Optical-flow-based background-oriented schlieren technique for measuring a laser-induced underwater shock wave
Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda
EExperiments in Fluids manuscript No. (will be inserted by the editor)
Optical-flow-based background-oriented schlierentechnique for measuring a laser-induced underwatershock wave
Keisuke Hayasaka , Yoshiyuki Tagawa ∗ ,Tianshu Liu , , Masaharu Kameda , Received: date / Accepted: date
Abstract
The background-oriented schlieren (BOS) technique with the physics-based optical flow method (OF-BOS) is developed for measuring the pressure fieldof a laser-induced underwater shock wave. Compared to BOS with the conventionalcross-correlation method in PIV (called PIV-BOS), by using the OF-BOS, thedisplacement field generated by the small density gradient in water can be obtainedat the spatial resolution of one vector per pixel. The corresponding density andpressure fields can be further extracted. It is particularly demonstrated that thesufficiently high spatial resolution of the extracted displacement vector field isrequired in the tomographic reconstruction to correctly infer the pressure field ofthe spherical underwater shock wave. The capability of the OF-BOS is criticallyevaluated based on synchronized hydrophone measurements. Special emphasis isplaced on direct comparison between the OF-BOS with the PIV-BOS.
Keywords
Background-oriented schlieren (BOS) · Optical flow · Cross-correlation · Particle image velocimetry (PIV), Underwater shock wave
Non-contact measurements of underwater shock waves are crucial for understand-ing some important phenomena related to non-invasive medical treatments (Klase-boer et al (2007); Tagawa et al (2012, 2013)). The back-ground-oriented schlieren(BOS) technique can provide global non-contact diagnostics for measuring the
1: Department of Mechanical Systems Engineering, Tokyo University of Agriculture andTechnology, Japan2: Department of Mechanical and Aerospace Engineering, Western Michigan University,Kalamazoo, MI 49008, USA3: Institute of Global Innovation Research, Tokyo University of Agriculture and Technology ∗ Correspondent author: Y. TagawaE-mail: [email protected].: +8142-388-7407Fax: +8142-388-7407 a r X i v : . [ phy s i c s . f l u - dyn ] O c t Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda displacement field induced by the density change in fluid, which is used to in-fer the density and pressure fields. The BOS technique has been used in variousmeasurements particularly in compressible air flows and gas-mixing flows (Raf-fel et al (2000); Meier (2002); Venkatakrishnan and Meier (2004); Murphy andAdrian (2011)). A review of the BOS technique is given by Raffel (2015). TheBOS technique is an extension of the classical schlieren technique traditionallyused in high-speed wind tunnel for aerodynamics measurements. Due to the sig-nificant advance of digital cameras and image processing, the experimental setupof the BOS system is much simpler than its classical counterpart with complexarrangements of optics. In general, a background pattern plate placed behind ameasurement domain of fluid is imaged by a camera in the cases with and with-out the fluid density gradient. The background pattern image is disturbed by thedensity gradient that deflects the light rays radiated from the background pat-tern plate. The disturbed image appears shifted relative to the undisturbed imagetaken in the case with the homogenous fluid density. The displacement field in thedisturbed image is related to the path integral of the density gradient. When thedisplacement field is measured, the density and pressure fields could be inferred.To determine the displacement field from the undisturbed and disturbed images,the cross-correlation method in particle image velocimetry (PIV) can be applied.However, the application of the BOS technique in liquid is more difficult com- pared to its application in gas flows since the light-ray deflection due to the density change in liquids is much smaller. In particular, the measurement of an underwa-ter shock wave by using the BOS technique is scanty. Recently, using the BOStechnique with the cross-correlation method, we obtained the displacement fieldinduced by the local density gradient of the underwater shock wave (Yamamotoet al (2015)). Based on these results, we tried to further reconstruct the pressurefield from the displacement field. However, it is found that the pressure field couldnot be correctly reconstructed due to the insufficient number of the displacementvectors extracted by the typical correlation algorithm in PIV (Hayasaka et al(2016)). In order to overcome this problem, we propose a high-resolution BOStechnique by incorporating the optical flow method, which is simply referred toas OF-BOS. Atcheson et al (2009) suggested that the typical optical flow meth-ods in computer vision, such as the Horn-Schunck method and the Locas-Kanademethod, could be used for the BOS technique. They found that these optical flowmethods particularly the Horn-Schunck method performed better than the corre-lation method in terms of the accuracy and the spatial resolution.In this paper, we adopt the physics-based optical flow method in BOS, whichis