Irregular reflection of spark-generated shock pulses from a rigid surface: Mach-Zehnder interferometry measurements in air

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1 Irregular reflection of sark-generated shock ulses from a rigid surface: Mach-Zehnder interferometry measurements in air Maria M. Karzova, 1,a) Thomas Lechat, 2 Sebastien Ollivier, 3 Didier Dragna, 2 Petr V. Yuldashev, 1 Vera A. Khokhlova, 1 and Philie Blanc-Benon 4 1 Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow , Russia 2 Universite de Lyon, Ecole Centrale de Lyon, Laboratoire de Mecanique des Fluides et d Acoustique, Unite Mixte de Recherche, Centre National de la Recherche Scientifique 5509, Ecully, F-69134, France 3 Universite de Lyon, Universite Lyon 1, Laboratoire de Mecanique des Fluides et d Acoustique, Unite Mixte de Recherche, Centre National de la Recherche Scientifique 5509, Ecully, F-69134, France 4 Universite de Lyon, Ecole Centrale de Lyon, Centre National de la Recherche Scientifique, Laboratoire de Mecanique des Fluides et d Acoustique, Unite Mixte de Recherche 5509, F-69134, Ecully Cedex, France (Received 6 August 2018; revised 26 November 2018; acceted 3 December 2018; ublished online 3 January 2019) The irregular reflection of weak acoustic shock waves, known as the von Neumann reflection, has been observed exerimentally and numerically for sherically diverging waves generated by an electric sark source. Two otical measurement methods are used: a Mach-Zehnder interferometer for measuring ressure waveforms and a Schlieren system for visualizing shock fronts. Pressure waveforms are reconstructed from the light hase difference measured by the interferometer using the inverse Abel transform. In numerical simulations, the axisymmetric Euler equations are solved using finite-difference time-domain methods and the sark source is modeled as an instantaneous energy injection with a Gaussian shae. Waveforms and reflection atterns obtained from the simulations are in good agreement with those measured by the interferometer and the Schlieren methods. The Mach stem formation is observed close to the surface for incident ressures within the range of 800 to 4000 Pa. Similarly, as for strong shocks generated by blasts, it is found that for sherical weak shocks the Mach stem length increases with distance following a arabolic law. This study confirms the occurrence of irregular reflections at acoustic ressure levels and demonstrates the benefits of the Mach-Zehnder interferometer method when microhone measurements cannot be alied. VC 2019 Acoustical Society of America. htts://doi.org/ / [JDM] Pages: I. INTRODUCTION Irregular reflection is a classical henomenon of shock wave hysics first observed by Ernst Mach in In Mach s reorts, reflection of shock waves is categorized in two tyes: the two-shock configuration (regular reflection) and the three-shock configuration (Mach reflection). In the second case, the reflection attern consists of three shocks: the incident and reflected shocks, which intersect above the surface, and the Mach stem, which connects the oint of intersection to the surface. In 1943, these exerimental observations were theoretically studied by von Neumann, who formulated two- and three-shock theories. 2 Nowadays, a whole field of research is devoted to the study of irregular reflection of shock waves. The main feature of irregular reflection is the number of shocks in the reflection attern, which is not equal to two. Basically, irregular reflection can be broadly divided into two tyes: Mach reflection and von Neumann reflection. 3 The shock strength can be characterized by the acoustic Mach number M a, a) Also at: Universite de Lyon, Ecole Centrale de Lyon, Centre National de la Recherche Scientifique, Laboratoire de Mecanique des Fluides et d Acoustique, Unite Mixte de Recherche 5509, Ecully, F-69134, France. Electronic mail: masha@acs366.hys.msu.ru defined as the ratio of the maximum article velocity of the acoustic wave to the seed of the ambient sound in the roagation medium. The Mach reflection effect is tyical for strong shocks (M a > 0.47) and has broad classification of ossible reflection configurations. 3,4 The von Neumann reflection concerns weak acoustic shocks and occurs in the framework of the von Neumann aradox, whereas the threeshock theory either has no hysically accetable solutions or no solution at all. 3,5 The von Neumann reflection differs from the Mach reflection by the absence of sloe discontinuity between the reflected shock and the Mach stem. 6 In the following, the Mach stem refers to the single shock that forms between the surface and the oint where the incident shock and the reflected shock start to searate. Note that irregular reflections may exist for all shock ressure ranges and can be observed in gases, liquids, and even in solids. 7,8 In this aer, we focus on the reflection of weak acoustic shock waves in the framework of the von Neumann aradox. In Ref. 6 the different reflection tyes of weak lane ste-shock, saw-tooth wave, and N-wave have been studied in detail using numerical simulations based on the 2D Khokhlov-Zabolotskaya (KZ) equation. 9 Three ossible tyes of nonlinear reflection were reorted for weak acoustic shocks: nonlinear regular reflection, characterized by the resence of two shocks with different angles of incidence 26 J. Acoust. Soc. Am. 145 (1), January /2019/145(1)/26/10/$30.00 VC 2019 Acoustical Society of America

