Philippe Meliga, E. Hachem. Time-accurate calculation and bifurcation analysis of the incompressible flow over a square cavity using variational multiscale modeling. Journal of Computational Physics, Elsevier, 2019, 376, pp.952-972. 〈hal-01946893〉 Plus de détails...
A thorough variational multiscale (VMS) modeling of the Navier-Stokes equations is used to compute numerical solutions of the incompressible flow over an open cavity. This case features several competing instabilities, and is highly challenging for VMS methods with regard to frequency and pattern selection, because of the non-normality of the linearized Navier-Stokes operator. The relevance of the approach is thus carefully assessed by comparing to direct numerical simulation (DNS) data benchmarked at several Reynolds numbers, and highly accurate time advancing methods are shown to predict relevant evolutions of the transient and saturated solutions. The VMS reduces substantially the computational cost, by similar to 35% (resp. similar to 60%) in terms of CPU time using a semi-implicit discretization scheme based on backward differentiation formula (resp. the implicit Crank-Nicholson scheme), and by similar to 80% in terms of memory requirement. Eventually, the highly efficient semi-implicit VMS numerical framework is used to unravel the onset of the flow oscillations and the selection of the limit cycle frequency, that happens to involve a subcritical Neimark-Sacker bifurcation.
Philippe Meliga, E. Hachem. Time-accurate calculation and bifurcation analysis of the incompressible flow over a square cavity using variational multiscale modeling. Journal of Computational Physics, Elsevier, 2019, 376, pp.952-972. 〈hal-01946893〉
Jessica Sari, Francesco Cremonesi, Mehdi Khalloufi, François Cauneau, Philippe Meliga, et al.. Anisotropic adaptive stabilized finite element solver for RANS models. International Journal for Numerical Methods in Fluids, Wiley, 2018, 86 (11), pp.717-736. 〈10.1002/fld.4475〉. 〈hal-01629274〉 Plus de détails...
Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A Variational MultiScale stabilized finite element method is employed to solve the incompressible Navier-Stokes equations. Finally, the Spalart-Allmaras turbulence model is solved using the Streamline Upwind Petrov-Galerkin (SUPG) method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases. This article is protected by copyright. All rights reserved.
Jessica Sari, Francesco Cremonesi, Mehdi Khalloufi, François Cauneau, Philippe Meliga, et al.. Anisotropic adaptive stabilized finite element solver for RANS models. International Journal for Numerical Methods in Fluids, Wiley, 2018, 86 (11), pp.717-736. 〈10.1002/fld.4475〉. 〈hal-01629274〉
Journal: International Journal for Numerical Methods in Fluids
Philippe Meliga, E. Boujo, M. Meldi, F. Gallaire. Revisiting the drag reduction problem using adjoint-based distributed forcing of laminar and turbulent flows over a circular cylinder. European Journal of Mechanics - B/Fluids, Elsevier, 2018, 72, pp.123-134. 〈10.1016/j.euromechflu.2018.03.009〉. 〈hal-01789049〉 Plus de détails...
Philippe Meliga, E. Boujo, M. Meldi, F. Gallaire. Revisiting the drag reduction problem using adjoint-based distributed forcing of laminar and turbulent flows over a circular cylinder. European Journal of Mechanics - B/Fluids, Elsevier, 2018, 72, pp.123-134. 〈10.1016/j.euromechflu.2018.03.009〉. 〈hal-01789049〉
Philippe Meliga. Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2017, 826, pp.503 - 521. 〈10.1017/jfm.2017.439〉. 〈hal-01585331〉 Plus de détails...
The flow over an open cavity is an example of supercritical Hopf bifurcation leading to periodic limit-cycle oscillations. One of its distinctive features is the existence of strong higher harmonics, which results in the time-averaged mean flow being strongly linearly unstable. For this class of flows, a simplified formalism capable of unravelling how exactly the instability grows and saturates is lacking. This study builds on previous work by Mantic-Lugo et al. (Phys. Rev. Lett., vol. 113, 2014, 084501) to fill in the gap using a parametrized approximation of an instantaneous, phase-averaged mean flow, coupled in a quasi-static manner to multiple linear harmonic disturbances interacting nonlinearly with one another and feeding back on the mean flow via their Reynolds stresses. This provides a self-consistent modelling of the mean flow-fluctuation interaction, in the sense that all perturbation structures are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations. The first harmonic is sought as the superposition of two components, a linear component generated by the instability and aligned along the leading eigenmode of the mean flow, and a nonlinear orthogonal component generated by the higher harmonics, which progressively distorts the linear growth rate and eigenfrequency of the eigenmode. Saturation occurs when the growth rate of the first harmonic is zero, at which point the stabilizing effect of the second harmonic balances exactly the linear instability of the eigenmode. The model does not require any input from numerical or experimental data, and accurately predicts the transient development and the saturation of the instability, as established from comparison to time and phase averages of direct numerical simulation data.
