Measurement of a laser-induced underwater shock wave by the optical-flow-based background-oriented schlieren technique

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1 Measurement of a laser-induced underwater shock wave by the optical-flow-based background-oriented schlieren technique K. Hayasaka 1, Y. Tagawa 1*, T. Liu 2, M. Kameda 1 1: Dept. of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Japan 2: Dept. of Mechanical and Aerospace Engineering, Western Michigan University, Kalamazoo, MI 49008, USA * Correspondent author: tagawayo@cc.tuat.ac.jp Keywords: Background-Oriented Schlieren, PIV, Optical flow, Shock wave ABSTRACT We propose a high-resolution BOS technique based on the optical flow (OF-BOS) for measuring a laser-induced underwater shock wave for the first time. BOS technique obtains the pressure field by detecting the displacement vectors of local density gradient of water. We previously utilized PIV-based BOS (PIV-BOS). However it was difficult to reconstruct the pressure field of the underwater shock wave due to poor number density of the vector field. The advantage of OF-BOS is that it obtains a lot of vectors, number of which is the same as number of pixels of the original image. In our experiments, OF-BOS obtains 1,700 1,700 vectors whereas PIV-BOS obtains only vectors from the same original image. We find that the number density of vectors strongly affects the accuracy of the pressure field. On one hand, in the case of many vectors, we reasonably reconstruct the pressure field, which is similar to the shape of the shock front. On the other hand, in the case of less vectors, it is difficult to reconstruct the pressure field. Furthermore, we compare quantitatively the pressure distribution obtained from OF-BOS, PIV-BOS and a hydrophone. Remarkably, the pressure distribution obtained from OF-BOS with 3,922 3,922 vectors fairly agrees with that obtained with a hydrophone. We find that the pressure gradient of the shock front of 21MPa/mm can be detected with 150 vectors/mm. 1. Introduction Non-contact measurement of an underwater shock wave is crucial for investigating important phenomena such as non-invasive medical treatments (Klaseboer et al., 2007, Tagawa et al., 2012, Tagawa et al., 2013). Background-oriented schlieren (BOS) is a contact-less technique for measuring the displacement field of fluids, which is related to the pressure field (Meier et al., 2002). Raffel et al. (Raffel et al., 2000) applied BOS to an airflow around a helicopter. Murphy and Adrian (Murphy and Adrian, 2011) quantify the density gradient field of a shock in the air. However, as far as we know, the measurement of an underwater shock wave using BOS is scarcely reported. The measurement is challenging since the density gradient of water is small. Recently we have quantified the displacement field of the local density gradient of the

2 underwater shock wave using BOS (Yamamoto et al., 2015). Based on these results we have tried to reconstruct the pressure field. However, the pressure field is not correctly reconstructed owing to poor number of vectors. One of the main causes is the cross-correlation algorithm (PIV) included in BOS procedure since PIV algorithm provides the vector field with much lower number of vectors compared to the number of pixels of original image. In order to overcome this problem, in this paper, we propose a high-resolution BOS technique based on the optical flow (OF-BOS). Its advantage is that it obtains the vector field with the same spatial resolution as the original image. We apply the optical flow to BOS technique and investigate the effect of number of vectors on the pressure field. Furthermore, we compare the pressure distribution obtained from both the OF-BOS, the PIV-BOS and hydrophone. 2. BOS technique and analysis method We explain BOS technique in this section. We display the principle of BOS in Fig.1 (Yamamoto et al., 2015, Venkatakrishnan et al., 2004). It consists of a camera, a density gradient and a background of random-dot. The BOS compares a non-disturbed image and disturbed image due to the presence of the density gradient as indicated by the red line in Fig.2. The BOS then calculate the displacement w, which is related to the integral of the density gradient as (Hinsberg et al., 2014) w = # $ c(c + 2l *) #, -.,./ (1) n = Kρ + 1 (2) where c (= 200 "m) is the thickness of the density gradient, l 0 (= 5 mm) is the distance from the density gradient to the background, n 0 is the refractive index of water, n is the refractive index, z is the coordinate along the direction of w, K (= kg/m 3 ) is the Gladstone-Dale constant, and ρ is the density of the fluid. We calculate the displacements (u of x-coordinate and w of z-coordinate) using both the optical flow and PIV. A snapshot of the shock wave (shadowgraph) is shown in Fig.2 (i). The shock wave propagates spherically around a laserinduced bubble. The high-pressure region is expected at a shock front. Fig.2 (ii) and (iii) show a non-disturbed image and disturbed image of the background, respectively. It is difficult for us to see directly the shock wave in Fig.2 (iii) since the density change of water is small. Fig.3 (a) and (b) show the schematic diagrams of PIV and optical flow, respectively. A right column displays the result of calculating a displacement. A black square means a dot of a background. PIV calculates the displacements by the cross-correlation algorithm. The correlation window is shown by thick-blue line in Fig.3 (a). In our experiment, PIV setting is 50 % overlap

