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
growth in the presence of surface diffusion, and we find that the type of singularity,
which has been found in numerical simulations, cannot exist.
V. Putkaradze, T. Bohr, (J. Krug) and (M. Bazhenov)
Numerical Simulation of the Boundary Layer Equations with a Free Surface
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.
T. Bohr, T. Petersen and V. Putkaradze
Advection of Particles and Passive Scalars in Nonlinear Field Theories
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.
T. Bohr and (A. Pikovsky)
Growth Shapes of Turbulent Spots in Unstable Systems
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.
T. Bohr and (C. Conrado)
Space-Time Correlations and Lyapunov Spectra in Turbulent Systems
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.
T. Bohr, (G. Grinstein), (C. Jayaprakash), (W.
van de Water) and (E. Bosch)
Experimental and Theoretical Studies of Pattern Formation and
Turbulence in Capillary Waves
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
P. Alstrøm, J. Sparre Andersen, (W. Goldburg), (G. Huber),
M. T. Levinsen and E. Schröder
Phase Turbulence in 2D
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.
E. Schröder, P. Alstrøm and T. Bohr
Bound States of Vortices in the 2D Complex Ginzburg-Landau Equation
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.
T. Bohr, (G. Huber) and (E. Ott)
Experimental and Theoretical Studies of Turbulence in a Soap Film,
or in a Pipe
T. Bohr, (G. Huber) and (E. Ott) truecm
Experimental and Theoretical Studies of Turbulence in a Soap Film, or in a Pipe truecm
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.
P. Alstrøm, C. Ellegaard, C. Ghidaglia, M. T. Levinsen,
M. B. Nielsen, T. Rasmussen, E. Schröder and M. Tibian
Experimental Study of Waves in Thin Film Flows
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.
Experimental Study of the Circular Hydraulic Jump
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.
S. Hansen, S. Hørlück, (D. Zauner), P. Dimon, C. Ellegaard and S.C. Creagh
Instabilities and Transitions in the Circular Hydraulic Jump
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.
C. Ellegaard and T. Bohr
Turbulence in Inhomogeneous Chemical Reactions
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.
P. Alstrøm, T. Bohr, P. Graae Sørensen and F. Hynne
Vortex Interactions in 2D Turbulence
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.
T. Bohr, (X. He), (J.J. Rasmussen) and (A. Nielsen)
Multidiffusion of Passive Particles in Strongly Turbulent Flows
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.
M.H. Jensen, K. Sneppen, (A. Brandenburg), (A. Vulpiani) and (G. Paladin)
M.H. Jensen, K. Sneppen, (A. Brandenburg), (A. Vulpiani) and (G. Paladin)