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## Turbulence

Theory of Separation of Boundary Layers and Hydraulic Jumps

Separation of boundary layers is an important classical problem, with applications to non-linear waves and hydraulic jumps. The standard approach to boundary layers, i. e. Prandtl's boundary layer equations, becomes singular at a separation point, which is a point at which the velocity along the bottom changes sign. We are currently investigating various methods to circumvent these singularities and predict various laminar flow structures including separation. The circular hydraulic jump occurs when a vertical jet impinges on a horizontal surface. As the fluid spreads out from the jetit decelerates, but it does so in a very non-uniform way. At a certain radius the flow suddenly slows down, with a corresponding increase in height. We have shown that such flows can be described by averaging methods (as introduced by von Karman and Pohlhausen). When such methods are applied to the boundary layer equations, the singularity at separation dissappears and we can account very accurately for the flow structure. we are currently trying to generalize these methods to include more complicated forms of the circular hydraulic jump seen experimentally (see the experimental project on this issue.)

Another situation, which to which we have applied the averaging techniques, is the free-surface flow down an inclined plane. Here hydraulic jumps also occur, either moving or stationary, and we can calculate flow structures including separation points. Finally, we study the flow in two-dimenssional closed channels with sudden expansion. This is another classical example, where separation is known to be very important.

T. Bohr, V. Putkaradze, S. Watanabe, K. Haste Andersen and L. Kjeldgaard

Simulation of Flows with Free Surfaces

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.

L. Kjeldgaard and T. Bohr

Particle Advection in Turbulent Field Theories

An important problem in the study of spatiotemporal chaos is how to design good diagnostic methods, which can be used also in experimental situations, for quantifying the nature and strength of the chaotic fluctuations.We have introduced analogs of ``Lagrangian" variables by studying advection of fictitious ``particles" in generalizations of the Kuramoto-Sivashinsky equation in 1 and 2 dimensions, such that the velocity of the particle is given by the gradient of the field. It has been found that the diffusive motion, which such particles undergo, is anomalous in 1 dimension. Also, in 1 dimension, particles always coalsce, which means that the Lagrangian Lyapunov exponent is negative. In two dimensions one can generate Hamiltonian flows by relating the velocity to the gradient of the field rotated by 90 degrees. By linear combinations of the gradient flow and the rotated one we obtain dissipative flows and if one Lagrangian Lyapunov exponent is positive, the particles lie on a strange attractor.

If the field theory (in 1 dimension) contains a linear term in the gradient (which would typically always be present if there is no exact left-right symmetry), this term can be removed by going into a moving frame. This creates an extra ``force" on the particle and we are investigating how this influences the drift velocity for the particle.

T. Bohr, J. Lundbek Hansen and M. van Hecke

Turbulent Spots and Pulses in Generalizations of the Kuramoto-Sivashinsky Equation

The Kuramoto-Sivashinsky equation is perhaps the simplest deterministic nonlinear partial differential equation exhibiting spatio-temporal chaos in 1+1 dimensions, and has been derived in a variety of physical situations. If the initial conditions are given as a small hump in an otherwise flat state, the large scale structure of the growing solution can be predicted by looking at the velocity of the linear instability i.e. the velocity with respect to the lab frame at which the absolute instability turns into a convective instability. This theory breaks down if the equation is generalized with a sufficiently large third order term. This happens because of the existence of pulses (solitary waves) in the solutions which are spontaneously created. These pulses become stable when their velocity is larger than the linear velocity.

The new growth shapes can thus be predicted by using the velocities of pulses instead of the linear velocities. Pulses are created nearly at rest with respect to the lab frame on the right side of the growing spot and are in time accelerated towards the left side.

T. Bohr and J. Lundbek Hansen

Ripple Formation in Sand under Water

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 compare our results to experiments on sand ripples under water done at ISVA (DTU).

K. Andersen, T. Bohr and (J. Fredsøe)

Intermittency in the Dynamics of Passive Scalars

The dynamics of a passive scalar appears to be even more intermittent than the underlying turbulent motion. We study such effects by formulating a shell model for a passive scalar incorporating the correct conservation laws. Calculating higher moments of the structure functions we observe large corrections to the Kolmogorov solution. We are studying the stability properties of this Kolmogorov solution using the shell equations and calculating the corresponding eigenvalues. We also formulate a simplified version of the Kraichnan equation in which the velocity field is assumed to be correlated in scape but have no time correlations. Preliminary results indicate strong corrections to the Kolmogorov scaling but some care must be taken to handle the corresponding stochastic differential equation in a correct way.

F. Okkels, M.H. Jensen and P. Olesen

Turbulent Binary Fluid Mixtures

When binary fluids approaches their critical point the effective diffusivity vanishes leading to the Prandtl number to diverge. At this point one might observe that the mixing time diverges when the fluids are in a turbulent state. This causes a peak in the spectrum of the scalar. We study those effects by formulating the hydrodynamic equations on shells in k-space. In this way we can study the system at very high values of the Reynolds and Prandtl numbers. Applying continuum limits of the equations we obtain analytic solutions in various limits. From the equations we predict that by separating the two fluids initially keeping the fluids at rest, a spontaneous motion should emerge due to an "active" coupling in the velocity equation. We are in the process of studying this experimentally using binary fluids at quite similar density and with a critical temperature close to room temperature.

