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Simulation of Flows with Free Surfaces
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We are developing a scheme for numerical simulation of the Navier- Stokes equations with a free surface. We are trying two approaches:

1. A surface marker method in which the surface is followed
dynamically by a set of "markers" - fictitious fluid particles in the
fluid.
2. Method of successive approximations for the shape of the surface
for a stationary flow. Here we try, for a given surface shape, to
fulfill all boundary conditions but one. The the surface shape is
relaxed to improve the last condition.
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L. Kjeldgaard and T. Bohr
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Separation in Free Surface Flows
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We study generic properties of hydrodynamical flows with a free
surface undergoing *separation* - such as hydraulic jumps. At a
separation point the *skinfriction* changes sign, since backflow is
generated very close to the bottom. Separation in boundary layers
inside a fluid was studied in a famous paper by L. Prandtl in 1904.
There he derived the so-called boundary layer approximation for the
Navier-Stokes equations. They describe the boundary layer well as
long as no separation occurs. At the separation point they break
down and the vertical velocity diverges.

In the circular hydraulic jump a free surface abruptly changes its height. It can be seen experimentally that, although the flow can be fully laminar, separation, in the form of one or several localized regions of backflow, occur. We try to model the velocity profile as a low order polynomium and to write down a truncated set of equations for the various coefficients. We would like to

1. Find a theory of separation without singularities.
2. Find how the size of the separated region scales with viscosity and
flux.
3. Explain the various types of hydraulic jumps found when the
external
height of the fluid layer is varied.
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T. Bohr, V. Putkaradze, S. Watanabe
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Ripple Formation in Sand under Water
truecm**

We study various models for the formation of sand ripples under
water with surface waves. We are interested in the laminar case
where predictions for the characteristic lenght of the ripples exist
and depend crucially on the separation occurring near the maxima of
the ripples. For practical application the tubulence in the flow must
be modeled, and this is done numerically. We shall compare our
results to the forthcoming series of experiments on sand ripples
under water by C. Ellegaard.
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K. Andersen, T. Bohr and (J. Fredsøe)
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Particle Motion under Linear Surface Waves
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It is well known that the motion of fluid particles can be chaotic even
when the flow is laminar. The particle motion under a single
harmonic surface wave is known to be almost elliptical, except for a
slow "Stokes drift" coming from the slight variation of the velocity
with the height of the particle. For a linear superposition of two or
more harmonic surface waves the motion is not known, but could in
principle be chaotic. We show that, indeed, even for two surface
waves in resonnance, the particle motion can be chaotic.
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T. Bohr and J.L. Hansen.
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Finite Time Singularities in Partial Differential Equations
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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.
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V. Putkaradze, T. Bohr, (J. Krug) and (M. Bazhenov)
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Computer Simulation of the Formation of "Viscous Trees"
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The name "viscous trees" has been used recently for the tree-like
patterns, which occur in a thin layer of a viscous fluid sandwiched
between two plates, when the plates are slowly separated. When two
plates are squeezed together the trapped fluid forms a circular spot,
but when they are separated, which requires considerable force for a
thin layer, the circular shape is unstable and leads to the formation
of complicated tree-like structures. We have generated a simple
numerical method to simulate such structures in which we follow the
evolution of the interface. The instability is driven by the lowering
of the pressure in the center which we model by a radially dependent
pressure gradient. The motion in a given branch is stopped when the
width becomes too small (of the order of the distance between the
plates).
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T. Bohr and (M. Ernebjerg)
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Vortex Merging and Spectral Cascade in Two-Dimensional Flows
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Merging of two identical vortex patches is studied using a spectral
code. It is identified that the enstrophy cascade is most active on the
distorted vortex boundaries, with a Kolmogorov-like spectrum , , developed at high
wavenumbers. The inverse energy cascade is completed when the
vortices merge into one of larger size.
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T. Bohr, (A.H. Nielsen), (J.J. Rasmussen) and (X. He)
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Multidiffusion of Passive Particles in Strongly Turbulent Flows
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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 generated by
converting results from shell models to real space. Both compressible
and incompressible field are used. The diffusive behavior is
surprisingly found to follow the Richardson law, with intermittency
corrections in the incompressible case. We plan to compare these
results with direct simulations on the Navier-Stokes equations.
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M.H. Jensen, A. Brandenburg, J. Bundgaard, (A. Vulpiani) and (G. Paladin)
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Intermittent Activity, Multiscaling and Self-Organized Criticality
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We characterize the spatio-temporal behavior of self-organized
critical phenomena by means of an activity function. The activity
function is equivalent to a roughening front and shows intermittent
behavior in space and time. Due to the intermittency, the moments of
the activity function exhibits multiscaling in time with a continuous
spectrum of exponents. We try to compare these results with Levy
flights models where some analytic calculations can be performed.
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M.H. Jensen, K. Sneppen and M. Sellitto
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Dynamics of Granular Material - Dune, Ripple and Vibrating Bed
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We study various phenomena in spatially extended dissipative systems. Especially, i)the formation and the evolution process of sand ripples and dunes, and ii)the dynamics of vibrating granular bed, are investigated. These systems are made up of granular material which has macroscopically discrete elements. Thus,in the systems, very complex behaviors are rather easily realized than in usual continuum systems.

