Unsteady and Transitional Flows team (EIT)

The team studies flow instability, transition to turbulence and flow control. This very complex topic aims to understand and control shear flows through numerical simulation and structural studies based on stability analyses. The team relies on a strong numerical component, but also on theoretical tools that help interpret the results.
The team's scientific project is organised around two complementary sub-themes:
1. Stability of complex flows
2. Ensteady near-wall flows

Theme 1: Stability of complex flows


This research theme focuses on the dynamics of flows that can become chaotic at Reynolds numbers below their critical value, in particular shear flows (plane Couette flow, boundary layers, plane channel, etc.). The aim is to understand the stages and mechanisms leading to transition, with the ultimate goal of controlling or delaying subcritical transition. Part of the team's research addresses the early stages of transition, especially the transient growth of perturbations caused by the non-normality of stable eigenvalues. At DynFluid these early stages are studied for several shear flows, including non-parallel ones, by computing linear optimal perturbations [1,2], i.e. the small-amplitude perturbations that produce maximum energy amplification in the flow, or through sensitivity studies to harmonic forcing [3]. The optimal perturbations or forcings are followed through their nonlinear evolution with direct numerical simulations, to detect the coherent structures that populate the flow during transition (e.g. vortices, hairpin et streaks). The nonlinear stages are studied in detail thanks to recently developed nonlinear optimisation and minimisation techniques, which compute nonlinear optimal perturbations [4], as well as the defects or minimal perturbations that have the largest effect on flow transition [5,6]. These nonlinear optimal perturbations differ greatly from linear optima and drive the flow to transition very quickly and efficiently [7,8]. At DynFluid these techniques are also used to control the flow optimally and delay its transition in the presence of finite-amplitude perturbations [9]. For the final stages of transition to turbulence, we study coherent structures that can sustain themselves nonlinearly, by computing 'exact coherent structures', i.e. steady, periodic or even chaotic coherent states with very few unstable directions. These states (edge states, periodic orbits or travelling waves), which act as relative attractors and therefore shape the dynamics of shear flows, are computed and studied for various parallel and non-parallel shear flows [10,11]. 

 

This research theme studies the transient and asymptotic dynamics of strongly spatially inhomogeneous flows, including the study and tracking of a flow's successive bifurcations. For several years DynFluid has focused on developing methods for, and analysing, global linear and nonlinear instabilities. Thanks to improved computing resources, the linear stability of a fixed point of the three-dimensional Navier-Stokes equations has become tractable within the last decade. A matrix-free method for computing the eigenvalues of the Jacobian operator was developed in the NEK5000 code (Argonne National Laboratory) and applied to the flow around a cylindrical roughness element. The results show that the dynamics depend strongly on the roughness aspect ratio and on the Reynolds number. For flat roughness elements, we showed that above a critical Reynolds number a varicose global instability arises, leading to nonlinear dynamics with coherent structures of the hairpin type. Conversely, when the roughness is tall and narrow enough, a sinuous global instability resembling a von Kármán vortex street develops. The flow around a roughness element is of interest because it allows the study of receptivity and transition mechanisms of a finite-amplitude perturbation in a  boundary layer. For more than ten years, DynFluid has studied transition mechanisms in separated flows, in particular shock-wave/boundary-layer interactions. When the interaction is strong enough, it causes the boundary layer to separate; the separated flow has complex dynamics and in some cases develops a self-sustained low-frequency oscillation. The laboratory aims to understand the physical mechanism behind these oscillations. Such flows are mainly characterised by instability waves developing in the shear layer bounding the recirculation region. This instability is convective and produces strong transient energy growth. When the interaction is intense enough, a large-scale nonlinear synchronisation mechanism appears and generates a self-sustained low-frequency oscillation.  

Theme 2: Unsteady near-wall flows

This research theme is the temporal counterpart of the previous one: it studies the transient and asymptotic dynamics of flows that are unsteady but spatially very simple. The difficulty comes from multiple time scales, which lead to so-called 'stiff' problems. In particular, the combination of oscillations, walls and viscous effects (no-slip) gives rise to atypical linear instabilities (T.1) and to quasi-linear rectified-flow or streaming effects (T.2).

Oscillating flows.
Transition to turbulence in a flow oscillating around a zero mean is an open problem in fluid mechanics. This configuration must be distinguished from the case where the oscillations are 'carried' by a non-zero mean flow; there, unsteadiness has only a secondary effect and can be modelled with a multiple-scale method. The Stokes layer is the simplest example. The control parameter depends on the viscosity and on the period and amplitude of the oscillations. With no imposed length scale, a single Reynolds number Re is the only control parameter. Transition to turbulence is observed from Re > 900, whereas linear stability analysis gives a critical value above 2500, so the transition is subcritical. The question is still open and has attracted renewed interest, as shown by the number of recent publications on it. This work aims to identify the linear mechanisms that amplify perturbations, based on computing the maximum transient energy growth. The results show strong transient growth through the Orr mechanism, with an energy gain that increases exponentially with the Reynolds number. As a result, the modal transition scenario is very unlikely to be observed. Nonlinear simulations show that at a Reynolds number of 1000, a perturbation with a low initial energy, consistent with the linear assumption, is enough to trigger turbulence.

Acoustic streaming in thermoacoustic machines.
Thermoacoustic machines convert heat into acoustic energy (engines) or vice versa (refrigerators). They consist of an acoustic resonator (usually cylindrical) containing a stack with heat exchangers at both ends. An acoustic wave set up in the resonator interacts with the walls and produces a secondary flow of lower order in Mach number: acoustic streaming. This flow is currently identified as one of the main sources of heat loss in thermoacoustic machines, reducing their efficiency, so studying it in depth is crucial to understand and quantify its impact on machine operation. Depending on the acoustic amplitude, streaming can be linear (slow streaming) or nonlinear (fast streaming). Slow streaming is fairly well understood and has been widely studied, notably analytically. In a resonator it consists of two kinds of recirculation cells: inner cells near the wall (Schlichting streaming) and outer cells in the central region (Rayleigh streaming). We focus on nonlinear Rayleigh streaming at high acoustic amplitude, which is poorly understood and has received little attention. The study is numerical (with LIMSI, Orsay) and experimental (with Institut Pprime, Poitiers). The simulations solve the compressible Navier-Stokes equations, which resolves the acoustic field and the mean streaming flow simultaneously.