3 with the Fast Fourier Transform (FFT) multi-pass interrogation (from 32 to 4 pixel) (Raffel et al., 2000). The optical flow is based on a form typical of the transport equation (Horn and Schunck, 1981, Liu and Shen, 2008), which determines the displacement of each pixel in the image plane. Liu and Shen (Liu and Shen, 2008) derive the physics-based optical flow equation; gu = f(x, z, g) (3) where g = g(i) is a function of the normalized image intensity I, = ( / x, / z), u = (u, w) is the velocity vector in the image plane, x and z are the image coordinates. The velocity u multiplied by the unit time t is the displacement w. The displacements obtained from PIV and the optical flow show the projected information along y-direction. To calculate the pressure field of x-y surface, we reconstruct the displacement field on x-y surface from the displacements field on x-z surface. We utilize the back projection method without filtering procedure to reconstruct the displacement. In two-dimensions space, the equation (1) and (2) are described as the Poisson equation (Eq. (4)). We then calculate the density field applying the Gauss-Seidel method to equation (4) and the pressure field applying the Tait equation (5).. >? +.>? = $ #CD? - (.G +.H ) (4).@ >.A > ED EC$F -.@.A ICJ I - CJ = (?? - ) K (5) where p 0 is the hydrostatic pressure, B is 314 MPa, ρ 0 is the fluid density under hydrostatic pressure and α is 7. Fig. 1 Principle of BOS technique

4 Fig. 2 (i) A snapshot of the laser induced shock wave by shadowgraph. A setup is the same as Fig.4 expect for the background. The pulsed laser is illuminated from the upper. (ii) Nondisturbed image by BOS technique. (iii) Disturbed image by BOS technique. Laser condition is the same as Fig.2 (i). Fig. 3 Image analysis method of detecting displacements. (i) PIV-BOS (ii) OF-BOS 3. Experimental setup Fig.4 shows the experimental setup of BOS for photographing a shock wave. A laser pulse (Nd:YAG laser Nano S PIV, Litron Lasers, wavelength: 532 nm, pulse width: 6 ns, laser energy: 1.9 mj) is focused throughout a 20 microscope objective to a point inside a distilled-water-filled glass tank ( mm). The underwater shock wave originates at a laser-focused point and propagates spherically. The background is placed behind the laser-focused point inside the tank. We record the shock wave using a camera with 2,048 2,048 pixels (FASTCAM Mini WX100,

