{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:03:16Z","timestamp":1760241796425,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2018,9,7]],"date-time":"2018-09-07T00:00:00Z","timestamp":1536278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The immersed boundary method has attracted considerable interest in the last few years. The method is a computational cheap alternative to represent the boundaries of a geometrically complex body, while using a cartesian mesh, by adding a force term in the momentum equation. The advantage of this is that bodies of any arbitrary shape can be added without grid restructuring, a procedure which is often time-consuming. Furthermore, multiple bodies may be simulated, and relative motion of those bodies may be accomplished at reasonable computational cost. The numerical platform in development has a parallel distributed-memory implementation to solve the Navier-Stokes equations. The Finite Volume Method is used in the spatial discretization where the diffusive terms are approximated by the central difference method. The temporal discretization is accomplished using the Adams-Bashforth method. Both temporal and spatial discretizations are second-order accurate. The Velocity-pressure coupling is done using the fractional-step method of two steps. The present work applies the immersed boundary method to simulate a Newtonian laminar flow through a three-dimensional sudden contraction. Results are compared to published literature. Flow patterns upstream and downstream of the contraction region are analysed at various Reynolds number in the range     44 \u2264 R  e D  \u2264 993     for the large tube and     87 \u2264 R  e D  \u2264 1956     for the small tube, considerating a contraction ratio of     \u03b2 = 1.97    . Comparison between numerical and experimental velocity profiles has shown good agreement.<\/jats:p>","DOI":"10.3390\/computation6030050","type":"journal-article","created":{"date-parts":[[2018,9,7]],"date-time":"2018-09-07T11:47:41Z","timestamp":1536320861000},"page":"50","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Immersed Boundary Method Application as a Way to Deal with the Three-Dimensional Sudden Contraction"],"prefix":"10.3390","volume":"6","author":[{"given":"Jonatas E.","family":"Borges","sequence":"first","affiliation":[{"name":"School of Food Engineering, Federal University of Mato Grosso, Cuiab\u00e1 78600-000, Brazil"}]},{"given":"Marcos","family":"Louren\u00e7o","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Technologic Federal University of Parana, Apucarana 86300-000, Brazil"}]},{"given":"Elie L. M.","family":"Padilla","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Federal University of Uberl\u00e2ndia, Uberl\u00e2ndia 38400-902, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0407-2082","authenticated-orcid":false,"given":"Christopher","family":"Micallef","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of Malta, Msida, MSD 2080, Malta"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ferziger, J.H., and Peric, M. (2002). Computational Methods for Fluid Dynamics, 3rd., Springer.","DOI":"10.1007\/978-3-642-56026-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/0021-9991(72)90065-4","article-title":"Flow patterns around heart valves: A numerical method","volume":"10","author":"Peskin","year":"1972","journal-title":"J. Comput. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/0021-9991(77)90100-0","article-title":"Numerical analysis of the blood flow in the heart","volume":"25","author":"Peskin","year":"1977","journal-title":"J. Comput. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0021-9991(92)90307-K","article-title":"A front tracking method for viscous, incompressible, multi-fluid flows","volume":"100","author":"Unverdi","year":"1992","journal-title":"J. Comput. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1006\/jcph.1993.1081","article-title":"Modeling a no-slip flow with an external force field","volume":"105","author":"Goldstein","year":"1993","journal-title":"J. Comput. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/S0021-9991(03)00214-6","article-title":"Numerical Simulation of two-dimensional flows over a circular circular cylinder using the immersed boundary method","volume":"189","author":"Damasceno","year":"2003","journal-title":"J. Comput. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1146\/annurev.fluid.37.061903.175743","article-title":"Immersed boundary methods","volume":"37","author":"Mittal","year":"2005","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/j.jcp.2005.03.017","article-title":"An immersed boundary method with direct forcing for the simulation of particulate flows","volume":"209","author":"Uhlmann","year":"2005","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1016\/j.jcp.2006.10.032","article-title":"An immersed boundary method for complex incompressible flows","volume":"224","author":"Choi","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/j.ijmultiphaseflow.2007.10.004","article-title":"Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles","volume":"34","author":"Wang","year":"2008","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1146\/annurev.fl.14.010182.001315","article-title":"The Fluid Dynamics of Heart Valves: Experimental, Theoretical, and Computational Methods","volume":"14","author":"Peskin","year":"1982","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1016\/0021-9991(89)90213-1","article-title":"A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid","volume":"81","author":"Peskin","year":"1989","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0021-9991(89)90050-8","article-title":"A three-dimensional computational method for blood flow in the heart. II. contractile fibers","volume":"82","author":"McQueen","year":"1989","journal-title":"J. Comput. Phys."},{"key":"ref_14","first-page":"331","article-title":"Immersed boundary technique for turbulent flow simulations","volume":"56","author":"Iaccarino","year":"2003","journal-title":"Am. Soc. Mech. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1006\/jcph.1998.5965","article-title":"A Cartesian Grid Embedded Boundary Method for Poisson\u2019s Equation on Irregular Domains","volume":"147","author":"Johansen","year":"1998","journal-title":"J. Comput. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1080\/1061856021000025184","article-title":"Direct Numerical Simulation of Wakes with Virtual Cylinders","volume":"16","author":"Silvestrini","year":"2002","journal-title":"Int. J. Comput. Fluid Dyn."},{"key":"ref_17","first-page":"1","article-title":"A new incompressible Navier-Stokes solver combining Fourier pseudo-spectral and immersed bondary methods","volume":"1589","author":"Mariano","year":"2010","journal-title":"CMES Comput. Model. Eng. Sci."},{"key":"ref_18","unstructured":"Iaccarino, G., and Kalitzin, G. (2003, January 6\u20139). Towards an immersed boundary rans flow solver. Proceedings of the 41st Aerospace Sciences Meeting and Exhibit, Reno, NV, USA."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1006\/jcph.2001.6778","article-title":"An immersed-boundary finite-volume method for simulations of flow in complex geometries","volume":"171","author":"Kim","year":"2001","journal-title":"J. Comput. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1006\/jcph.2000.6483","article-title":"An Immersed Boundary Method with Formal Second-Order Accuracy and Reduced Numerical Viscosity","volume":"160","author":"Lai","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1017\/S0962492902000077","article-title":"The immersed boundary method","volume":"11","author":"Peskin","year":"2002","journal-title":"Acta Numer."},{"key":"ref_22","unstructured":"Louren\u00e7o, M.A.S. (2012). Desenvolvimento De Uma Plataforma Para Computa\u00e7\u00e3o de Alto Desempenho De Escoamentos Transientes, Utilizando Estrutura Octon\u00e1ria. [Ph.D. Thesis, Faculdade de Engenharia Mec\u00e2nica, Universidade Federal de Uberlandia]."},{"key":"ref_23","unstructured":"Pienaar, V. (2004). Viscous Flow Through Sudden Contractions. [Ph.D. Thesis, Faculty of Engineering Cape Technikon]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s10404-008-0277-5","article-title":"Simulations of extensional flow in microrheometric devices","volume":"5","author":"Oliveira","year":"2008","journal-title":"Microfluid. Nanofluid."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0045-7930(85)90030-1","article-title":"Investigations of Laminar Flow in a Pipe with Sudden Contraction of Cross Sectional Area","volume":"13","author":"Durst","year":"1985","journal-title":"Comput. Fluids"},{"key":"ref_26","unstructured":"S\u00e1nchez, F.M.P. (2011). Estudo Experimental do Escoamento De Fluido Newtoniano Em Contra\u00e7\u00c3o Abrupta Axissim\u00e9trica Com a T\u00e9cnica De Velocimetria Por Imagem De Part\u00edcula. [Ph.D. Thesis, UTFPR, Universidade Tecnol\u00f3gica Federal do Paran\u00e1]."},{"key":"ref_27","unstructured":"Patankar, S. (1980). Numerical Heat Transfer and Fluid Flow, McGraw Hill Book Company."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2182","DOI":"10.1063\/1.1761178","article-title":"Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface","volume":"8","author":"Harlow","year":"1965","journal-title":"AIP Phys. Fluids"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/0021-9991(85)90148-2","article-title":"Application of a fractional-step method to incompressible Navier-Stokes equations","volume":"59","author":"Kim","year":"1985","journal-title":"J. Comput. Phys."},{"key":"ref_30","unstructured":"Heroux, M., Bartlett, R., Hoekstra, V.H.R., Hu, J., Kolda, T., Lehoucq, R., Long, K., Pawlowski, R., Phipps, E., and Salinger, A. (2003). An Overview of Trilinos."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"856","DOI":"10.1137\/0907058","article-title":"GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems","volume":"7","author":"Saad","year":"1986","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/0956-0521(91)90014-V","article-title":"Partitioning of Unstructured Problems for Parallel Processing","volume":"2","author":"Applied","year":"1991","journal-title":"Comput. Syst. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1016\/j.jcp.2007.06.002","article-title":"A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulate flow past a circular cylinder","volume":"226","author":"Shu","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.compfluid.2005.09.004","article-title":"An immersed boundary technique for simulating complex flows with rigid boundary","volume":"36","author":"Su","year":"2007","journal-title":"Comput. Fluids"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"6617","DOI":"10.1016\/j.jcp.2009.06.003","article-title":"A moving-least-squares reconstruction for embedded-boundary formulations","volume":"228","author":"Vanella","year":"2009","journal-title":"J. Comput. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Liu, G.R. (2002). Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press. [1st ed.].","DOI":"10.1201\/9781420040586"},{"key":"ref_37","unstructured":"Fox, R.W., McDonald, A.T., and Pritchard, P.J. (2011). Introduction to Fluid Mechanics, Wiley. [8th ed.]."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1002\/aic.690160617","article-title":"Laminar flow in the entrance region of a cylindrical tube: Part I. Newtonian fluids","volume":"16","author":"Sylvester","year":"1970","journal-title":"Am. Inst. Chem. Eng. J."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/BF00413071","article-title":"Flow of a Newtonian fluid through a sudden contraction","volume":"28","author":"Vrentas","year":"1973","journal-title":"Appl. Sci. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1021\/i160025a005","article-title":"Excess pressure drop in laminar flow through sudden contraction\u2014Newtonians liquids","volume":"7","author":"Astarita","year":"1968","journal-title":"Ind. Eng. Chem. Fundam."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.cej.2008.03.003","article-title":"Energy losses of non-Newtonian fluids in sudden pipe contractions","volume":"145","author":"Fester","year":"2008","journal-title":"Chem. Eng. J."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/6\/3\/50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:19:27Z","timestamp":1760195967000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/6\/3\/50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,7]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,9]]}},"alternative-id":["computation6030050"],"URL":"https:\/\/doi.org\/10.3390\/computation6030050","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2018,9,7]]}}}