A freely rotating sphere in a solid-body rotating flow is experimentally investigated. When the sphere is buoyant, it reaches an equilibrium position from which drag and lift coefficients are determined over a wide range of particle Reynolds numbers (2 ≤ Re ≤ 1060). The wake behind the sphere is visualized and appears to deflect strongly when the sphere is close to the cylinder axis. The spin rate of the sphere is recorded. In fluids with low viscosity, spin rates more than twice as large as the angular velocity of the cylinder can be observed. By comparing numerical results for a fixed but freely spinning sphere with a fixed non-spinning sphere for Re ≤ 200, the effect of the sphere spin on the lift coefficient is determined. The experimentally and numerically determined lift and drag coefficients and particle spin rates all show excellent agreement for Re ≤ 200. The combination of the experimental and numerical results allows for a parameterization of the lift and drag coefficients of a freely rotating sphere as function of the Reynolds number, the particle spin and the location of the particle with respect to the cylinder axis. Although the effect of the flow rotation on the particle spin is different in shear flow and solid-body rotating flow, the effect of spin on lift is found to be comparable for both types of flow.

The available experimental results have shown that in time-periodic motion the rheology of fluid mud displays complex viscoelastic behaviour. Based on the measured rheology of fluid mud from two field sites, we study the interaction of water waves and fluid mud by a two-layered model in which the water above is assumed to be inviscid and the mud below is viscoelastic. As the fluid-mud layer in shallow seas is usually much thinner than the water layer above, the sharp contrast of scales enables an approximate analytical theory for the interaction between fluid mud and small-amplitude waves with a narrow frequency band. It is shown that at the leading order and within a short distance of a few wavelengths, wave pressure from above forces mud motion below. Over a much longer distance, waves are modified by the accumulative dissipation in mud. At the next order, infragravity waves owing to convective inertia (or radiation stresses) are affected indirectly by mud motion through the slow modulation of the short waves. Quantitative predictions are made for mud samples of several concentrations and from two different field sites.

Planar Rayleigh scattering provides quantitative mixing measurements in the developing region of axisymmetric turbulent helium jets issuing into air. The measurements focus on the relatively near field, in which the jets are primarily momentum driven. The imaging parameters are specified to ensure high spatial resolution. The mean jet fluid concentration fields attain self-similarity within the measurement region, though the forms of the mole fraction profiles indicate a reduction in turbulent transport at the jet outer boundary, arising from the reduced jet fluid density. Nevertheless, jet-like scaling pertains for the concentration fields. Mass fraction fluctuations on the jet centreline attain the expected asymptotic value of ≈23% of the centreline mass fraction values. The scalar dissipation rates, however, show an axial decay rate that is slower than theoretical predictions. The two-dimensional extent of the measurements also allows spatial filtering similar to that inherent in large-eddy simulations (LESs). The results confirm that fluctuation levels and scalar dissipation rates determined for the filtered fields are reduced as the effective resolution is reduced, but while the fluctuation profiles for the filtered fields are similar for the different filter sizes, the forms of the scalar dissipation profiles are highly dependent on filter size. These latter results in particular are of a form that will be useful for grid-dependent assessments of LES results.

Numerical investigation of the compressible flow past an 18% thick circular-arc aerofoil was carried out using detached-eddy simulation for a free-stream Mach number M∞ = 0.76 and a Reynolds number Re = 1.1 × 107. Results have been validated carefully against experimental data. Various fundamental mechanisms dictating the intricate flow phenomena, including moving shock wave behaviours, turbulent boundary layer characteristics, kinematics of coherent structures and dynamical processes in flow evolution, have been studied systematically. A feedback model is developed to predict the self-sustained shock wave motions repeated alternately along the upper and lower surfaces of the aerofoil, which is a key issue associated with the complex flow phenomena. Based on the moving shock wave characteristics, three typical flow regimes are classified as attached boundary layer, moving shock wave/turbulent boundary layer interaction and intermittent boundary layer separation. The turbulent statistical quantities have been analysed in detail, and different behaviours are found in the three flow regimes. Some quantities, e.g. pressure-dilatation correlation and dilatational dissipation, have exhibited that the compressibility effect is enhanced because of the shock wave/boundary layer interaction. Further, the kinematics of coherent vortical structures and the dynamical processes in flow evolution are analysed. The speed of downstream-propagating pressure waves in the separated boundary layer is consistent with the convection speed of the coherent vortical structures. The multi-layer structures of the separated shear layer and the moving shock wave are reasonably captured using the instantaneous Lamb vector divergence and curl, and the underlying dynamical processes are clarified. In addition, the proper orthogonal decomposition analysis of the fluctuating pressure field illustrates that the dominated modes are associated with the moving shock waves and the separated shear layers in the trailing-edge region. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.

