The hydroelastic waves in a channel covered by an ice sheet, without or with crack and subject to various edge constraints at channel banks, are investigated based on the linearized velocity potential theory for the fluid domain and the thin-plate elastic theory for the ice sheet. An effective analytical solution procedure is developed through expanding the velocity potential and the fourth derivative of the ice deflection to a series of cosine functions with unknown coefficients. The latter are integrated to obtain the expression for the deflection, which involves four constants. The procedure is then extended to the case with a longitudinal crack in the ice sheet by using the Dirac delta function and its derivatives at the crack in the dynamic equation, with unknown jumps of deflection and slope at the crack. Conditions at the edges and crack are then imposed, from which a system of linear equations for the unknowns is established. From this, the dispersion relation between the wave frequency and wavenumber is found, as well as the natural frequency of the channel. Extensive results are then provided for wave celerity, wave profiles and strain in the ice sheet. In-depth discussions are made on the effects of the edge condition, and the crack.

This paper proposes a theoretical model to describe previous laboratory observations of the dynamics of debris accumulations around bridge piers of cylindrical shape. The model is based on the assumption that the observed dynamics is mainly governed by dynamic changes of the point of application of the drag force exerted on a solid body formed by debris accumulated around a pier. A phase-plane analysis of the resulting nonlinear system of ordinary differential equations shows that the model captures the main patterns observed in previous laboratory experiments, including an oscillatory motion and the removal of debris from the pier by the flow. The model provides a theoretical basis for the analysis of the conditions required for debris jams to remain stable over long periods of exposure to impinging flow. Namely, the model indicates that the stability of debris accumulations primarily depends on geometrical asymmetry and on the length of the extension downstream of the pier. The former induces the torque required to rotate the jam about the pier, while the latter produces a stabilising effect after the body rotates.

Front-propagating systems provide some of the most fundamental physical examples of interfacial instability and pattern formation. However, their nonlinear dynamics is rarely addressed. Here, we present an experimental study of air displacing a viscous fluid within a collapsed, compliant channel – a model for pulmonary airway reopening. Air injected at a constant flow rate from one end of the liquid-filled, collapsed channel results in the propagation of a reopening finger. Depending on the imposed flow rate, we observe a wide variety of finger-tip morphologies, which occur persistently or evolve transiently as the finger propagates. Persistent fingers are stable in the sense that they propagate with approximately constant bubble pressure. We find that their pressure increases monotonically as a function of bubble speed along two disconnected lines. Although the line associated with higher bubble speed exhibits a minimum pressure value, the low-speed line supports finger propagation down to the smallest flow rates investigated. We present evidence that the lower and higher-speed fingers are dominated by viscous and elastic forces, respectively. We also find a range of bubble speeds separating the two pressure lines where no stable finger propagation is observed. Instead, complex transient dynamics leads to the long-term selection of stable fingers depending on initial conditions. The early transient evolution of these fingers is characterised by an increase in bubble pressure alongside a reduction in bubble speed. We hypothesise the existence of a weakly unstable, steady mode, which orchestrates the transient evolution of the finger towards either low- or high-speed modes of propagation.

When open-cut mines are eventually abandoned, they leave a large hole with sloping sides. The hole fills with rain water, and there is also contaminated run-off from surrounding land, that moves through the rock and eventually through the sloping sides of the abandoned mine. This paper considers a two-dimensional unsteady model motivated by this leaching flow through the rock and into the rain-water reservoir. The stability of the interface between the two fluids is analysed in the inviscid limit. A viscous Boussinesq model is also presented, and a closed-form solution is presented to this problem, after it has been linearized in a manner consistent with Boussinesq theory. That solution suggests that the interfacial zone is effectively neutrally stable as it evolves in time. However, an asymptotic theory in the interfacial region shows the interface to be unstable. In addition, the nonlinear Boussinesq model is solved using a spectral method. Interfacial travelling waves and roll-up are observed and discussed, and compared against the predictions of asymptotic Boussinesq theory.

