Direct numerical simulations of a turbulent channel flow with passive scalar transport are used to examine the relationship between small-scale velocity and scalar fields. The Reynolds number based on the friction velocity and the channel half-width is equal to 180, 395 and 640, and the molecular Prandtl number is 0.71. The focus is on the interrelationship between the components of the vorticity vector and those of the scalar derivative vector. Near the wall, there is close similarity between different components of the two vectors due to the almost perfect correspondence between the momentum and thermal streaks. With increasing distance from the wall, the magnitudes of the correlations become smaller but remain non-negligible everywhere in the channel owing to the presence of internal shear and scalar layers in the inner region and the backs of the large-scale motions in the outer region. The topology of the scalar dissipation rate, which is important for small-scale scalar mixing, is shown to be associated with the organized structures. The most preferential orientation of the scalar dissipation rate is the direction of the mean strain rate near the wall and that of the fluctuating compressive strain rate in the outer region. The latter region has many characteristics in common with several turbulent flows; viz. the dominant structures are sheetlike in form and better correlated with the energy dissipation rate than the enstrophy.

We study the classical problem of planar shock refraction at an oblique density discontinuity, separating two gases at rest. When the shock impinges on the density discontinuity, it refracts, and in the hydrodynamical case three signals arise. Regular refraction means that these signals meet at a single point, called the triple point. After reflection from the top wall, the contact discontinuity becomes unstable due to local Kelvin–Helmholtz instability, causing the contact surface to roll up and develop the Richtmyer–Meshkov instability (RMI). We present an exact Riemann-solver-based solution strategy to describe the initial self-similar refraction phase, by which we can quantify the vorticity deposited on the contact interface. We investigate the effect of a perpendicular magnetic field and quantify how its addition increases the deposition of vorticity on the contact interface slightly under constant Atwood number. We predict wave-pattern transitions, in agreement with experiments, von Neumann shock refraction theory and numerical simulations performed with the grid-adaptive code AMRVAC. These simulations also describe the later phase of the RMI.

The paper examines, by means of highly resolved large-eddy simulations, the fluid mechanic behaviour of an incompressible, turbulent flow separating from an axisymmetric, hill-shaped obstacle. The hill is subjected to a turbulent flat-plate boundary layer of thickness half of the hill height, developing within a large duct in which the hill is placed. The Reynolds number based on the hill height and the free-stream velocity is 130000, and the momentum-thickness Reynolds number of the boundary layer is 7300. Extensive and detailed experimental data are available for these same conditions, and these are compared with the simulations. Two entirely independent simulations are reported, undertaken by different groups with different codes and different grids. They are shown to agree closely in most respects, and any differences are carefully identified and discussed. Mean-flow properties, pressure distributions and turbulence characteristics are reported and compared with the experimental data, with particular emphasis being placed on the separation region that is formed in the leeward of the hill and on the near wake. The connection between the wall topology of the mean flow and the secondary motions in the wake is discussed in detail. Finally, some aspects of the unsteady flow are analysed, including the unsteadiness of the separation process and structural features and coherent motions in the wake.

We investigate experimentally the transition to turbulence of a yield stress shear-thinning fluid in Hagen–Poiseuille flow. By combining direct high-speed imaging of the flow structures with Laser Doppler Velocimetry (LDV), we provide a systematic description of the different flow regimes from laminar to fully turbulent. Each flow regime is characterized by measurements of the radial velocity, velocity fluctuations and turbulence intensity profiles. In addition we estimate the autocorrelation, the probability distribution and the structure functions in an attempt to further characterize transition. For all cases tested, our results indicate that transition occurs only when the Reynolds stresses of the flow equal or exceed the yield stress of the fluid, i.e. the plug is broken before transition commences. Once in transition and when turbulent, the behaviour of the yield stress fluid is somewhat similar to a (simpler) shear-thinning fluid. Finally, we have observed the shape of slugs during transition and found their leading edges to be highly elongated and located off the central axis of the pipe, for the non-Newtonian fluids examined.

