shown in Fig.
It is interesting that there exist disturbance growings on both sides of the target although initially the disturbance machined only on the inner surface. We believe that it is belong to the spallation of the target which would induce disturbance on the other side. Figure
It can be note from the dynamic frames in Fig.
An analytical model for the disturbance evolution in cylindrical geometry was given by Mikaelian[41] under some assumptions of incompressible, irrotational, and inviscid flows with perturbations in linear regime. The governing equation is shown in below
Mikaelian[32] analyzed the viscous effect on shock-induced interface instability in planner geometry and derived the governing equation. He found some similarity between metal strength and fluid viscous under the following relation:
A series of RMI experiments in solid tin material were initiated on FP-1 facility. Shock wave was generated by magnetic driving method for its advantage in lots of aspects. The 1D MHD code predicted well the impactor velocity which means that the driving condition could be derived from this code and also the desired impact condition could be designed beforehand using this code. With the limitation that the radiography diagnostic could only generate one pulse in one shot, excellent repeatability property of the FP-1 facility guarantees the reliability of the RMI experiment in our study. Five individual times of the disturbance development were obtained by radiography. The spikes of the disturbance existed in both sides of the target which maybe induced by spallation of the tin target on impact. The amplitude of the spikes is derived from the radiographic frames and compared with linear analytical model. The model containing strength effects shows excellent agreement to experimental results. This conclusion makes confidence to the application of using the Richtmyer–Meshkov instability to infer material strength. The results in this paper have validated the magnetic driving method which is appropriate for studying the solid state RMI problem. But lack of sufficient time interval frames makes the results dissatisfy. So, much individual time of the disturbance development will be investigated in the future.
Under impulse assumption, as Richtmyer[42] showed, the
The authors would thank Yuesong Jia, Xiaoming Zhao and Weidong Qin for their kindly help to maintain the FP-1 facility. The authors would also thank Yang Zhang, Zheng Fu and Lili Wang for their fruitful discussion of the experiment results.
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