Hostname: page-component-76d6cb85b7-rxvq6 Total loading time: 0 Render date: 2026-07-13T23:24:03.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Global instability in a plane jet with flexible nozzle

Published online by Cambridge University Press:  22 January 2026

Simone Cruciani ,
Vincenzo Citro ,
Michel Fournié  and
Franco Auteri Open the ORCID record for Franco Auteri [Opens in a new window]
Simone Cruciani
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy Fédération ENAC ISAE-SUPAERO ONERA, Département d’Ingénierie des Systèmes Complexes (DISC), Université de Toulouse, 10 avenue Edouard-Belin, 31055 Toulouse, France
Vincenzo Citro
Affiliation:
Dipartimento di Ingegneria Industriale (DIIN), Universitá degli Studi di Salerno, Fisciano 84084, Italy
Michel Fournié
Affiliation:
Fédération ENAC ISAE-SUPAERO ONERA, Département d’Ingénierie des Systèmes Complexes (DISC), Université de Toulouse, 10 avenue Edouard-Belin, 31055 Toulouse, France
Franco Auteri*
Affiliation:
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, via La Masa 34, 20156 Milano, Italy
*
Corresponding author: Franco Auteri, franco.auteri@polimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of free jets is one of the fundamental problems that has driven the development of new theoretical and numerical methods in fluid mechanics. Extensive research has focused on the convective instabilities that characterise their elusive dynamics. However, in real-world configurations, free jets are often confined by solid walls which may exhibit different degrees of flexibility. The present paper presents, for the first time, evidence that even slightly flexible nozzles can lead to global instabilities. To show it, we adopted the classical tools of linear stability analysis, solving the fluid–structure interaction (FSI) problem by an arbitrary Lagrangian–Eulerian method, formulating a monolithic three-field problem. The investigation of the base flow properties reveals the effect of the Reynolds number, based on the bulk velocity and channel height, in the range $[50,200]$ and of the plate stiffness on the nozzle deformation and on the jet flow development. Exploiting an idea first proposed by Luchini and Charru, we develop an ad hoc quasi-one-dimensional model capable of predicting the displacement of elastic boundaries even for large displacements. The stability and sensitivity analysis shows that the interaction of the flow with the flexible structure leads to two categories of globally unstable modes: sinuous (in-phase) modes and varicose (out-of-phase) modes. All the results presented have been cross-checked with direct numerical simulations of the nonlinear FSI system, revealing that the instabilities correspond to supercritical bifurcations. This work has significant implications for many natural and industrial phenomena where a jet is produced by a compliant nozzle.

JFM classification

Aerodynamics: Flow-structure interactions Instability: Absolute/convective instability Wakes/Jets: Jets

