Philippe Carrière

Convective versus absolute instability in mixed Rayleigh–Bénard–Poiseuille convection

Transition from convective to absolute instability in Rayleigh–Benard convection in the presence of a uni-directional Poiseuille flow is studied. The evaluation of the long-time behaviour of the Green function in the horizontal plane allows the determination of regions of convective and absolute instability in the Rayleigh–Reynolds number plane as a function of Prandtl number. It is found that the mode reaching zero group velocity at the convective–absolute transition always corresponds to transverse rolls, while the system remains convectively unstable with respect to pure streamwise (longitudinal) rolls for all non-zero Reynolds numbers. Finally, the roll pattern within the entire wave packet and in particular near its centre is elucidated and possible connections between experiments and our findings are discussed.

Spectral decay of a passive scalar in chaotic mixing

In this paper we take a closer look at the decay phase of a passive, diffusing, scalar field undergoing steady, three-dimensional chaotic advection. The energy spectrum of the scalar is obtained by numerical simulation of the advection–diffusion equation at high Peclet number. At large times, the spectral decay is found to be exponential and self-similar. It is emphasized that the asymptotic decay-time is an important measure of mixing efficiency, alongside the time required for diffusion to first become effective. The large-wavenumber spectral form, representing the distribution of scalar energy over small scales, is analyzed. Power-law behavior is found at scales intermediate between the large ones, comparable in size with the entire flow volume, and the smallest ones, at which diffusion is effective and the spectrum falls off exponentially with increasing wavenumber. Fitting of the numerical results allows the exponent of the power-law to be estimated. It is observed to vary with the parameters of the ...

Study of a chaotic mixing system for DNA chip hybridization chambers

Numerical simulations of a micromixing system based on chaotic advection for improved deoxyribonucleic acid (DNA) chip hybridization are presented. To attain best chip performance, homogeneous dispersion of DNA molecules throughout the chamber in which the chip is placed is of primary importance. Poincare sections of a simple time-periodic flow, based on numerical simulations of the flow, are compared with visualizations in a scaled-up experiment, with good agreement. The influence on mixing efficiency of varying the period of the flow at fixed volume flow rate is studied and a trade off is found between the absence of regular islands and a small enough total sample volume. The results illustrate the potential for optimization of such devices based on numerical flow simulations.

A numerical Eulerian approach to mixing by chaotic advection

Results of numerical simulation of the advection‐diffusion equation at large Peclet number are reported, describing the mixing of a scalar field under the action of diffusion and of a class of steady, bounded, three‐dimensional flows, which can have chaotic streamlines. The time evolution of the variance of scalar field is calculated for different flow parameters and shown to undergo modulated exponential decay, with a decay rate which is a maximum for certain values of the flow parameters, corresponding to cases in which the streamlines are chaotic everywhere. If such global chaos is present, the decay rate tends to oscillate, whereas the presence of regular regions produces a more constant decay rate. Significantly different decay rates are obtained depending on the detailed properties of the chaotic streamlines. The relationship between the decay rate and the characteristic Lyapunov exponents of the flow is also investigated.

Towards better DNA chip hybridization using chaotic advection

Numerical studies for two protocols of micromixing based on chaotic advection to improve DNA chip hybridization are presented. The first protocol uses syringes; the other one, pumps. For both protocols, numerical Poincare sections and Lyapunov exponents of the three-dimensional, time-periodic flow are investigated as functions of the period. Model experiments also confirm numerical results. Homogeneity of the dispersion of particles inside the chamber is of primary importance to achieve best chip reliability: although global chaos was obtained for both protocols, we find that the one employing the pumps is more likely to achieve better and more rapid hybridization.

Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection

enard–Poiseuille (RBP) system are analysed for the case of non-uniform heating of the lower wall. Specifically, a single two-dimensional ‘hot spot’ or ‘temperature bump’ giving rise to a finite region of local instability is considered. For the case of the lower wall temperature varying slowly on the scale of the RBP cell height, i.e. for a gentle temperature bump, WKBJ asymptotics are used to construct an analytical approximation of the linear global mode. At the same time, an analytical selection criterion for the critical global mode is derived from the breakdown of the WKBJ expansion at a double turning point located at the top of the temperature bump. The analytical construction and the underlying assumptions are supported by comparison with direct numerical simulations for Gaussian temperature bumps of elliptical planform not necessarily aligned with the mean flow. From these comparisons, it is concluded that the proposed analytical construction indeed yields the most amplified global mode, which is characterized by an essentially transverse orientation (normal to the mean flow direction) of the convection rolls, independent of the planform of the temperature bump. The paper concludes with preliminary DNS results on the saturated global mode shape and a discussion of possible connections to the ‘steep’ fully nonlinear global modes found for one-dimensional inhomogeneities of the basic state.

Transverse-roll global modes in a Rayleigh-Bénard-Poiseuille system with streamwise variable heating

Abstract The effect of a spatially inhomogeneous heating of the bottom wall in Rayleigh–Benard–Poiseuille convection is studied for slow streamwise variations of the temperature profile. The problem is defined by the constant Reynolds number of the Poiseuille through flow, assumed to be low (typically ⩽10), the constant Prandtl number, and the spatial evolution of the Rayleigh number R (X) , assumed to be subcritical everywhere except in a limited region around its single maximum R t . In this initial study, all spanwise inhomogeneities such as side walls or spanwise variable heating are neglected to obtain two-dimensional (transverse roll) global mode solutions by means of WKBJ asymptotics. The resulting frequency selection yields, at leading order, a global mode frequency equal to the local absolute frequency ωt at the streamwise location where the Rayleigh number is maximum, with higher-order corrections for non-parallelism. These allow the determination of critical values of R t for global instability as a function of the profile of the local Rayleigh number R (X) and of Prandtl and Reynolds numbers.

Envelope equations for the Rayleigh-Bénard-Poiseuille system. Part 1. Spatially homogeneous case

Envelope equations are derived for the convection rolls in the Rayleigh-Benard-Poiseuille system, taking into account both their slow streamwise and transverse variations. At finite O(1) Reynolds numbers, the stability of finite-amplitude longitudinal roll patterns is accessible to analysis in a moving frame of reference and stability is predicted provided a generalized Eckhaus criterion is satisfied. At lower Reynolds numbers, the analysis allows the analytical determination of the Green function for arbitrary orientations of the instability pattern. It clarifies previous results concerning the purely convective nature of all modes of instability except transverse rolls (for which a convective absolute transition exists), as soon as the Reynolds number is non-zero

Envelope equations for the Rayleigh–Bénard–Poiseuille system. Part 2. Linear global modes in the case of two-dimensional non-uniform heating

Linear global modes in the Rayleigh–Benard–Poiseuille system, for the case of two-dimensional non-uniform heating in the form of a single hot spot, are analysed in the framework of the envelope equation formalism. Global mode solutions are sought by means of WKBJ asymptotics. As for the one-dimensional case, an analytical selection criterion for the frequency may be derived from the breakdown of the WKBJ expansion at a two-dimensional double turning point located at the maximum of the local Rayleigh number. The analytical results, including the behaviour of the mode in the vicinity of the turning point, are compared with results obtained from numerical simulations of the envelope equation. Finally, the issue of the selection of the wavevector branches in the WKBJ expansion is discussed.

The distribution of “time of flight” in three dimensional stationary chaotic advection

The distributions of “time of flight” (time spent by a single fluid particle between two crossings of the Poincare section) are investigated for five different three dimensional stationary chaotic mixers. Above all, we study the large tails of those distributions and show that mainly two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. When no-slip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to −3.