Philippe Meliga. Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2017, 826, pp.503 - 521. 〈10.1017/jfm.2017.439〉. 〈hal-01585331〉
Eddy Constant, Julien Favier, Marcello Meldi, Philippe Meliga, Eric Serre. An immersed boundary method in OpenFOAM : Verification and validation. Computers and Fluids, Elsevier, 2017, 157, pp.55 - 72. 〈10.1016/j.compfluid.2017.08.001〉. 〈hal-01591562〉 Plus de détails...
The present work proposes a modified Pressure-Implicit Split-Operator (PISO) solver integrating the recent Immersed Boundary Method (IBM) proposed by Pinelli et al. [1] in order to perform reliable simulations of incompressible flows around bluff bodies using the open source toolbox OpenFOAM version 2.2 (ESI-OpenCFD [2]). The (IBM) allows for a precise representation of fixed and moving solid obstacles embedded in the physical domain, using uniform or stretched Cartesian meshes. Owing to this feature, the maximum level of accuracy and scalability of the numerical solvers can be systematically achieved. An iterative scheme based on sub-iterations between (IBM) and pressure correction has been implemented in the native (PISO) solver of OpenFOAM. This allows one to use fast optimized Poisson solvers while satisfying simultaneously the divergence-free flow state and the no-slip condition at the body surface. To compute the divergence of the momentum equation (in the PISO loop) and the interpolation of the fluxes, we propose an hybrid calculation with an analytical resolution (using the kernel function equation) of the quantities involving the force term (singular quantities). A careful and original verification study has been carried out which allows to estimate three different errors related to the discretization and to the (IBM). Various 2D and 3D well-documented test cases of academic flows around fixed or moving cylinders have been simulated and carefully validated against existing data from the literature in a large range of Reynolds numbers, Re = 30 − 3900 and in the frame of DNS and DDES OpenFOAM native models.
Eddy Constant, Julien Favier, Marcello Meldi, Philippe Meliga, Eric Serre. An immersed boundary method in OpenFOAM : Verification and validation. Computers and Fluids, Elsevier, 2017, 157, pp.55 - 72. 〈10.1016/j.compfluid.2017.08.001〉. 〈hal-01591562〉
Philippe Meliga. Computing the sensitivity of drag and lift in flow past a circular cylinder: Time-stepping versus self-consistent analysis. Physical Review Fluids, American Physical Society, 2017, 2 (7), pp.073905. 〈10.1103/PhysRevFluids.2.073905〉. 〈hal-01698605〉 Plus de détails...
We provide in-depth scrutiny of two methods making use of adjoint-based gradients to compute the sensitivity of drag in the two-dimensional, periodic flow past a circular cylinder (Re≲189): first, the time-stepping analysis used in Meliga et al. [Phys. Fluids 26, 104101 (2014)] that relies on classical Navier-Stokes modeling and determines the sensitivity to any generic control force from time-dependent adjoint equations marched backwards in time; and, second, a self-consistent approach building on the model of Mantič-Lugo et al. [Phys. Rev. Lett. 113, 084501 (2014)] to compute semilinear approximations of the sensitivity to the mean and fluctuating components of the force. Both approaches are applied to open-loop control by a small secondary cylinder and allow identifying the sensitive regions without knowledge of the controlled states. The theoretical predictions obtained by time-stepping analysis reproduce well the results obtained by direct numerical simulation of the two-cylinder system. So do the predictions obtained by self-consistent analysis, which corroborates the relevance of the approach as a guideline for efficient and systematic control design in the attempt to reduce drag, even though the Reynolds number is not close to the instability threshold and the oscillation amplitude is not small. This is because, unlike simpler approaches relying on linear stability analysis to predict the main features of the flow unsteadiness, the semilinear framework encompasses rigorously the effect of the control on the mean flow, as well as on the finite-amplitude fluctuation that feeds back nonlinearly onto the mean flow via the formation of Reynolds stresses. Such results are especially promising as the self-consistent approach determines the sensitivity from time-independent equations that can be solved iteratively, which makes it generally less computationally demanding. We ultimately discuss the extent to which relevant information can be gained from a hybrid modeling computing self-consistent sensitivities from the postprocessing of DNS data. Application to alternative control objectives such as increasing the lift and alleviating the fluctuating drag and lift is also discussed.