developed by Liu and Shen (2008) for various flow visualizations. This methodcould achieve the spatial resolution of one vector per pixel and the better accuracy(Liu and Shen (2008)), which is important for the application of the BOS techniquein water.Following this instruction, we will describe the optical-flow-based BOS tech-nique, tomographic reconstruction of the field of the divergence of the displacement vector, and determination of the density field by solving the Poisson’s equation.Then, the experimental setup will be described for BOS measurements of the laser-induced shock wave. Next, the results will be discussed, focusing on the effectsof the spatial resolution of the extracted displacement field on reconstruction ofthe density and pressure fields. In particular, we will quantitatively compare the itle Suppressed Due to Excessive Length 3 pressure distributions obtained from both the OF-BOS and the PIV-BOS withhydrophone data. x, z ) is parallel tothe image plane. Thus, the image coordinates are equal to λ ( x, z ), where λ is aproportional constant in the orthographical projection. The light ray radiated fromthe background pattern plate is deflected from the original straight path througha fluid domain with the density gradient due to optical refraction, and thereforethe image of the background pattern has an apparent displacement (shift) field.The displacement on the image plane is proportional to the path integral ofthe small density gradient through a fluid domain. For BOS setting, van Hinsbergand R¨osgen (2014) gave the following relation for the in-plane displacement vector w (cid:48) in the background plate w (cid:48) = 12 c ( c + 2 l ) 1 n ∇ n, (1)where c (100 µ m in this case) is the thickness of the density gradient domain, l (8 mm in this case) is the distance from the density gradient domain to thebackground plate, n is the refractive index of water at the condition withouta shock wave, n is the refractive index of water in the test condition, and ∇ isthe gradient operator on the coordinate plane ( x, z ). The relation between therefractive index and the fluid density is given by the Gladstone-Dale equation(Merzkirch (2012); Raffel (2015)) n = Kρ + 1 , (2) Fig. 1
Illustration of the principle of the BOS technique. Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda
Fig. 2
Typical images obtained the experimental setup in this work: (i) A shadowgraphsnapshot of the laser induced shock wave, (ii) non-disturbed image in BOS, and (iii) disturbedimage in BOS. where K is the Gladstone-Dale constant (3.34 × − m /kg for water), and ρ isthe density of the fluid. Substitution of Eq. (2) into Eq. (1) yields w (cid:48) = 12 c ( c + 2 l ) K Kρ ∇ ρ, (3) where ρ is the fluid density under hydrostatic pressure. It is emphasized that the displacement vector w (cid:48) obtained by the BOS technique is the path-integrated (orprojected) quantity across the measurement domain with the density gradient.In general, to extract the 3D field from the path-integrated quantity requirestomographic reconstruction from data at multiple viewing directions.Further, applying the dot product of ∇ to Eq. (3), we have the Poisson’sequation for the density ∇ ρ = S, (4)where the source term S is proportional to the divergence of the displacementvector w (cid:48) by S = 2(1 + Kρ ) cK ( c + 2 l ) ∇ · w (cid:48) . (5)In principle, when the displacement vector w (cid:48) is measured, the density fieldcan be determined by solving Eq. (4). The Gauss-Seidel method is used to solveEq. (4) numerically here. Then, the pressure field can be further determined byapplying the Tait equation p + Bp + B = (cid:18) ρρ (cid:19) α , (6)where p is the hydrostatic pressure, B is a constant of 314 MPa, and the expo-nent α is 7 (Brujan (2010); Yamamoto et al (2015)). In this study, we measure a spherical shock wave in water with the peak overpressure of about 1 MPa. In thiscase, the density change due to the overpressure is less than 1 kg/m . This meansthat the change in the refractive index is very small [O(10 − )].The BOS system acquires an undisturbed background pattern image and thecorresponding disturbed image. Figure 2 shows the typical images obtained in the itle Suppressed Due to Excessive Length 5 experimental setup in this work. Figure 2(i) shows a shadowgraph snapshot ofa shock wave propagating spherically around a laser-induced bubble, where thehigh-pressure region is expected at a shock front. Figure 2(ii) and 2(iii) show anundisturbed background reference image and the corresponding disturbed image,respectively. The zoomed-in image of the particle pattern is shown as well. Withoutimage processing, it is difficult to see directly the shock wave in Fig. 2(iii) sincethe change of the refractive index of water is so small.2.2 Optical flow methodThe key element in the BOS technique is to determine the displacement vec-tor field in the undisturbed background reference image and the correspondingdisturbed image. This computation is usually carried out by using the cross-correlation method in PIV. An effort has been made to adopt the classical opticalflow methods in computer vision science in BOS (Atcheson et al (2009)). The op-tical flow method can be further developed based