2 and of reflection, the von Neumann reflection, and the weak von Neumann reflection with no reflected shock at all. The tye of reflection deends on the shock amlitude, the grazing angle h of the lane incident wave, and the nonlinearity coefficient b of the roagation medium. Mathematically, this categorization is determined according to the critical arameter a introduced first in Ref. 10 as a ¼ sin h= ffiffiffiffiffiffiffiffiffiffiffi 2bM a. The transition from regular to irregular reflection tyes is reorted to occurwhen the value of the critical arameter decreases to a ¼ ffiffiffi 2 for the ste-shock and to about 0.8 for the lane N-wave. 6 These different reflection tyes have been exerimentally observed in water for lane ultrasound sawtooth waves. 7 The same categorization of reflection tyes has also been found for shear shock waves in soft elastic solids. 8 Excet those generated in shock tubes, shock waves are generally not lane and have a more comlicated morhology, often being closer to cylindrical or sherical waves. Examles would be focused ultrasound shock waves which are widely used in ultrasound theray, sonic booms roduced by aircraft, shock waves roduced by suersonic rojectiles, or blasts. For the last case, the shock wave has a sherical morhology and diverges sherically. However, to our knowledge, the irregular reflection of sherical weak acoustic shock waves has not been studied to the same extent as lane waves and strong shocks. The formation of the Mach stem for sherically diverging shock waves roduced by a sark source was first observed numerically in Ref. 11. In our revious works, the reflection of sherical N-waves generated by sarks over flat rigid surfaces was studied numerically and exerimentally using the Schlieren technique. 12,13 In this aer, revious exerimental and numerical observations are imroved and extended to longer roagation distances. In addition, waveforms in irregular reflection atterns are measured for the first time in air at ressure levels u to 10 kpa. Since microhones disturb the ressure field and do not measure shocks accurately, otical methods must be used to observe reflection atterns and wavefront geometry. In aeroacoustics, otical methods are commonly used to visualize the structure of shock waves. 14 The basic rincile of these methods is that acoustic waves induce variations in air density and consequently variations in the otical refractive index. As a result, the light beam deviates from its initial trajectory and acquires an additional otical hase difference when assing an acoustic wave. In revious studies on shock wave reflections, otical methods were mainly used only for visualizing reflection atterns to determine the tye of reflection Irregular reflection atterns and Mach stem formation can even be observed in one hundred-years-old hotograhs of ressure waves roagating in small room models. 19 However, in most cases, otical methods only rovide a visualization of the location of shocks and do not rovide a quantitative measure of ressure waveforms. It is recalled that the resonse of microhones does not allow for the detection of very raid ressure variations during the assage of shocks. Also note that using microhones in a field with shocks will distort it and create additional reflections. Recently, it has been shown that in a homogeneous atmoshere it is ossible to accurately reconstruct ressure signals from an electric sark source using otical techniques, in articular from Schlieren images or from interferometer measurements. 20,21 In addition, the temoral resolution of the interferometric method is 0.4 ls, which is 6 times higher than the resolution of a 1/8-in. condenser microhone (2.5 ls). This ability to reconstruct the ressure signature of acoustic shock waves using otical methods allows the study of waveforms at oints of reflection, where usual microhone-based techniques cannot be alied. The goal of this aer is to demonstrate that the interferometric method is caable of measuring waveforms in regular and irregular reflection atterns. The method is alied to a laboratory exeriment where short shock ulses of about 35 ls duration are generated by an electric sark source. Validation of the interferometric method is erformed by comarison with the results of numerical simulations based on the axisymmetric Euler equations and with reflection atterns obtained using the Schlieren technique. According to the categorization given in Ref. 6, three nonlinear reflection tyes shown in Fig. 1(a) are observed. These tyes are the regular reflection with unequal incident and reflected angles, the von Neumann reflection, and the weak von Neuman reflection. The interest here is focused on the von Neumann reflection tye in which the Mach stem formation occurs. In this aer, we also ay attention to the evolution of the Mach stem length in the reflection rocess. The content of the aer is organized as described as follows. Two exerimental configurations based on the interferometry and Schlieren techniques are described first (Secs. II A and II B, corresondingly). Then, the numerical model is FIG. 1. (Color online) (a) Scheme illustrating the transition of nonlinear reflection tyes for weak acoustic shocks 1: regular reflection, 2: von Neumann reflection (irregular reflection), 3: weak von Neumann reflection (irregular reflection). (b), (c) Diagrams of exerimental arrangements for measurements of reflection atterns in air: Mach-Zehnder interferometry method (b) and Schlieren technique (c). J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al. 27