The effects of polymer additives on Rayleigh–Taylor (RT) instability of immiscible fluids is investigated using the Oldroyd-B viscoelastic model. Analytic results obtained exploiting the phase-field approach show that in polymer solution the growth rate of the instability speeds up with elasticity (but remains slower than in the pure solvent case). Numerical simulations of the viscoelastic binary fluid model confirm this picture.

A numerical and experimental investigation is reported for the flow around a rolling sphere when moving adjacent to a plane wall. The dimensionless rotation rate of the sphere is varied from forward to reversed rolling and the resulting wake modes are found to be strongly dependent on the value of this parameter. Results are reported for the Reynolds number range 100 < Re < 350, which has been shown to capture the unsteady transitions in the wake. Over this range of Reynolds number, both steady and unsteady wake modes are observed. As the sphere undergoes forward rolling, the wake displays similarities to the flow behind an isolated sphere in a free stream. As the Reynolds number of the flow increases, hairpin vortices form and are shed over the surface of the sphere. However, for cases with reversed rotation, the wake takes the form of two distinct streamwise vortices that form around the sides of the body. These streamwise structures in the wake undergo a transition to a new unsteady mode as the Reynolds number increases. During the evolution of this unsteady mode, the streamwise vortices form an out-of-phase spiral pair. Four primary wake modes are identified and a very good qualitative agreement is observed between the numerical and experimental results. The numerical simulations also reveal the existence of an additional unsteady mode that is found to be unstable to small perturbations in the flow.

Wall-bounded turbulence in pressure gradients is studied using direct numerical simulation (DNS) of a Couette–Poiseuille flow. The motivation is to include adverse pressure gradients, to complement the favourable ones present in the well-studied Poiseuille flow, and the central question is how the scaling laws react to a gradient in the total shear stress or equivalently to a pressure gradient. In the case considered here, the ratio of local stress to wall stress, namely τ+, ranges from roughly 2/3 to 3/2 in the ‘wall region’. By this we mean the layer believed not to be influenced by the opposite wall and therefore open to simple, universal behaviour. The normalized pressure gradients p+ ≡ dτ+/dy+ at the two walls are −0.00057 and +0.0037. The outcome is in broad agreement with the findings of Galbraith, Sjolander & Head (Aeronaut. Quart. vol. 27, 1977, pp. 229–242) relating to boundary layers (based on measured profiles): the logarithmic velocity profile is much more resilient than two other, equally plausible assumptions, namely universality of the mixing length ℓ = κy and that of the eddy viscosity νt = uτκy. In pressure gradients, with τ+ ≠ 1, these three come into conflict, and our primary purpose is to compare them. We consider that the Kármán constant κ is unique but allow a range from 0.38 to 0.41, consistent with the current debates. It makes a minor difference in the interpretation. This finding of resilience appears new as a DNS result and is free of the experimental uncertainty over skin friction. It is not as distinct in the (rather strong) adverse gradient as it is in the favourable one; for instance the velocity U+ at y+ = 50 is lower by 3% on the adverse gradient side. A plausible cause is that the wall shear stress is small and somewhat overwhelmed by the stress and kinetic energy in the bulk of the flow. The potential of a correction to the ‘law of the wall’ based purely on p+ is examined, with mixed results. We view the preference for the log law as somewhat counter-intuitive in that the scaling law is non-local but also as becoming established and as highly relevant to turbulence modelling.

Previous studies have shown that a dilute dispersion of fine particles can either augment or attenuate the gas-phase turbulent kinetic energy (TKE). However, such turbulence modification is not accurately captured by numerical simulation models. A critical reason is that the models do not incorporate flow distortion occurring at small scales on the order of the particle diameter or the Kolmogorov scale. These scales are too small to be resolved by most experiments and simulations, so the small-scale effects remain poorly understood. The main objective of this study is to investigate experimentally the small-scale turbulence structures that affect the overall turbulence modification to improve understanding and prediction of the macroscopic turbulence modification. A high resolution particle image velocimetry (PIV) system was developed that provided two-dimensional velocity field measurements with a sub-Kolmogorov vector spacing of 60 μm. Measurements of gas-phase isotropic turbulence were performed in the facility developed by Hwang & Eaton (Exp. Fluids, vol. 36 (3), 2004a, p. 444) in the presence of dispersed 500 μm glass, 250 μm glass or 250 μm polystyrene particles at mass loading ratio up to 0.45. The Reynolds number based on the Taylor microscale was 130 for the unladen case. The TKE was attenuated by up to 25 % in the presence of particles. The high-resolution measurements of the dissipation rate show that changes in the dissipation rate are smaller than changes to the TKE, in contrast to previous underresolved experiments. An analysis of a large set of PIV images allowed calculation of the average turbulence distortion around particles. The measurements also showed strong damping of the TKE and strong augmentation of the dissipation rate in a roughly spherical region surrounding the particles.

