The random sweeping decorrelation hypothesis was analysed theoretically and experimentally in terms of the higher-order velocity structure functions $D_{u_i}^{(m)}(r) = \left< [u_i^m(x + r) - u_i^m(x)]^2\right>$. Measurements in two high Reynolds number laboratory shear flows were used: in the return channel (Rλ ≈ 3.2 × 103) and in the mixing layer (Rλ ≈ 2.0 × 103) of a large wind tunnel. Two velocity components (in the direction of the mean flow, u1, and in the direction of the mean shear, u2) were processed for m = 1−4. The effect of using Taylor's hypothesis was estimated by a specially developed method, and found to be insignificant. It was found that all the higher-order structure functions scale, in the inertial subrange, as r2/3. Such a scaling has been argued as supporting evidence for the sweeping hypothesis. However, our experiments also established a strong correlation between energy- and inertial-range excitation. This finding leads to the conclusion that the sweeping decorrelation hypothesis cannot be exactly valid.The hypothesis of statistical independence of large- and small-scale excitation was directly checked with conditionally averaged moments of the velocity difference $\left< [u_i(x + r) - u_i(x)]^l\right>_{u_i^*}, l = 2-4$, at a fixed value of the large-scale parameter u*i. Clear dependence of the conditionally averaged moments on the level of averaging was found. In spite of a strong correlation between the energy-containing and the inertial-scale excitation, universality of the intrinsic structure of the inertial subrange was shown.

In connection with the recent investigations of the instability of unbounded elliptical flows, some methods are discussed for the study of the instability of bounded flows. The stability of a ‘basic flow’ which is two-dimensional and rotating, with elliptical streamlines similar to the elliptical section of an experimentally studied cavity, is investigated in the framework of linear theory (for circular rotation, the flow discussed is stable). The regions of instability for three-dimensional disturbances are found in the plane of the parameters defining the geometry of the system (the height of the ellipsoidal cavity and the degree of ellipticity). It is shown that two types of instability exist, characterized by either monotone or oscillatory growth of the amplitudes of small disturbances.The influence of the Coriolis force field on this instability mechanism is also studied. Rotation of the system as a whole changes the regions of instability in parameter space characterizing the geometry of the cavity and the wavenumbers of unstable disturbances. As a result, the Coriolis force may stabilize or destabilize the basic flow for a given geometry.The instability of rotating density-stratified flow with elliptical streamlines is also considered.

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The stability conditions of the flow between two concentric cylinders with the inner one rotating (circular Couette flow) have been investigated experimentally and theoretically for a fluid with axial, stable linear density stratification. The behaviour of the flow, therefore, depends on the Froude number Fr = Ω/N (where Ω is the angular velocity of the inner cylinder and N is the buoyancy frequency of the fluid) in addition to the Reynolds number and the non-dimensional gap width ε, here equal to 0.275.Experiments show that stratification has a stabilizing effect on the flow with the critical Reynolds number depending on N, in agreement with linear stability theory. The selected, most amplified, vertical wavelength at onset of instability is reduced by the stratification effect and is for the geometry considered only about half the gap width. Furthermore, the observed instability is non-axisymmetric. The resulting vortex motion causes some mixing and this leads to layer formation, clearly visible on shadowgraph images, with the height of the layer being determined by the vertical vortex size. This regime of vertically reduced vortex size is referred to as the S-regime.For larger Reynolds and Froude numbers the role of stratification decreases and the most amplified vertical wavelength is determined by the gap width, giving rise to the usual Taylor vortices (we call this the T-regime). The layers which emerge are determined by these vortices. For relatively small Reynolds number when Fr ≈ 1 the Taylor vortices are stable and the layers have a height h equal to the gap width; for larger Reynolds number or Fr ≈ 2 the Taylor vortices interact in pairs (compacted Taylor vortices, regime CT) and layers of twice the gap width are predominant. Stratification inhibits the azimuthal wavy vortex flow observed in homogeneous fluid. By further increasing the Reynolds number, turbulent motions appear with Taylor vortices superimposed like in non-stratified fluid.The theoretical analysis is based on a linear stability consideration of the axisymmetric problem. This gives bounds of instability in the parameter space (Ω, N) for given vertical and radial wavenumbers. These bounds are in qualitative agreement with experiments. The possibility of oscillatory-type instability (overstability) observed experimentally is also discussed.

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