on rational physical foundationsfor various flow visualizations (Liu and Shen (2008)). The optical flow is describedby the generic equation in the image plane, i.e., ∂g∂t + ∇ · ( g u ) = f, (7)where u is the velocity in the image plane referred to as the optical flow, g is thenormalized image intensity, ∇ is the spatial gradient in the image plane, and f isa term related to the diffusion and boundary fluxes which are negligibly small inmost cases. In velocimetry, the optical flow in Eq. (7) has a clear physical meaning,that is, the optical flow is proportional to the light-ray-path-averaged velocity offluid or particles in flows. We call this method as the physics-based optical flowtechnique. In a special case where ∇· u = 0 and f = 0, Eq. (7) is reduced tothe Horn-Schunck brightness constraint equation which is the foundation of theclassical optical flow method developed by Horn and Schunck (1981). It is noted Fig. 3
Illustration of measurements of displacement vectors. (i) PIV-BOS (ii) OF-BOS. Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda that in a general case the optical flow is not divergence-free and ∇· u = 0 is notphysically true. For BOS applications, the difference ∂g / ∂t ≈ ∆g / ∆t is used inEq. (7), where ∆g = g - g ref is the difference between the disturbed image andthe undisturbed reference image, and the nominal time interval is unitary ( ∆t =1). Therefore, the optical flow in Eq. (7) is interpreted as the displacement vectorfield in the image plane (i.e. u = w ) that is generated by the deflected light raythrough the density gradient domain. The displacement vector in the backgroundplane is related to that in the image plane by w = λ w (cid:48) .To determine the displacement vector field in the image plane, a variationalformulation with a smoothness constraint is typically used (Liu and Shen (2008)).By minimizing the functional, the Euler-Lagrange equation is given for the opticalflow. The standard finite difference method is used to solve the Euler-Lagrangeequation with the Neumann condition on the image domain boundary for theoptical flow. The optical flow algorithm used in this work has the routines: theHorn-Schunck estimator for an initial solution (Horn and Schunck (1981)) andthe Liu-Shen estimator for a refined solution of Eq. (7) (Liu and Shen (2008)).The relevant parameters in pre-processing and optical flow computation should besuitably selected. The main parameters are the Lagrange multipliers for the Horn-Schunck and Liu-Shen estimators. Other parameters are the number of iterationsin successive improvement of optical flow computation by using a coarse-to-fineiterative scheme, and the sizes of the Gaussian filters for correcting the effect of alocal illumination intensity change and removing small random noise in images. Amathematical analysis of the physics-based optical flow and an iterative numericalalgorithm are given by Wang et al (2015). Quantitative comparison between theoptical flow and cross-correlation methods for PIV images has been evaluated byLiu et al (2015). The variational optical flow method based on Eq. (7) is a dif-ferential approach that can achieve the theoretical spatial resolution of one vectorper pixel. Figure 3 illustrates the difference between the optical flow method asa differential approach and the cross-correlation method as a region-based inte-gral approach. The optical flow method is particularly suitable to detect smalldisplacements in narrow regions (such as shock wave).2.3 Tomographic reconstruction on a spherical surfaceThe tomographic reconstruction technique is used to reconstruct the relevant phys-ical quantity (such as the fluid density or the divergence of the displacement vector)on a spherical surface of a propagating shock wave from the path-integrated orprojected quantity obtained from the BOS measurements. In volumetric flow vi-sualizations, an image can be modeled as a set of 1D projections of a field of aquantity with the given projection angles through a 2D domain as illustrated inFig. 4. The basic tomographic problem is how to reconstruct the 2D field fromthese projections. The projection of the quantity q ( x, y ) can be expressed as anintegral along a ray, i.e., R ( q ) ≡ Q ( l, θ )= (cid:90) ∞−∞ (cid:90) ∞−∞ q ( x, y ) δ ( x cos θ + y sin θ − l ) dxdy, (8) itle Suppressed Due to Excessive Length 7 where l = x cos θ + y sin θ is the coordinate along a projection line, θ is the angledefining the projection lines, R ( q ) denotes the Radon transform, and δ denotes theDirac-delta function. An inversion of the Radon transform is sought for q ( x, y ).This tomographic problem has been studied in 3D flow measurements (Feng et al(2002); Venkatakrishnan and Meier (2004)). We introduce the Fourier transform S θ ( ξ ) = (cid:90) ∞−∞ Q ( l, θ ) e − j πξl dl. (9)Then, the solution of the integral equation Eq. (8) can be formally given by q ( x, y ) = (cid:90) π (cid:90) ∞−∞ S θ ( ξ ) e j πξl | ξ | dξdθ. (10)Clearly, the tomographic reconstruction of q ( x, y ) requires many projections in θ ∈ [0 , π ]. In actual computation, the inverse Radon transform in Matlab is used inthis work. In this process, we generate the sinogram digitally. Although the Abeltransform is simpler for the