Willem van de Water

Mappings of grazing-impact oscillators.

Impacting systems are found in a great variety of mechanical constructions and they are intrinsically nonlinear. In this paper it is shown how near-grazing systems, i.e. systems in which the impacts take place at low speed, can be described by discrete mappings. The derivation of this mapping for a harmonic oscillator with a stop is dealt with in detail. It is found that the resulting mapping for rigid obstacles is somewhat different from those presented earlier in the literature. The derivations are extended to systems with a compliant obstacle. We find that the map for impacts with a compliant obstacle is very similar to the one describing collisions with a rigid obstacle. A notable difference is a change of scale of the bifurcation parameter. We illustrate our findings in the limit of large damping, where the mechanism of period adding can be analysed exactly. The relevance of our results to experiments on practical impact systems is indicated.

Dynamics of vibrating atomic force microscopy

An atomic force microscope which is operated in the oscillating mode is an example of an impact oscillator. The description of such dynamical systems can be reduced to a mathematical mapping, which displays a square-root singularity. A direct consequence of this property is the emergence of an infinite series of period-adding bifurcations. This extremely characteristic phenomenon should be observed in atomic force microscopes. We consider an atomic force microscope in which the tip-substrate forces are modelled by a liquid-bridge interaction. By integrating the dynamical equations we show that the atomic force microscopy (AFM) dynamical behaviour has the same characteristic bifurcation scenario as the square-root map. We point to the remarkable role of the energy that is dissipated upon impact. We finally suggest ways to improve the operation of AFM.

Patterns of Faraday waves

Faraday waves are standing waves which arise through a parametric instability on the surface of a vertically oscillated fluid layer. They can emerge with various symmetries, simply square to

Spontaneous Rayleigh-Brillouin scattering of ultraviolet light in nitrogen, dry air and moist air,

N

Experiments on free-surface turbulence

-fold rotationally symmetric, which for

Turbulent parametric surface waves

N > 3

Rayleigh–Brillouin scattering profiles of air at different temperatures and pressures

are quasi-crystalline. In an experiment with a very large aspect ratio we determine the boundaries of the stability regions of waves with different rotational symmetries in the driving frequency–amplitude parameter plane. We find a remarkable agreement with a recent theory by Chen & Vinals (1999) who predict the stability boundaries at the onset amplitude. We argue why such agreement can only be observed in a very large experiment. The main nonlinear damping mechanism of the waves is a three-wave resonance. We devise a simple model that captures this mechanism and that can explain quantitatively the change of the symmetry of the waves with fluid depth. Detailed information about the surface is obtained by scanning the wave field and measuring the phase of subharmonic and harmonic components. Also the results of these measurements compare very favourably to the theoretical predictions.

Measuring droplet size distributions from overlapping interferometric particle images

Atmospheric lidar techniques for the measurement of wind, temperature, and optical properties of aerosols rely on the exact knowledge of the spectral line shape of the scattered laser light on molecules. We report on spontaneous Rayleigh-Brillouin scattering measurements in the ultraviolet at a scattering angle of 90 degrees on N(2) and on dry and moist air. The measured line shapes are compared to the Tenti S6 model, which is shown to describe the scattering line shapes in air at atmospheric pressures with small but significant deviations. We demonstrate that the line profiles of N(2) and air under equal pressure and temperature conditions differ significantly, and that this difference can be described by the S6 model. Moreover, we show that even a high water vapor content in air up to a volume fraction of 3.6vol.% has no influence on the line shape of the scattered light. The results are of relevance for the future spaceborne lidars on ADM-Aeolus (Atmospheric Dynamics Mission) and EarthCARE (Earth Clouds, Aerosols, and Radiation Explorer).

