Interaction of weak shock waves with highly porous materials

. The paper presents the results of an experimental study of the interaction of weak shock waves with two types of highly porous materials: cellular porous nickel and mesh packets. The experiments were carried out in a shock tube in the range of Mach numbers 1.1-1.8. For the high-porosity cellular materials a numerical simulation of the interaction problem using a toroidal model of the porous medium was performed. It was shown in experiments and calculations that regardless of material type, decreasing cell size of porous material increases intensity of shock waves reflected from front layers of porous insert but suppresses shock wave reflected from rear end of shock tube. A unified mechanism of shockwave interaction with all types of highly porous materials is revealed.


Introduction
The study of shockwave (SW) interaction with various materials is of considerable scientific and practical interest.The main research in this area has focused on the attenuation of shockwaves as they pass through the material column, which is related to the problem of protection against the effects of shock waves.Hard and flexible foam coatings [1,2], coatings from complexly curved channel [3,4], multilayer mesh packages and perforated plates [5,6] are used for this purpose.Nevertheless, there are tasks where minimising the reflection of SWs is required.In particular, the use of coatings with a minimum SW reflection coefficient reduces the risk of detonation of a mixture of combustible dispersed particles or explosive gases with air in conditions of occurrence of focusing of reflected SW.In this connection it is of scientific interest to determine the characteristics of coating materials that reduce the intensity of reflected SW.
The purpose of this work is to experimental and numerical investigation of the influence of the structure of highly porous materials on the interaction with SW and to reveal the mechanism of attenuation of SW reflected from them.

Experiment
The experiments were carried out in the shock tube with a diameter of 0.05 m and a length of 2.1 m.In the measurements, the static pressure in front of the porous material insert was recorded with a PCB113B28 piezo transducer mounted on the pipe wall at a distance of 0.2 * Corresponding author: mironov@itam.nscrum from the rear end of the shock tube.The transducer boundary frequency was 500 kHz, which allowed the separation of the pressure jump at the SW.The piezoelectric transducer recorded the pressure as the incident and reflected SW passed by it.At a distance of 1 m from this sensor, closer to the front end of the shock tube, another similar piezo-sensor was installed, whose signal triggered the registration system with a sampling frequency of 2 MHz.Based on the difference in the transit times of the two piezo-sensors, the velocity and Mach number of the shock wave was calculated.
The rear end of the percussion tube was removable to allow porous inserts to be inserted into the tube.The inserts could be installed either close to the rear end of the shock tube or at some distance from it.The gaspermeable inserts used in the experiments were highporosity cellular material (HPCM) and mesh packs.
The HPCM material is foamed nickel with a porosity value of k = (95±1) %, with an average pore diameter of d = 1 and 3 mm.
The mesh packets were made of rectangular metal mesh with mesh sizes hh = 22 mm and 11 mm and a transparency of 75 %, and 0.20.2mm with a transparency of 40 %.The packets were formed as 4 layers of meshs separated by 5 mm spacing.Photographs and basic characteristics of the highly porous materials are shown in Table 1.
The front end of the shock tube was sealed with a thin plastic diaphragm before start-up and the tube channel was evacuated to a predetermined pressure by a vacuum pump.The start-up was carried out by piercing the diaphragm with atmospheric air flowing into the tube channel.The Mach number of SW in the experiments was varied in the range M = 1.11.8 by changing the initial pressure in the tube channel.The pressure-time dependences obtained in the experiments with the porous insert were compared with the time E3S Web of Conferences 459, 04007 (2023) https://doi.org/10.1051/e3sconf/202345904007XXXIX Siberian Thermophysical Seminar dependence of the pressure in the shock tube without the porous insert.Based on this, a conclusion was made about the effectiveness of suppression of reflected shock waves by one or another highly porous insert.

Numerical simulation
Numerical simulation of the shock wave propagation in the shock tube and interaction of shock wave with highly porous materials was performed using the ANSYS Fluent package based on the solution of unsteady Reynolds-averaged Navier-Stokes equations using the k- SST turbulence model.In the solution, explicit schemes of second order accuracy in space with the Roe-FDS splitting method of convective flows, explicit Runge-Kutta method in time, were applied.The calculations were performed under conditions corresponding to the shock tube experiments described in 2.1.The computational domain for solving the problem of interaction of SW with HPCM is a semi-closed cylinder with the length and diameter corresponding to the experimental tube.The computational area outside the porous insert was covered by a rectangular computational mesh with thickening to the insert.The porous insert area was covered by a predominantly irregular quadrilateral grid.At the left inlet boundary the values of total pressure and temperature were set at the pipe walls, including the back end and the elements of the porous insert, the no-slip condition and adiabatic wall temperature were set.
The gas-permeable cellular porous HPCM insert was simulated within the toroidal skeleton model, which has previously proven itself in the problems of supersonic flowing of cylindrical bodies with front porous inserts [7].The three-dimensional toroidal skeleton model (Fig. 2) consists of system of coaxial toroidal elements of different diameters arranged in a staggered arrangement.In the plane of the axial section this ring system is a set of impermeable circular elements staggered with a distance d equal to the pore diameter between the elements in the radial and axial directions.The main objective of the numerical simulation was to obtain data on the pressure and velocity fields in the porous material necessary to determine the mechanism of interaction of the SW with the HPCM.Parametric calculations were performed for different cell sizes and porosity of the HPCM, porous material thicknesses, as well as the distance from the porous insert to the back wall of the shock tube.The ultimate goal of the studies was to find the optimal spatial structure of the HPCM to minimize the intensity of the reflected SW.

