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| Titel |
Comparing attractor frequency bands with eigen frequencies for a confined rotating fluid |
| VerfasserIn |
I. D. Borcia, U. Harlander, T. Seelig, C. Egbers |
| Konferenz |
EGU General Assembly 2012
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 14 (2012) |
| Datensatznummer |
250068050
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| Zusammenfassung |
| Due to its simplicity, trapezoidal cross section was extensively use for studying wave
attractors mainly for gravity waves [1,2]. For inertial waves a trapezoidal prism tank filled
with a homogenous liquid (water) was investigated [3]. Due to the fact that inertial waves are
essentially 3D, systems with rotational symmetry are adequate for the study of the inertial
waves. The rotating spherical shell case [4] however, gives quite complicate attractor
solutions. In order to combine the rotational symmetry and the trapezoidal cross section, the
first geometry that occurs to us is a system consisting on a liquid confined between an outer
vertical cylinder and a co-rotating coaxial inner cone. In the axial direction the
liquid is bounded by horizontal plates. The cylinder and the cone are subject to
the same average angular velocity. The inertial waves are forced by modulating
the rotation rate of the inner cone. At small viscosities (or high angular velocity)
inertial waves show multiple reflections at the walls before the waves are damped by
diffusion.
From the mathematical point of view, the description of the waves that collapse to an
attractor is given by a hyperbolic PDE. Generally one deals with ill-posed boundary value
problems in the meaning that hyperbolic partial differential equations are combined with
elliptic boundary conditions. However, for such problems one can find eigenmode solutions
only for very limited number of domains. One of these domains is obtained for non-inclined
walls of the inner cone. When the inner cylinder wall is inclined, the hyperbolicity leads to
internal shear layers corresponding to singularities for the inviscid case (attractor solutions).
The geometrical structure of the shear layers can be explained by inertial waves, trapped on
limit cycles. The shape of the limit cycle defines the structure of the internal shear layers
[1,5].
In fact, the spectrum of regular modes, existing for the case of vertical cylinder
walls, vanishes when the inner wall is inclined. We will show that the ’spectrum’ of
limit cycles for weakly inclined walls still captures some part of the eigenvalue
spectrum of trajectories for the vertical walls. To answer this question we compute
the attractor frequency ’spectrum’ for a small cylinder inclination angle and we
compare it with the eigen spectrum for the case of vertical cylinder walls. If m is
the number of reflection points of the attractor solution on the radial axis and n
the number of the reflection points on the vertical axis one can observe that only
(m-even, n-odd) and (m-odd, n-odd) attractors are obtained. It appears that the
odd-even solutions are the eigenmodes surviving from the vertical wall problem.
The intervals where attractors can be found collapse to distinct single frequencies
when the wall inclination goes to zero. These values are the same as the eigenmode
frequencies for small curvature (when cylinder radius is much larger than the cylindrical
gap).
References
[1] L.R.M. Maas and F.-P.A., J. Fluid Mech. 300, 1-41,1995.
[2] U. Harlander, L.R.M. Maas, J. Fluid Mech., 588, 331-351, 2007.
[3] L.R.M. Maas, J. Fluid Mech., 437, 13-28, 2001.
[4] M. Rieutord B. Georgeot, L. Valdettaro, J. Fluid Mech. 435, 103-144, 2001.
[5] U. Harlander, L.R.M. Maas, Meteorol. Z. 15(4), 439-450, 2006. |
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