Motivated by the need to characterize the spatio-temporal structure of turbulence in wall-bounded flows, we study wavenumber–frequency spectra of the streamwise velocity component based on large-eddy simulation (LES) data. The LES data are used to measure spectra as a function of the two wall-parallel wavenumbers and the frequency in the equilibrium (logarithmic) layer. We then reformulate one of the simplest models that is able to reproduce the observations: the random sweeping model with a Gaussian large-scale fluctuating velocity and with additional mean flow. Comparison with LES data shows that the model captures the observed temporal decorrelation, which is related to the Doppler broadening of frequencies. We furthermore introduce a parameterization for the entire wavenumber–frequency spectrum $E_{11}(k_{1},k_{2},{\it\omega};z)$, where $k_{1}$, $k_{2}$ are the streamwise and spanwise wavenumbers, ${\it\omega}$ is the frequency and $z$ is the distance to the wall. The results are found to be in good agreement with LES data.

The hydrodynamics of the dense granular flow of rough inelastic particles down an inclined plane is analysed using constitutive relations derived from kinetic theory. The basic equations are the momentum and energy conservation equations, and the granular energy conservation equation contains a term which represents the dissipation of energy due to inelastic collisions. A fundamental length scale in the flow is the ‘conduction length’ δ=(d/(1-en)1/2), which is the length over which the rate of conduction of energy is comparable to the rate of dissipation. Here, d is the particle diameter and en is the normal coefficient of restitution. For a thick granular layer with height h ≫ δ, the flow in the bulk is analysed using an asymptotic analysis in the small parameter δ/h. In the leading approximation, the rate of conduction of energy is small compared to the rates of production and dissipation, and there is a balance between the rate of production due to mean shear and the rate of dissipation due to inelastic collisions. A direct consequence of this is that the volume fraction in the bulk is a constant in the leading approximation. The first correction due to the conduction of energy is determined using asymptotic analysis, and is found to be O(δ/h)2 smaller than the leading-order volume fraction. The numerical value of this correction is found to be negligible for systems of practical interest, resulting in a lack of variation of volume fraction with height in the bulk.

The flow in the ‘conduction boundary layers’ of thickness comparable to the conduction length at the bottom and top is analysed. Asymptotic analysis is used to simplify the governing equations to a second-order differential equation in the scaled cross-stream coordinate, and the resulting equation has the form of a diffusion equation. However, depending on the parameters in the constitutive model, it is found that the diffusion coefficient could be positive or negative. Domains in the parameter space where the diffusion coefficients are positive and negative are identified, and analytical solutions for the boundary layer equations, subject to appropriate boundary conditions, are obtained when the diffusion coefficient is positive. There is no boundary layer solution that matches the solution in the bulk for parameter regions where the diffusion coefficient is negative, indicating that a steady solution does not exist. An analytical result is derived showing that a boundary layer solution exists (diffusion coefficient is positive) if, and only if, the numerical values of the viscometric coefficients are such that volume fraction in the bulk decreases as the angle of inclination increases. If the numerical values of the viscometric coefficients are such that the volume fraction in the bulk increases as the angle of inclination increases, a boundary layer solution does not exist.

The results are extended to dense flows in thin layers using asymptotic analysis. Use is made of the fact that the pair distribution function is numerically large for dense flows, and the inverse of the pair distribution function is used as a small parameter. This approximation results in a nonlinear second-order differential equation for the pair distribution function, which is solved subject to boundary conditions. For a dissipative base, it is found that a flowing solution exists only when the height is larger than a critical value, whereas the temperature decreases to zero and the flow stops when the height becomes smaller than this critical value. This is because the dissipation at the base becomes a larger fraction of the total dissipation as the height is decreased, and there is a minimum height below which the rate of production due to shear is not sufficient to compensate for the rate of dissipation at the base. The scaling of the minimum height with dissipation in the base, the bulk volume fraction and the parameters in the constitutive relations are determined. From this, the variation of the minimum height on the angle of inclination is obtained, and this is found to be in qualitative agreement with previous experiments and simulations.

