The wake transition of the flow around two circular cylinders placed in staggered arrangements with fixed streamwise separation of 5D and cross-stream separation varying from 0 to 3D has been studied. The wake transition is compared to that of a single isolated cylinder. Linear stability analysis utilizing Floquet theory and direct numerical simulations using a spectral/hp element spatial discretization were carried out. The unstable modes that first appear in the wake transition of the flow around a single cylinder, which are the long-spanwise-wavelength mode A and the short-spanwise-wavelength mode B, are also found in the flow around the staggered arrangements. However, a third mode, referred to as mode C, is also present in the wake transition of the flow around staggered arrangements, depending on the relative positioning of the cylinders. This mode has an intermediate spanwise wavelength and period-doubling character. The structure and onset characteristics of mode C are analysed and the nonlinear character of the bifurcation for this mode is investigated.

Zero-pressure-gradient turbulent boundary-layer drag reduction by polymer injection has been studied with particle image velocimetry. Flow fields ranging from low to maximum drag reduction have been investigated. A previously developed technique – the (1 − y/δ) fit to the total shear stress profile – has been used to evaluate the skin friction, drag reduction and polymer stress. Current results agree well with the semi-log plot of drag reduction vs. normalized polymer flux which has been used by previous workers and can be used as a guide to optimize the use of polymer from a single injector. Detailed flow-field statistics show many special features that pertain to polymer flow. It is shown that the mean velocity responds quickly to the suddenly reduced wall shear stress associated with polymer injection. However, it takes a much longer time for the entire Reynolds shear stress profile to adjust to the same change. The Reynolds shear stress profiles in wall units can be higher than unity and this unique feature can be used to further judge whether the flow is in equilibrium. The streamwise evolution of drag reduction magnitude is used to divide the flow into three regions: development region; steady-state region; and depletion region. The polymer stress is estimated and found to be proportional to drag reduction in the depletion region, but not necessarily so in the other regions. The interaction between injected polymer and turbulent activity in a developing boundary-layer flow is dependent upon the flow history and it produces an equally complex relationship between polymer stress and drag reduction. The stress balance in the boundary layer and the dynamical contribution of the various stresses to the total stress are evaluated and it is seen that the polymer stresses can account for up to 25% of the total stress. This finding is in contrast to channel flows with homogeneous polymer injection where the polymer stress is found to account for up to 60% of the total stress.

This paper presents a comparative analysis of the budgets of acoustic energy and Myers' second-order ‘disturbance energy’ in a simple inhomogeneous flow with heat communication. The flow considered is non-diffusive and one-dimensional, with excitation by downstream-travelling acoustic and entropic disturbances. Two forms of heat communication are examined: a case with only steady heat communication and another in which unsteady heat addition cancels the generation of entropy disturbances throughout the inhomogeneous region.

It is shown that significant entropic disturbances are usually generated at low frequency when a flow with steady heat communication is excited either acoustically or entropically. However, for acoustic excitation and regardless of the form of heat communication, entropic disturbances are not created at high frequency, inferring that all source terms create mainly sound in this limit. A general method is therefore proposed for determining an approximate frequency beyond which the generation of entropy disturbances can be ignored, and the disturbance energy flux then approximates the acoustic energy flux. This frequency is shown to depend strongly on the problem under investigation, which is expected to have practical significance when studying sound generation and propagation in combusting flows in particular. Further, sound is shown to be generated by fluid motion experiencing only steady heat communication, which is consistent with the known mechanism of sound generation by the acceleration of density disturbances.

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cylinder and, for large ones, pools at the cylinder's bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

The goal of this study is to demonstrate numerically that certain hydrodynamic systems, derived from inelastic kinetic theory, give fairly good descriptions of rapid granular flows even if they are way beyond their supposed validity limits. A numerical hydrodynamic solver is presented for a vibrated granular bed in two dimensions. It is based on a highly accurate shock capturing state-of-the-art numerical scheme applied to a compressible Navier–Stokes system for granular flow. The hydrodynamic simulation of granular flows is challenging, particularly in systems where dilute and dense regions occur at the same time and interact with each other. As a benchmark experiment, we investigate the formation of Faraday waves in a two-dimensional thin layer exposed to vertical vibration in the presence of gravity. The results of the hydrodynamic simulations are compared with those of event-driven molecular dynamics and the overall quantitative agreement is good at the level of the formation and structure of periodic patterns. The accurate numerical scheme for the hydrodynamic description improves the reproduction of the primary onset of patterns compared to previous literature. To our knowledge, these are the first hydrodynamic results for Faraday waves in two-dimensional granular beds that accurately predict the wavelengths of the two-dimensional standing waves as a function of the perturbation's amplitude. Movies are available with the online version of the paper.

