{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T23:17:32Z","timestamp":1772839052973,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,11]],"date-time":"2024-05-11T00:00:00Z","timestamp":1715385600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12102028"],"award-info":[{"award-number":["12102028"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Schlieren technique is widely adopted for visualizing supersonic jets owing to its non-invasiveness to the flow field. However, extending the classical Schlieren method for quantitative refractive index measurements is cumbersome, especially for three-dimensional supersonic flows. Background-oriented Schlieren has gained increasing popularity owing to its ease of implementation and calibration. This study utilizes multi-view-based tomographic background-oriented Schlieren (TBOS) to reconstruct axisymmetric and weakly non-axisymmetric supersonic jets, highlighting the impact of flow axisymmetry breaking on TBOS reconstructions. Several classical TBOS reconstruction algorithms, including FDK, SART, SIRT, and CGLS, are compared quantitatively regarding reconstruction quality. View spareness is identified to be the main cause of degraded reconstruction quality when the flow experiences axisymmetry breaking. The classic visual hull approach is explored to improve reconstruction quality. Together with the CGLS tomographic algorithm, we successfully reconstruct the weakly non-axisymmetric supersonic jet structures and confirm that increasing the nozzle bevel angle leads to wider jet spreads.<\/jats:p>","DOI":"10.3390\/sym16050596","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T08:59:37Z","timestamp":1715677177000},"page":"596","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Tomographic Background-Oriented Schlieren for Axisymmetric and Weakly Non-Axisymmetric Supersonic Jets"],"prefix":"10.3390","volume":"16","author":[{"given":"Tong","family":"Jia","sequence":"first","affiliation":[{"name":"Key Laboratory of Fluid Mechanics of Ministry of Education & Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0284-6515","authenticated-orcid":false,"given":"Jiawei","family":"Li","sequence":"additional","affiliation":[{"name":"Key Laboratory of Fluid Mechanics of Ministry of Education & Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jie","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering, Huazhong University of Science & Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-5144-6649","authenticated-orcid":false,"given":"Yuan","family":"Xiong","sequence":"additional","affiliation":[{"name":"Key Laboratory of Fluid Mechanics of Ministry of Education & Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University, Beijing 100191, China"},{"name":"Tianmushan Laboratory, Xixi Octagon City, Yuhang District, Hangzhou 310023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"570","DOI":"10.2514\/1.989","article-title":"Numerical Investigation of Shock-Reflection Phenomena in Overexpanded Supersonic Jets","volume":"42","author":"Hadjadj","year":"2004","journal-title":"AIAA J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2559","DOI":"10.1007\/s12206-011-0730-3","article-title":"Experimental study on hysteresis phenomena of shock wave structure in an over-expanded axisymmetric jet","volume":"25","author":"Matsuo","year":"2011","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s00193-011-0335-x","article-title":"Numerical simulation of Mach reflection in steady flows","volume":"21","author":"Schmisseur","year":"2011","journal-title":"Shock Waves"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1260\/1757-2258.2.4.221","article-title":"Numeric study of flow separation and shock reflection hysteresis in planar nozzles","volume":"2","author":"Shimshi","year":"2010","journal-title":"Int. J. Aerosp. Innov."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1146\/annurev.fl.27.010195.000313","article-title":"Supersonic Jet Noise","volume":"27","author":"Tam","year":"1995","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Tam, C., Golebiowski, M., and Seiner, J. (1996, January 6\u20138). On the two components of turbulent mixing noise from supersonic jets. Proceedings of the Aeroacoustics Conference, State College, PA, USA.","DOI":"10.2514\/6.1996-1716"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1006\/jsvi.1999.2181","article-title":"Supersonic jet screech: Half-century from powell to the present","volume":"225","author":"Raman","year":"1999","journal-title":"J. Sound Vib."