3 resented (Sec. II C). In the results section, reconstructed waveforms from interferometric measurements (Sec. III A) and reflection atterns (Sec. III B) are analyzed and comared with simulated ones. Finally, the evolution of the Mach stem over the roagation distance is shown u to 70 cm from the sark source (Sec. III C). The results are summarized in Sec. IV. II. MATERIALS AND METHODS A. Mach-Zehnder interferometry method A Mach-Zehnder otical interferometer is used to measure the ressure signature of shock waves. The method has been resented in detail in Ref. 21, here we only describe the exerimental setu and summarize the otical signal rocessing erformed for reconstructing the ressure waveforms. The exerimental setu includes acoustical and otical arts [Fig. 1(b)]. The sound source is an electrical sark ositioned at a height of z s ¼ 21 mm above a lane rigid surface made of PVC. Considering the ratio of air and PVC imedances, the transmission is negligible and the surface is considered as erfectly reflecting. The sark source is made of two tungsten electrodes searated by a ga of 20 mm and sulied by an electrical voltage of kv. The sark generates sherically divergent ressure waves that are similar to N-waves or blast waves. The otical art is the Mach-Zehnder interferometer. As shown in Fig. 1(b), the main comonents are: a laser source (He-Ne laser, wavelength k ¼ nm, nominal ower 10 mw); two beam slitters (one searates the laser beam into two beams, and the second one collimates the two beams back into one); two mirrors; and a hotodiode sensitive to the light intensity that results from the interference of the two beams (NT53 372, Edmund Otics, surface 3.2 mm 2, bandwidth 16 MHz). The rincile of the interferometer is to analyze the hase difference between the robing beam A, which crosses the acoustic wave, and the undisturbed reference beam B. The resulting light intensity measured by the hotodiode is I ¼ I A þ I B þ 2 ffiffiffiffiffiffiffiffi I A I B cos u; (1) where I A and I B are intensities of the robing and the reference beams, corresondingly, and u is the otical hase difference between the two beams. The interferometer is first stabilized in the absence of the acoustic wave in such a way that the hase between the robing beam and the reference beam is u 0 ¼ =2. Neglecting light refraction and taking into account the radial symmetry of the wavefront, the otical hase difference u ac induced by the acoustic wave can be written as the direct Abel transform of the erturbation of the otical refractive index n (Ref. 21): u ac ðr; tþ ¼ 4 ð 1 nr ð s ; tþr s dr s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (2) k R rs 2 R2 where R is the distance between the sark and the robing beam. The refractive index n is related to the erturbation of the air density q via the Gladstone Dale constant G (Ref. 22): n þ n 0 ¼ 1 þ Gðq þ q 0 Þ, where q 0 is the ambient density and n 0 is the ambient refractive index. The exerimental conditions are such that the acoustic ressure is three orders of magnitude lower than the ambient atmosheric ressure. The density erturbation can therefore be considered as a linear function of the acoustic ressure : q ¼ =c 2 0, where c 0 is the seed of sound. Thus, the refractive index is exressed as n ¼ G=c 2 0 : (3) The measurement rotocol is described below for the osition of the robing beam above the rigid surface uon which acoustic waves reflect. Before each series of measurement, the system is calibrated. First, two outut voltages u A and u B, which are resectively roortional to the intensities I A and I B, are measured searately in the absence of the acoustic wave. This is done by stoing the reference beam with a screen when measuring u A and in the same way by stoing the robing beam when measuring u B. Then, the bias voltage on the hotodiode is shifted from zero to the value ð u A u B Þ. Once the system is calibrated, the outut voltage of the hotodiode u(t) is recorded when the robing beam is crossed by the acoustic wave. The outut voltage signal u(t) is induced by the interference of the reference and robing beams. It is exressed using the otical hase difference u ¼ u 0 þ u ac as follows: uðtþ ¼2 ffiffiffiffiffiffiffiffiffiffi u A u B sin fu ac ðtþg: (4) By combining Eqs. (1) (4) and alying the inverse Abel transform to Eq. (2), one obtains the following relation between the acoustic ressure (t) and the outut voltages u A, u B, and u: 0 1 ðtþ¼ c2 0 k ð 1 d 2 2 G dr s R u t s R r s c arcsin 0 2 ffiffiffiffiffiffiffiffiffiffi u A u B C A dr s ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; rs 2 R2 (5) where t s is a constant time shift corresonding to the osition of the front shock in the signal u(t). In exeriments, 140 waveforms are recorded at each oint in order to allow for statistical analysis of data. Because of small variations in the arameters of the generated waveforms (arrival time, eak ositive ressure, and eak negative ressure) and because these arameters are not linearly related, it is counterintuitive to comute an average waveform from the 140 records. A better solution is therefore to statistically analyze waveform arameters (arrival time, eak of ositive ressure, and negative eak) and to choose, among the 140 waveforms, the one whose arameters values are the closest to the average values comuted over all waveforms. The selected waveform is hereinafter referred to as the average waveform. Technically, the average waveform is such that the sum of the relative error for the three reviously mentioned arameters is the smallest. Since the method gives access to the waveform (t) ata single altitude above the reflecting surface, this rocedure is 28 J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al.