A system of shallow water equations and a bed evolution equation are used to examine the evolution of perturbations on an erodible, initially plane beach subject to normal wave incidence. Both a permeable (under Darcy's law) and an impermeable beach are considered. It is found that alongshore-periodic morphological features reminiscent of swash beach cusps form after a number of incident wave periods on both beaches. On the permeable (impermeable) beach these patterns are accretional (erosional). In both cases flow is ‘horn divergent’. Spacings of the cusps are consistent with observations, and are close to those provided by a standing synchronous linear edge wave. An analysis of the processes leading to bed change is presented. Two physical mechanisms are identified: concentration gradient and flow divergence, which are dominant in the lower and upper swash respectively, and their difference over a wave cycle leads to erosion or deposition on an impermeable beach. Infiltration enters this balance in the upper swash. A bed wave of elevation is shown to advance up the beach at the tip of the uprush, with a smaller wave of depression on the backwash. It is found that cusp horns can grow by a positive feedback mechanism stemming from decreased (increased) backwash on positive (negative) bed perturbations.

We consider the nonlinear interactions between the steady Kelvin waves behind an advancing ship and an (unsteady) ambient wave. It is shown that, for moderately steep ship waves and/or ambient waves, third-order (quartet) resonant interaction among the two wave systems could occur, leading to the generation of a new propagating wave along a specific ray in the Kelvin wake. The wave vector of the generated wave as well as the angle of the resonance ray are determined by the resonance condition and are functions of the ship forward speed and the wave vector of the ambient wave. To understand the resonance mechanism and the characteristics of the generated wave, we perform theoretical analyses of this problem using two related approaches. To obtain a relatively simple model in the form of a nonlinear Schrödinger (NLS) equation for the evolution of the resonant wave, we first consider a multiple-scale approach assuming locally discrete Kelvin wave components, with constant wave vectors but varying amplitudes along the resonance ray. This NLS model captures the key resonance mechanism but does not account for the detuning effect associated with the wave vector variation of Kevin waves in the neighbourhood of the resonance ray. To obtain the full quantitative features and evolution characteristics, we also consider a more complete model based on Zakharov's integral equation applied in the context of a continuous wave vector spectrum. The resulting evolution equation can be reduced to an NLS form with, however, cross-ray variable coefficients, on imposing a narrow-band assumption valid in the neighbourhood of the resonance ray. As expected, the two models compare well when wave vector detuning is small, in the near wake close to the ray. To verify the analyses, direct high-resolution simulations of the nonlinear wave interaction problem are obtained using a high-order spectral method. The simulations capture the salient features of the resonance in the near wake of the ship, with good agreements with theory for the location of the resonance and the growth rate of the generated wave.

We consider, as a case study, the optimization of mixing protocols for a two-dimensional, piecewise steady, nonlinear flow, the sine flow, for both the advective–diffusive and purely advective cases. We use the mix-norm as the cost function to be minimized by the optimization procedure. We show that the cost function possesses a complex structure of local minima of nearly the same values and, consequently, that the problem possesses a large number of sub-optimal protocols with nearly the same mixing efficiency as the optimal protocol. We present a computationally efficient optimization procedure able to find a sub-optimal protocol through a sequence of short-time-horizon optimizations. We show that short-time-horizon optimal mixing protocols, although sub-optimal, are both feasible and efficient at mixing flows with and without diffusion. We also show that these optimized protocols can be derived, at lower computational cost, for purely advective flows and successfully transported to advective–diffusive flows with small molecular diffusivity. We characterize our results by discussing the asymptotic properties of the optimized protocols both in the pure advection and in the advection–diffusion cases. In particular, we quantify the mixing efficiency of the optimized protocols using the Lyapunov exponents and Poincaré sections for the pure advection case, and the eigenvalue–eigenfunction spectrum for the advection–diffusion case. Our results indicate that the optimization over very short-time horizons could in principle be used as an on-line procedure for enhancing mixing in laboratory experiments, and in future engineering applications.

