Paul Manneville

Dissipative structures and weak turbulence

We present a brief overview of the current understanding of temporal and spatio-temporal chaos, both termed weak turbulence according to the context [1]. The process which allows one to reduce the primitive problem to a low-dimensional dynamical system is discussed. It turns out to be appropriate as long as confinement effects are sufficiently strong to freeze the space dependence of unstable modes, hence temporal chaos only. Otherwise modulated patterns arise, yielding genuine space-time chaos. The corresponding theory rests on envelope equations providing a useful framework for weak turbulence in a globally super-critical setting. spatio-temporal intermittency analyzed next is the relevant scenario in the sub-critical case. Finally, the connection with hydrodynamic turbulence and the more general relevance of some of the ideas developed here are examined.

Different ways to turbulence in dissipative dynamical systems

Abstract The Lorenz model is studied in details for σ = 10, b = 8 3 and 145 r r = 145 and r = 148.4 the Lore nz attractor disaggregates itself into a limit cycle through a cascade of bifurcation with successive undoubling of periods. At r = 166.07 this limit cycle looses its stability through “intermittency”, giving rise to a second aperiodic attractor. We give a semi-quantitative interpretation of these processes and discuss their relation with the different transitions to turbulence observed experimentally.

Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations

We study the reliability of two-dimensional models of film flows down inclined planes obtained by us [Ruyer-Quil and Manneville, Eur. Phys. J. B 15, 357 (2000)] using weighted-residual methods combined with a standard long-wavelength expansion. Such models typically involve the local thickness h of the film, the local flow rate q, and possibly other local quantities averaged over the thickness, thus eliminating the cross-stream degrees of freedom. At the linear stage, the predicted properties of the wave packets are in excellent agreement with exact results obtained by Brevdo et al. [J. Fluid Mech. 396, 37 (1999)]. The nonlinear development of waves is also satisfactorily recovered as evidenced by comparisons with laboratory experiments by Liu et al. [Phys. Fluids 7, 55 (1995)] and with numerical simulations by Ramaswamy et al. [J. Fluid Mech. 325, 163 (1996)]. Within the modeling strategy based on a polynomial expansion of the velocity field, optimal models have been shown to exist at a given order in the long-wavelength expansion. Convergence towards the optimum is studied as the order of the weighted-residual approximation is increased. Our models accurately and economically predict linear and nonlinear properties of film flows up to relatively high Reynolds numbers, thus offering valuable theoretical and applied study perspectives.

Wave patterns in film flows: Modelling and three-dimensional waves

In a previous work, two-dimensional film flows were modelled using a weightedresidual approach that led to a four-equation model consistent at order e2. A twoequation model resulted from a subsequent simplification but at the cost of lowering the degree of the approximation to order e only. A Pade approximant technique is applied here to derive a refined two-equation model consistent at order e2. This model, formulated in terms of coupled evolution equations for the film thickness h and the flow rate q, accounts for inertia effects due to the deviations of the velocity profile from the parabolic shape, and closely follows the asymptotic long-wave expansion in the appropriate limit. Comparisons of two-dimensional wave properties with experiments and direct numerical simulations show good agreement for the range of parameters in which a two-dimensional wavy motion is reported in experiments. The stability of two-dimensional travelling waves to three-dimensional perturbations is investigated based on the extension of the models to include spanwise dependence. The secondary instability is found to be not very selective, which explains the widespread presence of the synchronous instability observed in the experiments by Liu et al. (1995) whereas Floquet analysis predicts a subharmonic scenario in most cases. Three-dimensional wave patterns are computed next assuming periodic boundary conditions. Transition from two- to three-dimensional flows is shown to be strongly dependent on initial conditions. The herringbone patterns, the synchronously deformed fronts and the three-dimensional solitary waves observed in experiments are recovered using our regularized model, which is found to be an excellent compromise between the complete model, which has seven equations, and the simplified model, which does not include the second-order inertia corrections. Those corrections are found to play a role in the selection of the type of secondary instability as well as of the spanwise wavelength of the emerging pattern.

Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow

Elongated streamwise structures are considered as a key element of the transition to turbulence in various wall flows. In pure plane Couette flow (pCf), longitudinal streaks originating from pairs of streamwise counter-rotating vortices are clearly identified surrounding growing turbulent spots or at late stages of spot relaxation. The same structures bifurcate subcritically from a slightly modified Couette flow where a thin spanwise wire has been introduced in the zero-velocity plane. The basic flow profile, as measured by laser Doppler velocimetry, is shown to approach continuously the original linear velocity profile as the radius of the wire is decreased. On the other hand, the vortices remain almost unchanged and the bifurcation threshold remains bounded from above by the global stability threshold below which turbulent spots relax spontaneously. This supports the conjecture that a related nontrivial nonlinear solution exists in the pure pCf limit. These observations are compared to numerical stabili...

Phase diagram of the two-dimensional complex Ginzburg-Landau equation

After a brief introduction to the complex Ginzburg-Landau equation, some of its important features in two space dimensions are reviewed. A comprehensive study of the various phases observed numerically in large systems over the whole parameter space is then presented. The nature of the transitions between these phases is investigated and some theoretical problems linked to the phase diagram are discussed.

Spatiotemporal perspective on the decay of turbulence in wall-bounded flows

By use of a reduced model focusing on the in-plane dependence of plane Couette flow, it is shown that the turbulent-->laminar relaxation process can be understood as a nucleation problem similar to that occurring at a thermodynamic first-order phase transition. The approach, apt to deal with the large extension of the system considered, challenges the current interpretation in terms of chaotic transients typical of temporal chaos. The study of the distribution of the sizes of laminar domains embedded in turbulent flow proves that an abrupt transition from sustained spatiotemporal chaos to laminar flow can take place at some given value of the Reynolds number Rlow, whether or not the local chaos lifetime, as envisioned within low-dimensional dynamical systems theory, diverges at finite R beyond Rlow.

Modeling transitional plane Couette flow

Abstract.The Galerkin method is used to derive a realistic model of plane Couette flow in terms of partial differential equations governing the space-time dependence of the amplitude of a few cross-stream modes. Numerical simulations show that it reproduces the globally sub-critical behavior typical of this flow. In particular, the statistics of turbulent transients at decay from turbulent to laminar flow displays striking similarities with experimental findings.

Rayleigh-Bénard Convection: Thirty Years of Experimental, Theoretical, and Modeling Work

A brief review of Rayleigh-Benard studies performed during the twentieth century is presented, with an emphasis on the transition to turbulence and the appropriate theoretical framework, relying on the strength of confinement effects and the distance to threshold, either dynamical systems for temporal chaos in the strongly confined case, or models of space-time chaos when confinement effects are weak.

Long-range order with local chaos in lattices of diffusively coupled ODEs

Abstract The existence of non-trivial collective behavior in lattices of diffusively coupled differential equations is investigated. For a two-dimensional square lattice of Rossler systems, a rotating long-range order is observed. This case is best described in terms of a complex Ginzburg-Landau (CGL) equation submitted to the local noise produced by the chaotic Rossler units. The parameters of this CGL equation are estimated to be in the so-called “Benjamin-Feir stable” region. The collective oscillation regime thus corresponds to the linearly-stable, spatially-homogeneous solution of the equivalent CGL equation. The possibility of more complex collective behavior in similar systems is discussed.