Marusic et al. (J. Fluid Mech., vol. 716, 2013, R3) show the first clear evidence of universal logarithmic scaling emerging naturally (and simultaneously) in the mean velocity and the intensity of the streamwise velocity fluctuations about that mean in canonical turbulent flows near walls. These observations represent a significant advance in understanding of the behaviour of wall turbulence at high Reynolds number, but perhaps the most exciting implication of the experimental results lies in the agreement with the predictions of such scaling from a model introduced by Townsend (J. Fluid Mech., vol. 11, 1961, pp. 97–120), commonly termed the attached eddy hypothesis. The elegantly simple, yet powerful, study by Marusic et al. should spark further investigation of the behaviour of all fluctuating velocity components at high Reynolds numbers and the outstanding predictions of the attached eddy hypothesis.

Spanwise-periodic roughness designed to excite selected wavelengths of stationary cross-flow modes was investigated in a three-dimensional boundary layer at Mach 3.5. The test model was a sharp-tipped $1{4}^{\circ } $ right-circular cone. The model and integrated sensor traversing system were placed in the Mach 3.5 supersonic low disturbance tunnel (SLDT) equipped with an axisymmetric ‘quiet design’ nozzle at NASA Langley Research Center. The model was oriented at a $4. {2}^{\circ } $ angle of attack to produce a mean cross-flow velocity component in the boundary layer over the cone. The research examined both passive and active surface roughness. The passive roughness consisted of indentations (dimples) that were evenly spaced around the cone at an axial location that was just upstream of the first linear stability neutral growth branch for cross-flow modes. The active roughness consisted of an azimuthal array of micrometre-sized plasma actuators that were designed to produce the effect of passive surface bumps. Two azimuthal mode numbers of the passive and active patterned roughness were examined. One had an azimuthal mode number that was in the band of initially amplified stationary cross-flow modes. This was intended to represent a controlled baseline condition. The other azimuthal mode number was designed to suppress the growth of the initially amplified stationary cross-flow modes and thereby increase the transition Reynolds number. The results showed that the stationary cross-flow modes were receptive to both the passive and active patterned roughness. Only the passive roughness was investigated at a unit Reynolds number where transition would occur on the cone. Transition front measurements using the Preston tube approach indicated that the transition Reynolds number had increased by 35 % with the subcritical wavenumber roughness compared with the baseline smooth tip cone, and by 40 % compared with the critical wavenumber roughness. Based on the similarities in the response of the stationary cross-flow modes with the active roughness, we expect it would produce a similar transition delay.

We study the mechanisms of centrifugal instability and its eventual self-limitation, as well as regenerative instability on a vortex column with a circulation overshoot (potentially unstable) via direct numerical simulations of the incompressible Navier–Stokes equations. The perturbation vorticity (${\boldsymbol{\omega} }^{\prime } $) dynamics are analysed in cylindrical ($r, \theta , z$) coordinates in the computationally accessible vortex Reynolds number, $\mathit{Re}({\equiv }\mathrm{circulation/viscosity} )$, range of 500–12 500, mostly for the axisymmetric mode (azimuthal wavenumber $m= 0$). Mean strain generates azimuthally oriented vorticity filaments (i.e. filaments with azimuthal vorticity, ${ \omega }_{\theta }^{\prime } $), producing positive Reynolds stress necessary for energy growth. This ${ \omega }_{\theta }^{\prime } $ in turn tilts negative mean axial vorticity, $- {\Omega }_{z} $ (associated with the overshoot), to amplify the filament, thus causing instability. (The initial energy growth rate (${\sigma }_{r} $), peak energy (${G}_{\mathit{max}} $) and time of peak energy (${T}_{p} $) are found to vary algebraically with $\mathit{Re}$.) Limitation of vorticity growth, also energy production, occurs as the filament moves the overshoot outward, hence lessening and shifting $\vert {- }{\Omega }_{z} \vert $, while also transporting the core $+ {\Omega }_{z} $, to the location of the filament. We discover that a basic change in overshoot decay behaviour from viscous to inviscid occurs at $Re\sim 5000$. We also find that the overshoot decay time has an asymptotic limit of 45 turnover times with increasing $\mathit{Re}$. After the limitation, the filament generates negative Reynolds stress, concomitant energy decay and hence self-limitation of growth; these inviscid effects are enhanced further by viscosity. In addition, the filament transports angular momentum radially inward, which can produce a new circulation overshoot and renewed instability. Energy decays at the $\mathit{Re}$ studied, but, at higher $\mathit{Re}$, regenerative growth of energy is likely due to the renewed mean shearing. New generation of overshoot and Reynolds stress is examined using a helical ($m= 1$) perturbation. Regenerative energy growth, possibly resulting in even vortex breakup, can be triggered by this new overshoot at practical $\mathit{Re}$ (${\sim }1{0}^{6} $ for trailing vortices), which are currently beyond the computational capability.

