Abstracts - KTH Mechanics

Waves above turbulence
R. Savelsberg ∗ , G.J.F. van Heijst ∗ ,andWillem van de Water ∗
Surprisingly little is known about the statistical nature of the shape of a free
surface above turbulence and how these statistics are connected to those of the subsurface
turbulence itself. Naively one would expect surface wrinkles to be primarily
associated with low pressure in the cores of vortices attached to the surface.
We study this in a free surface water channel in which turbulence is generated by
means of an active grid. The grid produces turbulence with a Taylor-based Reynolds
number up to Reλ = 250. The surface slope is measured in space and time by means
of a novel technique, based on measuring the deflection of a laser beam that is swept
along a line on the surface. By combining this with simultaneous Particle Image
Velocimetry measurements of the sub-surface velocity field, we find that part of the
surface shape is indeed correlated with large sub-surface structures. This is also clear
from spectra of the surface slope in space and time. Such a spectrum, measured
along a streamwise line, is shown in figure 1 (a). The structures directly connected
to the turbulence are represented by a branch in this spectrum that corresponds to
the mean-stream velocity. However, the same spectrum also shows the presence of
gravity-capillary waves. Far more energy is present in a branch that corresponds to
the dispersion relation for such waves, which in this streamwise spectrum is Dopplershifted
due to the mean stream velocity. The corresponding spectrum in the spanwise
direction, shown in figure 1 (b), also shows the presence of gravity capillary waves.
These waves are radiated from the large scale structures attached to the turbulence
and travel in all directions across the surface. As a consequence the wave-number
spectrum of the surface slope is far steeper than the spectrum of the sub-surface
turbulence. Furthermore, the anisotropy of the surface shape is directly connected to
the anisotropy of the sub-surface turbulence.
ω/(2π) (Hz)
∗ Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
200
150
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0
ky /(2π) (m -1 -300 -200 -100 0 100 200 300
)
(a)
1e-06
1e-07
1e-08
1e-09
1e-10
ω/(2π) (Hz)
kx /(2π) (m -1 0
0 50 100 150 200 250 300 350
)
Figure 1: (a) Wavenumber-frequency spectrum of the surface slope along a streamwise
line and (b) the corresponding spectrum in the spanwise direction. The solid lines
correspond to the dispersion relation for gravity-capillary waves. The dotted line
corresponds to the mean-stream velocity in the water-channel.
200
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(b)
1e-06
1e-07
1e-08
1e-09
1e-10
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