4 reeated for a series of robing beam ositions above the surface in order to determine the ressure field. The horizontal distance r between the sark source and the robe laser beam is varied from 10 u to 37 cm. Thus, for a given sark osition (z s ¼ 21 mm) the angle of incidence is in the range from 3.2 to 11.9 and the corresonding value of the acoustic Mach number is in the range from to Considering the exerimental conditions (relative humidity: 49%, temerature: 292 K, atmosheric ressure 0 : Pa), the Gladstone-Dale constant at nm is G ¼ m 3 /kg, the seed of sound is c 0 ¼ 343 m/s, the nonlinearity coefficient b is 1.2, and the ambient density q 0 is 1.22 kg/m 3. B. Schlieren visualization Otical visualization of the satial distribution of the acoustic field is erformed by a Z-tye Schlieren system [Fig. 1(c)]. 13,14 The acoustical art of the exerimental setu is the same as in the case of interferometric measurements. The source is 21 mm above the surface and r is in the range from 10 to 33 cm. The otical art includes two off-axis arabolic mirrors (15, 108 mm diameter, 864 mm focal length), a continuous halogen white light source at the focus of the first mirror, and a knife edge at the focus of the second mirror in front of a high-seed Phantom V12 CMOS camera. Schlieren images are recorded with a resolution of ixels, a frame rate of 18 khz and an exosure time of 1 ls. The brightness on images corresonds to the modulation of the light intensity and is roortional to the ressure gradient. 14 In order to imrove the contrast of the Schlieren images, an averaged background image is calculated for each set of measurements and then is subtracted from each image containing reflection atterns. The satial accuracy of the Schlieren measurement is limited by both the resolution of the image and the exosure time of the high-seed camera. The satial resolution is 0.15 mm er ixel. The exosure time is 1 ls, which corresonds to the roagation distance of 0.34 mm for the acoustic wave. These two uncertainties lead to a satial error of 0.51 mm. C. Numerical model In the resent work, the axisymmetric Euler equations are solved using a high order finite-difference time-domain (FDTD) algorithm develoed in the field of comutational aeroacoustics. These equations can be written in cylindrical coordinates (r, z) ð ~qv rþ ð ~qv zþ þ ~qv r ð~qv r Þ ~qv2 r þ ~ ð ~qv rv z Þ þ ~qv2 r >< @z ð~qv z Þ ð ~qv rv z Þ ~qv2 z þ ~ þ ~qv rv z @z ð~qe Þ ð v r f~qe þ ~ gþ ð v z f~qe þ ~ gþ þ v r f~qe þ ~ gþ >: @z r (6) where ~ is the total ressure, ~q is the total density, and v r and v z are velocity comonents in the r and z directions, resectively [see Fig. 1(a)]. The total energy E is defined for a erfect gas as ~qe ¼ ~ v2 þ ~q c 1 2 ; (7) where c ¼ 1:4 is the secific heat ratio for diatomic air and v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2 r þ v2 z : The time integration of Eqs. (6) is erformed by an otimized second-order low-storage Runge-Kutta algorithm. 23 Satial derivatives are evaluated by fourth order 11-oint stencil finite-difference schemes. At the interior oints, the centered schemes given in Ref. 23 are used. At the oints near the boundaries of the comutational domain, noncentered schemes 24 are used excet at the r ¼ 0 boundary, where the reviously mentioned centered schemes are alied to enforce the axisymmetric condition. Both satial and temoral methods are otimized in the Fourier sace to minimize numerical disersion and dissiation of the acoustic signal during long-range roagation. In addition, at each time ste, a selective filter with an 11-oint stencil is alied to suress grid-to-grid oscillations. 24,25 The shock caturing methodology roosed in Ref. 25 and imroved for the acoustic case in Ref. 26 is emloyed. It consists of a second order filter with a magnitude significant only around the shock and negligible everywhere else. The aim is to suress signal comonents of excessively high frequencies that are generated by nonlinear effects. The strength of this filter and the selective filter are both set to 0.1. The boundary condition given at the flat surface, z ¼ 0, is zero normal velocity. Finally, the radiation boundary conditions derived from asymtotic solutions of the linearized Euler equations in Ref. 27 are used to reduce reflections of the acoustic waves at the outer boundaries. J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al. 29

5 Simulations are carried out for a mesh of oints with satial stes of Dz ¼ 0:1 mmanddr ¼ 0:05 mm. The temoral ste Dt is fixed at s corresonding to a Courant-Friedrichs-Lewy number (CFL) around 0.3. A moving window is used to reduce comutational costs. The sark source is modeled as an instantaneous energy release with a Gaussian shae centered at the location (r ¼ 0, z s ¼ 21 mm) in an atmoshere at rest where temerature, ressure, and density are elsewhere constant. The initial conditions for the numerical simulations are then given by 8 ~ ðr; z; t ¼ 0Þ ¼ s ex lnð2þ r2 þ ðz z s Þ 2 w 2 þ 0 ; >< ~v ðr; z; t ¼ 0Þ ¼ 0; ~q ðr; z; t ¼ 0Þ ¼ q 0 ; Er; ð z; t ¼ 0Þ ¼ ~r; ð z Þ >: q 0 ðc 1Þ : (8) The amlitude s and the half-width w of the Gaussian source have been first adjusted to fit the interferometric data measured in free field at different distances from the sark source, yielding s ¼ 0.75 MPa and w ¼ 2.5 mm. However, a better agreement between exerimental and numerical results in the resence of the rigid surface has been obtained with a lower amlitude of the model source than what was derived by fitting the data in free