The physical mechanism by which large-scale vortical structures augment convective heat transfer is a fundamental problem of turbulent flows. To investigate this phenomenon, two separate experiments were performed using simultaneous heat transfer and flow field measurements to study the vortex–wall interaction. Individual vortices were identified and studied both as part of a turbulent stagnation flow and as isolated vortex rings impacting on a surface. By examining the temporal evolution of both the flow field and the resulting heat transfer, it was observed that the surface thermal transport was governed by the transient interaction of the vortical structure with the wall. The magnitude of the heat transfer augmentation was dependent on the instantaneous strength, size and position of the vortex relative to the boundary layer. Based on these observations, an analytical model was developed from first principles that predicts the time-resolved surface convection using the transient properties of the vortical structure during its interaction with the wall. The analytical model was then applied, first to the simplified vortex ring model and then to the more complex stagnation region experiments. In both cases, the model was able to accurately predict the time-resolved convection resulting from the vortex interactions with the wall. These results reveal the central role of large-scale turbulent structures in the augmentation of thermal transport and establish a simple model for quantitative predictions of transient heat transfer.

The initial growth of a disturbance induced by a near-wall circular cylinder in a flat-plate boundary layer is experimentally investigated using both particle image velocimetry and hydrogen bubble visualization techniques. The secondary spanwise vortices appear in the near-wall region as a direct response to the outside passing wake vortices, consistent with previous studies on similar models (Pan et al., J. Fluid Mech., vol. 603, 2008, pp. 367–389; Mandal & Dey, J. Fluid Mech., vol. 684, 2011, pp. 60–84). The streamwise variation of the total disturbance energy within the boundary layer shows a two-stage growth, which characterizes the initial transition process: the first exponential growth stage, followed by a region with slower growth rate. It is revealed that these two stages of growth are related to the formation and the destabilization of the secondary vortex in the near-wall region. The technique of dynamic mode decomposition is used to decompose the total disturbance into temporally orthogonalized modes, and it shows that the first growth stage largely results from the increased disturbance at the same frequency as that of the wake vortex shedding, while the second growth stage comprises the disturbance growth in a number of frequencies, especially the lower ones.

The rapid granular plane Couette flow is known to be unstable to pure spanwise perturbations (i.e. perturbations having variations only along the mean vorticity direction) below some critical density (volume fraction of particles), resulting in the banding of particles along the mean vorticity direction: this is dubbed ‘vorticity banding’ instability. The nonlinear state of this instability is analysed using quintic-order Landau equation that has been derived from the pertinent hydrodynamic equations of rapid granular fluid. We have found analytical solutions for related modal/harmonic equations of finite-size perturbations up to quintic order in perturbation amplitude, leading to an exact calculation of both first and second Landau coefficients. This helped to identify the bistable nature of nonlinear vorticity-banding instability for a range of densities spanning from moderately dense to dense flows. For perturbations with small spanwise wavenumbers, the bifurcation scenario for vorticity banding unfolds, with increasing density from the dilute limit, as supercritical pitchfork $\rightarrow $ subcritical pitchfork $\rightarrow $ subcritical Hopf bifurcations. The transition from supercritical to subcritical pitchfork bifurcations is found to occur via the appearance of a degenerate/bicritical point (at which both the linear growth rate and the first Landau coefficient are simultaneously zero) that divides the critical line into two parts: one representing the first-order and the other the second-order phase transitions. Both subcritical oscillatory and stationary solutions have also been uncovered for dilute and dense flows, respectively, when the spanwise wavenumber is large. In all cases, the nonlinear solutions correspond to inhomogeneous states of shear stress and pressure along the vorticity direction, and hence are analogues of vorticity banding in other complex fluids. The quartic-order mean-flow resonance is evidenced in the parameter space for which the second Landau coefficient undergoes a jump discontinuity of infinite order. The importance of retaining higher-order terms to calculate the second Landau coefficient and their possible effects on the nature of bifurcations are elucidated.

