{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:27:20Z","timestamp":1760146040782,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2024,9,14]],"date-time":"2024-09-14T00:00:00Z","timestamp":1726272000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Czech Science Foundation","award":["GA 22-25799S"],"award-info":[{"award-number":["GA 22-25799S"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper deals with CFD analyses of the difference in the nature of the shock waves in supersonic flow under atmospheric pressure and pressure conditions at the boundary of continuum mechanics for electron microscopy. The first part describes the verification of the CFD analyses in combination with the experimental chamber results and the initial analyses using optical methods at low pressures on the boundary of continuum mechanics that were performed. The second part describes the analyses on an underexpanded nozzle performed to analyze the characteristics of normal shock waves in a pressure range from atmospheric pressure to pressures at the boundary of continuum mechanics. The results obtained by CFD modeling are prepared as a basis for the design of the planned experimental sensing of density gradients using optical methods, and for validation, the expected pressure and temperature courses from selected locations suitable for the placement of temperature and pressure sensors are prepared from the CFD analyses.<\/jats:p>","DOI":"10.3390\/s24185968","type":"journal-article","created":{"date-parts":[[2024,9,16]],"date-time":"2024-09-16T11:36:37Z","timestamp":1726486597000},"page":"5968","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["CFD Analyses of Density Gradients under Conditions of Supersonic Flow at Low Pressures"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8528-4430","authenticated-orcid":false,"given":"Robert","family":"Bayer","sequence":"first","affiliation":[{"name":"Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick\u00e1 10, 616 00 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9793-9767","authenticated-orcid":false,"given":"Petr","family":"Ba\u010da","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick\u00e1 10, 616 00 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0640-0406","authenticated-orcid":false,"given":"Ji\u0159\u00ed","family":"Maxa","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick\u00e1 10, 616 00 Brno, Czech Republic"},{"name":"Institute of Scientific Instruments of the CAS, Kr\u00e1lovopolsk\u00e1 147, 612 64 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3908-5120","authenticated-orcid":false,"given":"Pavla","family":"\u0160aback\u00e1","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick\u00e1 10, 616 00 Brno, Czech Republic"},{"name":"Institute of Scientific Instruments of the CAS, Kr\u00e1lovopolsk\u00e1 147, 612 64 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4426-2857","authenticated-orcid":false,"given":"Tom\u00e1\u0161","family":"Binar","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick\u00e1 10, 616 00 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4160-8031","authenticated-orcid":false,"given":"Petr","family":"Vyroubal","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Communication, Brno University of Technology, Technick\u00e1 10, 616 00 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"112954","DOI":"10.1016\/j.ultramic.2020.112954","article-title":"Simulation-based optimisation of thermodynamic conditions in the ESEM for dynamical in-situ study of spherical polyelectrolyte complex particles in their native state","volume":"211","author":"Maxa","year":"2020","journal-title":"Ultramicroscopy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2385","DOI":"10.5194\/tc-13-2385-2019","article-title":"The morphology of ice and liquid brine in an environmental scanning electron microscope: A study of the freezing methods","volume":"13","author":"Heger","year":"2019","journal-title":"Cryosphere"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.nbt.2018.05.005","article-title":"Effects of copper and arsenic stress on the development of Norway spruce somatic embryos and their visualization with the environmental scanning electron microscope","volume":"48","author":"Trojan","year":"2019","journal-title":"New Biotechnol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"014503","DOI":"10.1063\/1.5100852","article-title":"Vitrification and increase of basicity in between ice Ih crystals in rapidly frozen dilute NaCl aqueous solutions","volume":"151","author":"Gasser","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/s10535-012-0062-x","article-title":"Imaging of early conifer embryogenic tissues with the environmental scanning electron microscope","volume":"56","year":"2012","journal-title":"Biol. Plant."},{"key":"ref_6","first-page":"39","article-title":"The impact of critical flow on the primary electron beam passage through differentially pumped chamber","volume":"6","author":"Maxa","year":"2011","journal-title":"Adv. Mil. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1593","DOI":"10.1007\/s00706-012-0861-y","article-title":"Pressure dependence of the tin\u2013phosphorus phase diagram","volume":"143","author":"Ritscher","year":"2012","journal-title":"Monatsh. Chem."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Stelate, A., Tihla\u0159\u00edkov\u00e1,  E., Schwarzerov\u00e1, K., Ned\u011bla, V., and Petr\u00e1\u0161ek, J. (2021). Correlative Light-Environmental Scanning Electron Microscopy of Plasma Membrane Efflux Carriers of Plant Hormone Auxin. Biomolecules, 11.","DOI":"10.3390\/biom11101407"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1007\/s12010-014-1174-x","article-title":"Physical and Bioengineering Properties of Polyvinyl Alcohol Lens-Shaped Particles Versus Spherical Polyelectrolyte Complex Microcapsules as Immobilisation Matrices for a Whole-Cell Baeyer\u2013Villiger Monooxygenase","volume":"174","author":"Gemeiner","year":"2014","journal-title":"Appl. Biochem. Biotechnol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1017\/S1431927615013483","article-title":"The Simulation of Energy Distribution of Electrons Detected by Segmental Ionization Detector in High Pressure Conditions of ESEM","volume":"21","author":"Konvalina","year":"2015","journal-title":"Microsc. Microanal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"063307","DOI":"10.1063\/1.3086622","article-title":"Modeling and experiments on differential pumping in linear plasma generators operating at high gas flows","volume":"105","author":"Koppers","year":"2009","journal-title":"J. Appl. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1088\/0950-7671\/10\/12\/304","article-title":"Improvements in the schlieren method","volume":"10","author":"Taylor","year":"1933","journal-title":"J. Sci. Instrum."},{"key":"ref_13","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_14","first-page":"303","article-title":"Schlieren methods and their applications","volume":"20","author":"Schradin","year":"1942","journal-title":"Ergeb. Exakten Naturewiss"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1016\/j.micron.2011.10.023","article-title":"Velocity and ejector-jet assisted differential pumping: Novel design stages for environmental SEM","volume":"43","author":"Danilatos","year":"2012","journal-title":"Micron"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1111\/j.1365-2818.2011.03521.x","article-title":"Figure of merit for environmental SEM and its implications","volume":"244","author":"Danilatos","year":"2011","journal-title":"J. Microsc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1111\/j.1365-2818.2009.03148.x","article-title":"Optimum beam transfer in the environmental scanning electron microscope","volume":"234","author":"Danilatos","year":"2009","journal-title":"J. Microsc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1111\/j.1365-2818.2010.03455.x","article-title":"Beam transfer characteristics of a commercial environmental SEM and a low vacuum SEM","volume":"242","author":"Danilatos","year":"2011","journal-title":"J. Microsc."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Xue, Z., Zhou, L., and Liu, D. (2023). Accurate Numerical Modeling for 1D Open-Channel Flow with Varying Topography. Water, 15.","DOI":"10.3390\/w15162893"},{"key":"ref_20","first-page":"1","article-title":"Proud\u011bn\u00ed plyn\u016f a par tryskami","volume":"2","year":"2006","journal-title":"Transform. Technol."},{"key":"ref_21","unstructured":"Moran, M.J., and Shapiro, H.N. (1998). Fundamentals of Engineering Thermodynamics, John and Wiley and Sons. [3rd ed.]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s00193-008-0173-7","article-title":"Shock structure in separated nozzle flows","volume":"19","author":"Nasuti","year":"2008","journal-title":"Shock. Waves"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"102572","DOI":"10.1016\/j.rinp.2019.102572","article-title":"The effect of nozzle configuration on the evolution of jet surface structure","volume":"15","author":"Gong","year":"2019","journal-title":"Results Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"026126","DOI":"10.1063\/5.0187688","article-title":"Effects of inflow Mach numbers on shock train dynamics and turbulence features in a backpressured supersonic channel flow","volume":"36","author":"Yuan","year":"2024","journal-title":"Phys. Fluids"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Liu, Q., and Feng, X.-B. (2020). Numerical Modelling of Microchannel Gas Flows in the Transition Flow Regime Using the Cascaded Lattice Boltzmann Method. Entropy, 22.","DOI":"10.3390\/e22010041"},{"key":"ref_26","unstructured":"Salga, J., and Ho\u0159en\u00ed, B. (1997). Tabulky Proud\u011bn\u00ed Plynu, UNOB."},{"key":"ref_27","unstructured":"Dan\u011bk, M. (1990). Aerodynamika a Mechanika Letu, VVL\u0160 SNP."