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1027 , 25 January 2026 , A38
Check for updates
Copyright
© The Author(s), 2026. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Benjamin, T.B. 1960 Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9 (4), 513–532.10.1017/S0022112060001286CrossRefGoogle Scholar
Benjamin, T.B. & Batchelor, G.K. 1961 a Dynamics of a system of articulated pipes conveying fluid - ii. Experiments. Proc. R. Soc. Lond. Series A. Math. Phys. Sci. 261 (1307), 487–499.Google Scholar
Benjamin, T.B. & Batchelor, G.K. 1961 b Dynamics of a system of articulated pipes conveying fluid - i. Theory. Proc. R. Soc. Lond. Series A. Math. Phys. Sci. 261 (1307), 457–486.Google Scholar
Bickley, W.G. 1937 The plane jet. Lond. Edin. Dublin Phil. Mag. J. Sci. 23 (156), 727–731.10.1080/14786443708561847CrossRefGoogle Scholar
des Boscs, P.-E. & Kuhlmann, H.C. 2024 Flow stability and sequence of bifurcations in a cubic cavity driven by a constant shear stress. J. Fluid Mech. 978, A28.10.1017/jfm.2023.946CrossRefGoogle Scholar
Brown, G.B. 1935 On vortex motion in gaseous jets and the origin of their sensitivity to sound. Proc. Phys. Soc. 47 (4), 703.10.1088/0959-5309/47/4/314CrossRefGoogle Scholar
Choi, D. & Park, H. 2022 Flow–structure interaction of a starting jet through a flexible circular nozzle. J. Fluid Mech. 949, A39.10.1017/jfm.2022.781CrossRefGoogle Scholar
Curle, N. 1957 On hydrodynamic stability in unlimited fields of viscous flow. Proc. R. Soc. Lond. Series A Math. Phys. Sci. 238 (1215), 489–501,Google Scholar
Das, P., Govardhan, R.N. & Arakeri, J.H. 2018 Unsteady two-dimensional jet with flexible flaps at the channel exit. J. Fluid Mech. 845, 462–498.10.1017/jfm.2018.230CrossRefGoogle Scholar
Davies, C. & Carpenter, P.W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.10.1017/S0022112097007313CrossRefGoogle Scholar
de Langre, E. 2000 Ondes variqueuses absolument instables dans un canal élastique. Comptes Rendus de l’Académie des Sciences - Series IIB - Mechanics-Physics-Astronomy 328 (1), 61–65.Google Scholar
De Langre, E. 2001 Fluides et Solides. Editions Ecole Polytechnique.Google Scholar
Fabre, D., Citro, V., Ferreira Sabino, D., Bonnefis, P., Sierra, J., Giannetti, F. & Pigou, M. 2019 A practical review on linear and nonlinear global approaches to flow instabilities. Appl. Mech. Rev. 70 (6), 060802.10.1115/1.4042737CrossRefGoogle Scholar
Farnell, D.J.J., David, T. & Barton, D.C. 2004 Coupled states of flapping flags. J. Fluid. Struct. 19 (1), 29–36.10.1016/j.jfluidstructs.2003.10.001CrossRefGoogle Scholar
Formaggia, L. & Nobile, F. 1999 Stability analysis for the arbitrary lagrangian eulerian formulation with finite elements. East–West J. Num. Math. 7 (2), 105–131.Google Scholar
Garg, V.K. 1981 Spatial stability of the non-parallel bickley jet. J. Fluid Mech. 102, 127–140.10.1017/S0022112081002577CrossRefGoogle Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.10.1017/S0022112007005654CrossRefGoogle Scholar
Grotberg, J.B. & Jensen, O.E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.10.1146/annurev.fluid.36.050802.121918CrossRefGoogle Scholar
Hecht, F. 2012 New development in freefem++. J. Numer. Math. 20 (3–4), 251–265.10.1515/jnum-2012-0013CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473–537.10.1146/annurev.fl.22.010190.002353CrossRefGoogle Scholar
Jia, L.-B., Li, F., Yin, X.-Z. & Yin, X.-Y. 2007 Coupling modes between two flapping filaments. J. Fluid Mech. 581, 199–220.10.1017/S0022112007005563CrossRefGoogle Scholar
Juniper, M.P. 2006 The effect of confinement on the stability of two-dimensional shear flows. J. Fluid Mech. 565, 171–195.10.1017/S0022112006001558CrossRefGoogle Scholar
Kumaran, V. 2021 Stability and the transition to turbulence in the flow through conduits with compliant walls. J. Fluid Mech. 924, P1.10.1017/jfm.2021.602CrossRefGoogle Scholar
Landahl, M.T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13 (4), 609–632.10.1017/S002211206200097XCrossRefGoogle Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.10.1146/annurev-fluid-010313-141253CrossRefGoogle Scholar
Luchini, P. & Charru, F. 2010 Consistent section-averaged equations of quasi-one-dimensional laminar flow. J. Fluid Mech. 656, 337–341.10.1017/S0022112010002594CrossRefGoogle Scholar
Michelin, S. & Llewellyn Smith, S.G. 2009 Linear stability analysis of coupled parallel flexible plates in an axial flow. J. Fluid. Struct. 25 (7), 1136–1157.10.1016/j.jfluidstructs.2009.06.002CrossRefGoogle Scholar