Now we are planning;
i)to pick up various behaviors of the above systems which characterize
them, e.g.solitary wave of dune, strange type of capillary action in
the vibrating bed, using simulation methods and experimental methods,
ii)to construct the general method to describe such discrete systems
theoretically.
Moreover, the above investigation will be extended
to treat a wider range of phenomena which consist of
discrete elements, e.g.traffic systems, the collective behavior
of animals.
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H. Nishimori and (M. Yamasaki)
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Experimental and Theoretical Studies of Turbulence in Pipe Flow
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We investigate, theoretically as well as experimentally, turbulence produced in a pipe flow behind a grid or an obstacle. The velocity fluctuations are measured using laser Doppler anemometry, and distributions and their moments of these fluctuations are determined at different temporal scales.

Theoretically, we have developed a new type of cascade model. This is derived from Navier-Stokes equations, or more precisely from the Kolmogorov equation for the energy dissipation that follows from Navier-Stokes equations. The derivation relies on assumptions that experimentally are easy to access. Experiments (from other groups) show that the assumptions are valid in the inertial range. Experimentally, we investigate the validity of the assumptions in greater detail. Theoretically, we study the scaling properties of the cascade model and its variants. The cascade model considered is very different from traditional cascade models (like the -model), which rely on a fractal cascade picture with `eddies' as elements, and with no relation to Navier-Stokes equations. In our cascade picture the basic quantities are the energy flows in and out of a volume in the moving frame around a fluid particle. Also, our cascade model is dynamical, not static.

In a related project, we consider the use of a product of a
Gaussian and a Lorentzian distribution as a fit to the
distribution of velocity differences over a given scale.
It turns out that the form of the fit is determined by the
ratio of the characteristic velocities for the Gaussian and
the Lorentzian distributions. Experimentally, we investigate
how this ratio changes with scale. Theoretically, we are
able to relate this behavior to the scaling structure
described by the moments. Moreover, in the Gaussian-Lorentzian
description, the multifractal scaling behavior (exponents) for
the moments can be derived from the second and the third moments.
Our studies of the Gaussian-Lorentzian description goes beyond
the inertial range, and include the energy-injection scales,
where the distribution of velocity differences becomes more
Gaussian like.
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P. Alstrøm, J. Borg, C. Ghidaglia, M. S. Johansen, M. T. Levinsen,
M.-B. Nielsen, T. Rasmussen and E. Schröder
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Experimental and Theoretical Studies of Pattern Formation and
Turbulence in Capillary Waves
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Eight years after quasicrystals were observed in solid-state physics in 1984, we observed the first quasicrystal pattern in a fluid. The quasicrystalline 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. We have worked out a third order perturbation theory which describe the system right above the onset of surface waves, under the assumption that the system is spatially extended (boundary conditions not important), and that the viscosity is low (as for water and alcohol). This shows that the square pattern is subcritical, and that the earlier observed quasicrystalline pattern is the most stable among the supercritical patterns.

At higher drive, the flow becomes turbulent. In this regime we have
carried out measurements of self-diffusion and relative diffusion of
particles placed on the surface.
This is done using image processing and particle tracking programs.
For self-diffusion, we have shown that there is a crossover from
a more ballistic motion at distances less than a wavelength, to
a more diffusive motion at longer distances. Moreover, the
anomalous scaling behavior observed at different forcing amplitudes
can be collapsed into one curve by a forcing-dependent rescaling in time.
For relative diffusion, we have investigated the scaling properties
in the light of weak turbulence theory. Also, we have discussed our
results in terms of the Constantin-Procaccia relation, which relates
the exponent for the velocity difference to the fractal dimension of
isoconcentration contours formed by a passive scalar.
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P. Alstrøm, J. Sparre Andersen, (W. Goldburg), P. Lyngs Hansen,
M. T. Levinsen and E. Schröder
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Row-Switching States in 2D Josephson Junction Arrays
truecm**

We study the dynamics of a 2D Josephson junction array that
can be viewed as a large system of coupled pendula.
In underdamped arrays it was observed experimentally
that the flux-flow regime suddenly jumps to a state called
the row-switched states as the driving current is increased.
In these states, several active rows dominate all the voltage drop
while the other rows are ``silent'' and superconducting.
We attempt to obtain analytical expression of these states.
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S. Watanabe, (H.S.J. van der Zant), (S.H. Strogatz) and (T.P. Orlando)
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Fri Mar 29 00:13:44 MET 1996