We investigate the flow in a spherical shell subject to a time harmonic oscillation of its rotation rate, also called longitudinal libration, when the oscillation frequency is larger than twice the mean rotation rate. In this frequency regime, no inertial waves are directly excited by harmonic forcing. We show, however, that, through nonlinear interactions in the Ekman layers, it can generate a strong mean zonal flow in the interior. An analytical theory is developed using a perturbative approach in the limit of small libration amplitude $\varepsilon $ and small Ekman number $E$. The mean flow is found to be at leading order an azimuthal flow that scales as the square of the libration amplitude and depends only on the cylindrical radius coordinate. The mean flow also exhibits a discontinuity across the cylinder tangent to the inner sphere. We show that this discontinuity can be smoothed through multi-scale Stewartson layers. The mean flow is also found to possess a weak axial flow that scales as $O({\varepsilon }^{2} {E}^{5/ 42} )$ in the Stewartson layers. The analytical solution is compared to axisymmetric numerical simulations, and a good agreement is demonstrated.

When a pair of tandem cylinders is immersed in a flow the downstream cylinder can be excited into wake-induced vibrations (WIV) due to the interaction with vortices coming from the upstream cylinder. Assi, Bearman & Meneghini (J. Fluid Mech., vol. 661, 2010, pp. 365–401) concluded that the WIV excitation mechanism has its origin in the unsteady vortex–structure interaction encountered by the cylinder as it oscillates across the wake. In the present paper we investigate how the cylinder responds to that excitation, characterising the amplitude and frequency of response and its dependency on other parameters of the system. We introduce the concept of wake stiffness, a fluid dynamic effect that can be associated, to a first approximation, with a linear spring with stiffness proportional to $\mathit{Re}$ and to the steady lift force occurring for staggered cylinders. By a series of experiments with a cylinder mounted on a base without springs we verify that such wake stiffness is not only strong enough to sustain oscillatory motion, but can also dominate over the structural stiffness of the system. We conclude that while unsteady vortex–structure interactions provide the energy input to sustain the vibrations, it is the wake stiffness phenomenon that defines the character of the WIV response.

We study the stability of a viscous incompressible fluid ring on a partially wetting substrate within the framework of long-wave theory. We discuss the conditions under which a static equilibrium of the ring is possible in the presence of contact angle hysteresis. A linear stability analysis (LSA) of this equilibrium solution is carried out by using a slip model to account for the contact line divergence. The LSA provides specific predictions regarding the evolution of unstable modes. In order to describe the evolution of the ring for longer times, a quasi-static approximation is implemented. This approach assumes a quasi-static evolution and takes into account the concomitant variation of the instantaneous growth rates of the modes responsible for either collapse of the ring into a single central drop or breakup into a number of droplets along the ring periphery. We compare the results of the LSA and the quasi-static model approach with those obtained from nonlinear numerical simulations using a complementary disjoining pressure model. We find remarkably good agreement between the predictions of the two models regarding the expected number of drops forming during the breakup process.

We investigate phase-averaged equations describing the spectral evolution of dispersive water waves subject to weakly nonlinear quartet interactions. In contrast to Hasselmann’s kinetic equation, we include the effects of near-resonant quartet interaction, leading to spectral evolution on the ‘fast’ $O({\epsilon }^{- 2} )$ time scale, where $\epsilon $ is the wave steepness. Such a phase-averaged equation was proposed by Annenkov & Shrira (J. Fluid Mech., vol. 561, 2006b, pp. 181–207). In this paper we rederive their equation taking some additional higher-order effects related to the Stokes correction of the frequencies into account. We also derive invariants of motion for the phase-averaged equation. A numerical solver for the phase-averaged equation is developed and successfully tested with respect to convergence and conservation of invariants. Numerical simulations of one- and two-dimensional spectral evolution are performed. It is shown that the phase-averaged equation describes the ‘fast’ evolution of a spectrum on the $O({\epsilon }^{- 2} )$ time scale well, in good agreement with Monte-Carlo simulations using the Zakharov equation and in qualitative agreement with known features of one- and two-dimensional spectral evolution. We suggest that the phase-averaged equation may be a suitable replacement for the kinetic equation during the initial part of the evolution of a wave field, and in situations where ‘fast’ field evolution takes place.

