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Institute
BTU
Singular vector growth in different flow regimes of the differentially heated rotating annulus
(2011)
Effect of the Initial Conditions on the Growth of Thermoelectric Instabilities During Parabolic
(2019)
Laboratory experiments with a rotating cylindrical annulus arereported that reveal a prograde jet, which is adjacent to a (longitu-dinally) librating inner straight cylindrical wall. Here, wall libration isrealised as a time-harmonic modulation of the inner cylinder’s rota-tion rate. The outer cylindrical wall and bottom and top lids rotatewith constant angular velocity. The main purpose of our study is tocontribute to a qualitative and quantitative understanding of non-linearities that are present in oscillating, but centrifugally stable,vertical boundary layers frequently encountered in rotating wall-bounded flows. We consider a problem that is in a sense comple-mentary to that of previous works that focused on oscillating Ekmanlayers but neglected the vertical Stokes−Stewartson layers. A sim-ple analytical model is proposed that is able to predict the magni-tude and spatial structure of the emerging prograde near-wall jet interms of nonlinearity inherent in the inner cylinder’s boundary layerdynamics.
Direct numerical simulations (DNS) of inertial wave attractors have been carried out in a librating Taylor-Couette system with broken mirror symmetry in the radial-axial cross-section. The inertial wave excitation mechanism and its localisation at the edges was clarified by applying boundary layer theory. Additional resonance peaks in the simulated response spectra were found to agree with low-order wave attractors obtained by geometric ray tracing. Numerics and theory are in qualitative agreement with recent lab experiments.
Inertial waves and wave attractors in a rotating annulus with inner or outer cylinder libration
(2013)
The mechanism of localized inertial wave excitation and its efficiency is investigated for an annular cavity rotating with Ω0 . Meridional symmetry is broken by replacing
the inner cylinder with a truncated cone (frustum). Waves are excited by individual longitudinal libration of the walls. The geometry is non-separable and exhibits wave
focusing and wave attractors. We investigated laboratory and numerical results for the Ekman number E ≈ 10−6. inclination α = 5.71◦ and libration amplitudes ε 0.2 within the inertial wave band 0 < ω < 2Ω0 . Under the assumption that the inertial waves do not essentially affect the boundary-layer structure, we use classical boundary-layer analysis to study oscillating Ekman layers over a librating wall that is at an angle α = 0 to the axis of rotation. The Ekman layer erupts at frequency ω = f∗, where f∗ ≡ 2Ω0 sin α is the effective Coriolis parameter in a plane tangential to the wall. For
the selected inclination this eruption occurs for the forcing frequency ω/Ω0 = 0.2. For the librating lids eruption occurs at ω/Ω0 = 2. The study reveals that the frequency dependence of the total kinetic energy Kω of the excited wave field is strongly connected to the square of the Ekman pumping velocity wE (ω) that, in the linear limit, becomes singular when the boundary layer erupts. This explains the frequency dependence of non-resonantly excited waves. By the localization of the forcing, the two configurations investigated, (i) frustum libration and (ii) lids together with outer cylinder in libration, can be clearly distinguished by their response spectra. Good agreement was found for the spatial structure of low-order wave attractors and periodic orbits (both characterized by a small number of reflections) in the frequency windows predicted by geometric ray tracing. For ‘resonant’ frequencies a significantly increased total bulk energy was found, while the energy in the boundary layer remained nearly constant. Inertial wave energy enters the bulk flow via corner beams, which are parallel to the characteristics of the underlying Poincaré problem. Numerical simulations revealed a mismatch between the wall-parallel mass fluxes
near the corners. This leads to boundary-layer eruption and the generation of inertial waves in the corners.