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Seiner, J., and Norum, T. (1980, January 4\u20136). Aerodynamic aspects of shock containing jet plumes. Proceedings of the 6th Aeroacoustics Conference, Hartford, CT, USA.","DOI":"10.2514\/6.1980-965"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"453","DOI":"10.2514\/1.44896","article-title":"Corrugated Tabs for Supersonic Jet Control","volume":"48","author":"Phanindra","year":"2010","journal-title":"AIAA J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"084107","DOI":"10.1063\/1.3194306","article-title":"Influence of nozzle sharpness on the flow fields of V-notched nozzle jets","volume":"21","author":"New","year":"2009","journal-title":"Phys. Fluids"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.1007\/s00348-013-1614-3","article-title":"Some observations in the vortex-turning behaviour of noncircular inclined jets","volume":"54","author":"Shi","year":"2013","journal-title":"Exp. Fluids"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1017\/S0022112099003833","article-title":"Spreading characteristics of compressible jets from nozzles of various geometries","volume":"383","author":"Zaman","year":"1999","journal-title":"J. Fluid Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1146\/annurev.fl.27.010195.002111","article-title":"Mixing Enhancement in Supersonic Free Shear Flows","volume":"27","author":"Gutmark","year":"1995","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2731","DOI":"10.1063\/1.870132","article-title":"Mixing enhancement via nozzle trailing edge modifications in a high speed rectangular jet","volume":"11","author":"Kim","year":"1999","journal-title":"Phys. Fluids"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1016\/j.ast.2017.01.003","article-title":"An investigation on supersonic bevelled nozzle jets","volume":"63","author":"Wu","year":"2017","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Rice, E., and Raman, G. (1993, January 25\u201327). Mixing noise reduction for rectangular supersonic jets by nozzle shaping and induced screech mixing. Proceedings of the 15th Aeroacoustics Conference, Long Beach, CA, USA.","DOI":"10.2514\/6.1993-4322"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1017\/S0022112096003801","article-title":"Screech tones from rectangular jets with spanwise oblique shock-cell structures","volume":"330","author":"Raman","year":"1997","journal-title":"J. Fluid Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1576","DOI":"10.1088\/0957-0233\/12\/9\/325","article-title":"Principle and applications of the background oriented schlieren (BOS) method","volume":"12","author":"Richard","year":"2001","journal-title":"Meas. Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1007\/s003480050391","article-title":"Whole-field density measurements by \u2018synthetic schlieren\u2019","volume":"28","author":"Dalziel","year":"2000","journal-title":"Exp. Fluids"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1017\/S0001924000008447","article-title":"Density measurements for rectangular free jets using background-oriented schlieren","volume":"117","author":"Tipnis","year":"2013","journal-title":"Aeronaut. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s12650-015-0297-7","article-title":"Quantitative density measurement of the lateral jet\/cross-flow interaction field by colored-grid background oriented schlieren (CGBOS) technique","volume":"18","author":"Ota","year":"2015","journal-title":"J. Vis."},{"key":"ref_22","unstructured":"Wong, P. (2001). Investigation of Axi-Symmetric Laminar and Turbulent Jets with Background Oriented Schlieren (BOS), von Karman Institute. VKI Stagiaire Report 6."},{"key":"ref_23","unstructured":"Kirmse, T. (2003). Weiterentwicklung des Messsystems BOS (Background Oriented Schlieren) zur Quantitativen Bestimmung axialsymmetrischer Dichtefelder. [Ph.D. Thesis, Institute of Aerodynamics and Flow Technology]. Available online: https:\/\/elib.dlr.de\/49602\/."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s00348-004-0807-1","article-title":"Density measurements using the Background Oriented Schlieren technique","volume":"37","author":"Venkatakrishnan","year":"2004","journal-title":"Exp. Fluids"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1007\/s00348-015-2076-6","article-title":"Measurement of density in axisymmetric jets using a novel background-oriented schlieren (BOS) technique","volume":"56","author":"Tan","year":"2015","journal-title":"Exp. Fluids"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1007\/s00348-020-03007-4","article-title":"Towards robust BOS measurements for axisymmetric flows","volume":"61","author":"Xiong","year":"2020","journal-title":"Exp. Fluids"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kirby, R. (2018, January 8\u201312). Tomographic Background-Oriented Schlieren for Three-Dimensional Density Field Reconstruction in Asymmetric Shock-containing Jets. Proceedings of the 2018 AIAA