Yohann Duguet

Transition in pipe flow: the saddle structure on the boundary of turbulence

The laminar–turbulent boundary Σ is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognizing approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on Σ are organized around only a few travelling waves and their heteroclinic connections. If the flow is suitably constrained to a subspace with a discrete rotational symmetry, it is possible to find locally attracting travelling waves embedded within Σ. Four new types of travelling waves were found using this approach.

Formation of turbulent patterns near the onset of transition in plane Couette flow

The formation of turbulent patterns in plane Couette flow is investigated near the onset of transition, using numerical simulation in a very large domain of size 800 h x 2h x 356 h. Based on a maxi ...

Localized edge states in plane Couette flow

The dynamics at the threshold of transition in plane Couette flow is investigated numerically in a large spatial domain for a certain type of localized initial perturbation, for Re between 350 and 1000. The corresponding edge state is an unsteady spotlike structure, localized in both streamwise and spanwise directions, which neither grows nor decays in size. We show that the localized nature of the edge state is numerically robust, and is not influenced by the size of the computational domain. The edge trajectory appears to transiently visit localized steady states. This suggests that basic spatiotemporally intermittent features of transition to turbulence, such as the growth of a turbulent spot, can be described as a dynamical system.

Relative periodic orbits in transitional pipe flow

A dynamical system description of the transition process in shear flows with no linear instability starts with knowledge of exact coherent solutions, among them traveling waves (TWs) and relative periodic orbits (RPOs). We describe a numerical method to find such solutions in pipe flow and apply it in the vicinity of a Hopf bifurcation from a TW which looks to be especially relevant for transition. The dominant structural feature of the RPO solution is the presence of weakly modulated streaks. This RPO, like the TW from which it bifurcates, sits on the laminar-turbulent boundary separating initial conditions which lead to turbulence from those which immediately relaminarize.

Highly symmetric travelling waves in pipe flow

The recent theoretical discovery of finite-amplitude travelling waves (TWs) in pipe flow has reignited interest in the transitional phenomena that Osborne Reynolds studied 125 years ago. Despite all being unstable, these waves are providing fresh insight into the flow dynamics. We describe two new classes of TWs, which, while possessing more restrictive symmetries than previously found TWs of Faisst & Eckhardt (2003 Phys. Rev. Lett. 91, 224502) and Wedin & Kerswell (2004 J. Fluid Mech. 508, 333–371), seem to be more fundamental to the hierarchy of exact solutions. They exhibit much higher wall shear stresses and appear at notably lower Reynolds numbers. The first M-class comprises the various discrete rotationally symmetric analogues of the mirror-symmetric wave found in Pringle & Kerswell (2007 Phys. Rev. Lett. 99, 074502), and have a distinctive double-layered structure of fast and slow streaks across the pipe radius. The second N-class has the more familiar separation of fast streaks to the exterior and slow streaks to the interior and looks like the precursor to the class of non-mirror-symmetric waves already known.

Minimal transition thresholds in plane Couette flow

Subcritical transition to turbulence requires finite-amplitude perturbations. Using a nonlinear optimisation technique in a periodic computational domain, we identify the perturbations of plane Couette flow transitioning with least initial kinetic energy for Re ⩽ 3000. We suggest a new scaling law Ec = O(Re−2.7) for the energy threshold vs. the Reynolds number, in quantitative agreement with experimental estimates for pipe flow. The route to turbulence associated with such spatially localised perturbations is analysed in detail for Re = 1500. Several known mechanisms are found to occur one after the other: Orr mechanism, oblique wave interaction, lift-up, streak bending, streak breakdown, and spanwise spreading. The phenomenon of streak breakdown is analysed in terms of leading finite-time Lyapunov exponents of the associated edge trajectory.

Complexity of localised coherent structures in a boundary-layer flow

We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar-turbulent separatrix is investigated in a spanwise-extended domain that allows for robust localisation of all edge states. We work at fixed Reynolds number and study the edge states as a function of the streamwise period. We demonstrate the complex spatio-temporal dynamics of these localised states, which exhibits multistability and undergoes complex bifurcations leading from periodic to chaotic regimes. It is argued that in all regimes the dynamics restricted to the edge is essentially low-dimensional and non-extensive.Graphical abstract

Oscillatory jets and instabilities in a rotating cylinder

The viscous flow inside a closed rotating cylinder of gas subject to periodic axial compression is investigated numerically. The numerical method is based on a spectral Galerkin expansion of the velocity field, assuming axisymmetry of the flow. If the forcing amplitude is weak and the angular forcing frequency is less than twice the rotation rate, inertial waves emanate from the corners, forming conical oscillatory jets which undergo reflections at the walls. Their thickness is O(E1∕3), or O(E1∕4) for particular forcing frequencies, where E is the Ekman number. For larger forcing amplitudes, the conical pattern breaks down. When the forcing frequency is resonant with a low-order inertial mode, the flow can undergo two types of parametric instabilities: a mode-triad resonance, and a subharmonic instability. The combination of both these mechanisms provides a possible route to quasiperiodicity of the flow.

Splitting of a turbulent puff in pipe flow

The transition to turbulence of the flow in a pipe of constant radius is numerically studied over a range of Reynolds numbers where turbulence begins to expand by puff splitting. We first focus on the case where splitting occurs as discrete events. Around this value only long-lived pseudo-equilibrium puffs can be observed in practice, as typical splitting times become very long. When is further increased, the flow enters a more continuous puff splitting regime where turbulence spreads faster. Puff splitting presents itself as a two-step stochastic process. A splitting puff first emits a chaotic pseudopod made of azimuthally localized streaky structures at the downstream (leading) laminar–turbulent interface. This structure can later expand azimuthally as it detaches from the parent puff. Detachment results from a collapse of turbulence over the whole cross-section of the pipe. Once the process is achieved a new puff is born ahead. Large-deviation consequences of elementary stochastic processes at the scale of the streak are invoked to explain the statistical nature of splitting and the Poisson-like distributions of splitting times reported by Avila et al (2011 Science 333 192–6).

Bypass transition and spot nucleation in boundary layers

The spatio-temporal aspects of the transition to turbulence are considered in the case of a boundary layer flow developing above a flat plate exposed to free-stream turbulence. Combining results on the receptivity to free-stream turbulence with the nonlinear concept of a transition threshold, a physically motivated model suggests a spatial distribution of spot nucleation events. To describe the evolution of turbulent spots a probabilistic cellular automaton is introduced, with all parameters directly fitted from numerical simulations of the boundary layer. The nucleation rates are then combined with the cellular automaton model, yielding excellent quantitative agreement with the statistical characteristics for different free-stream turbulence levels. We thus show how the recent theoretical progress on transitional wall-bounded flows can be extended to the much wider class of spatially developing boundary-layer flows.