The boundary value problem for the nonlinear shallow-water equations with a beach source term is solved by direct use of physical variables, so that solutions are more easily inspected than those obtained by means of hodograph transformations. Beyond an overall description of the near-shoreline flows in terms of the nonlinear shallow-water equations, significant results are provided by means of a perturbation approach which enables much of the information on the flow to be retained. For sample waves of interest (periodic and solitary), first-order solutions of the shoreline motion and of the near-shoreline flows are computed, illustrated and successfully compared with the equivalent ones obtained through a hodograph transformation method previously developed by the authors. Wave–wave interaction, both at the seaward boundary and within the domain, is also accurately described. Analytical conditions for wave breaking within the domain are provided. These, compared with the authors' hodograph model, show that the first-order condition of the present model is comparable to the second-order condition of that model.

A numerical study based on large eddy simulation is performed to investigate a bottom boundary layer under an oscillating tidal current. The focus is on the boundary layer response to an external stratification. The thermal field shows a mixed layer that is separated from the external stratified fluid by a thermocline. The mixed layer grows slowly in time with an oscillatory modulation by the tidal flow. Stratification strongly affects the mean velocity profiles, boundary layer thickness and turbulence levels in the outer region although the effect on the near-bottom unstratified fluid is relatively mild. The turbulence is asymmetric between the accelerating and decelerating stages. The asymmetry is more pronounced with increasing stratification. There is an overshoot of the mean velocity in the outer layer; this jet is linked to the phase asymmetry of the Reynolds shear stress gradient by using the simulation data to examine the mean momentum equation. Depending on the height above the bottom, there is a lag of the maximum turbulent kinetic energy, dissipation and production with respect to the peak external velocity and the value of the lag is found to be influenced by the stratification. Flow instabilities and turbulence in the bottom boundary layer excite internal gravity waves that propagate away into the ambient. Unlike the steady case, the phase lines of the internal waves change direction during the tidal cycle and also from near to far field. The frequency spectrum of the propagating wave field is analysed and found to span a narrow band of frequencies clustered around 45°.

We consider the propagation of a non-Boussinesq gravity current in an axisymmetric configuration (full cylinder or wedge). The current of density ρc is released from rest from a lock of radius r0 and height h0 into an ambient fluid of density ρa in a container of height H. When the Reynolds number is large, the resulting flow is governed by the parameters ρc/ρa and H* = H/h0. We show that the one-layer shallow-water model, carefully combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1, and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio, H*. We obtain finite-difference solutions for the propagation and show that a self-similar behaviour develops for large times. This reveals the main features, in particular: (a) The heavy current propagates faster and its front is thinner than that for the light counterpart; (b) For large time, t, both the heavy and light currents spread like t1/2, but the thickness profiles display significant differences; (c) The energy-constrained propagation with the thickness of half-ambient-depth (when H* is close to 1) is a very limited occurrence, in contrast to the rectangular geometry counterpart in which this effect plays a major role. The predictions of the simple model are supported by some axisymmetric Navier–Stokes finite-difference simulations.

We report direct numerical simulation (DNS) and large-eddy simulation (LES) of statistically stationary buoyancy-driven turbulent mixing of an active scalar. We use an adaptation of the fringe-region technique, which continually supplies the flow with unmixed fluids at two opposite faces of a triply periodic domain in the presence of gravity, effectively maintaining an unstably stratified, but statistically stationary flow. We also develop a new method to solve the governing equations, based on the Helmholtz–Hodge decomposition, that guarantees discrete mass conservation regardless of iteration errors. Whilst some statistics were found to be sensitive to the computational box size, we show, from inner-scaled planar spectra, that the small scales exhibit similarity independent of Reynolds number, density ratio and aspect ratio. We also perform LES of the present flow using the stretched-vortex subgrid-scale (SGS) model. The utility of an SGS scalar flux closure for passive scalars is demonstrated in the present active-scalar, stably stratified flow setting. The multi-scale character of the stretched-vortex SGS model is shown to enable extension of some second-order statistics to subgrid scales. Comparisons with DNS velocity spectra and velocity-density cospectra show that both the resolved-scale and SGS-extended components of the LES spectra accurately capture important features of the DNS spectra, including small-scale anisotropy and the shape of the viscous roll-off.