Philippe Meliga. Computing the sensitivity of drag and lift in flow past a circular cylinder: Time-stepping versus self-consistent analysis. Physical Review Fluids, American Physical Society, 2017, 2 (7), pp.073905. 〈10.1103/PhysRevFluids.2.073905〉. 〈hal-01698605〉
Yosuke Anzai, Koji Fukagata, Philippe Meliga, Edouard Boujo, François Gallaire. Numerical simulation and sensitivity analysis of a low-Reynolds-number flow around a square cylinder controlled using plasma actuators. Physical Review Fluids, American Physical Society, 2017, 2 (4), 〈10.1103/PhysRevFluids.2.043901〉. 〈hal-01585359〉 Plus de détails...
Flow around a square cylinder controlled using plasma actuators (PAs) is numerically investigated by direct numerical simulation in order to clarify the most effective location of actuator installation and to elucidate the mechanism of control effect. The Reynolds number based on the cylinder diameter and the free-stream velocity is set to be 100 to study the fundamental effect of PAs on two-dimensional vortex shedding, and three different locations of PAs are considered. The mean drag and the root-mean-square of lift fluctuations are found to be reduced by 51% and 99% in the case where two opposing PAs are aligned vertically on the rear surface. In that case, a jet flow similar to a base jet is generated by the collision of the streaming flows induced by the two opposing PAs, and the vortex shedding is completely suppressed. The simulation results are ultimately revisited in the frame of linear sensitivity analysis, whose computational cost is much lower than that of performing the full simulation. A good agreement is reported for low control amplitudes, which allows further discussion of the linear optimal arrangement for any number of PAs.
Yosuke Anzai, Koji Fukagata, Philippe Meliga, Edouard Boujo, François Gallaire. Numerical simulation and sensitivity analysis of a low-Reynolds-number flow around a square cylinder controlled using plasma actuators. Physical Review Fluids, American Physical Society, 2017, 2 (4), 〈10.1103/PhysRevFluids.2.043901〉. 〈hal-01585359〉
Philippe Meliga, Olivier Cadot, Eric Serre. Experimental and Theoretical Sensitivity Analysis of Turbulent Flow Past a Square Cylinder. Flow, Turbulence and Combustion, Springer Verlag (Germany), 2016, 97 (4, SI), pp.987-1015. 〈10.1007/s10494-016-9755-0〉. 〈hal-01461791〉 Plus de détails...
We assess experimentally and theoretically the ability of a small control cylinder to alter vortex shedding in turbulent flow past a square cylinder at R e = 22,000. Results are presented in terms of sensitivity maps showing the flow regions where the shedding frequency and amplitude are most affected by the control cylinder. Experimental results are obtained for a ratio 0.02 of the cylinder diameters, over an extended domain covering the wake, the shear layers and the free stream. The shedding frequency can be either decreased or increased, the largest effects being obtained placing the control cylinder at the outer edge of the detached shear layers (associated with frequency decrease) or upstream of the square cylinder (associated with frequency increase, in contrast with previous results obtained for a D-shaped geometry of the main cylinder). In contrast, the oscillation amplitude is rarely decreased, meaning that any variation of the shedding frequency comes at the expense of more intense vortex shedding. These findings are revisited in the frame of a theoretical, linear sensitivity analysis of the time-averaged mean flow, performed using adjoint methods in the frame of Reynolds-averaged Navier-Stokes modeling. We show that the retained approach carries valuable information in view of guiding efficient control strategy, as it allows identifying the main regions yielding either a decrease or an increase of the shedding frequency in striking agreement with the experiments. This is a tremendous timesaving in so far as the controlled states need not be computed, the overall computational cost being roughly that of computing the mean flow. In contrast, performing the sensitivity analysis on the underlying unstable steady state yields flawed predictions, hence stressing the need to encompass some level of mean coherent-coherent perturbations interaction in the linear model.
Philippe Meliga, Olivier Cadot, Eric Serre. Experimental and Theoretical Sensitivity Analysis of Turbulent Flow Past a Square Cylinder. Flow, Turbulence and Combustion, Springer Verlag (Germany), 2016, 97 (4, SI), pp.987-1015. 〈10.1007/s10494-016-9755-0〉. 〈hal-01461791〉
François Gallaire, Edouard Boujo, Vladislav Mantic-Lugo, Cristobal Arratia, Benjamin Thiria, et al.. Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake. Fluid Dynamics Research, IOP Publishing, 2016, 48 (6), 〈10.1088/0169-5983/48/6/061401〉. 〈hal-01461792〉 Plus de détails...