M.H. Jensen, P. Olesen and C. Ellegaard

Multidiffusion of Passive Particles in Strongly Turbulent Flows

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.

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

Perturbation Theory of Parametrically Driven Capillary Waves

We have worked out the Hamiltonian theory for parametrically driven capillary waves. The theory neglects boundary conditions and is valid to first order in viscosity. Our theory has been compared to previous Hamiltonian and Lagrangian theories, pointing out their successes and failures. From the theory, the stability of standing wave patterns has been derived. The theory shows that the square pattern has a subcriticality, and that the eightfold quasicrystalline pattern observed experimentally is more stable than both roles and the hexagonal pattern.

P. Alstrøm and (P. Lyngs Hansen)

Particle Motion in Capillary Waves

We have followed the motion of particles floating on a water surface in a cylindrical dish which was oscillated vertically so as to create capillary waves. Both the relative motion for pair of particles and the single-particle motion was recorded at various values of the driving amplitude.

For the single-particle motion, a cross-over in the diffusion motion is observed, from a strongly anomalous diffusion at length scales below the wave length to a motion being closer to Brownian at larger length scales. Surprisingly, the diffusion curves obtained for different driving amplitudes can be data-collapsed into one universal scaling function. A similar universal behavior is found in upper-ocean studies. Also the fractal dimension of the particle trajectories has been determined and related to the diffusivity results.

For the relative particle motion, the non-Brownian character of the flow is very pronounced. We have determined the relative diffusivity and studied its dependence on distance. The results differs from those obtained for Brownian motion and for strong turbulence, but are consistent with the theory of weak turbulence and results obtained for drifters in the upper ocean.

P. Alstrøm, J. Sparre Andersen, (W. I. Goldburg), A. Espe Hansen, M. T. Levinsen and E. Schröder

We have extended the dynamic cascade model of turbulence proposed by C. Bech 1994, in order to make the theoretical scale dependence of the velocity difference distribution in agreement with that observed experimentally. We have shown that the crucial parameter is the momentum loss from one level to the next, and how it depends on the scale.

P. Alstrøm, M.-B. Brun Nielsen, E. Schröder and T. Rasmussen

Gaussian and Lorentzian Distributions for Turbulence

A product of a Gaussian and a Lorentzian distribution has been used to fit the velocity difference distributions obtained at intermediate Reynolds numbers. We have carried out an analytical study of the n'th moments of the velocity difference distribution in the range where the Gaussian-Lorentzian product fits well. In this range all scaling exponents only depend on the scaling exponents for the second and the third moment. We have derived the exact relationship between these exponents. In the Gaussian limit, this relationship is shown to be equal the parabolic relationship, proposed by Kolmogorov and Obukhov in 1962. The relative weight of the Gaussian and the Lorentzian distribution plotted versus the third moment of the product distribution is a parameter-free plot, and can be compared with experimental results obtained from turbulent pipe flows.

P. Alstrøm, J. Borg, M. Stoklund Johansen and M.T. Levinsen

Extended Self-Similarity at Intermediate Reynolds Numbers

Turbulence is produced in a pipe flow behind a grid, and the velocity fluctuations are measured using laser Doppler anemometry. The distributions and their moments of these fluctuations are determined at various temporal scales and Reynolds number. At intermediate Reynolds numbers the moments do not follow a power law in scale, but they still follow a power law when one moment is plotted versus another. We investigate how the exponents obtained this way depend on the Reynolds number.

P. Alstrøm, M. Stoklund Johansen and M.T. Levinsen

Bifurcations and Chaos in One-Dimensional Map Lattices

In order to better understand the various types of dynamics that can arise in systems of many coupled self-sustained oscillators and in unstable spatially extended systems, we have performed a qualitative study of one-dimensional coupled map lattices with arbitrary nonlinearity of the local map and with space-shift as well as with diffusion coupling. The conditions for overall synchronization of the map lattice were derived, and the bifurcation arising under the transition from spatiotemporal chaos to chaotic synchronization were discussed. Finally, we analyzed some of the peculiarities of coupled map lattices with specific symmetries.

(V. Belykh) and E. Mosekilde

Pinning of Fronts between Hexagons and Squares

One of the topics that deserves attention in the study of pattern formation in nonequilibrium systems is the dynamics of defects and of fronts separating domains of different symmetry. Stationary fronts have been observed in the transition between rolls and hexagonal patterns in convection under non-Boussinesq conditions as well as in chemical reaction-diffusion systems. We have studied pinning effects for domain walls separating regions with hexagonal and squared patterns through numerical simulations of a generalized Swift-Hohenberg model. The conditions for pinning effects and stable fronts have been determined. The results agree qualitatively with recent observations in convection and in ferrofluid instabilities.

C. Kubstrup, (H. Herrero and C. Pérez-Garcia)

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

Klaus Lindemann
Fri Feb 21 15:17:28 MET 1997