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Simulation of Flows with Free Surfaces truecm

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. truecm

L. Kjeldgaard and T. Bohr truecm

Separation in Free Surface Flows truecm

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. truecm

T. Bohr, V. Putkaradze, S. Watanabe truecm

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. truecm

K. Andersen, T. Bohr and (J. Fredsøe) truecm

Particle Motion under Linear Surface Waves truecm

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. truecm

T. Bohr and J.L. Hansen. truecm

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

Computer Simulation of the Formation of "Viscous Trees" truecm

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). truecm

T. Bohr and (M. Ernebjerg) truecm

Vortex Merging and Spectral Cascade in Two-Dimensional Flows truecm

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. truecm

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

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 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. truecm

M.H. Jensen, A. Brandenburg, J. Bundgaard, (A. Vulpiani) and (G. Paladin) truecm

Intermittent Activity, Multiscaling and Self-Organized Criticality truecm

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. truecm

M.H. Jensen, K. Sneppen and M. Sellitto truecm

Dynamics of Granular Material - Dune, Ripple and Vibrating Bed truecm

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. truecm

H. Nishimori and (M. Yamasaki) truecm

Experimental and Theoretical Studies of Turbulence in Pipe Flow truecm

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. truecm

P. Alstrøm, J. Borg, C. Ghidaglia, M. S. Johansen, M. T. Levinsen, M.-B. Nielsen, T. Rasmussen and E. Schröder truecm

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

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. truecm

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

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. truecm

S. Watanabe, (H.S.J. van der Zant), (S.H. Strogatz) and (T.P. Orlando)

next up previous contents
Next: Fractalscritical phenomena Up: RESEARCH PROJECTS Previous: Chaos experiments

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Fri Mar 29 00:13:44 MET 1996