We perform a direct numerical simulation (DNS) investigation of the incompressible temporally developing turbulent boundary layer. The approach is inspired by temporal simulations of flows which are generally thought of as developing in space, such as wakes and mixing layers. Compressible boundary layers have previously been studied in this manner yet the temporal approach appears to be under-exploited in the literature concerning incompressible boundary layers. The flow is the turbulent counterpart to the laminar Rayleigh problem or Stokes’ first problem, in which a fluid at rest is set into motion by a wall moving at constant velocity. An initial profile that models the effect of a wall-mounted trip wire is implemented and allows the characterisation of initial conditions by a trip Reynolds number. For the current set-up, a trip Reynolds number of 500 based on the trip-wire diameter successfully triggers transition yet only mildly perturbs the flow so it assumes a natural development at the lowest possible Reynolds number based on momentum thickness. A systematic trip study reveals that as the ratio of momentum thickness to trip-wire diameter approaches unity, our flow approaches a state free from the effects of its starting trip Reynolds number. The transport of a passive scalar by this flow is also simulated. The role played by domain size is investigated with two boxes, sized to accommodate two chosen final Reynolds numbers. Comparisons of the skin friction coefficient, velocity and scalar statistics demonstrate that the temporally developing boundary layer is a good model for the spatially developing boundary layer once initial conditions can be neglected. Analysis of similarity solutions suggests such a rapprochement of the spatial and temporal boundary layers may be expected at high Reynolds numbers given that the only terms that asymptotically persist are those common to both cases. If one seeks statistics for the turbulent boundary layer, the temporal boundary layer is therefore a viable method if modest convergence is sufficient. We suggest that such a temporal set-up could prove useful in the study of turbulence dynamics.

The Hartmann air jet generator (Hartmann 1939) is sound wave generator in which the sound is generated by oscillations of a shock developed in an over-expanded air jet by means of a blunt body: a resonator. A theory for the instability mechanism is advanced, and it is found that the result calculated from the theory are in good agreement with the experimental observations.

In a stably stratified shear layer, thin vorticity layers (‘baroclinic layers’) are produced by buoyancy effects and strain in between the Kelvin–Helmholtz vortices. A two-dimensional numerical study is conducted, in order to investigate the stability of these layers. Besides the secondary Kelvin–Helmholtz instability, expected but never observed previously in two-dimensional numerical simulations, a new instability is also found.

The influence of the Reynolds number (Re) upon the dynamics of the baroclinic layers is first studied. The layers reach an equilibrium state, whose features have been described theoretically by Corcos & Sherman (1976). An excellent agreement between those predictions and the results of the numerical simulations is obtained. The baroclinic layers are found to remain stable almost up to the time the equilibrium state is reached, though the local Richardson number can reach values as low as 0.05 at the stagnation point. On the basis of the work of Dritschel et al. (1991), we show that the stability of the layer at this location is controlled by the outer strain field induced by the large-scale Kelvin–Helmholtz vortices. Numerical values of the strain rate as small as 3% of the maximum vorticity of the layer are shown to stabilize the stagnation point region.

When non-pairing flows are considered, we find that only for Re ≤ 2000 does a secondary instability eventually amplify in the layer. (Re is based upon half the initial vorticity thickness and half the velocity difference at the horizontally oriented boundaries.) This secondary instability is not of the Kelvin–Helmholtz type. It develops in the neighbourhood of convectively unstable regions of the primary Kelvin–Helmholtz vortex, apparently once a strong jet has formed there, and moves along the baroclinic layer while amplifying. It next perturbs the layer around the stagnation point and a secondary instability, now of the Kelvin–Helmholtz type, is found to develop there.

