Jerry P. Gollub

Many routes to turbulent convection

Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5–5·0), in fluid layers of small horizontal extent. Contour maps of the structure of the time-averaged velocity field, in conjunction with high-resolution power spectral analysis, demonstrate that several mean flows are stable over a wide range in the Rayleigh number R , and that the sequence of time-dependent instabilities depends on the mean flow. A number of routes to non-periodic motion have been identified by varying the geometrical aspect ratio, Prandtl number, and mean flow. Quasi-periodic motion at two frequencies leads to phase locking or entrainment, as identified by a step in a graph of the ratio of the two frequencies. The onset of non-periodicity in this case is associated with the loss of entrainment as R is increased. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. A third route contains a well-defined regime with three generally incommensurate frequencies and no broadband noise. The spectral analysis used to demonstrate the presence of three frequencies has a precision of about one part in 10 4 to 10 5 . Finally, we observe a process of intermittent non-periodicity first identified by Libchaber & Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in R . Several of these processes are also manifested by rather simple mathematical models, but the complicated dependence on geometrical parameters, Prandtl number, and mean flow structure has not been explained.

Dynamical instabilities and the transition to chaotic Taylor vortex flow

We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/R c = 10·1, where R c is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/R c ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/R c = 10·1 disappears at R/R c = 19·3, and the remaining sharp component disappears at R/R c = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landaus picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.

Measurements of the primary instabilities of film flows

We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h ( x , y , t ) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number R c for the onset of waves as a function of angle β. The measurements of R c (β) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f ). A new phase boundary f * s ( R ) is measured, which separates the regimes of saturated finite amplitude waves (at high f ) from multipeaked solitary waves (at low f ). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.

Chaotic mode competition in parametrically forced surface waves

Vertical forcing of a fluid layer leads to standing waves by means of a subharmonic instability. When the driving amplitude and frequency are chosen to be near the intersection of the stability boundaries of two nearly degenerate modes, we find that they can compete with each other to produce either periodic or chaotic motion on a slow timescale. We utilize digital image-processing methods to determine the time-dependent amplitudes of the competing modes, and local-sampling techniques to study the onset of chaos in some detail. Reconstruction of the attractors in phase space shows that in the chaotic regime the dimension of the attractor is fractional and at least one Lyapunov exponent is positive. The evidence suggests that a theory incorporating four coupled slow variables will be sufficient to account for the mode competition.

Solitary wave dynamics of film flows

The development and interaction of solitary wave pulses is critical to understanding wavy film flows on an inclined (or vertical) surface. Sufficiently far downstream, the wave structure consists of a generally irregular sequence of solitary waves independent of the conditions at the inlet. The velocity of periodic solitary waves is found to depend on their frequency and amplitude. Larger pulses travel faster; this property, plus a strong inelasticity, causes larger pulses to absorb others during interactions, leaving a nearly flat interface behind. These wave interactions lead to the production of solitary wave trains from periodic small amplitude waves. The spacings between solitary waves can be irregular for several different reasons, including the amplification of ambient noise, and the interaction process itself. On the other hand, this irregularity is suppressed by the addition of periodic forcing.

Experimental Measurements of Stretching Fields in Fluid Mixing

The mixing of an impurity into a flowing fluid is an important process in many areas of science, including geophysical processes, chemical reactors, and microfluidic devices. In some cases, for example periodic flows, the concepts of nonlinear dynamics provide a deep theoretical basis for understanding mixing. Unfortunately, the building blocks of this theory, i.e. the fixed points and invariant manifolds of the associated Poincare map, have remained inaccessible to direct experimental study, thus limiting the insight that could be obtained. Using precision measurements of tracer particle trajectories in a two-dimensional fluid flow producing chaotic mixing, we directly measure the time-dependent stretching and compression fields. These quantities, previously available only numerically, attain local maxima along lines coinciding with the stable and unstable manifolds, thus revealing the dynamical structures that control mixing. Contours or level sets of a passive impurity field are found to be aligned parallel to the lines of large compression (unstable manifolds) at each instant. This connection appears to persist as the onset of turbulence is approached.

Chaos and threshold for irreversibility in sheared suspensions

Systems governed by time reversible equations of motion often give rise to irreversible behaviour. The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions. Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science, are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk. By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.

Persistent patterns in transient chaotic fluid mixing

Chaotic advection of a fluid can cause an initially inhomogeneous impurity (a passive scalar field) to develop complex spatial structure as the elements of the fluid are stretched and folded, even if the velocity field is periodic in time. The effect of chaotic advection on the transient mixing of impurities—the approach to homogeneity—has been explored theoretically and numerically. A particularly intriguing prediction is the development of persistent spatial patterns, whose amplitude (contrast) decays slowly with time but without change of form. Here we investigate these phenomena using an electromagnetically driven two-dimensional fluid layer in which one half is initially labelled by a fluorescent dye (the passive scalar). We observe the formation of structurally invariant but slowly decaying mixing patterns, and we show how the various statistical properties that characterize the dye concentration field evolve with time as mixing proceeds through many cycles. These results show quantitatively how advective stretching of the fluid elements and molecular diffusion work together to produce mixing of the impurity. We contrast the behaviour of time-period; c flows and identically forced but weakly turbulent flows at lower viscosity, where mixing is much more efficient.

Particle dynamics in sheared granular matter

The particle dynamics and shear forces of granular matter in a Couette geometry are determined experimentally. The normalized tangential velocity V(y) declines strongly with distance y from the moving wall, independent of the shear rate and of the shear dynamics. Local rms velocity fluctuations deltaV(y) scale with the local velocity gradient to the power 0.4+/-0.05. These results agree with a locally Newtonian, continuum model, where the granular medium is assumed to behave as a liquid with a local temperature [deltaV(y)](2) and density dependent viscosity.

The transition to turbulence

Fluid flows have been studied systematically for more than a century and their equations of motion are well known, yet the transition from laminar flow to turbulent flow remains an enigma. The difficulty lies in the intractability of the nonlinear hydrodynamic equations that express the conservation of mass, momentum and energy for a fluid continuum. Although these equations can be linearized and readily solved for a system near thermodynamic equilibrium, the solutions of the nonlinear equations—required to describe fluids far from equilibrium—are generally neither unique nor obtainable.