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Finite Time Singularities in Partial Differential Equations truecm

There are many claims in the literature of finite time singularities in various non- linear PDEs. Often it is not clear, whether these singularities are not simply due to numerical problems. We develop new methods which can rule out finite time singularities in a large class of systems. In particular, we have applied these methods to a conserving variant of the Kardar-Parisi-Zhang equation relevant for growth in the presence of surface diffusion, and we find that the type of singularity, which has been found in numerical simulations, cannot exist. truecm

V. Putkaradze, T. Bohr, (J. Krug) and (M. Bazhenov) truecm

Numerical Simulation of the Boundary Layer Equations with a Free Surface truecm

We are developing numerical methods for solving stationary flow problems with a free surface. In connection with experimental and theoretical investigations of hydraulic jumps, it has become clear that no good numerical results exist for such flows, when both non-linearities and friction are important. We have succeeded in finding stationary solutions of Prandtl's boundary layer equations with a hydrostatic pressure distribution. The results are in agreement with our earlier predictions of the necessity of singularities in such flows. truecm

T. Bohr, T. Petersen and V. Putkaradze truecm

Advection of Particles and Passive Scalars in Nonlinear Field Theories truecm

We have developed a new method for studying the statistical properties of the turbulent states of non-linear PDE's especially the so-called Kuramoto-Sivashinsky equation and its generalizations. We seed the flow with fictitious particles (or a whole passively advected field). In this way one can study a wide class of transport phenomena in a setting which is much simpler than real 3d hydrodynamical turbulence, but has many of the same properties. We look at scaling properties (e.g. anomalous diffusion) and qualitative transitions, e.g. between pinned and flowing states in the presence of symmetry breaking terms. truecm

T. Bohr and (A. Pikovsky) truecm

Growth Shapes of Turbulent Spots in Unstable Systems truecm

In a convectively unstable system, a localized disturbance will move "down-stream", while amplifying and spreading. We have studied the asymptotic form of such local disturbances in a very general setting of a PDE in one or two dimensions for a scalar field. In one dimension we have found analytically the form of the asymptotic field and predicted a novel type of instability for a generalized Kuramoto-Sivashinsky equation. In two dimensions we show how to find the spatial shape of the spreading spots and note the transitions between different shapes and the appearance of non-convex growth forms. truecm

T. Bohr and (C. Conrado) truecm

Space-Time Correlations and Lyapunov Spectra in Turbulent Systems truecm

Turbulent, or space-time chaotic, systems are characterized by a set of Lyapunov exponents of which the number of positive ones are believed to be an extensive quantity (i.e. proportional to the system volume). This implies that distant parts are decorrelated. Often, however, the systems can have long range correlations e.g. like the scale invariant structure functions of fully developed fluid turbulence or interface models. The chaotic fluctuations should show the lack of correlations and we are investigating the best ways of showing this for various systems. In the complex Ginzburg-Landau equation and in coupled map lattices we have shown that the so called mutual information functions are extremely effective in detecting exponential decay. We have conjectured, based on exact solutions for certain classes of coupled map lattices, that the Lyapunov spectra of conserving 1-d systems should show an anomaly at 0, and we believe that this constitutes an new way of looking for conservation laws or for effectively one-dimensional behaviour. truecm

T. Bohr, (G. Grinstein), (C. Jayaprakash), (W. van de Water) and (E. Bosch) truecm

Experimental and Theoretical Studies of Pattern Formation and Turbulence in Capillary Waves truecm

One of our most exciting recent discoveries in physics is the observation of liquid quasicrystals. Quasicrystals were observed in solid-state physics in 1984. Eight years later, we observed the first quasicrystal pattern in a fluid. The quasicrystal pattern was made out of capillary waves, formed on a liquid surface undergoing vertical oscillations at high frequencies. In our laboratory a cell containing a fluid (water or alcohol) is vertically forced by a vibration exciter. At sufficient forcing capillary waves are formed with a frequency-dependent wavelength. Besides the quasicrystal pattern, square and hexagonal patterns are observed. The problem is to understand under what conditions the patterns are formed. At higher drive, the flow becomes turbulent. In this regime we study diffusion and relative diffusion of particles placed on the surface. This is done using image processing and particle tracking programs. For comparison, simulations on particle diffusion in chaotic fields are carried out. truecm

P. Alstrøm, J. Sparre Andersen, (W. Goldburg), (G. Huber), M. T. Levinsen and E. Schröder truecm

Phase Turbulence in 2D truecm

We investigate the stability of the "phase turbulent" regime in the complex Ginzburg-Landau equation. One can argue that this phase should be instable in 2 dimensions and this seems to be what we find, although we need more simulations to prove it. truecm

E. Schröder, P. Alstrøm and T. Bohr truecm

Bound States of Vortices in the 2D Complex Ginzburg-Landau Equation truecm

We study the geometric properties of the glass-like bound states of vortices in the complex Ginzburg-Landau equation. We have shown that the shock lines that separate the vortices (or spiral waves) are segments of hyperbola joined together at corners where three lines meet, and that the microscopic structure of the shocks can be obtained easily and be shown to change from oscillatory to monotonic as the parameters are varied. We are currently investigating the effective interactions between vortices by the same methods. truecm