Results
Fig. 2 shows the results of the experiment and the numerical simulation of the inserts in the HPCM.The time dependences of static pressure at the piezoelectric transducer are shown for the case without inserts and with inserts with pore diameter d = 3 and 1 mm, 20 mm thickness, located at 60 mm distance from the rear end of the shock tube.It should be noted that the calculation results are in agreement with the experiment.Fig. 2 shows the first sharp pressure jump (A), associated with the passage of the oncoming SW through the piezoelectric transducer.This is followed by a pressure jump (B) due to the partial reflection of the SW from the HPCM skeleton elements.Then, a pressure jump (C) is observed, formed by the shock tube's back wall being reflected by the incident shock tube.A relatively gradual increase in pressure (D) can then be seen due to the expansion of the shockwave-compressed air in the area between the porous material and the rear wall of the tube.It can be seen that the HPCM inserts create a weaker amplitude reflected pressure jump (B) from the porous insert compared to the one reflected from the back wall of the shock tube in the case without the insert.In addition, Fig. 2 shows that the porous insert does not create one strong reflected pressure spike as in the case without the insert, but converts it into a system of weaker jumps (B and C).
A HPCM with a smaller pore diameter, although it has a higher level of reflection (B), shows a slower pressure rise (D) when the gas flows between the insert and the back of the tube.The described mechanism of interaction of SW with the porous insert made of HPCM is illustrated by a series of calculated pictures of velocity fields obtained for different moments of time (Fig. 3).Fig. 3a shows the moment when the pressure jump (A) associated with the oncoming SW passing through the piezoelectric sensor reaches the surface of the porous insert.Fig. 3b,c,d shows how the front reflected from the porous medium (B) forms and moves in the opposite direction as the incident wave A enters the porous material.Fig. 3e shows the movement of the shock tube backward through the porous insert and Fig. 3f shows the interaction of the backward reflected shock wave with the porous insert.After it passes through the porous insert, a shock wave (C) is generated (Fig. 3g).The shock wave (C) is followed by the outflow (D) of compressed gas through the porous insert from the volume between the insert and the back wall of the shock tube (Fig. 3h).
The changes in the time dependences of static pressure were found for variations of Mach number value in the investigated range of Mach numbers.Since the dependences were obtained not only at different Mach numbers, but also at different initial pressures in the shock tube, a direct comparison is not possible.Therefore, a normalization of the values was carried out: P* = (P/PA-1)/(Pmax/PA-1) (PAthe pressure jump amplitude (A), Pmax -the maximum pressure value in the D area for each Mach number), t* = t/tA (tA -time of pressure jump arrival (A) on the piezo-sensor).
Fig. 4 shows the dependences of pressure P* versus time t* for a porous insert made of HPCM with pore diameter 3 mm, thickness 20 mm and located at distance 60 mm from the rear end of the shock tube at different Mach number values.It can be seen that the amplitude of the first reflected pressure jump (B) increases and the second jump (C) decreases with increasing Mach number.In this case, the mechanism of interaction of the shock wave with the porous insert at all Mach numbers is the same.Figure 5 shows the time dependencies of the static pressure at the piezo-sensor before inserting a pack of three square-mesh meshes of 22 mm and 11 mm with the same transparency value of 75%.In the pack, the meshes were separated by a distance of 5 mm and the pack was positioned 60 mm from the rear wall of the shock tube.It can be seen that the interaction of the SW with the mesh packs, as well as with the porous inserts made of HPCM, obeys the same mechanism, which includes the formation of a reflected wave (B) from the front layers of the porous insert, the formation of a compaction jump (C) reflected from the back wall of the tube, passed through the porous insert, and finally the outflow of compressed gas from the volume between the insert and the back wall of the shock tube (D).The dependence on the mesh size h is the same as for the HPCM with respect to the pore diameter d.

Conclusion
In the shock tube the experimental study of interaction of weak shock waves with two types of highly porous materials: cellular porous nickel and mesh packets has been performed.Dependences of static pressure on time, characterizing process of interaction of shock waves with porous materials, have been received.
The experiments for cellular porous nickel were supplemented with numerical simulations of the interaction of the porous insert with SW based on the toroidal skeleton model of the HPCM.The calculated fields of velocity, pressure and temperature inside and outside the porous material were obtained for all time points.The agreement between the calculated and experimental data was obtained.
Analysis of calculated and experimental data for all types of porous materials revealed the single interaction mechanism, which includes the formation of the reflected wave from the front layers of the porous insert, the formation of the reflected from the back wall of the tube compaction jump, passed through the porous insert, and the outflow of compressed gas from the volume between the insert and the back wall of the shock tube.
Experimental and computational results have shown that regardless of the material type, reducing the cell size of the porous material structure increases the intensity of the SW reflected from the front layers of the insert, but suppresses the SW reflected from the rear end of the shock tube more strongly.

Fig. 1 .
Fig. 1. 3D-image of a toroidal model of a porous insert from the HPCM.

Fig. 3 .
Fig. 3. Calculated velocity fields for different points in time.

Table 1 .
Characteristics of highly porous materials.