We next examine the influence of a pairing upon the flow behaviour. We show that this event promotes the occurrence of a secondary Kelvin–Helmholtz instability, which occurs for Re ≥ 400. Moreover, at high Reynolds number (≥ 2000), secondary Kelvin–Helmholtz instabilities develop successively in the baroclinic layer, at smaller and smaller scales, thereby transferring energy towards dissipative scales through a mechanism eventually leading to turbulence. Because the vorticity of such a two-dimensional stratified flow is no longer conserved following a fluid particle, an analogy with three-dimensional turbulence can be drawn.

The motion of a spherical particle along a rough bed in a simple shear viscous flow is studied experimentally for a wide range of parameters, varying the particle size and density, the fluid viscosity and the shear rate γ. The instantaneous particle velocity is calculated in the stream, transverse and vertical directions, using a high-speed video imaging system. It is found that the normalized streamwise mean particle velocity U/US, where US is the Stokes settling velocity, depends only on the dimensionless shear rate μ γ/(Δ ρ g d), this relationship being independent of the particle Reynolds number Rep. This result holds for small Rep, which was the case in our experiments (Rep < 10). The characteristic amplitude and frequency of the velocity fluctuations are also given and discussed. A model is then proposed for the mean streamwise velocity, based on ideas of Bagnold (Proc. R. Soc. Lond. A, vol. 332, 1973, p. 473) and calculations of Goldman et al. (Chem. Engng Sci., vol. 22, 1967b, p. 653) for the velocity of a particle close to a smooth plane. From this model an equivalent bed roughness and an effective friction coefficient are deduced.

Segregation of polydisperse granular materials occurs in many natural and industrial settings, but general theoretical modelling approaches with predictive power have been lacking. Here we describe a model capable of accurately predicting segregation for both discrete and continuous particle size distributions based on a generalized expression for the percolation velocity. The predictions of the model depend on the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The model is applied to heap and chute flow, and the resulting predictions are consistent with experimentally validated discrete element method (DEM) simulations. Several different continuous particle size distributions are considered to demonstrate the broad applicability of the approach.

The mechanism of reconfiguration of broad leaves subjected to wind loading is investigated. Circular plastic sheets cut along a radius are immersed in a water flow. They roll up into cones when held at their centres. The opening angle of the cone and the drag force exerted on the sheet are measured as a function of the flow velocity and of the sheet bending rigidity. The cone becomes sharper when the velocity increases or when the sheet stiffness decreases; the reconfiguration leads to a decrease in the drag coefficient. Scaling laws are derived from the mechanical equilibrium of the sheets – the balance between form drag and elastic forces – and the experimental data collapse onto master curves. Two models for the pressure field yield theoretical curves in semi-quantitative agreement with the experiments.

Slender laminar and turbulent, plane and axisymmetric jets emerging from orifices in plane or conical walls are studied at large distances from the orifices. The entrainment of momentum coupled with the entrainment of volume into a jet is determined, and its effect on the flow field is analysed by combining inner and outer expansions with a multiple scaling approach.

In turbulent (plane or axisymmetric) jets, the axial velocity decreases more rapidly than predicted by classical boundary-layer solutions, and the momentum flux vanishes as the distance from the orifice tends to infinity. The analysis unveils a source of discrepancies in previous experimental data on turbulent jets.

In a laminar plane jet, the momentum flux changes but little. In a laminar axisymmetric jet, the momentum flux changes slowly, yet considerably. When a critical distance from the orifice is approached, the momentum flux in the jet becomes very small, the jet diameter very large, and a toroidal viscous eddy is predicted. The structure of the flow field is briefly discussed.

