Modeling film flows down inclined planes

A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q, and a new variable that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier-Stokes equations. Received: 16 April 1998 / Revised: 29 June 1998 / Accepted: 2 July 1998     

Understanding the sub-critical transition to turbulence in wall flows

In contrast with free shear flows presenting velocity profiles with inflection points which cascade to turbulence in a relatively mild way, wall bounded flows are deprived of (inertial) instability modes at low Reynolds numbers and become turbulent in a much wilder way, most often marked by the coexistence of laminar and turbulent domains at intermediate Reynolds numbers, well below the range where (viscous) instabilities can show up. There can even be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille pipe flow (Ppf) that are currently the subject of intense research. Though the mechanisms involved in the transition to turbulence in wall flows are now better understood, statistical properties of the transition itself are yet unsatisfactorily assessed. A widely accepted interpretation rests on non-trivial solutions of the Navier-Stokes equations in the form of unstable travelling waves and on transient chaotic states associated to chaotic repellors. Whether these concepts typical of the theory of temporal chaos are really appropriate is yet unclear owing to the fact that, strictly speaking, they apply when confinement in physical space is effective while the physical systems considered are rather extended in at least one space direction, so that spatiotemporal behaviour cannot be ruled out in the transitional regime. The case of pCf will be examined in this perspective through numerical simulations of a model with reduced cross-stream (y) dependence, focusing on the in-plane (x, z) space dependence of a few velocity amplitudes. In the large aspect-ratio limit, the transition to turbulence takes place via spatiotemporal intermittency and we shall attempt to make a connection with the theory of first-order (thermodynamic) phase transitions, as suggested long ago by Pomeau.      

Nonperturbative renormalization group approach to turbulence

We suggest a new, renormalization group (RG) based, nonperturbative method for treating the intermittency problem of fully developed turbulence which also includes the effects of a finite boundary of the turbulent flow. The key idea is not to try to construct an elimination procedure based on some assumed statistical distribution, but to make an ansatz for possible RG transformations and to pose constraints upon those, which guarantee the invariance of the nonlinear term in the Navier-Stokes equation, the invariance of the energy dissipation, and other basic properties of the velocity field. The role of length scales is taken to be inverse to that in the theory of critical phenomena; thus possible intermittency corrections are connected with the outer length scale. Depending on the specific type of flow, we find different sets of admissible transformations with distinct scaling behaviour: for the often considered infinite, isotropic, and homogeneous system K41 scaling is enforced, but for the more realistic plane Couette geometry no restrictions on intermittency exponents were obtained so far. Received: 28 December 1997 / Accepted: 6 August 1998     

Spatio-temporal patterns in the thermoconvection of a planar nematic layer: I. Weakly nonlinear models

We study theoretically the formation of convection patterns in a laterally extended planar nematic layer heated from below, in the linear and weakly nonlinear regimes. By reformulating the viscous coupling terms of the basic nematohydrodynamic equations, a simple interpretation of the flow effects on the director dynamics can be proposed. A detailed linear analysis of the problem is presented. A systematic method to investigate nonlinear mechanisms is developed, and exemplified by the study of the nonlinear saturation in rolls. The extension of the roll amplitude equation with the envelope formalism is used to characterize the dynamics of the roll modulations near threshold. Coupled envelope equations are shown to describe the structure of the point defects in zig-zags observed experimentally. Finally the bifurcation to the bimodal varicose is studied. The secondary wavevector in the bimodal appears to be selected by a rotation of the director in the horizontal plane. Quantitative predictions concerning the amplitude of this rotation are given. Received: 1st December 1997 / Revised: 25 May 1998 / Accepted: 2 June 1998     

Pulses of tunable size near a subcritical bifurcation

We show that a nonlinear gradient term can be used to tune the width of pulse-like solutions to a generalized quintic Ginzburg-Landau equation. We investigate the dynamics of these solutions and show that weakly turbulent patches can persist for long times. Analogies with turbulent spots in plane Couette flows are discussed. Received: 29 May 1997 / Revised: 16 March 1998 / Accepted: 30 April 1998     

Transition to turbulence in a shear flow

We analyze the properties of a 19-dimensional Galerkin approximation to a parallel shear flow. The laminar flow with a sinusoidal shape is stable for all Reynolds numbers Re. For sufficiently large Re additional stationary flows occur; they are all unstable. The lifetimes of finite amplitude perturbations shows a fractal dependence on amplitude and Reynolds number. These findings are in accord with observations on plane Couette flow and suggest a universality of this transition scenario in shear flows.     