field. Therefore, the numerical source amlitude is reduced to s ¼ 0.46 MPa. The atmosheric ressure 0 and the ambient density q 0 are set from the exerimental data (Sec. II A). As in exeriments, the source is 21 mm above the surface, but a larger distance range is considered as r is in the range from 0 to 70 cm. III. RESULTS In this section, exerimental and numerical methods are alied to study the acoustic field resulting from the reflection of sark-generated waves from a lane rigid surface in air. Waveforms reconstructed from interferometric data are analyzed in detail and comared with those obtained by numerical simulations. Reflection atterns measured at different distances from the sark by the two otical methods are comared with those obtained in simulations. The evolution of the Mach stem with the roagation distance is studied u to 37 cm for the exeriments and u to 70 cm for the numerical simulations. A. Pressure waveforms: Irregular reflection The ressure waveforms obtained at different heights h above the rigid surface at the distance r ¼ 13 cm are shown in Fig. 2. Solid lines corresond to the waveforms reconstructed from interferometric measurements as described in Sec. II A. Recall that these waveforms are so-called average waveforms obtained from a series of 140 recordings for each measurement osition. The dashed lines are simulation results, which corresond to virtual microhone oututs. Note that the temoral resolution of the Mach- Zehnder interferometric method (0.4 ls) is much smaller than that of a microhone or a weak shock ressure sensor. Another great advantage of the Mach-Zehnder interferometric method is that it does not disturb the investigation area unlike a membrane sensor. The size of the Mach stem that will be discussed below is in the order of the diameter of a microhone. Thus, only such otical method is caable of accurately caturing the details of ressure waveforms close to reflective boundary. As seen in Fig. 2, at a height of 2 mm above the surface, the waveform shae is between the one of a blast wave and FIG. 2. (Color online) Pressure waveforms of sark-generated waves measured by the Mach-Zehnder interferometer (solid curves) and obtained from numerical simulations (dashed curves) at different heights h above the rigid surface at the distance r ¼ 13 cm along the surface. 30 J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al.

6 an N-wave. The ositive hase begins with a stee shock front (named front or leading shock), followed by a gradual ressure decrease to zero after the eak ressure. Next there is a negative hase and finally the ressure returns back to equilibrium with a less stee ressure rise than that of the front shock. At 4 mm above the surface, the searation of the direct and reflected shocks starts to be observed at the front shock. Additional measurements with a finer analysis resolution in height confirm that only one shock is formed u to 3 mm height above the surface. Thus, the shock observed at h ¼ 2 mm is the Mach stem. The reflection attern for this case therefore corresonds to the irregular reflection tye, and the height of the Mach stem is 3 6 1mm. The uncertainty is mostly related to the mechanical setu and signal analysis while the satial resolution of the interferometer is in the order of only 0.2 mm. Note that the height of the Mach stem observed for this source and receiver osition (r ¼ 13 cm, z s ¼ 21 mm) is in strong agreement with the equation given in Ref. 13: h M ¼ z s ð0:41=aþ 2 ; which was obtained from a series of simulations and exeriments with the same sark source. With the arameter values of a ¼ ( max ¼ 2 kpa, M a ¼ , h ¼ 10.9 ) and z s ¼ m this equation gives a Mach stem height of 3.4 mm. For waves measured farther from the surface, from h ¼ 4 to 32 mm, it can be noted that the reflected front shock is delayed until the delay corresonds to the duration of the wave, then it begins to merge with the ressure rise of the negative art (see frame at h ¼ 34 mm). Similar to what is observed with strong shocks, the interaction of these two shocks well above the surface can be nonlinear and also lead to the formation of a single shock front (see Fig. 2, frame at h ¼ 42 mm). Interestingly, at h ¼ 44 mm, the waveform finally resembles two cycles of a saw-tooth wave. The simulated waveforms (dashed lines in Fig. 2) are consistent with the measurements with good accuracy, esecially for the front shock. The negative art is slightly less identical. Concerning the simulations, the Gaussianenveloe injection of energy used to simulate the sark source is a simlified model that accurately reroduces the ositive imulse. However, it does not take into account neither the resence of electrodes nor hot gases generated during the discharge. Thus, the negative art of the initial ressure waveform is simulated less accurately. Above the surface, the initial difference in shae of the negative hase between the measured and simulated waveforms roagates and leads to a slight difference in the negative eak ressure (Fig. 2,seeh ¼ 2 mm and h > 36 mm). Concerning the exerimental method, one source of uncertainty is related to the value of distance R, which is used in the signal rocessing [Eq. (5)]. Actually, for measurement ositions above the surface, the ath lengths of the incident and reflected waves are different. However, this difference remains less than 10% for the configurations considered in the resent study, so when alying Eq. (5) the value of R is chosen as the length of the direct ath from the source to the measuring