{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:33:49Z","timestamp":1760240029171,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,2,20]],"date-time":"2019-02-20T00:00:00Z","timestamp":1550620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UID\/MAT\/00144\/2019"],"award-info":[{"award-number":["UID\/MAT\/00144\/2019"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["NORTE-01-0145-FEDER-000033"],"award-info":[{"award-number":["NORTE-01-0145-FEDER-000033"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007586","name":"Aix-Marseille Universit\u00e9","doi-asserted-by":"publisher","award":["LabEx MeC"],"award-info":[{"award-number":["LabEx MeC"]}],"id":[{"id":"10.13039\/100007586","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fluids"],"abstract":"<jats:p>With the aim of providing a first step in the quest for a reduction of the aerodynamic drag on the rear-end of a car, we study the phenomena of separation and reattachment of an incompressible flow by focusing on a specific aerodynamic geometry, namely a backward-slanted step at 25     \u2218     of inclination. The ensuing recirculation bubble provides the basis for an analytical and numerical investigation of streamwise-streak generation, lift-up effect, and turbulent-wake and Kelvin\u2013Helmholtz instabilities. A linear stability analysis is performed, and an optimal control problem with a steady volumic forcing is tackled by means of a variational formulation, adjoint methods, penalization schemes, and an orthogonalization algorithm. Dealing with the transient growth of spanwise-periodic perturbations, and inspired by the need of physically-realizable disturbances, we finally provide a procedure attaining a kinetic-energy maximal gain on the order of     10 6    , with respect to the power introduced by the external forcing.<\/jats:p>","DOI":"10.3390\/fluids4010033","type":"journal-article","created":{"date-parts":[[2019,2,20]],"date-time":"2019-02-20T11:45:39Z","timestamp":1550663139000},"page":"33","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Transient Growth in an Incompressible Flow past a Backward-Slanted Step"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3928-1482","authenticated-orcid":false,"given":"Marco","family":"Martins Afonso","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica (Faculdade de Ci\u00eancias) da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal"},{"name":"Aix Marseille Universit\u00e9, CNRS, Centrale Marseille, M2P2 UMR 7340, 13451 Marseille, France"}]},{"given":"Philippe","family":"Meliga","sequence":"additional","affiliation":[{"name":"Aix Marseille Universit\u00e9, CNRS, Centrale Marseille, M2P2 UMR 7340, 13451 Marseille, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3174-7727","authenticated-orcid":false,"given":"Eric","family":"Serre","sequence":"additional","affiliation":[{"name":"Aix Marseille Universit\u00e9, CNRS, Centrale Marseille, M2P2 UMR 7340, 13451 Marseille, France"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1017\/S0022112007005654","article-title":"Structural sensitivity of the first instability of the cylinder wake","volume":"581","author":"Giannetti","year":"2007","journal-title":"J. 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