Laser Doppler velocimetry analysis of transitional pipe flow

The objective of this study has been to experimentally analyze the correlation structure of the strong temporal intermittency which characterizes pipe flow close to the transition to turbulence. In doing so transitional pipe flow has been analyzed by Laser Doppler velocimetry and the Reynolds number dependence of the covariance function has been studied. The range which has been analyzed covers the transition to turbulence and moderately developed turbulence (Reynolds number from 1 500 to 5 000). The correlation structure which has been evidenced is generally in agreement with the deterministic, dynamical, interpretation of temporal intermittency which explains the intermittent behavior as a result of a saddle node bifurcation. However, the analysis has evidenced fluctuations even before the onset of turbulence. The structure of these fluctuations is perfectly autoregressive which leads us to conclude that the transition to turbulence can be viewed as a transition from linear randomness to (non-linear) homogeneity. Received 29 March 1999 and Received in final form 6 September 1999     

Long-wavelength modulation of turbulent shear flows

The reverse transition from turbulent to laminar flow is studied in very large aspect ratio plane Couette and Taylor–Couette experiments. We show that laminar-turbulence coexistence dynamics (turbulent spots, spiral turbulence, etc.) can be seen as the ultimate stage of a modulation of the turbulent flows present at higher Reynolds number leading to regular, long-wavelength, inclined stripes. This new type of instability, whose originality is to arise within a macroscopically fluctuating state, can be described in the framework of Ginzburg–Landau equations to which noise is heuristically added to take into account the intrinsic fluctuations of the basic state.     

Ginzburg–Landau description of laminar-turbulent oblique band formation in transitional plane Couette flow

Plane Couette flow, the flow between two parallel planes moving in opposite directions, is an example of wall-bounded flow experiencing a transition to turbulence with an ordered coexistence of turbulent and laminar domains in some range of Reynolds numbers [R _g, R _t] . When the aspect-ratio is sufficiently large, this coexistence occurs in the form of alternately turbulent and laminar oblique bands. As R goes up trough the upper threshold R _t, the bands disappear progressively to leave room to a uniform regime of featureless turbulence. This continuous transition is studied here by means of under-resolved numerical simulations understood as a modelling approach adapted to the long time, large aspect-ratio limit. The state of the system is quantitatively characterised using standard observables (turbulent fraction and turbulence intensity inside the bands). A pair of complex order parameters is defined for the pattern which is further analysed within a standard Ginzburg–Landau formalism. Coefficients of the model turn out to be comparable to those experimentally determined for cylindrical Couette flow.     

Identification of flow mechanisms for a soft crystal

We study the behavior under flow of soft spherical micelles forming a fcc phase at high volume fraction. Due to the size (300 ?) of the elementary objects, the structure can be investigated through X-rays and neutron scattering, at rest and under flow in a Couette cell. Using scattering in two perpendicular directions, different mechanisms of flow are identified. At intermediate shear (around 100 s^-1) the system exhibits the so called layer sliding mechanism where two dimensional hexagonal compact planes of spheres align themselves with the Couette cell walls. At lower shear rate, the fcc structure is locally preserved, and the flow is mediated by defects between crystallites. At high shear rate, we observe the melting of the structure and a liquid-like structure factor. Moreover, we were able to use the existence of the layer sliding regime to generate a fcc monocrystal by annealing the satcking faults between the decorrelated planes created by the layer sliding. Received: 7 July 1997 / Received in final form: 16 January 1998 / Accepted: 5 March 1998     

Modeling transitional plane Couette flow

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.     

Pattern Fluctuations in Transitional Plane Couette Flow

In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in opposite directions, displays alternatively turbulent and laminar oblique bands in a given range of Reynolds numbers R. We show that in periodic domains that contain a few bands, for given values of R and size, the orientation and the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially decreasing laws. This statistics is interpreted in terms of an activated process described by a Langevin equation whose deterministic part is a standard Landau model for two interacting complex amplitudes whereas the noise arises from the turbulent background.     

Spatiotemporal dynamics of shear induced bands en route to rheochaos

We show experimentally that the route to rheochaos in shear rate relaxation measurements is via Type-III intermittency and mixed mode oscillations in the shear-thinning wormlike micellar system of cetyltrimethylammonium tosylate in the presence of salt sodium chloride. Depolarised small angle light scattering measurements performed during flow show that scattered intensity temporally follows the shear rate/stress dynamics and portrays the crucial role played by nematic ordering. Direct visualization of the gap of the Couette cell, illuminated by an unpolarised laser sheet, in the (vorticity, velocity gradient) plane shows that the spatiotemporal dynamics of the shear induced structures is closely related to the temporal behaviour of shear rate/stress fluctuations.     