robe. However, as seen in Fig. 2, the difference between the simulated and measured ressure waveforms remains small, and does not exceed 10% for most art of the waveforms which is comarable to the reeatability error of the sark source. The simlified model of the sark source as a Gaussian energy injection combined with a nonlinear Euler equation roagation code yields time waveforms very close to the waveforms obtained from interferometric data and aroriate signal rocessing. Therefore, the methods are validated and alied to study the roblem of irregular reflections. B. Irregular reflection atterns Interferometric and Schlieren techniques lead to different reresentations of the reflection atterns. The Schlieren technique rovides hotograhs that show the location of the shock fronts in a lane, while the interferometric method rovides information in the time domain (Figs. 3 and 4). Each Schlieren hotograh catured by the high-seed camera is obtained with a single sark. In Schlieren images, the wave roagates from left to right and the wavefront is located on the right side of each image (Fig. 3). The contrast deends on the density gradient: the whitest areas corresond to the strongest ressure gradients. Thus, the different steeness of the front shock and the negative hase ressure rise leads to a different brightness. The reflection of the ressure rise of the negative hase is also visible, but as it is less stee, it is less bright. The reflection atterns obtained numerically corresond to the simulation of a single sark. The Schlieren ictures obtained numerically (Fig. 3, bottom) reresent the gradient of the density of air, but in both r and z directions, while in the exeriment, the gradient is only in the r direction. The Schlieren images in Fig. 3 show the reflection atterns catured at different distances r along the surface. They show the dynamic character of irregular reflection: the Mach stem formed by the interaction of the direct and reflected front shocks develos during its roagation along the surface. Unlike the front shock, the reflection of the negative hase is always regular because of its smaller amlitude and smoother ressure rise. As exected, Fig. 3 shows a distinctive feature of nonlinear reflection: the angle of incidence and the angle of reflection are not equal. 3 The Schlieren technique immediately rovides an image of reflection atterns. However, the images do not show the ressure field but rather the density gradient of the air in a direction determined by the orientation of the knife edge. The method described for the case of the free field in Ref. 20 to estimate the ressure from the image does not aly here due to the geometry of the wavefront. In addition, the resolution in the temoral and satial domains is limited by the settings of the camera and the otics. Thus, it is interesting to reconstruct the reflection atterns from interferometric data. In the same way that the reflection atterns are lotted from hydrohone measurements, 7 the reflection atterns can be lotted in the satio-temoral domain from the waveforms obtained with the Mach-Zehnder interferometer. The time domain reflection attern is obtained from the average waveforms measured at different heights above the surface by converting the ressure amlitude into color and tracing the resulting horizontal color bar at the corresonding height. J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al. 31

7 FIG. 3. Geometry of the reflection atterns visualized by Z-tye Schlieren system (to row) and simulated numerically as a gradient of the density field (bottom row). This means that each reflection attern in the left column of Fig. 4 is obtained from a series of about three thousand sarks: 140 waveforms at each values of height h ranging from 2 u to 44 mm with a 2 mm ste. Recalling that the sark source roduces ulses with good reeatability 28 (variations in eak ressure and duration of about 10%), a reflection attern obtained from average waveforms should be identical to that of a single sark. Unlike the reflection scheme obtained in the satial domain with the Schlieren technique, the front shock aears on the left side of the image since it corresonds to the shortest roagation time (Fig. 4). Note that the color scale in the atterns obtained from the interferometric measurements reresents the ressure, while the color scale of the Schlieren images reresents the density gradient of the air in one direction and does not give information about ressure levels. The numerical interferometric satio-temoral reflection atterns (Fig. 4, right column) are obtained from the ressure-time waveforms in Fig. 2 exactly as it was done for the exerimental data. For both otical methods, strong agreement is observed between the measured and simulated reflection atterns (Figs. 3 and 4). The front shock is reflected according to the irregular tye of reflection and the Mach stem is formed near the surface. At the r ¼ 13 cm osition, the Mach stem is short, only about 3 mm as reviously mentioned, but grows with the roagation distance and reaches the height of 13 mm at r ¼ 33 cm (Fig. 4). The highest ressure levels are located just behind the Mach stems near the surface. At r ¼ 13 cm, there is also an overressure zone formed at the to of the figure, where the front shock of the reflected wave interacts with the rear shock of the incident wave (at aroximately t ¼ 400 ls). As mentioned reviously in Sec. III A, nonlinear interaction of shocks can also occur in this area in the same manner as the formation of the Mach stem on the reflecting surface. C. Evolution of the Mach stem In the reflection rocess, the length of the Mach stem changes with the roagation distance. In the context of blast wave studies, emirical models have been develoed to redict the variation of the Mach stem length generated by the reflection of strong shock waves. 