The purpose of this review article is to push amplitude equations as far as possible from threshold. We focus on the Stuart-Landau amplitude equation describing the supercritical Hopf bifurcation of the flow in the wake of a cylinder for critical Reynolds number Re-c approximate to 46. After having reviewed Stuart's weakly nonlinear multiple-scale expansion method, we first demonstrate the crucial importance of the choice of the critical parameter. For the wake behind a cylinder considered in this paper, choosing. is an element of(2) = Re-c(1)-Re-c(-1) instead of is an element of'(2) = Re-Re-c/Re-c(2) considerably improves the prediction of the Landau equation. Although Sipp and Lebedev (2007 J. Fluid Mech 593 333-58) correctly identified the adequate bifurcation parameter is an element of, they have plotted their results adding an additional linearization, which amounts to using. is an element of' as approximation to is an element of'. We then illustrate the risks of calculating `running' Landau constants by projection formulas at arbitrary values of the control parameter. For the cylinder wake case, this scheme breaks down and diverges close to Re approximate to 100. We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuart's multiple-scale expansion method. We then briefly review a self-consistent model recently introduced in the literature and demonstrate a link between its properties and the above-mentioned failure.
François Gallaire, Edouard Boujo, Vladislav Mantic-Lugo, Cristobal Arratia, Benjamin Thiria, et al.. Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake. Fluid Dynamics Research, IOP Publishing, 2016, 48 (6), 〈10.1088/0169-5983/48/6/061401〉. 〈hal-01461792〉
Philippe Meliga, Edouard Boujo, François Gallaire. A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2016, 800, pp.327-357. 〈10.1017/jfm.2016.390〉. 〈hal-01461794〉 Plus de détails...
We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Benard von Karmzin vortex street in the wake of a circular cylinder. rfhe sensitivity, analysis is performed in the frame of the semi-linear self consistent model recently introduced by Mamie et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow-, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow- perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan
Philippe Meliga, Edouard Boujo, François Gallaire. A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2016, 800, pp.327-357. 〈10.1017/jfm.2016.390〉. 〈hal-01461794〉
Francois Gallaire, Philippe Meliga, Patrice Laure, Charles N. Baroud. Marangoni induced force on a drop in a Hele Shaw cell. Physics of Fluids, American Institute of Physics, 2014, 26 (6), pp.062105. 〈10.1063/1.4878095〉. 〈hal-01054669〉 Plus de détails...
We analyse the force balance on a cylindrical drop in a Hele-Shaw cell, subjected to a Marangoni flow caused by a surface tension gradient. Depth-averaged Stokes equations, called Brinkman equations, are introduced and a general closed form solution is obtained. The validity of the averaging procedure is ascertained by considering a linear surface tension gradient acting on a cylindrical flattened drop. The Marangoni-driven flow field and resulting force predicted by the Brinkman model are seen to match well a full three-dimensional direct numerical simulation. A closed form ex-pression of the force acting on the drop is obtained, calculated from contributions due to the normal viscous stress, tangential viscous stress, and pressure fields, integrated on the drop perimeter. This expression is used to predict the force balance when a stationary droplet is submitted to both a carrier flow and a Marangoni flow. We show that previous results in the literature had underestimated by a factor two the Marangoni-induced force.
Francois Gallaire, Philippe Meliga, Patrice Laure, Charles N. Baroud. Marangoni induced force on a drop in a Hele Shaw cell. Physics of Fluids, American Institute of Physics, 2014, 26 (6), pp.062105. 〈10.1063/1.4878095〉. 〈hal-01054669〉
Philippe Meliga, Edouard Boujo, Gregory Pujals, François Gallaire. Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder. Physics of Fluids, American Institute of Physics, 2014, 26, pp.26,104101. 〈10.1063/1.4896941〉. 〈hal-01082600v2〉 Plus de détails...