Series of concentration and velocity patterns are found for the rotating suspension of non-Brownian settling particles in a completely filled horizontal cylinder. Individual flow states, or phases, are studied using both side and cross-sectional imaging to examine the detailed flow structures. The overall steady-state phase diagram of the system is mapped over a wide range of the rotation rate and fluid viscosity. Effects of the particle radius a, volume fraction φ, and cylinder radius R on the transition boundaries are examined. It is found that the phase diagram of the rotating suspensions can be divided into three regions, in which the transition boundaries obey different scaling laws. A theoretical attempt is made to understand the scaling behaviour of the transition boundaries. The theoretical understanding is achieved at three different levels: a general dimensional consideration, a scaling analysis on the continuum equations of motion, and a specific instability calculation for the transition boundary at the centrifugal limit.

Experimental and theoretical investigations of undulation patterns in high-pressure inclined layer gas convection at a Prandtl number near unity are reported. Particular focus is given to the competition between the spatiotemporal chaotic state of undulation chaos and stationary patterns of ordered undulations. In experiments, a competition and bistability between the two states is observed, with ordered undulations most prevalent at higher Rayleigh number. The spectral pattern entropy, spatial correlation lengths and defect statistics are used to characterize the competing states. The experiments are complemented by a theoretical analysis of the Oberbeck–Boussinesq equations. The stability region of the ordered undulations as a function of their wave vectors and the Rayleigh number is obtained with Galerkin techniques. In addition, direct numerical simulations are used to investigate the spatiotemporal dynamics. In the simulations, both ordered undulations and undulation chaos were observed dependent on initial conditions. Experiment and theory are found to agree well.

The inviscid waves propagating on a Lamb–Oseen vortex in a rotating medium for an unstratified fluid and for a strongly stratified fluid are analysed using numerical and asymptotic approaches. By a local Lagrangian description, we first provide the characteristics of the local plane waves (inertia–gravity waves) as well as the local growth rate associated with the centrifugal instability when the vortex is unstable. A global WKBJ approach is then used to determine the frequencies of neutral core modes and neutral ring modes. We show that these global Kelvin modes only exist in restricted domains of the parameters. The consequences of these domain limitations for the occurrence of the elliptic instability are discussed. We argue that in an unstratified fluid the elliptic instability should be active in a small range of the Coriolis parameter which could not have been predicted from a local approach. The wavenumbers of the sinuous modes of the elliptic instability are provided as a function of the Coriolis parameter for both an unstratified fluid and a strongly stratified fluid.

Collective diffusivity in a suspension of rigid particles in steady linear viscous flows is evaluated by investigating the dynamics of the time correlation of long-wavelength density fluctuations. In the absence of hydrodynamic interactions between suspended particles in a dilute suspension of identical hard spheres, closed-form asymptotic expressions for the collective diffusivity are derived in the limits of low and high Péclet numbers, where the Péclet number with being the shear rate and D0 = kBT/6πη a is the Stokes–Einstein diffusion coefficient of an isolated sphere of radius a in a fluid of viscosity η. The effect of hydrodynamic interactions is studied in the analytically tractable case of weakly sheared (Pe ≪ 1) suspensions.

For strongly sheared suspensions, i.e. at high Pe, in the absence of hydrodynamics the collective diffusivity Dc = 6 Ds∞, where Ds∞ is the long-time self-diffusivity and both scale as , where φ is the particle volume fraction. For weakly sheared suspensions it is shown that the leading dependence of collective diffusivity on the imposed flow is proportional to D0 φPeÊ, where Ê is the rate-of-strain tensor scaled by , regardless of whether particles interact hydrodynamically. When hydrodynamic interactions are considered, however, correlations of hydrodynamic velocity fluctuations yield a weakly singular logarithmic dependence of the cross-gradient-diffusivity on k at leading order as ak → 0 with k being the wavenumber of the density fluctuation. The diagonal components of the collective diffusivity tensor, both with and without hydrodynamic interactions, are of O(φPe2), quadratic in the imposed flow, and finite at k = 0.

At moderate particle volume fractions, 0.10 ≤ φ ≤ 0.35, Brownian Dynamics (BD) numerical simulations in which there are no hydrodynamic interactions are performed and the transverse collective diffusivity in simple shear flow is determined via time evolution of the dynamic structure factor. The BD simulation results compare well with the derived asymptotic estimates. A comparison of the high-Pe BD simulation results with available experimental data on collective diffusivity in non-Brownian sheared suspensions shows a good qualitative agreement, though hydrodynamic interactions prove to be important at moderate concentrations.