reconstruction of an axial symmetrical field, we donot utilize it because of its high sensitivity to noise (Venkatakrishnan and Meier(2004)).In the BOS measurement of a spherical shock wave, the following proceduresare proposed as illustrated in Fig. 5.1. Image registration based on the affine transformation is applied to the refer-ence and disturbed images to correct any global misalignment between themcaused by the possible movement of the camera and other factors during mea-surements.2. The projected quantity is selected as the divergence of the projected displace-ment vector, i.e., Q ( l, θ ) = ∇ · w (cid:48) from the BOS measurement since it is thesource term of the Poisson’s equation for the fluid density. The quantity ∇ · w (cid:48) is mapped onto the image plane based on an assumption of the axial symmet-rical structure of the laser-induced shock wave. This is illustrated in Step (1-2)in Fig. 5. Fig. 4
Projections of light rays through a domain on the image plane, where l (cid:48) is an axis onthe image plane. Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda Fig. 5
Tomographic reconstruction procedures for a spherical shock wave.
3. The Radon transform Q ( l, θ ) = ∇ · w (cid:48) is given as a sinogram at section at agiven z -coordinate, as illustrated in Step (2-3). In the sinogram, we generate180 projected data in θ ∈ [0 , π ].4. The distribution of q ( x, y ) at that section is reconstructed by using the inverseRadon transform (symbolically expressed as q ( x, y ) = R − ( Q )), as illustratedin Step (3-4). In Matlab computation, we apply the spline interpolation to theback projection method and calculate the inverse Radon transform withoutfiltering.5. The field of ∇ · w (cid:48) on the spherical shock wave is obtained by superpositionof the reconstructed fields over a set of cross sections in the z -coordinate, asillustrated in Step (4-5). This superposition procedure for a spherical shockwave is further illustrated in Fig. 6. Fig. 6
Superposition of reconstructed fields at all cross sections in the z -coordinate.itle Suppressed Due to Excessive Length 9
6. The field of the density is obtained by solving the axisymmetric Poisson’s equa-tion for a given source term on the plane of symmetry and the correspondingpressure field is calculated by using Eq. (6), as shown in Step (5-6).It is necessary to comment the selection of the projected quantity for tomo-graphic reconstruction. In the BOS measurements by Venkatakrishnan and Meier(2004), the projected fluid density was obtained first by solving the Poisson’sequation, and then the local density was reconstructed by using the tomographictechnique. In contrast, the present procedures conduct the tomographic recon-struction before solving the Poisson’s equation. The projected quantity is the di-vergence of the projected displacement vector, i.e., Q ( l, θ ) = ∇ · w (cid:48) , and thus thedivergence of the local displacement vector is first reconstructed by using the to-mographic technique. Next, the local density is calculated by solving the Poisson’sequation. In principle, since the Poisson’s equation is valid for both the local andpath-integrated (projected) quantities, the approach used by Venkatakrishnan andMeier (2004) should be equivalent to the present approach. Nevertheless, the ra-tionale behind the present arrangement is to avoid the propagation of the errorin solving the Poisson’s equation into tomographic reconstruction that is moresensitive to the errors. Figure 7 is a schematic of the experimental setup of the BOS measurements ofa shock wave generated by a pulsed laser. A laser pulse with the wavelengthof 532 nm and pulse width of 6 ns (Nd:YAG laser Nano S PIV, Litron Lasers
Fig. 7
Experimental setup for BOS measurements of the laser-induced shock wave.0 Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda
Ltd.) was focused through a 20 × microscope objective lens to a point inside adistilled-water-filled glass tank (450 × ×
300 mm ). The underwater shock wavewas generated at a laser-focused point, propagating spherically. The backgroundrandom-dot-pattern plate was placed behind the laser-focused point inside thetank. The shock wave was recorded by using a camera with 2048 × µ mper pixel. The light source utilized for illuminating the background pattern wasa laser stroboscope (SI-LUX 640, Specialized Imaging Ltd.) with the pulse widthof 20 ns. The pulsed laser, camera, and light source were synchronized by usinga delay function generator (Model 575, BNC Co.). We utilized a PVDF pressuresensor (i.e. hydrophone) (Muller-Platte Needle Prove, Mueller Instruments) tovalidate the results obtained by using the OF-BOS and the PIV-BOS. The pressuresensor was set toward a center of the laser-induced shock. The distance from thecenter to the hydrophone was 5.0 ± A typical case with the laser power of 1.2 mJ is considered to compare the resultsobtained by using the OF-BOS and the PIV-BOS. Figure 8 shows the displacementmagnitude fields obtained by using the OF-BOS and the PIV-BOS. The OF-BOSgives 2008 × ×
138 vectors in the samedomain in Fig. 8. In optical flow computations, the Lagrange multipliers for theHorn-Schunck estimator and the Liu-Shen estimator are 20 and 2000, respectively.
