A model is proposed with which the statistics of the fluctuating streamwise velocity in the inner region of wall-bounded turbulent flows are predicted from a measured large-scale velocity signature from an outer position in the logarithmic region of the flow. Results, including spectra and all moments up to sixth order, are shown and compared to experimental data for zero-pressure-gradient flows over a large range of Reynolds numbers. The model uses universal time-series and constants that were empirically determined from zero-pressure-gradient boundary layer data. In order to test the applicability of these for other flows, the model is also applied to channel, pipe and adverse-pressure-gradient flows. The results support the concept of a universal inner region that is modified through a modulation and superposition of the large-scale outer motions, which are specific to the geometry or imposed streamwise pressure gradient acting on the flow.

By using the methods of the theory of two- and three-dimensional linear absolute and convective instabilities, we examine the nature of the instability at the onset of convection in a model of convection in an extended horizontal layer of a saturated porous medium with either horizontal or vertical salinity and inclined temperature gradients, and horizontal throughflow. First, normal modes are analysed and the critical values of the vertical thermal Rayleigh number, Rv, wavenumber vector, (k, l) and frequency, ω, are obtained for a variety of values of the horizontal thermal and salinity Rayleigh numbers, Rh and Sh, respectively, the vertical salinity Rayleigh number Sv and the horizontal Péclet number, Qh. In the computations, a high-precision pseudo-spectral Chebyshev-collocation method is used. In most of the cases of parameter combinations considered, the onset of convection occurs through a longitudinal mode. Most of the non-longitudinal critical modes are oscillatory. Further, it is revealed that there exists an absolute/convective instability dichotomy at the onset of three-dimensional convection in a set of the base states given by exact analytic solutions of the equations of motion in the model. This echoes the results of Brevdo (vol. 641, 2009, p. 475) for transverse modes in a model with inclined temperature gradient and vertical throughflow, but with no salinity. The dependence of the dichotomy on the inclined thermal gradient, and on the horizontal and the vertical salinity gradients is investigated, for the longitudinal modes treated both as two-dimensional as well as three-dimensional modes, and for the non-longitudinal modes. For a certain set of parameter cases, it was found that the destabilization through longitudinal modes treated as two-dimensional modes has the character of absolute instability whereas a three-dimensional analysis of these modes revealed that the instability is convective, with the group velocity vector of the emerging unstable wavepacket being parallel to the axis of the convection rolls. Since a similar effect was reported by Brevdo (vol. 641, 2009, p. 475) for a model with no salinity, we conclude that this effect is not a separate case. In most of the cases considered in which a marginally unstable base state is absolutely stable, but convectively unstable, the direction of propagation of the emerging unstable wavepacket is either parallel or perpendicular to the axis of the convection rolls. Only in the absolutely stable, but convectively unstable cases in which non-longitudinal modes are favourable, the angle, ϕ, between the group velocity vector of the unstable wavepacket and the axis of the rolls satisfies 0 < ϕ < 90°.

We present a semi-analytic solution for the non-axisymmetric oscillations of a viscous cylindrical free-standing liquid bridge formed between two coaxial discs of radius R. Even though a streamfunction does not exist, a Helmholtz decomposition is used to obtain an analytic representation of the velocity field. An eigenvalue problem is formulated by projecting the free-surface boundary conditions onto a suitable space of test functions. This is then solved iteratively along with the dispersion relation obtained from the satisfaction of endwall boundary conditions. Extensive comparison with previous theoretical and numerical results, for a range of Reynolds number and bridge slenderness ratio, shows very good agreement in most cases. The present solution generalises that of Tsamopoulos, Chen & Borkar (J. Fluid Mech., vol. 235, 1992, p. 579), which employed a streamfunction formulation and was for the axisymmetric case.

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibres in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibres that might be responsible for the deviation from theory, can at least partially be modelled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data are slightly non-uniform, in that the probability distribution of the orientation has small spatial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibres to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory.