par Bourcy, Sarah 
Président du jury Kozyreff, Gregory
Promoteur Knaepen, Bernard
Publication Non publié, 2022-09-02
Président du jury Kozyreff, Gregory
Promoteur Knaepen, Bernard
Publication Non publié, 2022-09-02
Thèse de doctorat
| Résumé : | Optimal disturbances and transition to turbulence in boundary layers over flat and concave walls under the influence of a spanwise magnetic fieldWe numerically study the instability and transition to turbulence of magnetohydrodynamic boundary layer flows over flat and concave walls in the presence of an external constant magnetic field in the spanwise direction. The linear stability of the system is explored through non-modal analysis (NMA): the optimal perturbations experiencing maximum transient growth are computed under the parallel-flow and the low magnetic Reynolds number approximations, by means of the linear operator that maps any initial condition to the solution of the initial-value problem at time t, the so-called propagator. NMA allows to take into account the non-normality of the linear operator governing the temporal evolution of disturbances and its eventual time-dependence. This dependence arises when we let the base flow, which is given by the Blasius profile, diffuse over time. This is what we do for both kinds of walls but we also consider a time-independent profile in the case of the flat plate.For the latter, the time diffusion of the base state is found to have little influence on the linear amplification and the structure of optimal perturbations, except in the hydrodynamic case. For both time-diffusing and time-independent profiles, NMA shows that optimal disturbances are increasingly oblique with growing Hartmann number Ha and thus magnetic field intensity. In contrast to the streamwise vortices that lead to maximum amplification in the absence of magnetic field, purely spanwise rolls are optimal for sufficiently strong magnetic fields, starting for Ha=10 and Ha=15 at Re=250 and Re=450. The nonlinear evolution of optimal modes and the transition to turbulence are investigated through temporal direct numerical simulations for Re=450 and diverse Hartmann numbers. Either streamwise, oblique or a superposition of symmetric oblique modes are used as initial conditions along with low three-dimensional noise. Transient growth and secondary instabilities of optimal perturbations are found to initiate the laminar-turbulent transition until Ha=10, which is large enough to prevent transition from occurring. Transition is therefore increasingly altered with growing magnetic field intensity. Unlike the linear evolution, which is little affected by the time diffusion except when Ha=0, the nonlinear evolution of optimal perturbations is strongly impacted by the temporal evolution of the mean state. Independently of the value of Ha, the kinetic energy of disturbances is smaller at any time and transition starts later. Starting from Ha=5, transition does not take place anymore. In the case of the concave wall, the Görtler number and the spanwise wavenumber of disturbances match those characterising the experiments of Swearingen and Blackwelder (1987), while several values of the Hartmann number are explored. The most unstable normal modes are also calculated. Increasing the Hartmann number is shown to suppress more the linear growth of normal modes than optimal ones. The nonlinear time evolution of the perturbations energy indicates that transition is triggered earlier by optimal disturbances but does not occur for Hartmann numbers greater than 3 in both cases. The examination of the velocity field allows to evaluate the impact of the magnetic field on the growth of the Görtler vortices that are generated by optimal perturbations, as well as on the secondary instabilities of the vortices. It is shown that secondary instabilities progressively switch from sinuous to varicose modes with increasing Hartmann number due to the growing suppression of the spanwise shear by the magnetic field. The spanwise shear, known to play a more important role than vertical shear in the transition process, is more suppressed than vertical shear. |
| The miscible interface separating two solutions of respective species A and B can be unstable due to buoyancy-driven effects. In the present work, we carry out a theoretical study within the context of flows in porous media. For such systems, drawing reliable information from linear stability analysis is complex as the underlying base states are time evolving and the linearised operators are also non-normal. Here, we analyse the stability problem through the non-modal approach that takes these two features into account, by defining and developing the tools used in this approach. The theoretical framework governing the flow dynamics in porous media is given by the Darcy's law, coupled with the advection-diffusion equations for the concentrations of solutes A and B. Two nondimensional parameters appear in this problem : the buoyancy-ratio R (that compares the density of the two solutions) and the ratio delta of the diffusion coefficients of the solutes. Non-modal analysis is performed by deriving the matrix that governs the temporal evolution of the concentrations perturbations, the so-called propagator, and by computing its Singular Value Decomposition (SVD), which provides the largest amplification disturbances can undergo and the corresponding optimal perturbations. Several buoyancy-driven instabilities are investigated by maximising amplification over all possible wavenumbers, namely the Rayleigh-Taylor instability, double diffusion, delayed-double diffusion and diffusive-layer convection. Special attention is paid to the delayed-double diffusive instability by examining the non-normality of the linearised operator and the influence of its time-dependence. For all the aforementioned instabilities, it is shown that the non-modal analysis predictions are significantly different from those of the linear stability analysis based on the quasi-steady-state approximation, especially at small times. |
