The flow produced in an enclosed cylinder of height-to-radius ratio of two by the counter-rotation of the top and bottom disks is numerically investigated. When the Reynolds number based on cylinder radius and disk rotation is increased, the axisymmetric basic state loses stability and different complex flows appear successively: steady states with an azimuthal wavenumber of 1; travelling waves; near-heteroclinic cycles; and steady states with an azimuthal wavenumber of 2. This scenario is understood in a dynamical system context as being due to a nearly codimension-two bifurcation in the presence of $O(2)$ symmetry. A bifurcation diagram is determined, together with the most dangerous eigenvalues as functions of the Reynolds number. Two distinct types of near-heteroclinic cycles are observed, with either two or four bursts per cycle. The physical mechanism for the primary instability could be the Kelvin–Helmholtz instability of the equatorial azimuthal shear layer of the basic state.

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The three-dimensional linear instability of axisymmetric flow between exactly counterrotating disks is studied numerically. The dynamics are governed by two parameters, the Reynolds number Re based on cylinder radius and disk rotation speed and the height-to-radius ratio Γ . The stability analysis performed for 0.5 6 Γ 6 3 shows that non-axisymmetric modes are dominant and stationary and that the critical azimuthal wavenumber is a decreasing function of Γ . The patterns of the dominant perturbations are analysed and a physical mechanism related to a shear layer instability is discussed. No evidence of complex dynamical behaviour is seen in the neighbourhood of the 1 : 2 codimension-two point when the m = 2 threshold precedes that of m = 1. Axisymmetric instabilities are also calculated; these may be stationary or Hopf bifurcations. Their thresholds are always higher than those of non-axisymmetric modes.

The starting flows past a two-dimensional NACA 0012 airfoil translating and oscillating at large incidences are investigated by visualization experiments and numerical calculations. The airfoil model is set in motion impulsively and subjected simultaneously to a constant translation and harmonic oscillation in pitch. The evolution of the vortex wake is followed in a sequence of streamline visualizations and the wake pattern generated is analysed. The parameters varied in the visualization experiment are the Reynolds number ranging from 1500 to 10000, the reduced frequency from 0.1 to 1.0, the mean incidence 30° or 15° and the angular amplitude 15° or 7°. There are also two additional parameters of special interest: the airfoil cross-section and the pitching axis. The effects of these parameters are discussed in relation to the resultant wake patterns. Some comparison is made with the results of earlier experiments.

SUMMARYThis paper presents a general methodology for studying instabilities of natural convection flows enclosed in cavities of complex geometry. Different tools have been developed, consisting of time integration of the unsteady equations, steady state solving, and computation of the most unstable eigenmodes of the Jacobian and its adjoint. The methodology is validated in the classical differentially heated cavity, where the steady solution branch is followed for vary large values of the Rayleigh number and most unstable eigenmodes are computed at selected Rayleigh values. Its effectiveness for complex geometries is illustrated on a configuration consisting of a cavity with internal heated partitions. We finally propose to reduce the Navier-Stokes equations to a differential system by expanding the unsteady solution as the sum of the steady state solution and of a linear combination of the leading eigenmodes. The principle of the method is exposed and preliminary results are presented.