Walter I. Goldburg

Two-dimensional turbulence: a review of some recent experiments

A review of recent experiments in two-dimensional turbulence is presented. Work on flowing soap films and on thin layers of fluid driven electromagnetically is covered. Theoretical notions of turbulence in two and three dimensions are introduced.

Two‐dimensional velocity profiles and laminar boundary layers in flowing soap films

In this study we examine laminar velocity profiles of freely suspended flowing soap films. We introduce a new device which supports large uniform films for indefinite periods of time. The geometry of the flow is two‐dimensional (2D), yet the measured velocity profiles depart from ideal 2D behavior. The main reason for this departure is that the soap film experiences an air drag force across its entire surface. Describing the air with Prandtl boundary layer theory, we predict the observed flow patterns with good accuracy. The downstream development of the profiles is self similar. Our models set an apparent upper limit on the film 2D viscosity of 5⋅10−6 surface poise for dilute soap concentrations. This measurement implies that the surfactant layers on the film may not contribute measurably to the 2D viscosity. For higher soap and glycerol concentrations the opposite appears to be true.

Spinodal decomposition in a binary liquid mixture

Light scattering was used to study phase separation near the critical temperature Tc in a critical mixture of 2,6‐lutidine and water. This system has an inverted coexistence curve, so that a quench into the two‐phase region is produced by an upward jump in temperature. The scattered intensity I (q,t) was recorded at various angles and at a number of quench depths ΔTf =Tf−Tc in the range 0.5≲ΔTf≲2.5 mK. Here Tf denotes the final temperature. The initial temperature was also varied, and no initial‐state effects were observed. One set of experiments employed a cell of very short optical path (0.1 mm) to minimize multiple scattering at the sacrifice of quenching speed. In another set, emphasis was placed on achieving a temperature jump in roughly 0.1 sec so that phase separation could be followed in its early stages. For ΔTf≲1 mK, the change in the measured ring diameter (qm−1) with time, is in fair agreement with the nonlinear theory of Langer, Bar‐on, and Miller. However, the intensity of the ring, I (qm,t)...

Eulerian and Lagrangian studies in surface flow turbulence

Experimental and numerical studies of turbulent fluid motion in a free surface are presented. The flow is realized experimentally on the surface of a tank filled with water stirred well below the surface. Numerically, it is modelled by free-slip boundary conditions. The surface flow is unconventional: it is not incompressible and neither kinetic energy nor enstrophy is conserved in the limit of zero fluid viscosity and in the absence of external driving as is the case for incompressible two-dimensional turbulent flows. The dynamics of passive Lagrangian tracers that are advected in such flows are dominated by rapidly changing patches of the surface flow divergence. Owing to compressibility, particles form clusters within multifractal mass distributions. Also studied is the motion of pairs and triplets of particles. The mean square separation shows an extended range with a reduced scaling exponent in comparison with the classical Richardson value. Clustering is also manifest in strongly deformed triangles spanned within triplets of tracers.

Electrical conductivity of binary mixtures near the critical point

The electrical conductivity σ of several liquid mixtures was measured near their respective critical points. The systems studied were pure phenol–water, KCl‐doped phenol–water, and isobutyric acid–water. The measurements spanned the reduced temperature range 10−5≲e≲10−2, where e≡ (T‐Tc)/Tc. All systems clearly showed the existence of a singular contribution to the conductivity. A least‐squares fit of the data to the equation, (σ‐σc)/σc∝eϑ+ background terms, showed that the singularity could be characterized by a critical exponent ϑ=0.70+0.15−0.10. A possible explanation of this singularity is offered which uses percolation theory in the mean field limit. This percolation approach yields ϑ=2β, in accord with our own observations. The exponent β (?1/3) characterizes the shape of the coexistence curve.

Spinodal decomposition in a binary liquid mixture near the critical point

Light scattering was used to study the nonlinear behavior of spinodal decomposition in a binary mixture of 2−6 lutidine−water near its lower critical point. We have photometrically analyzed the time evolution of the differential scattering cross section following a quench into the thermodynamically unstable spinodal region. Our results are in very good agreement with a mean field theory proposed by Langer. Using this theory we have determined the temperature dependence of the fourth concentration derivative of the free energy ∂4f/∂c4. The results are consistent with scaling if the nonclassical critical exponents are used.

Measurements of turbulent velocity fluctuations in a planar Couette cell

The anisotropy of turbulent velocity fluctuations in a planar Couette cell has been investigated by using homodyne photon correlation spectroscopy (HCS). We find that 〈δvq(l)〉, the mean velocity difference between two points separated by a distance l, is consistent with the scaling behavior 〈δvq(l)〉∝l0.55. These measurements were made at a Reynolds number Re much greater than Rec, where the subcritical transition first occurs. The HCS technique was also used to measure Rec itself.

Velocity fluctuations in a turbulent soap film: The third moment in two dimensions

Quasi-two-dimensional decaying turbulence is studied in a flowing soap film by measuring the moments of the probability density function P(δv(r)) for the longitudinal velocity differences δv(r) on a scale r. As in three-dimensional (3-D) turbulence, P becomes non-Gaussian with decreasing r. The third moment S3(r)≡〈(δv(r))3〉 is small and negative at small scales, but becomes positive at larger scales. The exact calculation of S3(r) for 2-D homogeneous isotropic turbulence relates this change in sign to the development of the velocity correlation function as the turbulence decays.

Modification of a vortex street by a polymer additive

A Karman vortex street is created in a flowing soap film by a rod penetrating the film. The velocity field generated by this rod is modified by the addition of the polymer polyethylene oxide having a molecular weight of 5×106 and a concentration of 30 wppm. The rms velocity fluctuations behind the rod are strongly suppressed by the polymer additive and the power spectrum of the velocity fluctuations is modified as well. The experiments show that the polymer additive decreases the rate at which energy is injected into the flow. The measurements further indicate that the polymer introduces an elongational viscosity term into the Navier–Stokes equation.

Transport Properties Near Magnetic Critical Points

In the last few years intense interest has developed in the nature of second‐order phase transitions. Experimental and theoretical investigations have centered primarily on equilibrium properties. Transport or nonequilibrium properties are only now beginning to be intensively studied. Thus far there exists only very limited theoretical work in this area, and little experimental data for comparison. We review the theoretical and the experimental situation (excluding neutron diffraction) as it applies to magnetic systems. Scaling‐law theory is summarized and the importance of the dynamic‐scaling hypothesis is emphasized. An attempt is made to relate existing experiments to these scaling laws. We conclude that the closest contact between scaling theory and experiment at present comes in the prediction and observation of transitional behavior between the hydrodynamic regime far from the critical point, and the critical regime in which the fluctuation range becomes large compared to other characteristic lengths.