Static magnetic fields are known to be suitable for damping mean flow and turbulent motion in an electrically conducting liquid. In this paper, an experimental study is presented considering the influence of a horizontal magnetic field on a bubble-driven flow of a liquid metal. The investigation is focused on the liquid circulation inside a liquid metal column driven by a central jet produced by gas injection. The fluid vessel has a circular cross-section and electrically insulating walls. Low gas flow rates were applied, resulting in a plume of separated bubbles rising inside a spot around the cylinder axis. This axisymmetric configuration is exposed to a horizontal magnetic field. We present detailed experimental data describing the spatial as well as the temporal structure of the velocity field. Measurements of the vertical and the radial velocity component, respectively, were performed using the ultrasound Doppler velocimetry (UDV), allowing for the first time a complete mapping of the liquid velocity distribution for a bubble-driven liquid metal flow. The magnetic field considerably modified the global and local properties of the flow field compared to an ordinary bubble plume. In the parameter range considered here we did not find a prior flow suppression, but, in fact, a restructuring of the convective motion. The original axisymmetric flow field became anisotropic with respect to the direction of the magnetic field lines. An upwards flow dominated in a plane parallel to the magnetic field, whereas the recirculating motion was enforced in the orthogonal plane. Contrary to usual expectations, the application of a moderate magnetic field (100 < Ha < 400, 1 ≲ N ≲ 10) destabilizes the global flow and gives rise to transient, oscillating flow patterns with predominant frequencies.

Sufficient conditions for the stability of steady solutions of a multi-layer model are found. The basic flow may be either parallel, axisymmetric or non-parallel. The lower boundary of the model may be either rigid, including the possibility of topography, or soft. The latter, ‘reduced gravity’, case represents an ideal situation in which the active layers are on top of an infinitely deep, motionless one.

Two conditions are sufficient to assure the stability of the basic flow. It is conjectured that unstable flows for which only the first or second condition is violated decay through Rossby-like or Poincaré-like growing perturbations, respectively.

In order to understand the meaning of both conditions, assume that a quite general O(α) ‘wave’ is superimposed on the basic flow: an O(α2) energy integral, δ2E can be calculated. This wave energy is neither conserved, because the wave might exchange energy with the O(a2) varying part of the the ‘mean flow’, nor positive definite, because the perturbation might lower the total energy by increasing the speed where it decreases the thickness, and vice versa. Now, the first condition determines that δ2E has an upper bound, and the second one implies that δ2E is positive definite; hence the stability of the basic solution. In the particular case of two-dimensional divergenceless flow, as well as for quasi-geostropic models, δ2E is a priori positive definite, and therefore the first condition suffices to guarantee the stability of the more basic solution. The conditions found here are indeed valid for more general perturbations, e.g. they prevent inertial (or 'symmetric’) instability, a phenomenon for which there is no distinction between wave and the varying part of the mean flow.

We propose a variational approach to the identification of an optimal nonlinear eddy viscosity as a subscale turbulence representation for proper orthogonal decomposition (POD) models. The ansatz for the eddy viscosity is given in terms of an arbitrary function of the resolved fluctuation energy. This function is found as a minimizer of a cost functional measuring the difference between the target data coming from a resolved direct or large-eddy simulation of the flow and its reconstruction based on the POD model. The optimization is performed with a data-assimilation approach generalizing the 4D-VAR method. POD models with optimal eddy viscosities are presented for a 2D incompressible mixing layer at $\mathit{Re}=500$ (based on the initial vorticity thickness and the velocity of the high-speed stream) and a 3D Ahmed body wake at $\mathit{Re}=300\,000$ (based on the body height and the free-stream velocity). The variational optimization formulation elucidates a number of interesting physical insights concerning the eddy-viscosity ansatz used. The 20-dimensional model of the mixing-layer reveals a negative eddy-viscosity regime at low fluctuation levels which improves the transient times towards the attractor. The 100-dimensional wake model yields more accurate energy distributions as compared to the nonlinear modal eddy-viscosity benchmark proposed recently by Östh et al. (J. Fluid Mech., vol. 747, 2014, pp. 518–544). Our methodology can be applied to construct quite arbitrary closure relations and, more generally, constitutive relations optimizing statistical properties of a broad class of reduced-order models.