We use adjoint-based gradients to analyze the sensitivity of the drag force on a square cylinder. At Re = 40, the flow settles down to a steady state. The quantity of interest in the adjoint formulation is the steady asymptotic value of drag reached after the initial transient, whose sensitivity is computed solving a steady adjoint problem from knowledge of the stable base solution. At Re = 100, the flow develops to the time-periodic, vortex-shedding state. The quantity of interest is rather the time-averaged mean drag, whose sensitivity is computed integrating backwards in time an unsteady adjoint problem from knowledge of the entire history of the vortex-shedding solution. Such theoretical frameworks allow us to identify the sensitive regions without com-puting the actually controlled states, and provide a relevant and systematic guideline on where in the flow to insert a secondary control cylinder in the attempt to reduce drag, as established from comparisons with dedicated numerical simulations of the two-cylinder system. For the unsteady case at Re = 100, we also compute an approxi-mation to the mean drag sensitivity solving a steady adjoint problem from knowledge of only the mean flow solution, and show the approach to carry valuable information in view of guiding relevant control strategy, besides reducing tremendously the re-lated numerical effort. An extension of this simplified framework to turbulent flow regime is examined revisiting the widely benchmarked flow at Reynolds number Re = 22 000, the theoretical predictions obtained in the frame of unsteady Reynolds-averaged Navier–Stokes modeling being consistent with experimental data from the literature. Application of the various sensitivity frameworks to alternative control objectives such as increasing the lift and reducing the fluctuating drag and lift is also discussed and illustrated with a few selected examples.
Philippe Meliga, Edouard Boujo, Gregory Pujals, François Gallaire. Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder. Physics of Fluids, American Institute of Physics, 2014, 26, pp.26,104101. 〈10.1063/1.4896941〉. 〈hal-01082600v2〉
Philippe Meliga, Gregory Pujals, Eric Serre. Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability. Physics of Fluids, American Institute of Physics, 2012, 24, pp.061701. 〈10.1063/1.4724211〉. 〈hal-01061797〉 Plus de détails...
We use adjoint-based gradients to analyze the sensitivity of turbulent wake past a D-shaped cylinder at Re = 13000. We assess the ability of a much smaller control cylinder in altering the shedding frequency, as predicted by the eigenfrequency of the most unstable global mode to the mean flow. This allows performing beforehand identification of the sensitive regions, i.e., without computing the actually controlled states. Our results obtained in the frame of 2-D, unsteady Reynolds-averaged Navier-Stokes compare favorably with experimental data reported by Parezanović and Cadot [J. Fluid Mech.693, 115 (2012)] and suggest that the control cylinder acts primarily through a local modification of the mean flow profiles.
Philippe Meliga, Gregory Pujals, Eric Serre. Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability. Physics of Fluids, American Institute of Physics, 2012, 24, pp.061701. 〈10.1063/1.4724211〉. 〈hal-01061797〉
Philippe Meliga, François Gallaire, Jean-Marc Chomaz. A weakly nonlinear mechanism for mode selection in swirling jets. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2012, 699, pp.216-262. 〈10.1017/jfm.2012.93〉. 〈hal-01067663〉 Plus de détails...
Global linear and nonlinear bifurcation analysis is used to revisit the spiral vortex breakdown of nominally axisymmetric swirling jets. For the parameters considered herein, stability analyses single out two unstable linear modes of azimuthal wavenumber m=1 and m=2, bifurcating from the axisymmetric breakdown solution. These modes are interpreted in terms of spiral perturbations wrapped around and behind the axisymmetric bubble, rotating in time in the same direction as the swirling flow but winding in space in the opposite direction. Issues are addressed regarding the role of these modes with respect to the existence, mode selection and internal structure of vortex breakdown, as assessed from the three-dimensional direct numerical simulations of Ruith et al. (J. Fluid Mech., vol. 486, 2003, pp. 331-378). The normal form describing the leading-order nonlinear interaction between modes is computed and analysed. It admits two stable solutions corresponding to pure single and double helices. At large swirl, the axisymmetric solution bifurcates to the double helix which remains the only stable solution. At low and moderate swirl, it bifurcates first to the single helix, and subsequently to the double helix through a series of subcritical bifurcations yielding hysteresis over a finite range of Reynolds numbers, the estimated bifurcation threshold being in good agreement with that observed in the direct numerical simulations. Evidence is provided that this selection is not to be ascribed to classical mean flow corrections induced by the existence of the unstable modes, but to a non-trivial competition between harmonics. Because the frequencies of the leading modes approach a strong 2:1 resonance, an alternative normal form allowing interactions between the m=2 mode and the first harmonics of the m=1 mode is computed and analysed. It admits two stable solutions, the double helix already identified in the non-resonant case, and a single helix differing from that observed in the non-resonant case only by the presence of a slaved, phase-locked harmonic deformation. On behalf of the finite departure from the 2:1 resonance, the amplitude of the slaved harmonic is however low, and the effect of the resonance on the bifurcation structure is merely limited to a reduction of the hysteresis range.
Philippe Meliga, François Gallaire, Jean-Marc Chomaz. A weakly nonlinear mechanism for mode selection in swirling jets. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2012, 699, pp.216-262. 〈10.1017/jfm.2012.93〉. 〈hal-01067663〉