29 These models describe a arabolic 29 or a cubic 30 growth of the Mach stem height. Recent studies reorting exeriments of weak shock reflection but at a larger scale than the resent exeriment showed a good agreement with a cubic law. 18 Although in our revious study with the same source a linear trajectory was reorted, 13 that study was done for a small range of roagation distances r from 7 u to 13 cm. In the resent study, the length of the Mach stem is analyzed from the three sets of data and at greater distances: with the Schlieren technique in the range from 10 u to 33 cm; with the interferometer in the range from 10 u to 37 cm, and with simulations u to 70 cm. The results are lotted in Fig. 5. Both otical methods and numerical simulations return aroximately the same length of the Mach stem. The extension of the study to greater distances showed that the evolution of height of the Mach stem actually follows a arabolic law as exected from Ref. 29 rather than a linear law as done in Ref. 13. With the source arameters and ositions considered in this study, a arabolic aroximation h M ¼ 0:104r 2 þ 8: r 4: is valid from r ¼ 0.1 m u to 0.7 m, the maximum roagation distance where the simulations were carried out (inset of Fig. 5). Note that the distance of 70 cm corresonds to about 60 wavelengths. 32 J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al.

8 FIG. 5. (Color online) Evolution of the length of the Mach stem over the roagation distance r. Data are obtained from otical exeriments using the Mach-Zehnder interferometer and Schlieren system as well as from numerical simulations. The arabolic aroximation 0:104r 2 þ 8: r 4: given in meters was found to fit the trajectory. FIG. 4. (Color online) Reflection atterns measured by the Mach-Zehnder interferometer (left column) and obtained in numerical simulations based on axisymmetric Euler equation (right column). Reflection atterns are shown at different distances r along the surface. The Mach stem starts to be formed at a distance r ¼ 10 cm. At this distance, the eak ositive ressure of the incident wave is 3 kpa. Estimating the value of the acoustic Mach number using its definition M a ¼ max =ðc 0 Þ for the lane wave gives M a ¼ The value of the critical arameter is estimated from a ¼ sin h= ffiffiffiffiffiffiffiffiffiffiffi 2bM a, where the angle h corresonds to the angle of incidence of the lane wave. The transition from the regular tye of reflection to the irregular one occurs for the value of the critical arameter a 0:92: This value agrees with our revious results 12,13 and also with corresonding transition for the lane N-wave obtained in numerical simulations based on the 2D KZ equation. 6 Although the critical arameter a introduced in Refs. 6, 10 determines the reflection tye only for lane waves, it includes two main hysical effects involved in the Mach stem formation for waves of non-lanar morhology. The first effect is the influence of the incident angle h: for smaller angles the incident and reflected shocks are closer to each other and their nonlinear interaction that results in formation of only one shock can occur at smaller shock amlitudes than for larger incident angles. The second effect is the influence of the acoustic Mach number M a which determines the strength of nonlinear effects. Since the Mach stem formation is a nonlinear henomenon, it resents itself at smaller distances r for stronger shock waves. Both decrease of the incident angle and increase of the shock wave amlitude lead eventually to the Mach stem formation. Thus, the evolution of the Mach stem is strongly deendent on the evolution of h and M a. For examle, in the case of idealized lane ste-shocks, both h and M a are constant along the roagation distance and hence the length of the Mach stem does not change. In the roagation of acoustic lane waves, like saw-tooth waves or N-waves, the shock amlitude decay occurs due to nonlinear absortion at the front shock while the incident angle h remains constant. For this case the length of the Mach stem gradually decreases to zero since the value of the critical arameter a monotonically increases. In the current aer, sherically diverging waves are considered. Such sherical morhology leads to continuous decrease of the incident angle h. The evolution of the Mach stem is therefore determined by the cometition between the influence of the incident angle h and the decrease in amlitude due to both attenuation at the shock front and sherical divergence. To estimate the evolution of the Mach stem over long distances r one can use an aroximation for grazing angles: sin h tg h ¼ z s =r: Pressure decay for the current sark source has been studied reviously in Refs. 20, 21, 28 and is described by the equation ¼ ref ðr=r ref Þ a ; where ref is the ressure amlitude at the roagation distance r ref and the coefficient a ¼ 1.2. In revious aers concerning sark sources, the coefficient a is reorted to be in the range of 1.2 to 1.4. Based on the knowledge of the ressure decay and angle aroximation at long distances r the critical arameter a can be estimated as a ¼ sin h= ffiffiffiffiffiffiffiffiffiffiffi 2bM qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a r a=2 1 z s c 0 =2b ref r a ; i.e., it is roortional to r a=2 1. ref For the given range of the coefficient a this deendence means that the critical arameter a decreases monotonically J. Acoust. Soc. Am. 145 (1), January 2019 Karzova et al. 33