In turbulent flows subject to strong background rotation, the advective mechanisms of turbulence are superseded by the propagation of inertial waves, as the effects of rotation become dominant. While this mechanism has been identified experimentally (Dickinson & Long, J. Fluid Mech., vol. 126, 1983, pp. 315–333; Davidson, Staplehurst & Dalziel, J. Fluid Mech., vol. 557, 2006, pp. 135–144; Staplehurst, Davidson & Dalziel, J. Fluid Mech., vol. 598, 2008, pp. 81–105; Kolvin et al.Phys. Rev. Lett., vol. 102, 2009, 014503), the conditions of the transition between the two mechanisms are less clear. We tackle this question experimentally by tracking the turbulent front away from a solid wall where jets enter an otherwise quiescent fluid. Without background rotation, this apparatus generates a turbulent front whose displacement recovers the $z(t)\sim t^{1/2}$ law classically obtained with an oscillating grid (Dickinson & Long, Phys. Fluids, vol. 21 (10), 1978, pp. 1698–1701) and we further establish the scale independence of the associated transport mechanism. When the apparatus is rotating at a constant velocity perpendicular to the wall where fluid is injected, not only does the turbulent front become mainly transported by inertial waves, but advection itself is suppressed because of the local deficit of momentum incurred by the propagation of these waves. Scale-by-scale analysis of the displacement of the turbulent front reveals that the transition between advection and propagation is local both in space and spectrally, and takes place when the Rossby number based on the considered scale is of order unity, or equivalently, when the scale-dependent group velocity of inertial waves matched the local advection velocity.

Zuriguel et al. (Phys. Rev. Lett., vol. 95, 2005a, 258002) and Pacheco-Vázquez and Ruiz-Suárez (Nat. Commun., vol. 1, 2010, p. 123) demonstrated that there exists a long-ranged force between intruders placed at a certain distance from each other in granular flow. The origin of these long-ranged forces, induced collectively by the grains, has not been fully understood. In our work, we provide a unified explanation for the origin of both attraction and repulsion between two intruders in terms of the building up of force chains and their subsequent buckling. The surface and shear zone of the other intruder makes a significant contribution to the strength or buckling of the force chains. Bernoulli’s effect used in earlier studies predicts the nature of these forces, viz., attraction or repulsion, correctly but is not well supported as observed in our study. The time-averaged flow fields around the intruders also support our explanation for the origin as evidenced by the burst in kinetic energy and granular temperature. The model proposed in this work predicts the qualitative trend of the sideways force with the separation between the intruders by combining Bernoulli’s equation with a minimum contact criterion of force chains. There exists an equilibrium at which the intruders neither attract nor repel each other and a certain separation distance where maximum attraction occurs between the two intruders. The effects of the static pressure, the velocity of the moving intruders and the friction coefficient on the attraction or repulsion force between the intruders have also been explored in our system.

Marangoni flow is the motion induced by a surface tension gradient along a fluid–fluid interface. In this study, we report a Marangoni flow generated when a bath of surfactant contacts a pre-wetted film of deionized water on a vertical substrate. The thickness profile of the pre-wetted film is set by gravitational drainage and so varies with the drainage time. The surface tension is lower in the bath due to the surfactant, and thus a liquid film climbs upwards along the vertical substrate due to the surface tension difference. Particle tracking velocimetry is performed to measure the dynamics in the film, where the mean fluid velocity reverses direction as the draining film encounters the front of the climbing film. The effect of the surfactant concentration and the pre-wetted film thickness on the film climbing is then studied. High-speed interferometry is used to measure the front position of the climbing film and the film thickness profile. As a result, higher surfactant concentration induces a faster and thicker climbing film. Also, for high surfactant concentrations, where Marangoni driving dominates, increasing the film thickness increases the rise speed of the climbing front, since viscous resistance is less important. In contrast, for low surfactant concentrations, where Marangoni driving balances gravitational drainage, increasing the film thickness decreases the rise speed of the climbing front while enhancing gravitational drainage. We rationalize these observations by utilizing a dimensionless parameter that compares the magnitudes of the Marangoni stress and gravitational drainage. A model is established to analyse the climbing front, either in the Marangoni-driving-dominated region or in the Marangoni-balanced drainage region. Our work highlights the effects of the gravitational drainage on the Marangoni flow, both by setting the thickness of a pre-wetted film and by resisting the film climbing.