Spatio-temporal patterns in the thermoconvection of a planar nematic layer: II. Experiments

We study experimentally the evolution of thermoconvection in a laterally extended planar nematic layer, at zero or weak stabilizing magnetic field. As the applied thermal gradient is increased, a cascade of symmetry breakings occurs, towards structures of increasing spatial complexity, and ultimately towards oscillating states. The patterns are characterized optically, and simple models for the distortion of the vertical (out of plane) component of the director field are proposed. Received: 1st December 1997 / Revised: 25 May 1998 / Accepted: 2 June 1998     

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

The transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. We here discuss a variety of different structures that are relevant for the transition, compare the critical Reynolds numbers and optimal wavelengths for their appearance, and explore the differences between flows operating at constant mass flux or at constant pressure drop. The Reynolds numbers quoted here are based on the mean flow velocity and refer to constant mass flux. Reynolds numbers based on constant pressure drop are always higher. The Tollmien–Schlichting (TS) waves bifurcate subcritically from the laminar profile at Re = 5772 at wavelength 6.16 and reach down to Re = 2610 at a different optimal wave length of 4.65. Their streamwise localised counter part bifurcates at the even lower value Re = 2334. Three-dimensional exact solutions appear at much lower Reynolds numbers. We describe one exact solutions that has a critical Reynolds number of 316. Streamwise localised versions of this state require higher Reynolds numbers, with the lowest bifurcation occurring near Re = 1018. The analysis shows that the various branches of TS-waves cannot be connected with transition observed near Re ≈ 1000 and that the exact coherent structures related to downstream vortices come in at lower Reynolds numbers and prepare for the transition.     

A model for ripple instabilities in granular media

We extend the model of surface granular flow proposed in [#!bcre!#] to account for the effect of an external `wind', which acts as to dislodge particles from the static bed, such that a stationary state of flowing grains is reached. We discuss in detail how this mechanism can be described in a phenomenological way, and show that a flat bed is linearly unstable against ripple formation in a certain region of parameter space. We focus in particular on the (realistic) case where the migration velocity of the instability is much smaller than the grains' velocity. In this limit, the full dispersion relation can be established. We relate the critical wave vector to the mean hopping length and to the ratio of the flight time to the `stick' time. We provide an intuitive interpretation of the instability. Received: 30 January 1998 / Revised: 12 May 1998 / Accepted: 8 June 1998     

Dynamics of density waves in a two dimensional funnel on an inclined plane

We report here experiments on two-dimensional funnel flow of diameter glass beads on an inclined plane. We have investigated the properties of the flow according to the outlet size D of the funnel and the gravity. We have identified three different regimes. For small funnel outlet sizes, there is no significant change in flow density: the flow is rather steady and homogeneous. For intermediate outlet sizes (), the flow is intermittent, consisting of spatially ordered density waves propagating upwards. At bigger outlet sizes, density waves do not exhibit any ordering and the flow dynamics becomes chaotic. In addition, we find that the flow dynamics is independent of the funnel opening angle except close to the channel flow configuration. Finally, it is stressed that the interactions between the beads and the inclined plane play a crucial role in the mechanism of formation of density waves. Received: 9 July 1998 / Received in final form and Accepted: 18 September 1998     

First and second order transition in frustrated XY systems

The nature of the phase transition for the XY stacked triangular antiferromagnet (STA) is a controversial subject at present. The field theoretical renormalization group (RG) in three dimensions predicts a first order transition. This prediction disagrees with Monte-Carlo (MC) simulations which favor a new universality class or a tricritical transition. We simulate by the Monte-Carlo method two models derived from the STA by imposing the constraint of local rigidity which should have the same critical behavior as the original model. A strong first order transition is found. Following Zumbach we analyze the second order transition observed in MC studies as due to a fixed point in the complex plane. We review the experimental results in order to clarify the critical behavior observed. Received: 18 February 1998 / Revised: 24 April 1998 / Accepted: 30 April 1998     

Momentum transport and torque scaling in Taylor-Couette flow from an analogy with turbulent convection

We generalize an analogy between rotating and stratified shear flows. This analogy is summarized in Table 1. We use this analogy in the unstable case (centrifugally unstable flow vs. convection) to compute the torque in Taylor-Couette configuration, as a function of the Reynolds number. At low Reynolds numbers, when most of the dissipation comes from the mean flow, we predict that the non-dimensional torque G = T/ν² L, where L is the cylinder length, scales with Reynolds number R and gap width η, G = 1.46η^3/2(1 - η)^-7/4 R ^3/2. At larger Reynolds number, velocity fluctuations become non-negligible in the dissipation. In these regimes, there is no exact power law dependence the torque versus Reynolds. Instead, we obtain logarithmic corrections to the classical ultra-hard (exponent 2) regimes: G = 0.50 . These predictions are found to be in excellent agreement with avail-able experimental data. Predictions for scaling of velocity fluctuations are also provided. Received 7 June 2001 and Received in final form 7 December 2001