9 with the roagation distance r. From this analysis one can conclude that the influence of the angle is more imortant than the decrease of the shock amlitude over the roagation distance. Thus, the reflection tye remains irregular at long distances r u to a distance after which nonlinear effects are negligible. In simulations and exeriments at the laboratory scale, we indeed observe only the growth of the Mach stem until it is difficult to be distinguished. At long range, the weak von Neumann reflection is therefore obtained. We believe that the Mach stem evolves in the same way as described for strong shocks roduced by exlosions, for which its height only increases with the distance before the Mach stem eventually disaears as the grazing angle tends to zero. IV. CONCLUSIONS In this work, otical techniques were alied to measure the irregular reflection atterns of weak shock waves above a rigid lane surface. The Schlieren otical method, which is sensitive to the density gradient, rovides the structure of the reflection attern in the satial domain. Mach-Zehnder interferometry with associated signal rocessing makes ossible the reconstruction of ressure-time waveforms with satial and temoral resolution that would not be achievable when using microhones. Having detailed knowledge of the characteristics of the source, it was ossible to calibrate a source model based on an initial energy deosition. This source model, combined with a solver based on axisymmetric Euler equations, simulates the exeriments with good recision. Both waveforms and reflection atterns obtained from otical measurements are in a good agreement with numerical simulations, roviding a cross validation of both techniques to investigate nonlinear henomena involving weak shocks. Thus, an instantaneous release of energy with a Gaussian shae is a suitable model for simulating sherical acoustic waves generated by a sark source, and a solver of the Euler equations is validated as a redictive tool to study nonlinear acoustic roagation and reflection henomena. This aroach can be generalized and alied to study roblems with more comlex geometries than a oint source above a reflecting lane rigid surface. Thanks to the temoral and satial resolution of the Mach-Zehnder interferometer, it was ossible to measure accurately the Mach stem height. The dynamics of irregular reflection were also observed over about 60 wavelengths, and the growth of the Mach stem was found to corresond to a arabolic law with the roagation distance along the surface in the same manner as reorted for stronger shocks. In the case of a sherical wave, it is exected that the height of the Mach stem, once formed, will increase until the incident wave and the reflected wave merge when the angle of incidence decreases to zero. The resent study was based on using a source that was well-known to the authors since it had been used in several revious exerimental works. This choice allowed for crosscomarison of exerimental and numerical data with already analyzed databases. However, the results are not limited to the current configuration but can be extended and generalized. From a metrological oint of view, this study demonstrates that the satial and temoral resolution of the Mach-Zehnder interferometric technique makes it ossible to study henomena at scales too small for microhones, in articular when shock waves are involved. From a hysics oint of view, these results show that irregular reflection occurs once the ressure waves are strong enough to generate shock waves and once the grazing angle is sufficiently low for a given eak ressure level. This henomenon, wellknown in the case of more intense shocks, can be observed for ressures on the order of 1 kpa, and must therefore be considered in nonlinear acoustics. Furthermore, given that it is imossible to observe this henomenon using microhones, it is generally not considered in acoustics. Therefore, further exerimental and numerical investigations with high temoral and satial resolution techniques must be carried out to study nonlinear shock interactions at acoustic levels (eak ressure level between 1 and 10 kpa), and in cases with more comlex geometries and surface characteristics, including rough and curved surfaces. Connecting the results obtained in the resent laboratory exeriment with exeriments at moderately larger scale 18 or with much strong blast waves is another ersective. ACKNOWLEDGMENTS Work suorted by RSF and by the Labex CeLyA of Universite de Lyon, oerated by the French National Research Agency (ANR-10-LABX-0060/ ANR-11- IDEX-0007). The numerical study was conducted within the framework of LETMA (Laboratoire ETudes et Modelisation Acoustique), a Contractual Research Laboratory shared between CEA, CNRS, Ecole Centrale de Lyon, C-Innov, and Sorbonne Universite. It was granted access to the HPC resources of IDRIS under the allocation made by GENCI. 1 P. Krehl and M. van der Geest, The discovery of the Mach reflection effect and its demonstration in an auditorium, Shock Waves 1(1), 3 15 (1991). 2 J. Von Neumann, Oblique reflection of shocks, in John von Neumann Collected Work, edited by A. H. Taub (MacMillan, New York, 1963), Vol. 6, G. Ben-Dor, Shock Wave Reflection Phenomena, 2nd ed. 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