This paper deals with the stability of water waves in a shear flow. Both temporal and spatial growth rates are derived. A carefully designed numerical solver enables us to extend the range of previous calculations, and to obtain results for larger wavelengths (up to 20 cm) and stronger winds (up to a friction-velocity of 1 m s−1). The main finding is the appearance of a second unstable mode which often turns out to be the dominant one. A comparison between results from the viscous model (Orr–Sommerfeld equations) and those of the inviscid model (Rayleigh equations), for 18 cm long waves, reveals some similarity in the structure of the eigenfunctions, but a significant difference in the imaginary part of the eigenvalues (i.e. the growth rate). It is found that the growth rate for the viscous model is 10 fold larger than that of the inviscid one.

A nonlinear analysis has been made to determine the effects of the free surface and transverse gravity field on the steady cavity flow past a shaped hydrofoil beneath the free surface. A closed cavity wake model has been proposed, and a method for the determination of an analytical function from its modulus and argument on the region boundary has been employed to derive the complex flow potential in a parameter plane. The boundary-value problem is reduced to a system of integral and integro-differential equations in the velocity modulus along the free boundaries and the velocity angle along the hydrofoil surface, both written as a function of parametric variables. The system of equations is solved through a numerical procedure, which is validated in the cases of a cavitating flat plate and non-cavitating shaped hydrofoils by comparison with data available in the literature. The results are presented in a wide range of Froude numbers and depths of submergence in terms of the cavity and free-surface shapes and force coefficients. The influences of the free surface and gravity on the aforementioned quantities are discussed. The limiting cavity size corresponding to zero cavitation number in the presence of gravity is found for various initial flow parameters.

In this paper we systematically investigate strong interactions between two like-signed quasi-geostrophic vortices containing different uniform potential vorticity. The interaction depends on six parameters: the potential vorticity ratio between the two vortices, their volume ratio, their individual height-to-width aspect ratio, their vertical offset, and their horizontal separation distance. We first determine the conditions under which a strong interaction may occur. To that end, we calculate equilibrium states using an asymptotic approach which models the vortices as ellipsoids and we additionally assess their linear stability. It is found that vortices having similar potential vorticity interact strongly (e.g. merge) at closer separation distances than do vortices with a dissimilar potential vorticity. This implies that interactions between vortices having significantly different potential vorticity may be more destructive, for a given separation distance. This is confirmed by investigating the nonlinear evolution of the vortices over a subset of the full parameter space, solving the full dynamical quasi-geostrophic equations. Many forms of interaction occur, but merger or partial merger (where the largest vortex grows in volume) is mostly observed for interactions between vortices of similar potential vorticity.

Two methods for calculating the available potential energy (APE) of an isolated feature in a density-stratified fluid, such as an internal solitary wave or an eddy, are compared. The first formulation calculates the APE by integrating the perturbation potential energy density Ew. The second uses an available potential energy density Ea. Both formulations are based on the reference density obtained by adiabatically rearranging the density field to a state of minimum energy. It is shown, under more general conditions than used previously, that (i) the integrals of Ew and Ea over a finite domain are identical; and (ii) that for an isolated feature in an unbounded domain, the far-field density can be used as the reference density if Ea is used to find the APE. This is not the case when Ew is used, hence use of the available potential energy density formulation is simpler in this situation.

We obtain approximate solutions to the equations that govern the shape of giant unilamellar vesicles (GUVs) with two fluid phases. The equations involve a dimensionless small parameter related to the resistance to changes in its local mean curvature. Asymptotic solutions for the shape are obtained up to and including terms of first order in the small parameter. At this order, we determine a relationship between the tangent angle at the interface and the difference in the Gaussian curvature stiffnesses of the co-existing phases. This relationship demonstrates that a difference in the Gaussian curvature stiffnesses moves the phase boundary away from the neck, as determined in previous numerical studies. The analytical expression for the tangent angle obtained here can be used to determine elastic parameters for the membranes from experimental data. Use of the analytical expression will eliminate the need for the repeated generation of numerical solutions in the estimation of the material parameters. Our analytical solution also reduces the number of measurements needed as inputs for an existing boundary layer analysis.

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.