Fig. 8
The displacement magnitude fields in the half domain: (i) OF-BOS, and (ii) PIV-BOS.itle Suppressed Due to Excessive Length 11
In PIV computations, the window size is iteratively reduced from the initial size of32 ×
32 pixels to 16 ×
16 pixels and then to 8 × z -direction is applied to reduce the noise. The displacements were estimatedfrom pressure data by using Eqs. (3)-(6) with an assumption of constant speed ofpropagation. It is indicated that both the OF-BOS and the PIV-BOS give resultsconsistent with that given by the hydrophone at that location. In particular, thedisplacement induced by the shock wave is well captured by both the techniques.Nevertheless, the OF-BOS achieves a much higher spatial resolution.Figure 10 shows the reconstructed pressure fields obtained by using the OF-BOS and the PIV-BOS. The OF-BOS capture correctly the sharp pressure changeacross the shock wave, while the PIV-BOS has a larger error in the reconstructedpressure field particularly near the ‘north pole’ and ‘south pole’ due to its lowspatial resolution. For quantitative comparison, as shown in Fig. 11, the pressureprofiles reconstructed based on the displacement vector fields given by the OF-BOS (2008 × ×
138 vectors) are plotted againstthe data obtained by the hydrophone. The pressure profile given by the OF-BOSalong a ray aligned with the hydrophone is consistent with that given by thehydrophone. In contrast, the pressure profile given by the PIV-BOS exhibits aconsiderably broader distribution extending to the inner region although its peakvalue at that location agrees with the data given by the hydrophone. Clearly,the OF-BOS provides the more accurate reconstruction of the pressure field ofthe shock wave. In contrast, the PIV-BOS has a much larger deviation from the
Fig. 9
The profiles of the measured displacement along a ray aligned with the hydrophone,where R is the radius of the shock wave, where the averaging operation over 50 lines for theOF-BOS and 20 lines for the PIV-BOS in the z -direction is applied to reduce the noise.2 Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda data given by the hydrophone, which is particularly contributed by the large errorsnear the ‘north pole’ and ‘south pole’ (as shown in Fig. 10). In this case, the majorissue of the PIV-BOS is its much lower spatial resolution that tends to corruptthe tomographic reconstruction of the shock wave.4.2 Effect of laser energyMeasurements at different levels of the laser power were conducted to furthercompare the data obtained by using the OF-BOS, the PIV-BOS and the hy-drophone. Figure 12 shows the typical displacement distributions induced by thelaser-induced shock wave at three levels of the laser energy, where the OF-BOSand PIV-BOS data are obtained along the ray aligned with the hydrophone. Over-all, the OF-BOS and PIV-BOS data are in good agreement with the hydrophonedata in the low and medium levels of the laser energy (0.6 and 1.4 mJ). In thecases of the higher levels of the laser energy (2.1 and 2.9 mJ) with larger dis-placements, a coarse-to-fine iterative scheme is adopted to improve the accuracyof optical flow computation (Liu et al (2012, 2015)). In this scheme, images areinitially downsampled by 2 for a coarse-grained velocity field and then a refinedvelocity field with the full image resolution is obtained in iterations for correctionof the large displacements. Two and three iterations are applied to the cases of 2.1and 2.9 mJ, respectively. Figure 13 shows the corresponding pressure distributionsof the laser-induced shock wave at three levels of the laser energy. The OF-BOS isable to detect the weak shock wave and the pressure peaks of the shock wave areconsistent with those given by the hydrophone. Figure 14 shows the peak pressurevalue of the shock wave as a function of the laser energy, indicating the favorablecomparison between the data obtained by using the OF-BOS and the hydrophone. Fig. 10
The reconstructed pressure fields obtained by using (i) the OF-BOS and (ii) thePIV-BOS.itle Suppressed Due to Excessive Length 13