T. Bohr, (G. Huber) and (E. Ott) truecm

Experimental and Theoretical Studies of Turbulence in a Soap Film, or in a Pipe truecm

Soap-film flows lasting up to 1 hour are produced in the laboratory. The flow velocity is determined by the viscosity of the solution, but it is also adjusted by the hight difference between the two containers between which the soap film is spanned. Turbulence is produced (e.g. behind an obstacle) at sufficient high velocities. The underlying problem is to describe the flow in terms of Navier-Stokes equations, and to describe the boundary layer of air surrounding the soap film, and how it is involved in stabilizing the flow at the selected mean velocity. More generally, we are interested in the velocity fluctuations in a turbulent state. In this connection, we investigate, theoretically as well as experimentally, turbulence produced in a pipe flow behind a grid. The velocity fluctuations are measured using laser Doppler anemometry. Usually the fluctuations are measured as changes in velocity over a short time scale. We have, however, recently developed equipment that with good precision measure velocity changes over short space scales. Traditionally, the mean velocity is used to replace spatial fluctuations with temporal fluctuations. This is the so-called Taylor hypothesis. A very important question is to what extent this assumption is applicable in obtaining the yet unresolved scaling properties for turbulence. truecm

P. Alstrøm, C. Ellegaard, C. Ghidaglia, M. T. Levinsen, M. B. Nielsen, T. Rasmussen, E. Schröder and M. Tibian truecm

Experimental Study of Waves in Thin Film Flows truecm

We study the dynamics of the flow of a thin liquid film on an inclined plane. A convective instability spontaneously creates waves in the film. We study the two- and three-dimensional development of both spontaneous and periodically forced waves. The nonlinear evolution of periodically forced waves depend strongly on the initial frequency. Two-dimensional solitary waves appear at low frequency while saturated finite-amplitude waves occur at high frequency. These periodic waves are usually unstable to further two- and three-dimensional instabilities and evolve into chaotic spatiotemporal patterns sufficiently far downstream. Corresponding studies will be made for granular flow. In a thin layer of fine sand on a plane with an inclination above the critical angle for the sand, wave patterns form that are analogous to the waves in the liquid. truecm

C. Ellegaard


Experimental Study of the Circular Hydraulic Jump truecm

When a vertical jet of fluid is directed upon a horizontal surface, it spreads out radially and then at a certain radius the height of the fluid increases abruptly. This is known as the circular hydraulic jump. As the flow rate of the jet is increased, surface waves are generated. The height fluctuations of the surface have been studied using the absorption of a light beam. The power spectrum of this signal shows the interference effects of outgoing waves with waves reflected from the boundaries of the system or from reflectors placed in the fluid. Using the dispersion relation for weakly damped gravity-capillary waves, it is possible to interpret these structures as the semiclassical orbits of a wave field. truecm

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

Instabilities and Transitions in the Circular Hydraulic Jump truecm

We investigate the forms of the hydraulic jump and the instabilities between them with a new experimental set-up. The fluid (ethylenglycol (anti-freeze) seeded with aluminum powder) is directed vertically down on a flat plate of variable height inside a dish of fluid. When the plate is high, the "normal" jump occurs in which the fluid just outside the jump rotates forward on the surface and backwards at the bottom. When the plate is lowered into the fluid this flow-pattern is reversed. Here the fluid just outside the jump rotates slowly backwards on the top and forwards near the bottom. Between these two states is a strongly unstable regime, where the jump looses its symmetry and intermittent jets shoot out. truecm

C. Ellegaard and T. Bohr truecm

Turbulence in Inhomogeneous Chemical Reactions truecm

We are performing experimental studies of the Belousov-Zhabotinsky reaction in order to find chemical turbulence. We are studying the reaction in a closed container,where the turbulent states would necessarily be transient. So far we have not found clear signs of turbulence. We see many spiral waves, but they are more immobile than expected. The quenching methods developed by P. Graae Sø rensen and F. Hynne give us a very accurate determination of the parameters, and some of the systems we have looked at should show at least transient turbulence. One problem is that we don't know how to vary the parameters in a systematic way and the second is that we don't know how important the deviations from the complex Ginzburg-Landau equation are. We hope to soon remedy both of these problems. truecm

P. Alstrøm, T. Bohr, P. Graae Sørensen and F. Hynne truecm

Vortex Interactions in 2D Turbulence truecm

We investigate the spectra obtained by looking at simple interacting groups of vortices. We hope in this way to be able to shed light on the origin of the energy spectrum of 2-D turbulence. truecm

T. Bohr, (X. He), (J.J. Rasmussen) and (A. Nielsen)


Multidiffusion of Passive Particles in Strongly Turbulent Flows truecm

We study the diffusion of particles subjected to a fully developed turbulent flow. The particles move completely passively subjected to the velocity field. The velocity field is strongly intermittent which causes large jumps in the diffusive motion and gives rise to multidiffusion where different moments in the distance scale with different exponents in time. We use a shell model to integrate the velocity field. We investigate various methods to transform the velocity field to a real space field. truecm

M.H. Jensen, K. Sneppen, (A. Brandenburg), (A. Vulpiani) and (G. Paladin)

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Next: Fractalscritical phenomena Up: RESEARCH PROJECTS Previous: Chaos experiments

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Mon Mar 6 19:42:06 MET 1995