An experimental investigation of an unsteady adverse pressure gradient turbulent boundary layer is described. It is demonstrated that the local flow physics is largely dominated by an inflectional instability which gives rise to an embedded shear layer contained within the boundary layer. Experimental measurements are presented which are fully consistent with the presence of clockwise spanwise-oriented coherent vorticity within the embedded shear layer. Using embedded shear layer scaling parameters in the form of the shear layer vorticity thickness and the velocity defect at the upper inflection point, both the mean and the phase-averaged boundary layer velocity profiles exhibit similarity in both space and time over a large wall-normal extent. In a similar manner, the profiles of the streamwise-component turbulence intensity and Reynolds stress also exhibit similarity when scaled with the embedded shear layer parameters. The embedded shear layer scaling of previously published adverse pressure gradient turbulent boundary layer measurements confirms its generic applicability in a wide range of flow-field geometries and extending to high Reynolds numbers.

Taylor–Couette flow with independently rotating inner ($i$) and outer ($o$) cylinders is explored numerically and experimentally to determine the effects of the radius ratio $\eta $ on the system response. Numerical simulations reach Reynolds numbers of up to $\mathit{Re}_i=9.5\times 10^3$ and $\mathit{Re}_o=5\times 10^3$, corresponding to Taylor numbers of up to $\mathit{Ta}=10^8$ for four different radius ratios $\eta =r_i/r_o$ between 0.5 and 0.909. The experiments, performed in the Twente Turbulent Taylor–Couette ($\mathrm{T^3C}$) set-up, reach Reynolds numbers of up to $\mathit{Re}_i=2\times 10^6$ and $\mathit{Re}_o=1.5\times 10^6$, corresponding to $\mathit{Ta}=5\times 10^{12}$ for $\eta =0.714\mbox{--}0.909$. Effective scaling laws for the torque $J^{\omega }(\mathit{Ta})$ are found, which for sufficiently large driving $\mathit{Ta}$ are independent of the radius ratio $\eta $. As previously reported for $\eta =0.714$, optimum transport at a non-zero Rossby number $\mathit{Ro}=r_i |\omega _i-\omega _o |/[2(r_o-r_i)\omega _o]$ is found in both experiments and numerics. Here $\mathit{Ro}_{opt}$ is found to depend on the radius ratio and the driving of the system. At a driving in the range between $\mathit{Ta}\sim 3\times 10^{8}$ and $\mathit{Ta}\sim 10^{10}$, $\mathit{Ro}_{opt}$ saturates to an asymptotic $\eta $-dependent value. Theoretical predictions for the asymptotic value of $\mathit{Ro}_{opt}$ are compared to the experimental results, and found to differ notably. Furthermore, the local angular velocity profiles from experiments and numerics are compared, and a link between a flat bulk profile and optimum transport for all radius ratios is reported.

The two-dimensional flow of an incompressible viscous fluid past a circular cylinder, symmetrically placed in a uniform shear field, is considered both theoretically and experimentally for small values of the shear Reynolds number. A series of angular rotational speeds is covered, each giving rise to a fundamentally different flow pattern. It is shown first that the Stokes solution to this problem is not entirely consistent everywhere with the linear shear boundary condition which presumably exists far from the body. Using the method of inner and outer expansions, this solution is then improved by properly taking into account the first-order effects of the inertia terms, but, surprisingly, the streamline structure in the outer region is still found to depart from that of the uniform shear sufficiently far away from the object.

In spite of the somewhat bizarre nature of the theoretical solution far from the cylinder, experimental studies clearly show, however, that it accurately represents the actual flow within the inner region over a wide range of cylinder rotation rates.