Miles, J.W. 1957 On the generation of surface waves by shear flows. J. Fluid Mech. 3 (2), 185–204.10.1017/S0022112057000567CrossRefGoogle Scholar
Mitchell, B. & Morris, S. 2025 Formation of multiple vortex rings from passively flexible nozzles. J. Fluid Mech. 1011, A3.10.1017/jfm.2025.378CrossRefGoogle Scholar
Murugappan, S., Gutmark, E.J., Lakhamraju, R.R. & Khosla, S. 2008 Flow-structure interaction effects on a jet emanating from a flexible nozzle. Phys. Fluids 20 (11), 117105.10.1063/1.3013634CrossRefGoogle ScholarPubMed
Paidoussis, M.P. 2004 Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 2. Academic Press.Google Scholar
Pfister, J.-L. 2019 Instabilities and Optimization of Elastic Strutures Interacting with Laminar Flows. Theses, Université Paris Saclay (COmUE).Google Scholar
Pfister, J.-L., Marquet, O. & Carini, M. 2019 Linear stability analysis of strongly coupled fluid–structure problems with the Arbitrary-Lagrangian–Eulerian method. Comput. Meth. Appl. Mech. Engng 355, 663–689.10.1016/j.cma.2019.06.024CrossRefGoogle Scholar
Poole, R. & Turner, M.R. 2023 Stability of jets and wakes confined by compliant walls. Phys. Rev. Fluids 8 (6), 063901.10.1103/PhysRevFluids.8.063901CrossRefGoogle Scholar
Poole, R. & Turner, M.R. 2024 The effect of asymmetry on the absolute instability of confined jets and wakes. Fluid Dyn. Res. 56 (5).10.1088/1873-7005/ad7aa0CrossRefGoogle Scholar
Rayleigh, L. 1878 On the instability of jets. Proc. Lond. Math. Soc. 1, 4–13.10.1112/plms/s1-10.1.4CrossRefGoogle Scholar
Renard, Y. & Poulios, K. 2020 Getfem: automated FE modeling of multiphysics problems based on a generic weak form language. ACM Trans. Math. Softw. 47 (1), 3412849.Google Scholar
Rognes, M.E. & Winther, R. 2010 Mixed finite element methods for linear viscoelasticity using weak symmetry. Math. Models Meth. Appl. Sci. 20 (06), 955–985.10.1142/S0218202510004490CrossRefGoogle Scholar
Sato, H. 1960 The stability and transition of a two-dimensional jet. J. Fluid Mech. 7 (1), 53–80.10.1017/S0022112060000049CrossRefGoogle Scholar
Sato, H. & Sakao, F. 1964 An experimental investigation of the instability of a two-dimensional jet at low Reynolds numbers. J. Fluid Mech. 20 (2), 337–352.10.1017/S0022112064001264CrossRefGoogle Scholar
Savic, P. 1941 On acoustically effective vortex motion in gaseous jets. Lond. Edin. Dublin Phil. Mag. J. Sci. 32 (212), 245–252.10.1080/14786444108520791CrossRefGoogle Scholar
Schlichting, H. 1933 Laminare strahlausbreitung. ZAMM — J. Appl. Math. Mechan. / Zeitschrift für Angewandte Mathematik und Mechanik 13 (4), 260–263.10.1002/zamm.19330130403CrossRefGoogle Scholar
Schouveiler, L. & Eloy, C. 2009 Coupled flutter of parallel plates. Phys. Fluids 21 (8), 081703.10.1063/1.3204672CrossRefGoogle Scholar
Schrader, L.-U., Mavriplis, C. & Brandt, L. 2012 Spatial linear disturbances in a plane wall jet. Phys. Fluids 24 (5), 054104.10.1063/1.4717954CrossRefGoogle Scholar
Shamanskiy, A. & Simeon, B. 2020 Mesh moving techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiency. Comput. Mech. 67 (2), 583–600.10.1007/s00466-020-01950-xCrossRefGoogle Scholar
Tang, L. & Païdoussis, M.P. 2009 The coupled dynamics of two cantilevered flexible plates in axial flow. J. Sound Vib. 323 (3-5), 790–801.10.1016/j.jsv.2009.01.028CrossRefGoogle Scholar
Tatsumi, T. & Kakutani, T. 1958 The stability of a two-dimensional laminar jet. J. Fluid Mech. 4 (3), 261–275.10.1017/S0022112058000422CrossRefGoogle Scholar
Vedeneev, V. & Nikitin, N. 2023 Absolute instability of plane incompressible jets. J. Fluid Mech. 962, A4.10.1017/jfm.2023.163CrossRefGoogle Scholar
Weaver, D.S. & Paidoussis, M.P. 1977 On collapse and flutter phenomena in thin tubes conveying fluid. J. Sound Vib. 50 (1), 117–132.10.1016/0022-460X(77)90555-7CrossRefGoogle Scholar
Yadykin, Y., Tenetov, V. & Levin, D. 2003 The added mass of a flexible plate oscillating in a fluid. J. Fluid. Struct. 17 (1), 115–123.10.1016/S0889-9746(02)00100-7CrossRefGoogle Scholar
Yu, M.-H. & Monkewitz, P.A. 1990 The effect of nonuniform density on the absolute instability of two-dimensional inertial jets and wakes. Phys. Fluids A 2 (7), 1175–1181.10.1063/1.857618CrossRefGoogle Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408 (6814), 835–839.10.1038/35048530CrossRefGoogle Scholar
Zhang, J., Ren, Z., Hu, W., Soon, R.H., Yasa, I.C., Liu, Z. & Sitti, M. 2021 Voxelated three-dimensional miniature magnetic soft machines via multimaterial heterogeneous assembly. Sci. Robotics 6 (53), eabf0112.10.1126/scirobotics.abf0112CrossRefGoogle ScholarPubMed