In this paper we consider the transient evolution of a swirling, recirculating flow in a truncated cylinder. In particular, we consider an initial time period during which the evolution of the flow is controlled by inertia. Such flows exhibit a mutual interaction between the swirl and the poloidal recirculation, whereby any axial gradient in swirl alters the recirculation, which, in turn, redistributes the swirl. This interaction may be visualized as a flexing of the poloidal vortex lines, the best known example of which is the inertial wave. Physical arguments and numerical experiments suggest that, typically, a strong, oscillatory recirculation will develop. We examine the exchange of energy between the swirl and recirculation, and show that the direction of transfer depends on the relative signs of ψ and ηuθ/ηz. In addition, there is a limit to the amount of energy that may be exchanged, since conservation of angular momentum imposes a lower bound on the kinetic energy of the swirl. The characteristic reversal time for the recirculation is estimated by considering the history of fluid particles on the endwalls. Its magnitude depends on the relative strengths of the swirl and recirculation. When the recirculation is large, the reversal time exceeds the turn-over time for a poloidal eddy and, consequently, the vortex lines accumulate at the stagnation points on the endwalls. This leads to accelerated local diffusion on the axis. An elementary one-parameter model is proposed for these nonlinear oscillations. In the limit of very weak recirculation, this model is consistent with the exact solution for inertial waves, while for strong recirculation, it confirms that the reversal time is greater than the turn-over time, and that the vortex lines accumulate on the axis.

It is well known that a system of N point vortices arranged in a circular row, so that the vortices are at the vertices of a regular polygon, is stable if N < 7, neutrally stable if N = 7 and unstable if N > 7 (Havelock 1931). The effect on this result of taking account of the finite size of the vortices is considered analytically. The vortices are considered to be uniform with small but finite core. Approximate equations for the shape and motion of a vortex subjected to an external velocity field are given and used to evaluate the shape and angular velocity of rotation of the system and to study its stability to plane infinitesimal disturbances. It is found that the system is stable if N < 7 and unstable if N ≥ 7. These asymptotic results for small core area are in general consistent with Dritschel (1985) where the motion and stability of up to N = 8 finite vortices is evaluated numerically; the steady configuration and the stability results for these values of N are in agreement except in a region of parameter space where a high degree of accuracy is required in the numerical calculation to resolve the growth rate of small disturbances. The case of a linear array of finite vortices is obtained as a special limiting case of the system. The growth rate of plane infinitesimal disturbances for this case is given.

In order to provide fundamental bases for incorporating the effects of favourable streamwise pressure gradients into models for internal turbulent flows and turbulent boundary layers, time series measurements with a cross-wire probe and a wall shear stress sensor were obtained simultaneously in an oil channel both for fully developed and laterally converging flows. Data were concentrated in the viscous layer and at the centreplane for slight to highly favourable pressure gradients. (Here the viscous layer is defined as the region where viscous effects are significant, but not necessarily dominant; it includes the ‘laminar’ and buffer sublayers in the terminology of some investigators). Results presented include comparisons of the profiles of the mean statistics, plus correlations and spectra, of streamwise and wall-normal components, their product and the wall shear stress. The key new data are the measurements of the fluctuating normal component and related statistics in the viscous layer for highly favourable pressure gradients. For the outer half of the viscous layer its root-mean-square fluctuations decrease as the pressure gradient is increased, consistent with heretofore unconfirmed predictions from direct numerical simulations. Based on examination of the probability density distributions, one may conclude that an effect of a strong pressure gradient is to reduce transport of momentum in the outer part of the viscous layer.

Hunt (1982) and Friedlander (1960a, b) used dimensional analysis to derive expressions for the steady-state particle-size distribution in aerosols and hydrosols. Their results were supported by the Monte Carlo simulation of a non-interacting coagulating population of suspended spherical particles developed by Pearson, Valioulis & List (1984). Here the realism of the Monte Carlo simulation is improved by accounting for the modification to the coagulation rate caused by van der Waals', electrostatic and hydrodynamic forces acting between particles. The results indicate that the major hypothesis underlying the dimensional reasoning, that is, collisions between particles of similar size are most important in determining the shape of the particle size distribution, is valid only for shear-induced coagulation. It is shown that dimensional analysis cannot, in general, be used to predict equilibrium particle-size distributions, mainly because of the strong dependence of the interparticle force on the absolute and relative size of the interacting particles.

This paper is the first of a pair that describe two-point velocity measurements made at various radial positions in water in fully developed pipe flow. Axial velocity fluctuations were measured with hot-film anemometers at two points sufficiently close together that the turbulence structure remained essentially unchanged while passing between them. Phases of the cross-spectra of these velocities were then determined and interpreted in terms of a wave model of the turbulence structure. The model assigns an axial velocity and streamwise inclination to the lines of equal phase of each frequency component of the spectra.

