FG Aerodynamik und Strömungslehre
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Time periodic variation of the rotation rate of an annulus induces in supercritical regime an unstable Stokes boundary layer over the cylinder side walls, generating
Görtler vortices in a portion of a libration cycle as a discrete event. Numerical results show that these vortices propagate into the fluid bulk and generate an azimuthal
mean flow. Direct numerical simulations of the fluid flow in an annular container with librating outer (inner) cylinder side wall and Reynolds-averaged Navier–Stokes
(RANS) equations as diagnostic equations are used to investigate generation mechanism of the retrograde (prograde) azimuthal mean flow in the bulk. First, we explain, phenomenologically, how absolute angular momentum of the bulk flow is mixed and changed due to the propagation of the Görtler vortices, causing a new vortex
of basin size. Then we investigate the RANS equations for intermediate time scale of the development of the Görtler vortices and for long time scale of the order of
several libration periods. The former exhibits sign selection of the azimuthal mean flow. Investigating the latter, we predict that the azimuthal mean flow is proportional to the libration amplitude squared and to the inverse square root of the Ekman number and libration frequency and then confirms this using the numerical data.
Additionally, presence of an upscale cascade of energy is shown, using the kinetic energy budget of fluctuating flow.
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.
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.