Stefan Albensoeder

Nonlinear three-dimensional flow in the lid-driven square cavity

The three-dimensional flow in a lid-driven cuboid is investigated numerically. The geometry is an extension to three dimensions of the lid-driven square cavity by translating the two-dimensional lid-driven cavity parallel to the third orthogonal direction. The incompressible Navier-Stokes equations are discretized by a pseudospectral Chebyshev-collocation method. The singularities caused by the discontinuous velocity boundary conditions are reduced by including asymptotic analytical solutions in the solution ansatz. The flow is computed for Reynolds numbers above the critical onset of Taylor-Gortler vortices. Nonlinear Taylor-Gortler cells are calculated for periodic and for realistic no-slip endwall conditions. For periodic boundary conditions the bifurcation is either sub- or supercritical, depending on the wavenumber. The limiting tricritical case arises near the critical wavenumber of the linear-stability problem. On the other hand, no-slip endwall conditions have a significant effect on the supercritical three-dimensional flow. In agreement with recent experimental results we find that Taylor-Gortler vortices are suppressed near no-slip endwalls.

Linear stability of rectangular cavity flows driven by anti-parallel motion of two facing walls

The flow in an infinite slab of rectangular cross-section is investigated numerically by a finite volume method. Two facing walls which move parallel to each other with the same velocity, but in opposite directions, drive a plane flow in the cross-section of the slab. A linear stability analysis shows that the two-dimensional flow becomes unstable to different modes, depending on the cross-sectional aspect ratio, when the Reynolds number is increased. The critical mode is found to be stationary for all aspect ratios. When the separation of the moving walls is larger than approximately twice the height of the cavity, the basic flow forms two vortices, each close to one of the moving walls. The instability of this flow is of centrifugal type and similar to that in the classical lid-driven cavity problem with a single moving wall. When the moving walls are sufficiently close to each other (aspect ratio less than 2) the two vortices merge and form an elliptically strained vortex. Owing to the dipolar strain this flow becomes unstable through the elliptic instability. When both moving walls are very close, the finite-length plane-Couette flow becomes unstable by a similar elliptic mechanism near both turning zones. The critical mode produces wide streaks reaching far into the cavity. For a small range of aspect ratios near unity the flow consists of a single vortex. Here, the strain field is dominated by a four-fold symmetry. As a result the instability process is analogous to the instability of a Rankine vortex in an quadripolar strain field, resulting from vortex stretching into the four corners of the cavity.

Stability of the steady three-dimensional lid-driven flow in a cube and the supercritical flow dynamics

The stability of the steady flow in a lid-driven cube is investigated by a collocation method making use of asymptotic solutions for the singular edges of the cavity up- and downstream of the moving wall. Owing to the rapid convergence of the method high-accuracy critical data are obtained. To determine the critical point subcritical growth rates of small perturbations are extrapolated to zero. We find the bifurcation to be of Hopf-type and slightly subcritical. Above the critical point, the oscillatory flow is symmetric with respect to the symmetric midplane of the cavity and characterized by nearly streamwise vortices in the boundary layer on the wall upstream of the moving wall. The oscillation amplitude grows slowly and seems to saturate. On a long time scale, however, the constant-amplitude oscillations are unstable. The periodic oscillations are interrupted by short bursts during which the oscillation amplitude grows substantially and the spatial structure of the oscillating streamwise vortices changes. Towards the end of each burst the mirror symmetry of the oscillatory flow is lost, the flow returns to the vicinity of the unstable steady state and the growth of symmetric oscillations starts again leading to an intermittent chaotic flow.