In general, the lines of equal phase for each frequency component are inclined to the wall in the flow direction, the lower frequencies being more inclined than the higher frequencies, though all lines of equal phase at points in the central region of the pipe tend towards the perpendicular. For points near the wall the inclinations are very pronounced.

In the central region, phase velocities of lower frequency components are lower than those for higher frequencies. All phase velocities could be normalized with respect to position by the local mean velocity. The group velocity of small-scale (large wavenumber) disturbances in the core region appears to be approximately constant and of the order of the local mean velocity. This leads to a modified form of Taylor's hypothesis.

The variance in all the measurements increases rapidly in the region y+ < 26. This may be due to the intermittent nature of the flow near the wall (which is discussed in part 2) or to a rotation of the ‘frozen’ pattern by the mean shear field between the two sensors. The magnitude of the latter effect is estimated in this paper and is significant very near the wall. The results in the central region are not affected.

Experiments are presented for three turbulent boundary layers generated by laterally converging, laterally diverging and parallel flow on a flat plate. A converging potential flow field outside the boundary layer was generated by superposing a parallel flow in the x-direction, a row of equally spaced line sources in the wall-normal (y) direction and an analogous row of sinks in the transversal (z) direction. This arrangement resulted in a velocity that was constant far upstream, far downstream and along the x-axis. The convergence – ∂W/∂z has its maximum in the plane of the source and sink rows. This flow field was realized with the test section shown in figure 1, based on streamlines intersecting a rectangular cross-section far upstream. The diverging flow was generated by reversing the flow direction through the test section.

The tests were conducted at about 42 m/s leading to a unit Reynolds number of 2.5 × 106/m and to a Reynolds number based on the momentum thickness of 4000 to 4700 at the inlet of the test sections, increasing up to 25000 at the outlet. In all three cases the velocity distribution near the wall agreed very well with the logarithmic law of the wall. The wake contribution in the outer layer was considerably increased by convergence and decreased by divergence. The Reynolds stresses, measured with crossed hot-wire probes, and the wall shear stress, measured with a floating-element balance, were generally increased by divergence and decreased by convergence and the same holds true for the mixing length and the turbulent viscosity.

A finite-difference boundary-layer code using a simple turbulence model was used to predict the experimental results. The comparison showed good agreement for the two-dimensional flow, reasonable agreement for the diverging flow and poor agreement for the converging one. Use of the experimentally determined turbulent viscosity as input into the computation did not systematically improve the agreement but excellent agreement was found if it was combined with anisotropy of the turbulent viscosity. It was much more difficult to predict the converging flow as small errors in the crossflow had a large effect on the flow in the plane of symmetry (z = 0).

The response of a contained rotating fluid to a small, abrupt change in the rotation rate is analysed by multi-scaling methods. The procedure makes use of the fact that three different physical processes (inertial oscillations, spin-up response, diffusion) give rise to three different time scales. Since the flow is known to have a boundary-layer character, the variables are derived into interior and boundary-layer parts. The pertinent parameter separating the magnitudes of the amplitudes and the different time scales is the square root of the Ekman number E½, so an expansion in powers of E½ is used. The solution for a homogeneous fluid is derived first and is shown to be consistent with the solution of Green-span & Howard (1963). The results are given in two forms: one is a direct deduction of the expansion method and is valid to O(E); the other is obtained by regrouping the terms to derive a form apparently valid for indefinitely long times. When the fluid is stratified, the physical structure of the system is substantially more complicated, and so is the analysis. Exact results can be obtained for the case where the buoyancy N and the rotational Ω frequencies are the same. For the general case F = N/Ω ≠ 1, results valid for t [Gt ] 1 can be obtained (where t is measured in units of Ω−1). In both cases the exact lowest-order solution for the interior can be derived since it is independent of short time t. For the stratified fluid the elementary spin-up solution of Holton (1965) is part of the solution at O(E½). The remaining part includes the long-time behaviour towards which the system tends as diffusive processes become dominant. The formulation of the long-time problem is complete a t O(E), but parts of it emerge from the analysis at lower order, and it is necessary to treat the lower-order system to obtain a consistent formulation at O(E). In particular, it is possible to show that the thermal boundary condition, which does not affect the elementary spin-up solution, should be satisfied only by the long-time part of the problem. The complete, lowest-order response of the system includes a diffusive part which is quantitatively significant even for times of the order of one spin-up time. It is suggested here that the diffusive contribution may be responsible for parts of the discrepancy between elementary spin-up theory and recent experiments.