An axisymmetric jet at a diameter-based Reynolds number of 1.1 × 104 is computed by a large eddy simulation (LES) in order to investigate its self-similarity region. The LES combines low-dissipation numerical schemes and explicit filtering of the flow variables to relax energy through the smaller scales discretized. The computational domain extends up to 150 jet radii in the downstream direction, which is found to be large enough to discretize a part of this region. Turbulence in the self-preserving jet is characterized by evaluating explicitly from the LES fields the second- and third-order moments of velocity, the pressure–velocity correlations as well as the budgets for the turbulent kinetic energy and for its components. Reference solutions are thus obtained. They agree well with the experimental data given by Panchapakesan & Lumley (J. Fluid Mech., vol. 246, 1963, p. 197) for a jet at the same Reynolds number. The distance required to achieve self-similarity in the LES, around 120 radii from the inflow, is particularly similar to that in the experiment. The discrepancies observed with respect to the data provided by Panchapakesan & Lumley and by Hussein, Capp & George (J. Fluid Mech., vol. 258, 1994, p. 31) for a jet at a higher Reynolds number, specially regarding the turbulence diffusion and the dissipation, are discussed. They appear largely resulting from the approximations made in the experiments to estimate the quantities that cannot be measured with accuracy. The role of the pressure terms in the energy redistribution is also clarified by the LES. Moreover, the turbulent energy budget is calculated in the jet from an equation derived from the filtered compressible Navier–Stokes equations, which includes the dissipation due to the explicit filtering. This has allowed us to assess the behaviour of the LES approach based on relaxation filtering (LES-RF) from the contributions of filtering and viscosity to energy dissipation. The filtering activity is particularly shown to adjust by itself to the grid and flow properties.

Waves of spanwise velocity imposed at the walls of a plane turbulent channel flow are studied by direct numerical simulations. We consider sinusoidal waves of spanwise velocity which vary in time and are modulated in space along the streamwise direction. The phase speed may be null, positive or negative, so that the waves may be either stationary or travelling forward or backward in the direction of the mean flow. Such a forcing includes as particular cases two known techniques for reducing friction drag: the oscillating wall technique (a travelling wave with infinite phase speed) and the recently proposed steady distribution of spanwise velocity (a wave with zero phase speed). The travelling waves alter the friction drag significantly. Waves which slowly travel forward produce a large reduction of drag that can relaminarize the flow at low values of the Reynolds number. Faster waves yield a totally different outcome, i.e. drag increase (DI). Even faster waves produce a drag reduction (DR) effect again. Backward-travelling waves instead lead to DR at any speed. The travelling waves, when they reduce drag, operate in similar fashion to the oscillating wall, with an improved energetic efficiency. DI is observed when the waves travel at a speed comparable with that of the convecting near-wall turbulence structures. A diagram illustrating the different flow behaviours is presented.

A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.

Lower branch travelling waves and equilibria computed in pipe flow and other shear flows appear intermediate between turbulent and laminar motions. We take a step towards connecting these lower branch solutions to transition by deriving a numerical method for finding certain special disturbances of the laminar flow in a short pipe. These special disturbances cause the disturbed velocity field to approach the lower branch solution by evolving along its stable manifold. If the disturbance were slightly smaller, the flow would relaminarize, and if slightly larger, it would transition to a turbulent state.

A wave basin experiment has been performed in the MARINTEK laboratories, in one of the largest existing three-dimensional wave tanks in the world. The aim of the experiment is to investigate the effects of directional energy distribution on the statistical properties of surface gravity waves. Different degrees of directionality have been considered, starting from long-crested waves up to directional distributions with a spread of ±30° at the spectral peak. Particular attention is given to the tails of the distribution function of the surface elevation, wave heights and wave crests. Comparison with a simplified model based on second-order theory is reported. The results show that for long-crested, steep and narrow-banded waves, the second-order theory underestimates the probability of occurrence of large waves. As directional effects are included, the departure from second-order theory becomes less accentuated and the surface elevation is characterized by weak deviations from Gaussian statistics.