A new model for Lagrangian particle-pair separation in turbulent flows is developed and compared with data from direct numerical simulations (DNS) of isotropic turbulence. The model formulation emphasizes (i) non-Gaussian behaviour in Eulerian and Lagrangian statistics, (ii) the occurrence of large separation velocities, (iii) the role of straining and streaming flow structure as recognized in kinematic simulations of turbulence, and (iv) the role of conditionally averaged accelerations in stochastic modelling of turbulent relative dispersion. Previous stochastic models of relative dispersion have produced unrealistic behaviour, particularly in the dissipation subrange where viscous effects are important, which have led to questions on the adequacy of stochastic modelling. However, this failure can now be recognized as inadequate detail in formulation, which is explained and rectified in this paper. The model is quasi-one-dimensional in nature, and is focused on the statistics of particle-pair separation distance and its rate of change, referred to as the separation speed. Detailed comparisons are presented at several Reynolds numbers using the DNS database reported in a companion paper (Part 1). Up to fourth-order moments for these quantities are examined, as well as the separation-distance probability density function, which is discussed in the context of recent claims of Richardson scaling in the literature. The model is able to account for the spatial representation of straining regions as well as incompressibility of the flow, and is shown to reproduce strong non-Gaussianity and intermittency in the Lagrangian separation statistics observed in DNS. Comparisons with recent physical experiments are also remarkably consistent. This work demonstrates that stochastic models when properly formulated are effective and efficient representations of the dispersion process and this general class of models therefore possess great utility for calculations of both one-particle and two-particle dispersion. The techniques developed in this paper will facilitate such general model development.

The stability of the viscometric motion of a viscoelastic fluid held between rotating parallel disks with large radii to small-amplitude perturbations is studied for the Oldroyd-B constitutive model. The disturbances are assumed to be radially localized and are expressed in Fourier form so that a separable eigenvalue problem results; these disturbances describe either axisymmetric or spiral vortices, depending on whether the most dangerous disturbance has zero or non-zero azimuthal wavenumber, respectively. The critical value of the dimensionless radius R* for the onset of the instability is computed as a function of the Deborah number De, a dimensionless time constant of the fluid, the azimuthal and radial wavenumbers, and the ratio of the viscosities of the solvent to the polymer solution. Calculations meant to match the experiments of McKinley et al. (1991) for a Boger fluid show that the most dangerous instabilities are spiral vortices with positive and negative angle that start at the same critical radius and travel outward and inward toward the centre of the disk; the axisymmetric mode also becomes unstable at only slightly greater values of R*, or De for fixed R*. The predicted dependence of the value of De for a fixed R* on the gap between the disks agrees quantitatively with the measurements of McKinley et al., when the longest relaxation time for the fluid at the shear rate corresponding to the maximum value of R* is used to define the time constant in the Oldroyd-B model.

An experimental investigation of forced and free oscillations of liquid bridges positioned between two rods of equal diameter is presented. Both the resonance frequencies and damping rates for different aspect ratios of the bridge are reported. The damping rate data of the liquid bridges are obtained by high-speed videography and are the first ever reported. Resonance frequencies for the three modified Reynolds numbers of 14, 295 and 1654, and damping rates for the two modified Reynolds numbers of 14 and 295 are reported. These values of modified Reynolds numbers are generated by using ethylene glycol, distilled water, and mercury in small bridges. Gravitational effects are kept small by reducing the size of the capillary bridge. The internal flow fields of several bridges for different modified Reynolds numbers are described based on high-speed visualization. Experimental results show good agreement with results of linear and nonlinear theory.

Zakharov's (1968) Hamiltonian formulation of water waves is used to prove analytically Tanaka's (1983) numerical result that superharmonic disturbances to periodic waves of permanent form exchange stability when the wave energy is an extremum as a function of wave height. Tanaka's (1985) explanation for the non-appearance of superharmonic bifurcation is also derived, and the non-existence of stability exchange when the wave speed is an extremum is explained.

Finite-amplitude solitary waves in water of arbitrary uniform depth are considered. A numerical scheme based on series truncation is presented to calculate the highest solitary wave. It is found that the ratio of the amplitude of the wave versus the depth is 0.83322. This value is compared with the values obtained by previous investigators. In addition, another numerical scheme based on an integral-equation formulation is derived to compute solitary waves of arbitrary amplitude. These calculations confirm and extend the calculations of Byatt-Smith & Longuet-Higgins (1976) for very steep waves.