Geometric Orbits of Surface Waves from a Circular Hydraulic Jump

A circular hydraulic jump is produced when a vertical jet of water strikes a horizontal surface. The flow spreads out in a thin layer, and then at a certain radius, the height of the flow increases abruptly. At low flow rates, the flow is stationary. At some critical flow rate, the flow becomes unstable, the circular symmetry is broken, and surface waves are generated. If a reflector is placed in the flow, then the power spectrum of the surface fluctuations shows an oscillatory behavior which can be interpreted as the interference of geometric orbits.

S.H. Hansen, S. Hørlück, D. Zauner, P. Dimon, C. Ellegaard, (and S.C. Creagh)

Two-Dimensional Granular Flow in a Small-Angle Funnel

We have investigated a granular flow consisting of a single layer of uniform balls in a two-dimensional funnel. The qualitative behavior of the flow depends in a sensitive way on the geometry. For a particular configuration in which only the funnel opening angle is varied, we find three regimes. When , the flow is dense and steady, and the flow rate is determined by the geometry at the outlet. When , the flow is intermittent, consisting of quasi-periodic kinematic shock waves propagating against the flow. The flow rate reaches its maximum in this regime. When , the waves become stationary, and the flow rate is now determined by the geometry at the inlet. We also measure the number density fluctuations of the flow and their power spectra. For all flows, the power spectra are white at low frequencies with structures at higher frequencies resulting from the kinematic waves and short-range correlations.

C. Veje and P. Dimon

Excited Granular Flow in a 2-dimensional Small-angle Funnel

The dynamics of granular flows are studied in the presence of excitations. It has already been found in previous work that a gravity-driven two-dimensional granular flow has both a steady flow regime and an intermittent regime resulting from quasi-periodic shock waves. We now shake the system to study the dynamics. For example, in a regime where the flow jams without an excitation, it will flow intermittently in the presence of one. In particular, there may be a critical acceleration at which the granular matter fluidizes.

K. Lindemann, C. Veje, and P. Dimon

Fluctuations in an Hourglass

We are studying granular flow in an hourglass as a function of grain size. We propose a simple stochastic process which can explain the 1/f noise in the density fluctuations observed in an older experiment. We are also exciting the system with an acoustic speaker to study the dynamics of the system.

C. Veje and P. Dimon

Shock Waves in a Two-dimensional Granular Flow

In a previous experiment, shock waves have been observed in a two-dimensional granular flow. They are responsible for the intermittent aspect of the flow in certain parameter regimes. Their behavior is being studied by video techniques which have revealed features not visible by eye. For example, it has been observed that although they are mostly produced at the outlet of the flow, they can also be created in the middle of the system, and that they interact in a repulsive manner. Quantitative measurements of the shock wave velocity and the density field will enable comparison with the various theoretical models that have been put forward.

S. Hørlück, C. Veje, and P. Dimon

Level Dynamics in Acoustics

We have studied parametric level dynamics in acoustic resonance spectra, where the parameter varied is the shape of the resonating object. We have studied hundreds of levels over a significant range of the shape parameter and observed the so-called spaghetti of anticrossings between levels. We have extracted velocity and curvative probability distributions and found them in good agreement with the universal functions predicted for quantum chaos.

C. Ellegaard, M. Oxborrow, P. Bertelsen and K. Schaadt

Exploring Symmetries in Acoustic Resonances of Plates

We measure acoustic resonance spectra on 2D plates with shapes varying from the very regular shape - a square - to a completely chaotic shpae - a Sinai Stadium. We find that even in the last case statistic measures such as the nearest neighbour distribution and distributions indicate that there is exactly one symmetry unbroken. The distributions correspond to exactly one GOE. It is concluded that even though the plates are thin (thickness < wavelength) there is still an odd-even symmetry in the direction of the thickness of the plate. This is proven by cutting the plate half through from one side. This yields distribution for exactly one GOE.

C. Ellegaard, M. Oxborrow, P. Bertelsen and K. Schaadt

**Figure 1:** Acoustic wave function of Sinai stadium.

Wavefunctions in Acoustics

We have measured the wavefunctions for acoustic resonances in various shape aluminium plates. We have extracted the statistics of the amplitude distribution and found a close relationship to the Porter Thomas distribution. We are also beginning to measure the transient wavefunctions in acoustic systems on imparting the object a sharp blow.

C. Ellegaard, M. Oxborrow, P. Bertelsen and K. Schaadt

Symmetry breaking in quartz crystals with acoustic resonances.

Because of the intrinsic anisotropy a rectangular block of single
crystal quartz has only one symmetry: the 180^{o} rotation about the
x-axis. This is reflected in the acoustic spectra.
The statistical statistcal measures: nearest neighbour
distributions (NND) and statistics show the distributions
corresponding to exactly two GOE. When the crystal is
deformed ever so slightly (an 0.5 mm radius octant is removed from one
corner of a 14x25x40 mm^{2} block) a significant change is observed in
the NND. The great sensitivity of this method to small
deformations would make it a good method for nondestructive testing of
mechanical parts.

C. Ellegaard, M. Oxborrow, P. Bertelsen and K. Schaadt

Experimental studies of transitions in the circular hydraulic jump

The circular hydraulic jump occurs when a vertical jet hits a horisontal surface. It is a very rich system, and we use it as a convienient system to study separation, onset of turbulence and very non-linear hydrodynamical pattern formation.

In the experiment the fluid (ethylenglycol (anti-freeze) seeded with aluminum powder) is directed vertically down on a flat plate inside a dish of fluid. We control the height of the fluid layer outside the jump. When the fluid id shallow, the "normal" jump occurs in which the fluid just outside the jump rotates forward on the surface and backwards at the bottom. When the fluid level is increased this flow-pattern is reversed. Here the fluid just outside the jump rotates backwards on the top and forwards near the bottom. We have in fact created a roller, as on a breaking wave, but stationary. This system has separation points both on the bottom and on the surface Systematic measurements are made of the onset of the various transitions as a function of height, flux and viscosity. By increasing the fluid level further and/or increasing the flux we observe that the circular symmetry is broken and stable or unstable polygons are created, depending on the flux and viscosity.

Simple numerical simulations of the Navier-Stokes equations where the surface is fixed have given results in rough agreement with the experiments. A model for the symmetry-broken states has yet to come.

C. Ellegaard, A. Espe Hansen, A. Haaning, T. Bohr.

Experimental Project in Acoustic Emission from Fracture and Failure Processes

The purpose of the project is to test experimentally a new working hypothesis
for failure and fracturing in heterogenous systems. Our approach consists of
viewing the final stage of rupture as a kind of critical point, which can be
formalized within a renormalization group scheme. Recent findings suggest that
quenched disorder in fracturing will under certain conditions initiate a
dynamical process leading to the formation of an approximately hierarchical
system of stress-interactions and *discrete* scale invariance.

C. Ellegaard, A. Johansen and (D. Sornette)

**Figure 2:** Example of stationary polygon in hydraulic jump.

Fri Feb 21 15:17:28 MET 1997