Propulsive flapping foils are widely studied in the development of swimming and flying animal-like autonomous systems. Numerical studies in this topic are mainly two-dimensional (2-D) studies, as they are quicker and cheaper, but this inhibits the three-dimensional (3-D) evolution of the shed vortices from leading and trailing edges. In this work, we examine the similarities and differences between 2-D and 3-D simulations through a case study in order to evaluate the efficacy and limitations of using 2-D simulations to describe a 3-D system. We simulate an infinite-span NACA0016 foil in both two and three dimensions at a Reynolds number of 5300 and an angle of attack of 10°. The foil is subject to prescribed heaving and pitching kinematics with varying trailing-edge deflection amplitude $A$. Our primary finding is that the flow and forces are effectively 2-D at intermediate amplitude-based Strouhal numbers ($St_{A}=2Af/U$, where $U$ is the free-stream velocity and $f$ is the flapping frequency), $St_{A}\approx 0.3$ for heaving, $St_{A}\approx 0.3{-}0.6$ for pitching and $St_{A}\approx 0.15{-}0.45$ for coupled motion, while 3-D effects dominate outside of these ranges. These 2-D regions begin when the fluid energy induced by the flapping motion overcomes the 3-D vortex shedding found on a stationary foil, and the flow reverts back to 3-D when the strength of the shed vortices overwhelms the stabilising influence of viscous dissipation. These results indicate that 3-D to 2-D transitions or vice versa are a balance between the strength and stability of leading/trailing-edge vortices and the flapping energy. However, 2-D simulations can still be used for flapping flight/swimming studies provided that the flapping amplitude/frequency is within a given range.

The weakly nonlinear stability analysis of plane Poiseuille flow (PPF) and plane Couette flow (PCF) when viscous dissipation is taken into account is considered. The impermeable lower boundary is considered adiabatic, while the impermeable upper boundary is isothermal. The linear stability of this problem has been performed by Barletta and Nield (J. Fluid Mech., vol. 662, 2010, pp. 475–492) for PCF and by Barletta et al. (J. Fluid Mech., vol. 681, 2011, pp. 499–514) for PPF. These authors found that longitudinal rolls are the preferred mode of convection and the onset of instability is described through the governing parameters $\unicode[STIX]{x1D6EC}=Ge\;Pe^{2}$ and $Pr$, where $Ge$, $Pe$ and $Pr$ are respectively the Gebhart number, the Péclet number and the Prandtl number. The current study focuses on the near-threshold behaviour of longitudinal rolls by using a weakly nonlinear analysis. We determine numerically up to third order the coefficients of the Landau amplitude equation and investigate in detail the influences on bifurcation characteristics of the different nonlinearities present in the system. The results indicate that for both PPF and PCF configurations (i) the inertial terms have no influence on the nonlinear evolution of the disturbance amplitude (ii) the nonlinear thermal advection terms act in favour of pitchfork supercritical bifurcations and (iii) the nonlinearities associated with viscous dissipation promote subcritical bifurcations. The global impact of the different nonlinear contributions indicate that, independently of the Gebhart number, the bifurcation is subcritical if $Pr<0.25$ ($Pr<0.77$) for PPF (PCF). Otherwise, for higher Prandtl number, there exists a particular value of Gebhart number, $Ge^{\ast }$ such that the bifurcation is supercritical (subcritical) if $Ge<Ge^{\ast }$ ($Ge>Ge^{\ast }$). Finally, for both PPF and PCF, the amplitude analysis indicates that, in the supercritical bifurcation regime, the equilibrium amplitude decreases on increasing $Pr$ and a substantial enhancement (reduction) in heat transfer rate is found for small $Pr$ (moderate or large $Pr$).

We present a theory for the evolution of a one-dimensional, steady-state detonation reaction zone to a two-dimensional reaction zone, when the explosive experiences a sudden loss of side-on confinement as a boundary of the explosive is impulsively withdrawn. Our focus is on condensed-phase explosives, which we describe as having a constant adiabatic gamma equation of state and an irreversible, state-independent reaction rate. We consider two detonation models: (i) the instantaneous reaction heat release Chapman–Jouguet (CJ)-limit and (ii) the spatially resolved reaction heat-release Zel’dovich–von Neumann–Döring (ZND) model, in the limit where only a small fraction of the energy release is resolved (the SRHR-limit). Two competing rarefaction waves are generated by this loss of confinement: (i) a smooth wave coming off the full length of the withdrawn boundary and (ii) a singular fan spreading out from the point where the detonation shock and the withdrawn boundary meet. For the CJ-limit, in all cases the singular rarefaction fan eventually dominates the competition to control the steady-state behaviour. For the SRHR-limit, the spatially resolved heat release moderates this competition. When the withdrawal speed is fast, the rarefaction fan dominates; when the withdrawal speed is slower, the smooth rarefaction eventually dominates, although the flow features a fan at early times. By examining the mathematical properties of the steady two-dimensional fan-based solution, we set down a mechanism for this transition in behaviours.