The PIV-BOS also detects the peak pressure in the lower energy cases, but thepressure distribution inside the shock front does not match that of hydrophone.4.3 Effect of spatial resolutionIt is conjectured that the spatial resolution of the extracted displacement fieldwould have a significant impact on the tomographic reconstruction of the pressurefield of the shock wave. This is because the spatial derivatives of the displacementvector field should be calculated accurately for the tomographic reconstruction ofthe divergence of ∇ · w (cid:48) , which will have the direct influences on the extracteddensity and pressure fields. To examine this problem, the displacement field of2008 × × ×
148 vectors. As shown in Fig. 17(a) the reconstructedpressure distribution exhibits that the pressure peak is broaden increasingly withthe decrease of the spatial resolution and the peak value deviates from the data
Fig. 11
Comparisons between the pressure profiles obtained by using the OF-BOS, the PIV-BOS and the hydrophone, where R is the radius of the shock wave.4 Keisuke Hayasaka, Yoshiyuki Tagawa, Tianshu Liu, Masaharu Kameda Fig. 12
The profiles of the measured displacement at three levels of the laser energy, wherethe OF-BOS and the PIV-BOS data are obtained along the ray aligned with the hydrophone.
Fig. 13
Typical pressure distributions of the laser-induced shock wave at three levels of thelaser energy, where the OF-BOS and the PIV-BOS data are obtained along the ray alignedwith the hydrophone and R is the radius of the shock wave. given by the hydrophone. In addition, the displacement field obtained by the PIV-BOS can be interpolated to generate a pseudo high resolution field of 2091 × In BOS measurements of a shock wave induced by a laser in water, it is critical to obtain the displacement vector field with the high spatial resolution. In thiswork, the physics-based optical flow method is incorporated into the BOS tech-nique, simply called the OF-BOS, which can obtain the displacement vector fieldfrom BOS images at the theoretical resolution of one vector per pixel. The tomo-graphic reconstruction for a spherical laser-induced shock wave is proposed, which itle Suppressed Due to Excessive Length 15
Fig. 14
The peak pressure value of the shock wave as a function of the laser energy, wherethe OF-BOS and the PIV-BOS data are obtained along the ray aligned with the hydrophone.
Fig. 15
The reconstructed pressure fields obtained by the OF-BOS based on the selectivelydownsampled displacement fields to show the effect of the spatial resolution on the tomographicreconstruction, where the field obtained by the PIV-BOS is shown for reference. is applied to the fields of the projected divergence of the displacement vector atcross sections in the vertical direction. Superposition of the tomographic solutionsat these cross sections yields the local field of the divergence of the displacementvector in the domain of the spherical shock wave. Then, the local density field isobtained by solving the Poisson’s equation and further the corresponding pressurefield is calculated.
Fig. 16
The effect of the spatial resolution of the field of the divergence of the displacementvector on the tomographic reconstruction on the pressure field of the shock wave.
Fig. 17
The effect of the spatial resolution of the displacement field on the pressure profilesalong the ray aligned with the hydrophone: (a) the OF-BOS, and (b) the PIV-BOS, where R is the radius of the shock wave. The OF-BOS gives the pressure field consistent with the result obtained by thehydrophone. In contrast, the BOS with the conventional cross-correlation methodin PIV (simply called the PIV-BOS) fails to correctly reveal the shock wave in itle Suppressed Due to Excessive Length 17 the reconstructed pressure field due to its much lower spatial resolution. It isfound that the quality of the tomographic reconstruction of the divergence ofthe displacement vector field significantly depends on the spatial resolution ofthe extracted displacement vector field. Therefore, the reconstructed density andpressure fields of the spherical shock wave are directly affected by the spatialresolution. Clearly, the OF-BOS has the advantages over the PIV-BOS in thisaspect.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 26709007from the Japan Society for the Promotion of Science. This work was financially supported byInstitute of Global Innovation Research at Tokyo University of Agriculture and Technology.
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