Aerospace Sciences Meeting, Kissimmee, FL, USA.","DOI":"10.2514\/6.2018-0008"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/s00348-017-2325-y","article-title":"3D reconstruction of a compressible flow by synchronized multi-camera BOS","volume":"58","author":"Nicolas","year":"2017","journal-title":"Exp. Fluids"},{"key":"ref_29","unstructured":"Ihrke, I. (2007). Reconstruction and Rendering of Time-Varying Natural Phenomena. [Ph.D. Thesis, Universitat des Saarlandes]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s00348-015-2100-x","article-title":"A direct approach for instantaneous 3D density field reconstruction from background-oriented schlieren (BOS) measurements","volume":"57","author":"Nicolas","year":"2016","journal-title":"Exp. Fluids"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Raffel, M., Willert, C.E., Scarano, F., K\u00e4hler, C.J., Wereley, S.T., and Kompenhans, J. (2018). Particle Image Velocimetry: A Practical Guide, Springer International Publishing.","DOI":"10.1007\/978-3-319-68852-7"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"101024","DOI":"10.1016\/j.pecs.2022.101024","article-title":"Volumetric emission tomography for combustion processes","volume":"94","author":"Grauer","year":"2023","journal-title":"Prog. Energy Combust. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Kak, A.C., and Slaney, M. (2001). Principles of Computerized Tomographic Imaging, Society for Industrial and Applied Mathematics. Number 33 in Classics in Applied Mathematics.","DOI":"10.1137\/1.9780898719277"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1364\/JOSAA.1.000612","article-title":"Practical cone-beam algorithm","volume":"1","author":"Feldkamp","year":"1984","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1177\/016173468400600107","article-title":"Simultaneous Algebraic Reconstruction Technique (SART): A Superior Implementation of the Art Algorithm","volume":"6","author":"Andersen","year":"1984","journal-title":"Ultrason. Imaging"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Sabat\u00e9 Landman, M., Biguri, A., Hatamikia, S., Boardman, R., Aston, J., and Sch\u00f6nlieb, C.B. (2023). On Krylov methods for large-scale CBCT reconstruction. Phys. Med. Biol., 68.","DOI":"10.1088\/1361-6560\/acd616"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"085302","DOI":"10.1088\/1361-6501\/ab1ca8","article-title":"PIV\/BOS synthetic image generation in variable density environments for error analysis and experiment design","volume":"30","author":"Rajendran","year":"2019","journal-title":"Meas. Sci. Technol."},{"key":"ref_38","unstructured":"CFX-Solver, A. (2006). Theory Guide, Release ll, ANSYS Inc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.ast.2010.02.001","article-title":"Numerical simulation of transverse side jet interaction with supersonic free stream","volume":"14","author":"Aswin","year":"2010","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"011202","DOI":"10.1115\/1.4040447","article-title":"Flow Characterization of Supersonic Jets Issuing From Double-Beveled Nozzles","volume":"141","author":"Wu","year":"2019","journal-title":"J. Fluids Eng."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s00348-017-2367-1","article-title":"Measurement of the fluctuating temperature field in a heated swirling jet with BOS tomography","volume":"58","author":"Lang","year":"2017","journal-title":"Exp. Fluids"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1409060.1409085","article-title":"Time-resolved 3d capture of non-stationary gas flows","volume":"27","author":"Atcheson","year":"2008","journal-title":"ACM Trans. Graph."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"104704","DOI":"10.1007\/s11433-015-5719-y","article-title":"Evaluating the accuracy performance of Lucas-Kanade algorithm in the circumstance of PIV application","volume":"58","author":"Pan","year":"2015","journal-title":"Sci. China Phys. Mech. Astron."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"105621","DOI":"10.1016\/j.ast.2019.105621","article-title":"Time-resolved three-dimensional imaging of flame refractive index via endoscopic background-oriented Schlieren tomography using one single camera","volume":"97","author":"Liu","year":"2020","journal-title":"Aerosp. Sci. Technol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/596\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:44:06Z","timestamp":1760107446000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/596"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,11]]},"references-count":44,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["sym16050596"],"URL":"https:\/\/doi.org\/10.3390\/sym16050596","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,11]]}}}