The primary instability of liquid film flow along periodically corrugated substrates is studied experimentally. Two different wall shapes, of the same wavelength and height, are tested for a wide range of inclinations. It is found that, beyond a specific inclination, a new instability mode occurs before the classical, convective, long-wave one. This is a short, travelling wave, which is highly regular and persistently two-dimensional, and appears to be a global mode. The exact shape of the corrugations has a leading-order effect on the inclination at which the new mode appears and on its wavelength at inception. Compared with the behaviour of film flow on a flat substrate, the presently tested periodic walls are found to delay very significantly, but each one to a different extent, the onset of the primary instability. This delay increases with inclination, and presents a distinct discontinuity when transition from the long- to the short-wave mode takes place.

Estimating the energetic costs of undulatory swimming is important for biologists and engineers alike. To calculate these costs it is crucial to evaluate the drag forces originating from skin friction. This topic has been controversial for decades, some claiming that animals use ingenious mechanisms to reduce the drag and others hypothesizing that undulatory swimming motions induce a drag increase because of the compression of the boundary layers. In this paper, we examine this latter hypothesis, known as the ‘Bone–Lighthill boundary-layer thinning hypothesis’, by analysing the skin friction in different model problems. First, we study analytically the longitudinal drag on a yawed cylinder in a uniform flow by using the approximation of the momentum equations in the laminar boundary layers. This allows us to demonstrate and generalize a result first observed semi-empirically by G.I. Taylor in the 1950s: the longitudinal drag scales as the square root of the normal velocity component. This scaling arises because the fluid particles accelerate as they move around the cylinder. Next we propose an analogue two-dimensional problem where the same scaling law is recovered by artificially accelerating the flow in a channel of finite height. This two-dimensional problem is then simulated numerically to assess the robustness of the analytical results when inhomogeneities and unsteadiness are present. It is shown that spatial or temporal changes in the normal velocity usually tend to increase the skin friction compared with the ideal steady case. Finally, these results are discussed in the context of swimming energetics. We find that the undulatory motions of swimming animals increase their skin friction drag by an amount that closely depends on the geometry and the motion. For the model problem considered in this paper the increase is of the order of 20 %.

A thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small-Mach-number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately $0. 5$. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased.

We analyse theoretically the interaction between water waves and a thin layer of fluid mud on a sloping seabed. Under the assumption of long waves in shallow water, weakly nonlinear and dispersive effects in water are considered. The fluid mud is modelled as a thin layer of viscoelastic continuum. Using the constitutive coefficients of mud samples from two field sites, we examine the interaction of nonlinear waves and the mud motion. The effects of attenuation on harmonic evolution of surface waves are compared for two types of mud with distinct rheological properties. In general mud dissipation is found to damp out surface waves before they reach the shore, as is known in past observations. Similar to the Eulerian current in an oscillatory boundary layer in a Newtonian fluid, a mean displacement in mud is predicted which may lead to local rise of the sea bottom.

The torque in turbulent Taylor–Couette flows for shear Reynolds numbers $R{e}_{S} $ up to $3\times 1{0}^{4} $ at various mean rotations is studied by means of direct numerical simulations for a radius ratio of $\eta = 0. 71$. Convergence of simulations is tested using three criteria of which the agreement of dissipation values estimated from the torque and from the volume dissipation rate turns out to be most demanding. We evaluate the influence of Taylor vortex heights on the torque for a stationary outer cylinder and select a value of the aspect ratio of $\Gamma = 2$, close to the torque maximum. The local transport resulting in the torque is investigated via the transverse current ${J}^{\omega } $ which measures the transport of angular momentum and can be computed from the velocity field. The typical spatial distribution of the individual convective and viscous contributions to the local torque is analysed for a turbulent flow case. To characterize the turbulent statistics of the transport, probability density functions (p.d.f.s) of local current fluctuations are compared with experimental wall shear stress measurements. P.d.f.s of instantaneous torques reveal a fluctuation enhancement in the outer region for strong counter-rotation. Moreover, we find for simulations realizing the same shear $R{e}_{S} \geq 2\times 1{0}^{4} $ the formation of a torque maximum for moderate counter-rotation with angular velocities ${\omega }_{o} \approx - 0. 4\hspace{0.167em} {\omega }_{i} $. In contrast, for $R{e}_{S} \leq 4\times 1{0}^{3} $ the torque features a maximum for a stationary outer cylinder. In addition, the effective torque scaling exponent is shown to also depend on the mean rotation state. Finally, we evaluate a close connection between boundary-layer thicknesses and the torque.