The modal structure of the swept Hiemenz flow, a model for the flow near the attachment line of a swept wing, consists of eigenfunctions which exhibit (super-)exponential or algebraic decay as the wall-normal coordinate tends to infinity. The subset of algebraically decaying modes corresponds to parts of the spectrum which are characterized by a significant sensitivity to numerical discretization. Numerical evidence further suggests that a continuous spectrum covering a two-dimensional range of the complex plane exists. We investigate the family of uniform swept Hiemenz modes using eigenvalue computations, numerical simulations and the concept of wave packet pseudo-modes. Three distinct branches of the family of algebraically decaying eigenmodes are identified. They can be superimposed to produce wavefronts propagating towards or away from the boundary layer and standing or travelling wave packets in the free stream. Their role in the exchange of information between the free stream and the attachment-line boundary layer for the swept Hiemenz flow is discussed. The concept of wave packet pseudo-modes has been critical in the analysis of this problem and is expected to lead to further insights into other shear flows in semi- or bi-infinite domains.

We compute the optimal response of the turbulent Couette mean flow to initial conditions, harmonic and stochastic forcing at Re = 750. The equations for the coherent perturbations are linearized near the turbulent mean flow and include the associated eddy viscosity. The mean flow is found to be linearly stable but it has the potential to amplify steamwise streaks from streamwise vortices. The most amplified structures are streamwise uniform and the largest amplifications of the energy of initial conditions and of the variance of stochastic forcing are realized by large-scale streaks having spanwise wavelengths of 4.4h and 5.2h respectively. These spanwise scales compare well with the ones of the coherent large-scale streaks observed in experimental realizations and direct numerical simulations of the turbulent Couette flow. The optimal response to the harmonic forcing, related to the sensitivity to boundary conditions and artificial forcing, can be very large and is obtained with steady forcing of structures with larger spanwise wavelength (7.7h). The optimal large-scale streaks are furthermore found proportional to the mean turbulent profile in the viscous sublayer and up to the buffer layer.

A study of the flow past an oscillatory rotating cylinder has been conducted, where the frequency of oscillation has been matched to the natural frequency of the vortex street generated in the wake of a stationary cylinder, at Reynolds number 300. The focus is on the wake transition to three-dimensional flow and, in particular, the changes induced in this transition by the addition of the oscillatory rotation. Using Floquet stability analysis, it is found that the fine-scale three-dimensional mode that typically dominates the wake at a Reynolds number beyond that at the second transition to three-dimensional flow (referred to as mode B) is suppressed for amplitudes of rotation beyond a critical amplitude, in agreement with past studies. However, the rotation does not suppress the development of three-dimensionality completely, as other modes are discovered that would lead to three-dimensional flow. In particular, the longer-wavelength mode that leads the three-dimensional transition in the wake of a stationary cylinder (referred to as mode A) is left essentially unaffected at low amplitudes of rotation. At higher amplitudes of oscillation, mode A is also suppressed as the two-dimensional near wake changes in character from a single- to a double-row wake; however, another mode is predicted to render the flow three-dimensional, dubbed mode D (for double row). This mode has the same spatio-temporal symmetries as mode A.

We investigate the properties of forced inertial modes of a rotating fluid inside a spherical shell. Our forcing is tidal like, but its main property is that it is on the large scales. By numerically solving the linear equations of this problem, including viscosity, we first confirm some analytical results obtained on a two-dimensional model by Ogilvie (J. Fluid Mech., vol. 543, 2005, p. 19); some additional properties of this model are uncovered like the existence of narrow resonances associated with periodic orbits of characteristics. We also note that as the frequency of the forcing varies, the dissipation varies drastically if the Ekman number E is low (as is usually the case). We then investigate the three-dimensional case and compare the results to the foregoing model. The three-dimensional solutions show, like their two-dimensional counterpart, a spiky dissipation curve when the frequency of the forcing is varied; they also display small frequency intervals where the viscous dissipation is independent of viscosity. However, we show that the response of the fluid in these frequency intervals is crucially dominated by the shear layer that is emitted at the critical latitude on the inner sphere. The asymptotic regime, where the dissipation is independent of the viscosity, is reached when an attractor has been excited by this shear layer. This property is not shared by the two-dimensional model where shear layers around attractors are independent of those emitted at the critical latitude. Finally, resonances of the three-dimensional model correspond to some selected least damped eigenmodes. Unlike their two-dimensional counter parts these modes are not associated with simple attractors; instead, they show up in frequency intervals with weakly contracting webs of characteristics. Besides, we show that the inner core is negligible when its relative radius is less than the critical value 0.4E1/5. For these spherical shells, the full sphere solutions give a good approximation of the flows.