We present a new method for generating robust guesses for unstable periodic orbits (UPOs) by post-processing turbulent data using dynamic mode decomposition (DMD). The approach relies on the identification of near-neutral, repeated harmonics in the DMD eigenvalue spectrum from which both an estimate for the period of a nearby UPO and a guess for the velocity field can be constructed. In this way, the signature of a UPO can be identified in a short time series without the need for a near recurrence to occur, which is a considerable drawback to recurrent flow analysis, the current state of the art. We first demonstrate the method by applying it to a known (simple) UPO and find that the period can be reliably extracted even for time windows of length one quarter of the full period. We then turn to a long turbulent trajectory, sliding an observation window through the time series and performing many DMD computations. Our approach yields many more converged periodic orbits (including multiple new solutions) than a standard recurrent flow analysis of the same data. Furthermore, it also yields converged UPOs at points where the recurrent flow analysis flagged a near recurrence but the Newton solver did not converge, suggesting that the new approach can be used alongside the old to generate improved initial guesses. Finally, we discuss some heuristics on what constitutes a ‘good’ time window for the DMD to identify a UPO.

A series of direct numerical simulations of a fully developed turbulent channel flow is conducted in order to clarify the effects of travelling wave-like wall blowing and suction on dissimilar heat transfer enhancement. While the wave form is kept sinusoidal and its amplitude is set to be 5 % of the bulk mean velocity, the wavelength and phase speed of the travelling wave are systematically changed in a wide parameter space. As a result, the global optimum of the parameter set for maximizing the analogy factor, which is defined as the ratio between the Stanton number and the skin-friction coefficient, is identified. Interestingly, the obtained globally optimal mode agrees well with that predicted from the optimal control theory taking into account the future dynamics within a limited time horizon by Yamamoto et al. (J. Fluid Mech., vol. 733, 2013, pp. 189–220). The instantaneous velocity and thermal fields are decomposed into coherent and random components in order to evaluate the contribution from each component to dissimilar heat transfer enhancement. The detailed mechanisms of dissimilarity are explained by the budget analyses of the coherent and random contributions. Also, their relationships with the near-wall turbulent structures modified by the applied control are discussed through flow visualization. It is found that the random component makes a dominant contribution to dissimilarity, and this can be explained by an indirect effect through the modification of the coherent field by the applied control. Based on the above mechanisms, we propose a simple unsteady Reynolds-averaged Navier–Stokes (URANS) approach, where the phase-averaged velocity and thermal fields are solved directly whereas the effects of the random component are modelled by the Boussinesq eddy viscosity and diffusivity hypothesis. It is shown that the present URANS can capture the overall trend of dissimilar heat transfer enhancement in a wide parameter range. The present results also explain why the optimal control theory with a limited time horizon succeeds in predicting the globally optimal control mode.

We present the results of a combined experimental and theoretical study of the spin coating of the inner surface of capillary tubes with viscous liquids, and the modified Rayleigh–Plateau instability that arises when the spinning stops. We show that during the spin coating, the thinning of the film is governed by the balance of viscous and centrifugal forces, resulting in the film thickness scaling as $h\sim t^{-1/2}$. We demonstrate that the method enables us to reach uniform micrometre-scale films on the tube walls. Finally, we discuss potential applications with curable polymers that enable precise control of film geometry and wettability.

The effect of spanwise heterogeneous surface geometry on turbulent boundary layer secondary flows and on skin friction is investigated experimentally. The surfaces consist of smooth streamwise-aligned ridges of different shapes and widths with spanwise wavelengths comparable to the boundary layer thickness ($S/\unicode[STIX]{x1D6FF}\approx O(1)$). Cross-stream stereoscopic particle image velocimetry combined with oil-film interferometry is used to investigate the flow field and assess the drag. Results show that the spanwise distribution of the skin friction varies as a consequence of the mean flow heterogeneity and is highly dependent on surface geometry. The swirling strength maps revealed remarkable changes in the secondary flow structures for different ridge shapes. For wide ridges, topological changes occur showing the appearance of tertiary vortices coexisting with the large-scale secondary structures. An imbalance in favour of these tertiary structures occurs over a certain width which take over the secondary structures, causing a swap in the locations of the low- and high-momentum pathways. Furthermore, the results indicate that the spanwise spacing alone is insufficient to characterise the surface heterogeneity. A new parameter ($\unicode[STIX]{x1D709}$), which is based on the ratio of the perimeter over and below the mean surface height, is shown to adequately capture the changes in skin friction and streamwise circulation of the secondary motions. Triple decomposition allowed the quantification of the dispersive stresses for all these cases, which can contribute up to $55\,\%$ of the total shear stress when strong secondary motions occur.