The oscillatory flow around a circular cylinder close to a plane wall is investigated numerically, by direct numerical simulation of the Navier–Stokes equations. The main aim of the research is to gain insight into the effect of the wall on the vorticity dynamics and the forces induced by the flow over the cylinder. First, two-dimensional simulations are performed for nine values of the gap-to-diameter ratio e. Successively, three-dimensional simulations are carried out for selected cases to analyse the influence of the gap on the three-dimensional organization of the flow. An attempt to explain the pressure distribution around the cylinder in terms of vorticity time development is presented. Generally, the time development of the hydrodynamic forces is aperiodic (i.e. changes from cycle to cycle). In one case (Re = 200), when the distance of the cylinder from the wall is reduced, the behaviour of the flow changes from aperiodic to periodic. When the cylinder approaches the wall the drag coefficient of the in-line force increases in a qualitative agreement with the results reported in literature. The transverse force is not monotonic with the reduction of the gap: it first decreases down to a minimum, and then increases with a further reduction of the gap. For intermediate values of the gap the decrease of the transverse force is due to the reduction of the angle of ejection of the shedding vortices caused by the closeness of the wall; for small gaps the increase of the transverse force is due to the strong interaction between the vortex system ejected from the cylinder and the shear layer generated on the wall.

Three-dimensional simulations show that the flow is unstable with respect to spanwise perturbations which cause the development of three-dimensional vortices and the distortion of the two-dimensional ones generated by flow separation.

In all the analysed cases, the three-dimensional effects on the hydrodynamic forces are clearly attenuated when the cylinder is placed close to the wall.

The spanwise modulation of the vortex structures induces oscillations of the sectional forces along the axis of the cylinder which in general are larger for the transverse sectional force. In the high-Reynolds-number case (Re = 500), the reduction of the gap produces a large number of three-dimensional vortex structures developing over a wide range of spatial scales. This produces homogenization of the flow field along the spanwise direction and a consequent reduction of the amplitudes of oscillation of the sectional forces.

The linear stability of fully developed Poiseuille flow of a Newtonian fluid in a deformable neo-Hookean tube is analysed to illustrate the shortcomings of extrapolating the linear elastic model for the tube wall outside its domain of validity of small strains in the solid. We show using asymptotic analyses and numerical solutions that a neo-Hookean description of the solid dramatically alters the stability behaviour of flow in a deformable tube. The flow-induced instability predicted to exist in the creeping-flow limit based on the linear elastic approximation is absent in the neo-Hookean model. In contrast, a new low-wavenumber (denoted by k) instability is predicted in the limit of very low Reynolds number (Re ≪ 1) with k ∝ Re1/2 for purely elastic (with ratio of solid to fluid viscosities ηr = 0) neo-Hookean tubes. The first normal stress discontinuity in the neo-Hookean solid gives rise to a high-wavenumber interfacial instability, which is stabilized by interfacial tension at the fluid–wall interface. Inclusion of dissipation (ηr ≠ 0) in the solid has a stabilizing effect on the low-k instability at low Re, and the critical Re for instability is a sensitive function of ηr. For Re ≫ 1, both the linear elastic extrapolation and the neo-Hookean model agree qualitatively for the most unstable mode, but show disagreement for other unstable modes in the system. Interestingly, for plane-Couette flow past a deformable solid, the results from the extrapolated linear elastic model and the neo-Hookean model agree very well at any Reynolds number for the most unstable mode when the wall thickness is not small. The results of this study have important implications for experimental investigations aimed at probing instabilities in flow through deformable tubes.