An experimental study is performed on the vortex induced vibrations of a rigid flexibly mounted circular cylinder placed in a crossflow. The cylinder is allowed to oscillate in combined crossflow and in-line motions, and the ratio of the nominal in-line and transverse natural frequencies is varied systematically. Experiments were conducted on a smooth cylinder at subcritical Reynolds numbers between 15 000 and 60 000 and on a roughened cylinder at supercritical Reynolds numbers between 320 000 and 710 000, with a surface roughness equal to 0.23 % of the cylinder diameter. Strong qualitative and quantitative similarities between the subcritical and supercritical experiments are found, especially when the in-line natural frequency is close to twice the value of the crossflow natural frequency. In both Reynolds number regimes, the test cylinder may exhibit a ‘dual-resonant’ response, resulting in resonant crossflow motion at a frequency fv, near the Strouhal frequency, and resonant in-line motion at 2 fv. This dual resonance is shown to occur over a relatively wide frequency region around the Strouhal frequency, accompanied by stable, highly repeatable figure-eight cylinder orbits, as well as large third-harmonic components of the lift force. Under dual-resonance conditions, both the subcritical and the supercritical response is shown to collapse into a narrow parametric region in which the effective natural-frequency ratio is near the value 2, regardless of the nominal natural-frequency ratio. Some differences are noted in the magnitudes of forces and the cylinder response between the two different Reynolds number regimes, but the dual-resonant response and the resulting force trends are preserved despite the large Reynolds number difference.

The generation and evolution of large-scale vortices with vertical axis (macro-vortices) in a straight compound channel under quasi-uniform flow conditions is investigated. We discuss possible similarities and clear differences with free shear layer flows induced by the meeting of shallow streams of different speeds. An experimental investigation based on particle image velocimetry (PIV) measurements of free-surface velocities forms the basis for an analysis of both the specific features of macro-vortices and of the related mean flow characteristics. Dynamical properties strongly depend on the ratio rh between the main channel flow depth (h*mc) and the floodplain depth (h*fp), and three flow classes can be identified. ‘Shallow flows’ (rh > 3) are dominated by strong shearing and large macro-vortices populating the transition region between the main channel and the floodplains. The mean streamwise velocity induced in ‘intermediate flows’ (2 ≤ rh ≤ 3) is characterized by a dip in the transition region, while it closely resembles that occurring in a rectangular channel in the case of ‘deep flows’ (rh < 2). For both the latter cases the shear in the transition region decreases and the macro-vortices are also generated in the wall boundary layer of the floodplains. The typical size of the quasi-two-dimensional macro-vortices, generated at the transition region, is found to be independent of the streamwise coordinate. This and the non-monotonic behaviour of the mean streamwise velocity suggest that in straight compound channels the topographic forcing is so dominant that conceptual models interpreting these flows as free shear layers may largely fail to describe the physics of compound channels flows.

We consider a class of higher order (quartet) Bragg resonance involving two incident wave components and a bottom ripple component (so called class III Bragg resonance). In this case, unlike class I/II Bragg resonance involving a single incident wave and one/two bottom ripple components, the frequency of the resonant wave, which can be reflected or transmitted, is a sum or difference of the incident wave frequencies. In addition to transferring energy across the spectrum leading to potentially significant spectral transformation, such resonances may generate long (infragravity) waves of special importance to coastal processes and engineering applications. Of particular interest here is the case where the incident waves are oblique to the bottom undulations (or to each other) which leads to new and unexpected wave configurations. We elucidate the general conditions for such resonances, offering a simple geometric construction for obtaining these. Perturbation analysis results are obtained for these resonances predicting the evolutions of the resonant and incident wave amplitudes. We investigate special cases using numerical simulations (applying a high-order spectral method) and compare the results to perturbation theory: infragravity wave generation by co- and counter-propagating incident waves normal to bottom undulations; longshore long waves generated by (bottom) oblique incident waves; and propagating–standing resonant waves due to (bottom) parallel incident waves. Finally, we consider a case of multiple resonance due to oblique incident waves on bottom ripples which leads to complex wave creation and transformations not easily tractable with perturbation theory. These new wave resonance mechanisms can be of potential importance on continental shelves and in littoral zones, contributing to wave spectral evolution and bottom processes such as sandbar formation.

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r−5/3, while the y component (in the direction of vorticity) decays like r−4/3. The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as yw ~ zw ~ |x|1/3. The maximum x and z components of the velocity are located in the wake cores; they scale like |x|−2/3 and |x|−1 respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.