Anomalous scaling and anisotropy in turbulence

Surprisingly little is known about the statistical nature of the shape of a free surface above turbulence and about how this shape depends on the properties of the turbulence. The main focus of this thesis is on experiments in which the statistical properties of both the surface and the turbulence are measured with a number of different techniques. The experiments are done in a free-surface water-channel, in which turbulence is generated with an active grid. This active grid consists of an array of horizontal and vertical rods through the channel, with small wings attached to them. The rods are individually driven by electric motors, according to a certain forcing protocol, thereby adding energy to the turbulence. A major advantage of an active grid is that, by changing this protocol, the properties, such as the intensity and the isotropy, of the generated turbulence can be changed. These properties were measured by means of Laser-Doppler Velocimetry. The turbulence behind the active grid is much more intense than turbulence generated by a more common static grid. The maximum Taylorbased Reynolds number reached with the active grid (at 40 times the mesh size behind the grid) was Re?= 256, compared to Re? = 70 with a similarly dimensioned static grid. Consequently, the active-grid-generated turbulence shows clear Kolmogorov scaling behaviour over a relatively wide range of scales. The stronger turbulence also leads to stronger surface deformations. In order to characterise the shape of the surface, it is essential to measure the surface shape with a high resolution both in space and time. In order to achieve this, a novel technique has been developed, based on refraction of a laser beam that shines through the surface. The de ection of the beam due to the local surface slope is measured by means of an optical position sensing device. The beam is swept along a line by means of a rapidly oscillating mirror (with a frequency of close to 2 kHz). This allows measurements of the surface slope at multiple points along the line as a function of time. This surface scanning technique can be combined with Particle Image Velocimetry (PIV), which provides snapshots of the velocity field and the vertical component of vorticity in horizontal planes just below the surface. This combination allows us to simultaneously measure the velocity field and the surface deformations above it. PIV is based on the cross-correlation of the intensity distributions in images of particles suspended in the ow, that are illuminated by a thin laser light sheet. When applying PIV to turbulence, it is important to realise that the velocity field that can be obtained with PIV is a spatially averaged representation of the actual velocity field. The effect this averaging has on measured turbulence properties is investigated by means of kinematic simulations, in which realistic turbulent velocity fields, with a prescribed energy spectrum, are generated. Synthetic particle images derived from these ??elds are evaluated by means of a PIV algorithm and the velocity spectrum is calculated. Comparing this to the prescribed spectrum clearly shows the averaging, and allows us to predict its in uence on other measured turbulence properties. The turbulence generated by our grid is not strong enough to lead to very large deformations of the surface. The measured changes in elevation are less than 1 mm. In that case, somewhat naively, one would expect the surface deformations to be primarily associated with sub-surface vortices. In the core of a vortex the magnitude of the vorticity is high, while the pressure is low. This low pressure causes a dimple in the surface above the vortex. This effect can, for instance, be seen when stirring a cup of tea or in the wake behind bridge pillars in a river. Consequently, in simultaneous measurements of the surface shape and the sub-surface velocity field one would expect to find a relatively large correlation between the vertical component of vorticity and the surface elevation. Indeed, our measurements show that relatively strong vortices in the turbulence do deform the surface. However, the measured correlation coefficients are low (<0:1). Spectra of the surface slope in space and time show that, instead of being connected directly to sub-surface structures, much of the surface actually consists of gravity-capillary waves, i.e. regular surface waves. For surface waves, there is a clear relation between their wavelength and their frequency. This relation can be identified in our spectra. The presence of these waves is somewhat surprising, since resonant wave growth can only be expected to occur if the uctuation velocities in the turbulence are larger than the minimum phase velocity of the waves (˜ 0:23 m/s), while the measured fluctuation velocities in our turbulence are an order of magnitude smaller. A remarkable feature of the waves above the turbulence is that they travel in all directions across the surface. In fact, provided that the turbulence far below the surface is isotropic, the surface shape itself is isotropic as well. In other words, statistically, the waves on the surface are the same in every direction. We can change this by changing the forcing protocol of the active grid such that the turbulence becomes anisotropic. In that case the surface shape becomes anisotropic as well. This is a clear indication that the surface waves are excited locally by the turbulence. We have found evidence to suggest that the waves are excited by the largest structures in the turbulence. As a consequence of this, the surface shape does not re ect the wide range of scales in the sub-surface turbulence, but instead exhibits waves primarily with wavelenghts close to the integral scale of the turbulence.

Charged colloidal systems

We study disordered capillary waves on the surface of a vertically oscillated fluid layer and try to establish their relation with weak wave turbulence. We measure the surface gradient in space and time and argue that gradient spectra are better suited for comparison to the predictions of weak wave turbulence theory than spectra of the surface elevation. Because the gradient is a vector quantity, we must distinguish longitudinal and transverse spectra. However, they prove to be related trivially through isotropy. From the measured wavenumber-frequency spectrum it appears that the dispersion relation is only satisfied approximately. In the wavenumber direction the spectral features are strongly broadened due to spatial disorder. This disagrees with weak wave turbulence theory where exact satisfaction of the dispersion relation is pivotal. We find approximate algebraic frequency and wavenumber spectra but with exponents that are different from those predicted by weak wave turbulence theory. However, other f...