We consider the effect of a wetting condition at the moving contact line on the frequency and damping of surface waves on an inviscid liquid in a circular cylinder. The velocity potential φ and the free surface elevation η are sought as complex eigenfunction expansions. The φ eigenvalues are the classical ones whereas the η eigenvalues are unknown and have to be computed so as to satisfy the wetting condition on the contact line and the other free surface conditions – these turn out to be complex in general. A projection of the latter conditions on to an appropriate basis leads to an eigenvalue problem, for the complex frequency Ω, which has to be solved iteratively with the wetting condition. The variation of Ω with liquid depth h, Bond number Bo, capillary coefficient λ and static contact angle θc0 is explored for the (1, 0),(2, 0),(0, 1),(3, 0) and (4, 0) modes. The damping vanishes for λ = 0 (pinned-end edge condition) and λ = ∞ (free-end edge condition) with a maximum in the interior while the frequency decreases with increasing λ, approaching limiting values at the endpoints. A comparison with the analytic results of Miles (J. Fluid Mech., vol. 222, 1991, p. 197) for the no-meniscus case and the experimental results of Cocciaro, Faetti, & Festa (J. Fluid Mech., vol. 246, 1993, p. 43), where a meniscus is present, is good. The study provides a simple procedure for calculating the inviscid capillary damping associated with the moving contact line in a circular cylinder of finite depth with meniscus effects also being considered.

Using the standard weak-field approximation, we analyse the steady-state electrokinetic flow about an uncharged ideally polarizable spherical particle for the case of a Debye thickness which is large compared with the particle size. The dimensionless problem is governed by two parameters: β, the applied field magnitude (normalized with the thermal scale), and λ, the Debye thickness (normalized with particle size). The double limit β ≪ 1 and λ ≫ 1 is singular, and the resolution of the flow field requires the use of inner–outer asymptotic expansions in the spirit of Proudman & Pearson (J. Fluid Mech., vol. 2, 1957, p. 237). Two asymptotic limits are identified: the ‘moderately thick’ limit βλ ≪ 1, in which the outer domain is characterized by the Debye thickness, and the ‘super-thick’ limit βλ ≫ 1, in which the outer domain represents the emergence of electro-migration in the leading-order ionic-transport process. The singularity is stronger in the comparable two-dimensional flow about a circular cylinder, where a switchback mechanism in the moderately thick limit modifies the familiar O(β2) leading-order flow to O(β2 ln λ).

Energy flow and radiation of linearized acoustic–gravity waves and propagation of boundary waves in a gravitationally stratified isothermal compressible inviscid semi-infinite fluid from a time-varying bottom boundary are investigated in the frequency–wavenumber domain. Impedance Z, the ratio of the bottom vertical displacement to the fluid pressure above it, is a function of the frequency and horizontal wavenumber (ω, k) of the bottom boundary undulation. The amplitude and phase of Z at the bottom boundary divide the (ω, k) coordinates into wave-type regimes. In contrast to the pure acoustic or gravity wave case, the phase of Z is continuous but changes quickly across the regime boundaries between the propagating waves and trapped waves at the bottom, except for the Lamb wave branch along which the amplitude is infinite and across which the phase jumps by π. The phase of Z determines the efficiency of the work against the fluid by the deforming bottom boundary, showing reduced upward wave-energy flow from the bottom near the regime boundaries in which the phase of Z approaches ±π/2. For precise modelling of pressure waves and the energy flow of acoustic and gravity waves in the fluid originating from a time-dependent bottom-surface deformation with an apparent phase velocity comparable to the speed of sound in the fluid, it is necessary to include the dependency on (ω, k) of impedance Z.

We investigate the dynamics of a non-Brownian sphere suspended in a quiescent fluid and moving through a periodic array of solid obstacles under the action of a constant external force by means of Stokesian dynamics simulations. We show that in the presence of non-hydrodynamic, short-range interactions between the solid obstacles and the suspended sphere, the moving particle becomes locked into periodic trajectories with an average orientation that coincides with one of the lattice directions and is, in general, different from the direction of the driving force. The locking angle depends on the details of the non-hydrodynamic interactions and could lead to vector separation of different species for certain orientations of the external force. We explicitly show the presence of separation for a mixture of suspended particles with different roughness, moving through a square lattice of spherical obstacles. We also present a dilute model based on the two-particle mobility and resistance functions for the collision between spheres of different sizes. This simple model predicts the separation of particles of different size and also suggests that microdevices that maximize the differences in interaction area between the different particles and the solid obstacles would be more sensitive for size separation based on non-hydrodynamic interactions.