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Baehr, H.D., and Kabelac, S. (2009). Thermodynamik, Springer. [14th ed.].","DOI":"10.1007\/978-3-642-00556-5"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Dutta, P.P., Benken, A.C., Li, T., Ordonez-Varela, J.R., and Gianchandani, Y.B. (2023). Passive Wireless Pressure Gradient Measurement System for Fluid Flow Analysis. Sensors, 23.","DOI":"10.3390\/s23052525"},{"key":"ref_30","unstructured":"(2022, October 21). Ansys Fluent Theory Guide. Available online: www.ansys.com."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Beer, M., Kudelas, D., and Ryb\u00e1r, R. (2023). A Numerical Analysis of the Thermal Energy Storage Based on Porous Gyroid Structure Filled with Sodium Acetate Trihydrate. Energies, 16.","DOI":"10.3390\/en16010309"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.cam.2005.04.071","article-title":"High order schemes based on upwind schemes with modified coefficients","volume":"195","author":"Yang","year":"2006","journal-title":"J. Comput. Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Barth, T., and Jespersen, D. (1989, January 9\u201312). The design and application of upwind schemes on unstructured meshes. Proceedings of the 27th Aerospace Sciences Meeting, Reno, NV, USA.","DOI":"10.2514\/6.1989-366"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1149\/08701.0377ecst","article-title":"Using the Ideal and Real Gas Model for the Mathematical\u2014Physics Analysis of the Experimental Chambre","volume":"87","author":"Maxa","year":"2018","journal-title":"ECS Trans."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Gab\u00e1niov\u00e1, \u013d., Kudelas, D., and Pr\u010d\u00edk, M. (2021). Modelling Ground Collectors and Determination of the Influence of Technical Parameters, Installation and Geometry on the Soil. Energies, 14.","DOI":"10.3390\/en14217153"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"\u0160aback\u00e1, P., Maxa, J., Bayer, R., Vyroubal, P., and Binar, T. (2022). Slip Flow Analysis in an Experimental Chamber Simulating Differential Pumping in an Environmental Scanning Electron Microscope. Sensors, 22.","DOI":"10.3390\/s22239033"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"\u0160aback\u00e1, P., Ned\u011bla, V., Maxa, J., and Bayer, R. (2021). Application of Prandtl\u2019s Theory in the Design of an Experimental Chamber for Static Pressure Measurements. Sensors, 21.","DOI":"10.3390\/s21206849"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"066109","DOI":"10.1063\/5.0151556","article-title":"Flow structure and parameter evaluation of conical convergent\u2013divergent nozzle supersonic jet flows","volume":"35","author":"Xiao","year":"2023","journal-title":"Phys. Fluids"},{"key":"ref_39","unstructured":"(2024, August 28). Dynamick\u00e1 Viskozita plyn\u016f, E-tabulky. Available online: https:\/\/uchi.vscht.cz\/studium\/tabulky\/viskozita-plyny."},{"key":"ref_40","unstructured":"(2024, August 20). The Engineering ToolBox, Nitrogen-Dynamic and Kinematic Viscosity vs. Temperature and Pressure. Available online: https:\/\/www.engineeringtoolbox.com\/nitrogen-N2-dynamic-kinematic-viscosity-temperature-pressure-d_2067.html."},{"key":"ref_41","unstructured":"Pitchard, P.J., Fox, R.W., and McDonald, A.T. (2011). Introduction to Fluid Mechanics, John and Wiley and Sons."},{"key":"ref_42","unstructured":"Hermann, R. (1956). Supersonic Inlet Diffusers, Minneapolis-Honeywell Regulator Co., Aeronautical Division."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Bayer, R., Maxa, J., and \u0160aback\u00e1, P. (2021). Energy Harvesting Using Thermocouple and Compressed Air. Sensors, 21.","DOI":"10.3390\/s21186031"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"036123","DOI":"10.1063\/5.0197296","article-title":"Gas dynamics at starting and terminating phase of a supersonic exhaust diffuser with a conical nozzle","volume":"36","author":"Afkhami","year":"2024","journal-title":"Phys. Fluids"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Belo, F.A., Soares, M.B., Lima Filho, A.C., Lima, T.L.d.V., and Adissi, M.O. (2023). Accuracy and Precision Improvement of Temperature Measurement Using Statistical Analysis\/Central Limit Theorem. Sensors, 23.","DOI":"10.3390\/s23063210"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Drexler, P., \u010c\u00e1p, M., Fiala, P., Steinbauer, M., Kadlec, R., Ka\u0161ka, M., and Ko\u010di\u0161, L. (2019). A Sensor System for Detecting and Localizing Partial Discharges in Power Transformers with Improved Immunity to Interferences. Sensors, 19.","DOI":"10.3390\/s19040923"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/18\/5968\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:56:20Z","timestamp":1760111780000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/18\/5968"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,14]]},"references-count":46,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["s24185968"],"URL":"https:\/\/doi.org\/10.3390\/s24185968","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2024,9,14]]}}}