The hairpin instability of a jet in a crossflow (JICF) for a low jet-to-crossflow velocity ratio is investigated experimentally for a velocity ratio range of $R\in (0.14,0.75)$ and crossflow Reynolds numbers $Re_{D}\in (260,640)$. From spectral analysis we characterize the Strouhal number and amplitude of the hairpin instability as a function of $R$ and $Re_{D}$. We demonstrate that the dynamics of the hairpins is well described by the Landau model, and, hence, that the instability occurs through Hopf bifurcation, similarly to other hydrodynamical oscillators such as wake behind different bluff bodies. Using the Landau model, we determine the precise threshold values of hairpin shedding. We also study the spatial dependence of this hydrodynamical instability, which shows a global behaviour.

We calculate the leading-order correction to the time period of rotation of a neutrally buoyant spheroid of arbitrary aspect ratio, in a simple shear flow ($\boldsymbol{u}^{\infty }=\dot{\unicode[STIX]{x1D6FE}}y\mathbf{1}_{1}$; $\mathbf{1}_{1}$ is the unit vector in the flow direction, $y$ being the coordinate along the gradient direction), in its long-time orbit set up by the weak fluid inertial drift at $O(Re)$. Here, $Re$ is the microscale Reynolds number, a dimensionless measure of the fluid inertial effects on the length scale of the spheroid, and is defined as $Re=\dot{\unicode[STIX]{x1D6FE}}L^{2}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D707}$, where $L$ is the semimajor axis of the spheroid, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70C}$ are respectively the viscosity and density of the fluid, and $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate. This long-time orbit is the tumbling orbit for prolate spheroids; for oblate spheroids, it is the spinning orbit for aspect ratios greater than $0.137$, and can be either the tumbling or the spinning orbit for oblate spheroids of aspect ratios less than $0.137$. We also calculate the leading-order correction to the time period of rotation of a neutrally buoyant triaxial ellipsoid in a simple shear flow, rotating with its intermediate principal axis aligned along the vorticity of the flow; the latter calculation is in light of recent evidence, by way of numerical simulations (Rosen, PhD dissertation, 2016, Stockholm), of the aforementioned rotation being stabilized by weak inertia. The correction to the time period for arbitrary $Re$ is expressed as a volume integral using a generalized reciprocal theorem formulation. For $Re\ll 1$, it is shown that the correction at $O(Re)$ is zero for spheroids (with aspect ratios of order unity) as well as triaxial ellipsoids in their long-time orbits. The first correction to the time period therefore occurs at $O(Re^{3/2})$, and has a singular origin, arising from fluid inertial effects in the outer region (distances from the spheroid or triaxial ellipsoid of the order of the inertial screening length of $O(LRe^{-1/2})$), where the leading-order Stokes approximation ceases to be valid. Since the correction comes from the effects of inertia in the far field, the rotating spheroid (triaxial ellipsoid) is approximated as a time-dependent point-force-dipole singularity, allowing for the reciprocal theorem integral to be evaluated in Fourier space. It is shown for all relevant cases that fluid inertia at $O(Re^{3/2})$ leads to an increase in the time period of rotation compared with that in the Stokes limit, consistent with the results of recent numerical simulations at finite $Re$. Finally, combination of the $O(Re^{3/2})$ correction derived here with the $O(Re)$ correction derived earlier by Dabade et al. (J. Fluid Mech., vol. 791, 2016, 631703) yields a uniformly valid description of the first effects of inertia for spheroids of all aspect ratios, including prediction of the arrest of rotation for extreme-aspect-ratio spheroids.

This experimental and numerical study considers the two-dimensional stability of a circular cylinder wake, whose axis is tilted with respect to a stable density gradient. When the Reynolds number increases, the wake transitions from a steady flow to a periodic von Kármán vortex street as in a homogeneous fluid. However, the presence of a moderate stratification delays the appearance of the von Kármán vortex street, in agreement with the stabilization of shear flows by a density gradient. This stabilization, which does not occur for a vertical cylinder, increases with the tilt angle of the cylinder and is maximum for a horizontal cylinder. The critical Reynolds number increases when the stratification increases and diverges at a Froude number of order one for a horizontal cylinder. This critical Reynolds number can be predicted using the Richardson number based on the projection of the gravity and the density gradient in the direction of the shear, as was proposed by Candelier (J. Fluid Mech., vol. 685, pp. 191–201) for a tilted stratified jet. This picture is completely different for a strongly stratified wake since a new unstable mode appears, creating a von Kármán vortex street with a smaller Strouhal number. This surprising result is due to the presence of tilted vortices with no vertical velocity, i.e. with horizontal elliptic streamlines. This mode occurs in a band of Froude numbers which becomes smaller and smaller when the tilt angle increases, and eventually disappears for a horizontal cylinder. The presence of the tilt has thus a large impact on the structure of the wake at small Froude numbers and might need to be taken into account in geophysical flows.