Within the same framework we calculate the mean induction of a magnetic field and the mean diffusivity of a passive scalar, for two families of flows in which the degree of spatial decorrelation can be systematically adjusted. We investigate the dependence of these quantities both on the spatial decoherence and on the molecular diffusivity. We demonstrate that for flows with similar global properties, the mean induction is dramatically reduced as the flows become less spatially correlated; the mean diffusivity, on the other hand, shows no significant or systematic variation.

Jets arising from rivers, streams and tidal flows entering still waters differ from most experimental studies of jets both in aspect ratio and in the presence of a solid bottom boundary and an upper free surface. Despite these differences, the applicability of experimental jet studies to these systems remains largely untested by either field or realistically scaled experimental studies. Here we present experimental results for a wall-bounded plane jet scaled to jets created by flow discharging into floodplain lakes. A characteristic feature of both our prototype and experimental jets is the presence of large-scale meandering turbulent structures that span the width of the jets. In our experimental jets, we observe self-similarity in the distribution of mean streamwise velocities by a distance of six channel widths downstream of the jet outlet. After a distance of nine channel widths the velocity decay and the spreading rates largely agree with prior experimental results for plane jets. The magnitudes and distributions of the cross-stream velocity and lateral shear stresses approach self-preserving conditions in the upper half of the flow, but decrease in magnitude, and deviate from self-preserving distributions with proximity to the bed. The presence of the meandering structure has little influence on the mean structure of the jet, but dominates the jet turbulence. A comparison of turbulence analysed at time scales both greater than and less than the period of the meandering structure indicates that these structures increase turbulence intensities by 3–5 times, and produce lateral shear stresses and momentum diffusivities that are one and two orders of magnitude greater, respectively, than turbulence generated by bed friction alone.

A model of hairy medium is developed using a homogenized approach, and the fluid flow around a circular cylinder partially coated with hair is analysed by means of numerical simulations. The capability of this coating to adapt to the surrounding flow is investigated, and its benefits are discussed in the context of separation control. This fluid–structure interaction problem is solved with a partitioned approach, based on the direct resolution of the Navier–Stokes equations together with a nonlinear set of equations describing the dynamics of the coating. A volume force, arising from the presence of a cluster of hair, provides the link between the fluid and the structure problems. For the structure part, a subset of reference elements approximates the whole layer. The dynamics of these elements is governed by a set of equations based on the inertia, elasticity, interaction and losses effects of articulated rods. The configuration chosen is that of the two-dimensional flow past a circular cylinder at Re = 200, a simple and well-documented test case. Aerodynamics performances quantified by the Strouhal number, the drag and the maximum lift in the laminar unsteady regime are modified by the presence of the coating. A set of parameters corresponding to a realistic coating (length of elements, porosity, rigidity) is found, yielding an average drag reduction of 15% and a decrease of lift fluctuations by about 40%, associated to a stabilization of the wake.

It is known that for finite Rossby numbers geostrophically balanced flows develop specific ageostrophic instabilities. We undertake a detailed study of the Rossby–Kelvin (RK) instability, previously studied by Sakai (J. Fluid Mech., vol. 202, 1989, pp. 149–176) in a two-layer rotating shallow-water model. First, we benchmark our method by reproducing the linear stability results obtained by Sakai (1989) and extend them to more general configurations. Second, in order to determine the relevance of RK instability in more realistic flows, simulations of the evolution of a front in a continuously stratified fluid are carried out. They confirm the presence of RK instability with characteristics comparable to those found in the two-layer case. Finally, these simulations are used to study the nonlinear saturation of the RK modes. It is shown that saturation is achieved through the development of small-scale instabilities along the front which modify the mean flow so as to stabilize the RK mode. Remarkably, the developing instability leads to conversion of kinetic energy of the basic flow to potential energy, contrary to classical baroclinic instability.