5 Photron Inc.) and another one with pixels (Kirana, Specialized Imaging Co.). As a light source we utilize a laser stroboscope (SI-LUX 640, Specialized Imaging Co.) with a pulse width of 20 ns. A delay generator (Model 575, BNC Co.) synchronizes the pulsed laser, the camera, and the light source. We utilize a pressure sensor (Muller-Platte Needle Prove, Mueller Instruments, Germany) to verify results obtained by PIV and the optical flow. The pressure sensor is set toward a center of a laser-induced shock. The distance from the center to the pressure sensor is 5.0 ± 0.1 mm. We convert the pressure to the displacements using equations (2), (3) and (4). 4. Results and Discussion Fig. 4 Experimental setup 4.1 Effect of the amount of vectors on reconstruction Fig.4 shows the pressure field in x-y surface obtained from PIV-BOS and OF-BOS. All results are calculated from the same original picture (1,700 1,700 pixel). Number of vectors before back projection procedure is also shown. An image of the left side is obtained from PIV-BOS. This image has much smaller number of vectors ( vectors) compared to number of pixels of the original image (1,700 1,700 pixels). The pressure is the highest in the center of image. This

6 result is totally inconsistent with pressure field expected from the shadowgraph image (Fig.1 (i)). In contrast, a result of 1,700 1,700 vectors shows the peak pressure on the spherical shock front. Remarkably, this pressure distribution is similar to the pressure field expected from the shadowgraph image (Fig.1 (i)). Both vectors and vectors are obtained by decreasing number of vectors from 1,700 1,700 vectors. The pressure field of 3,922 3,922 vectors is obtained by interpolating the displacement field. This result is the almost same pressure field as 1,700 1,700 vectors. These results indicate that number of vectors affects the pressure field. Here we show the effect of number of vectors on the pressure field in Fig.5. In the left order to right, results show the displacement field before the reconstruct procedure, the displacement field after the reconstruct procedure, the enlarge view of displacement field and pressure field, respectively. The upper line and the bottom line show results of the small number of vectors (OF- BOS, vectors) and results of the large number of vectors (OF-BOS, 1,700 1,700 vectors), respectively. Before the reconstruction procedure, vector field looks similar to 1,700 1,700 vector field. After the reconstruction procedure, vector field decreases the displacement compared to 1,700 1,700 vector field. The pressure field of vectors is different from the pressure field of 1,700 17,00 vectors. These results indicate that number of vectors affects the pressure field. 4.2 Comparison of results obtained from both OF-BOS, PIV-BOS and a hydrophone In this section, we compare results obtained from OF-BOS, PIV-BOS and a hydrophone. Fig.6 (i) and Fig.6 (ii) show the displacement vs. the distance from a center of a shock wave and the pressure vs. the distance from a center of a shock wave, respectively. A black solid line, a blue cross and a red cross mean the result obtained simultaneously from a hydrophone, PIV-BOS and OF-BOS, respectively. A circle plots, a triangle plots and a square plots represent the pressure obtained from vectors, vectors and 3,922 3,922 vectors, respectively. In Fig.6 (i), the peak displacements obtained from both PIV-BOS and OF-BOS fairly agree with those from a hydrophone. Therefore, these results indicate that OF-BOS and PIV-BOS successfully obtain displacements before the reconstruction procedure. In Fig.6 (ii), the pressure from PIV- BOS is close to the hydrostatic pressure at any distance. It means that PIV-BOS fails to calculate the pressure. This is caused by low number of vectors. OF-BOS with vectors does not reconstruct a shock front. OF-BOS with vectors obtains the peak pressure correctly although it does not reconstruct a shock front. These results suggest that the reconstruction with small number of vectors causes low accuracy of pressure field. In other words, sufficient number

7 of vectors in a shock front is necessary to solve Poisson equation. Therefore, the reconstruction with the small number of vectors does not obtain pressure field in BOS technique. The pressure obtained from OF-BOS with 1,500 1,500 vectors and 3,922 3,922 vectors show reasonable distribution of shock front. This indicates OF-BOS with 150 vectors/mm can detect the pressure gradient of 21MPa. These results indicate that OF-BOS with a lot of vectors is capable of measuring a laser-induced underwater shock wave. To improve the detectable range of pressure gradient and peak pressure, the following methods are recommended; (a) utilizing high spatial resolution camera, (b) adjusting the position of a camera and a background to pressure gradient and peak pressure and (c) interpolating vectors into the displacement field. The method (a) and the method (b) affect the equation (1) based on BOS principle. The method (c) is effective for reconstructing 2-D field from 3-D field. Fig. 4 The pressure field in x-y surface obtained by PIV-BOS and OF-BOS. All result is analyzed to same pictures. The number under the result means the size of vector field. The left side result is analyzed by PIV-BOS. The others are obtained from OF-BOS. Fig. 5 The effect of number of vectors on pressure field. In the left order to right, results show the displacement field before the reconstruct procedure, the displacement field after the reconstruct procedure, the enlarge view of displacement field and pressure field, respectively.

8 Fig. 6 (i) Distance from a center of a shock wave versus the displacement. (ii) Distance from a center of a shock wave versus pressure. Black solid line show results obtained from hydrophone measurements (Yamamoto et al. 2015). Blue plots and red plots indicate results obtained simultaneously from PIV-BOS and OF-BOS, respectively. Circle plots, triangle plots and square plots mean OF-BOS with vectors, vectors and 3,922 3,922 vectors in Fig4, respectively. 5. Conclusion We propose a high-resolution BOS technique based on the optical flow (OF-BOS) for measuring a laser-induced underwater shock wave. We find that the OF-BOS provides a lot of vectors of local density gradient of water than PIV-BOS. We indicate that number of vectors affects the accuracy of reconstruction. In the case of the large number of vectors, BOS is able to reconstruct the pressure field. It is difficult to reconstruct the pressure field with the small number of vectors since the small number of vectors causes the reconstruction to decrease its quality. Furthermore, we compare the displacement and the pressure obtained from OF-BOS, PIV-BOS and

9 hydrophone. The displacements obtained from both PIV-BOS and OF-BOS fairly agree with those from a hydrophone. Remarkably, the pressure distribution obtained from OF-BOS with large number of vectors agrees with that obtained with the hydrophone. This indicates that the OF-BOS is capable of measuring a laser-induced underwater shock wave in this condition. We currently investigate the detectable range of pressure gradient and peak pressure. References Van Hinsberg NP and Rösgen T (2014) Density measurements using near-field backgroundoriented Schlieren, Exp. Fluids 55(4) Horn BK, Schunck BG (1981) Determining optical flow. In 1981 Technical symposium east (pp ) International Society for Optics and Photonics. Klaseboer E, Fong SW, Turangan CK, Khoo BC, Szeri AJ, Calvisi ML, Zhong P (2007) Interaction of lithotripter shockwaves with single inertial cavitation bubbles. J Fluid Mech 593: Liu T, Shen L (2008) Fluid flow and optical flow. J Fluid Mech Meier G (2002) Computerized background-oriented schlieren. Exp Fluids 33(1) Murphy MJ, Adrian RJ (2011) PIV through moving shocks with refracting curvature. Exp Fluids 50(4) Raffel M, Richard H, Meier GEA (2000) On the applicability of background oriented optical tomography for large scale aerodynamic investigations. Exp Fluids 28(5) Tagawa Y, Oudalov N, Visser CW, Peters IR, van der Meer D, Sun C, Lohse D (2012) Highly focused supersonic microjets. Phys Rev X 2(3) Tagawa Y, Oudalov N, El Ghalbzouri A, Sun C, Lohse D (2013) Needle-free injection into skin and soft matter with highly focused microjets. Lab On a Chip 13(7) Venkatakrishnan L. Meier GEA (2004) Density measurements using the background oriented schlieren technique. Exp Fluids 37(2) Yamamoto S, Tagawa Y, Kameda M (2015) Application of background-oriented schlieren (BOS) technique to a laser-induced underwater shock wave. Exp Fluids 56(5) 1-7.