We consider Marangoni convection in a heated layer of a binary liquid. The solute is a surfactant, which is present in both surface and bulk phases; the bulk gradient of the concentration is formed due to the Soret effect. Linear stability analysis demonstrates a well-pronounced stabilization of the layer due to the adsorption kinetics and advection of the surface phase. We derive nonlinear amplitude equations for longwave perturbations in the case of fast sorption kinetics (small Langmuir number) and demonstrate that with increase in the effect of the adsorption, subcritical excitation occurs. In the case of a finite Langmuir number, the weakly nonlinear problem is ill-posed. A physical mechanism of subcritical bifurcation is discussed.

Vortices in stably stratified fluids generally have a pancake shape with a small vertical thickness compared with their horizontal size. In order to understand what mechanism determines their minimum thickness, the linear stability of an axisymmetric pancake vortex is investigated as a function of its aspect ratio $\alpha $, the horizontal Froude number ${F}_{h} $, the Reynolds number $\mathit{Re}$ and the Schmidt number $\mathit{Sc}$. The vertical vorticity profile of the base state is chosen to be Gaussian in both radial and vertical directions. The vortex is unstable when the aspect ratio is below a critical value, which scales with the Froude number: ${\alpha }_{c} \sim 1. 1{F}_{h} $ for sufficiently large Reynolds numbers. The most unstable perturbation has an azimuthal wavenumber either $m= 0$, $\vert m\vert = 1$ or $\vert m\vert = 2$ depending on the control parameters. We show that the threshold corresponds to the appearance of gravitationally unstable regions in the vortex core due to the thermal wind balance. The Richardson criterion for shear instability based on the vertical shear is never satisfied alone. The dominance of the gravitational instability over the shear instability is shown to hold for a general class of pancake vortices with angular velocity of the form $\tilde {\Omega } (r, z)= \Omega (r)f(z)$ provided that $r\partial \Omega / \partial r\lt 3\Omega $ everywhere. Finally, the growth rate and azimuthal wavenumber selection of the gravitational instability are accounted well by considering an unstably stratified viscous and diffusive layer in solid body rotation with a parabolic density gradient.

This work builds on the foundation laid by Benney & Timson (Stud. Appl. Maths, vol. 63, 1980, pp. 93–98), who examined the flow near a contact line and showed that, if the contact angle is $18{0}^{\circ } $, the usual contact-line singularity does not arise. Their local analysis, however, does not allow one to determine the velocity of the contact line and their expression for the shape of the free boundary involves undetermined constants. The present paper considers two-dimensional Couette flows with a free boundary, for which the local analysis of Benney & Timson can be complemented by an analysis of the global flow (provided that the slope of the free boundary is small, so the lubrication approximation can be used). We show that the undetermined constants in the solution of Benney & Timson can all be fixed by matching the local and global solutions. The latter also determines the contact line’s velocity, which we compute among other characteristics of the global flow. The asymptotic model derived is used to examine steady and evolving Couette flows with a free boundary. It is shown that the latter involve brief intermittent periods of rapid acceleration of contact lines.

Shock wave attenuation by means of rigid porous media is often applied when protective structures are dealt with. The passage of a shock wave through a layer of porous medium is accompanied by diffractions and viscous effects that attenuate and weaken the transmitted shock, thus reducing the load that develops on the target wall that is placed behind the protective layer. In the present study, the parameters governing the pressure build-up on the target wall are experimentally investigated using a shock tube facility. Different porous samples are impinged by normal shock waves of various strengths and the subsequent pressure histories that are developed on the target wall are recorded. In addition, different standoff distances from the target wall are investigated. Assuming that the flow through the porous medium is close to being isentropic enabled us to develop a general constitutive model for predicting the pressure history developed on the target wall. This model can be applied to predict the pressure build-up on the target wall for any pressure history that is imposed on the front face of the porous sample without the need to conduct numerous experiments. Results obtained by other investigators are found to be in very good agreement with the predictions of the presently developed constitutive model.

The viscously dominated, low-Reynolds-number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behaviour when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low-density fluid through a viscously deformable conduit. These flows are modelled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed-form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviours are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small- to moderate-amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented.