We theoretically investigate the pattern formation observed when a fluid flows over a solid substrate that can dissolve or melt. We use a turbulent mixing description that includes the effect of the bed roughness. We show that the dissolution instability at the origin of the pattern is associated with an anomaly at the transition from a laminar to a turbulent hydrodynamic response with respect to the bed elevation. This anomaly, and therefore the instability, disappears when the bed becomes hydrodynamically rough, above a threshold roughness-based Reynolds number. This suggests a possible mechanism for the selection of the pattern amplitude. The most unstable wavelength scales as observed in nature on the thickness of the viscous sublayer, with a multiplicative factor that depends on the dimensionless parameters of the problem.

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

There is a long-standing debate in the literature of stratified flows over topography concerning the correct dimensionless number to refer to as a Froude number. Common definitions using external quantities of the flow include $U/(ND)$, $U/(Nh_{0})$, and $Uk/N$, where $U$ and $N$ are, respectively, scales for the background velocity and buoyancy frequency, $D$ is the depth, and $h_{0}$ and $k^{-1}$ are, respectively, height and width scales of the topography. It is also possible to define an internal Froude number $Fr_{\unicode[STIX]{x1D6FF}}=u_{0}/\sqrt{g^{\prime }\unicode[STIX]{x1D6FF}}$, where $u_{0}$, $g^{\prime }$, and $\unicode[STIX]{x1D6FF}$ are, respectively, the characteristic velocity, reduced gravity, and vertical length scale of the perturbation above the topography. For the case of hydrostatic lee waves in a deep ocean, both $U/(ND)$ and $Uk/N$ are insignificantly small, rendering the dimensionless number $Nh_{0}/U$ the only relevant dynamical parameter. However, although it appears to be an inverse Froude number, such an interpretation is incorrect. By non-dimensionalizing the stratified Euler equations describing the flow of an infinitely deep fluid over topography, we show that $Nh_{0}/U$ is in fact the square of the internal Froude number because it can identically be written in terms of the inner variables, $Fr_{\unicode[STIX]{x1D6FF}}^{2}=Nh_{0}/U=u_{0}^{2}/(g^{\prime }\unicode[STIX]{x1D6FF})$. Our scaling also identifies $Nh_{0}/U$ as the ratio of the vertical velocity scale within the lee wave to the group velocity of the lee wave, which we term the vertical Froude number, $Fr_{vert}=Nh_{0}/U=w_{0}/c_{g}$. To encapsulate such behaviour, we suggest referring to $Nh_{0}/U$ as the lee-wave Froude number, $Fr_{lee}$.

Most commonly used models for turbulent mixing in the ocean rely on a background stratification against which turbulence must work to stir the fluid. While this background stratification is typically well defined in idealized numerical models, it is more difficult to capture in observations. Here, a potential discrepancy in ocean mixing estimates due to the chosen calculation of the background stratification is explored using direct numerical simulation data of breaking internal waves on slopes. Two different methods for computing the buoyancy frequency $N$, one based on a three-dimensionally sorted density field (often used in numerical models) and the other based on locally sorted vertical density profiles (often used in the field), are used to quantify the effect of $N$ on turbulence quantities. It is shown that how $N$ is calculated changes not only the flux Richardson number $R_{f}$, which is often used to parameterize turbulent mixing, but also the turbulence activity number or the Gibson number $Gi$, leading to potential errors in estimates of the mixing efficiency using $Gi$-based parameterizations.

The response of the free liquid surface in a Hele-Shaw cell subjected to a horizontal oscillation is investigated. We study the low-oscillation-amplitude regime and we show, by varying the fluid viscosity, $\unicode[STIX]{x1D708}$, and the forcing frequency, $\unicode[STIX]{x1D714}$, that the ratio between the Stokes viscous length, $\sqrt{2\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}}$, and the cell thickness greatly affects the amplitude and phase lag of the gravity waves. In particular, the sloshing system undergoes an underdamped/overdamped transition for sufficiently large viscosities. A consistent theoretical model, based on a modification of Darcy’s law to include unsteadiness, is then introduced to rationalize the experimental observations. Contrary to traditional sloshing wave theory, the viscous flow dissipation comes at leading order in the analysis, rather than as a higher-order asymptotic correction to the inviscid sloshing dynamics. The analytical expression for the resonance curves agrees well with experimental results without tunable parameters.

Accurate prediction of the spatial evolution of turbulent wake flows under pressure gradient conditions is required in some engineering applications such as the design of high-lift devices and wind farms over topography. In this paper, we aim to develop an analytical model to predict the evolution of a turbulent planar wake under an arbitrary pressure gradient condition. The model is based on the cross-stream integration of the streamwise momentum equation and uses the self-similarity of the mean flow. We have also made an experimentally supported assumption that the ratio of the maximum velocity deficit to the wake width is independent of the imposed pressure gradient. The asymptotic response of the wake to the pressure gradient is also investigated. After its derivation, the model is successfully validated against experimental data by comparing the evolution of the wake width and maximum velocity deficit. The inputs of the model are the imposed pressure gradient and the wake width under zero pressure gradient. The model does not require any parameter tuning and is deemed to be practical, computationally fast, accurate enough, and therefore useful for the scientific and engineering communities.

This experimental study focuses on the effect of horizontal boundaries with pyramid-shaped roughness elements on the heat transfer in rotating Rayleigh–Bénard convection. It is shown that the Ekman pumping mechanism, which is responsible for the heat transfer enhancement under rotation in the case of smooth top and bottom surfaces, is unaffected by the roughness as long as the Ekman layer thickness $\unicode[STIX]{x1D6FF}_{E}$ is significantly larger than the roughness height $k$. As the rotation rate increases, and thus $\unicode[STIX]{x1D6FF}_{E}$ decreases, the roughness elements penetrate the radially inward flow in the interior of the Ekman boundary layer that feeds the columnar Ekman vortices. This perturbation generates additional thermal disturbances which are found to increase the heat transfer efficiency even further. However, when $\unicode[STIX]{x1D6FF}_{E}\approx k$, the Ekman boundary layer is strongly perturbed by the roughness elements and the Ekman pumping mechanism is suppressed. The results suggest that the Ekman pumping is re-established for $\unicode[STIX]{x1D6FF}_{E}\ll k$ as the faces of the pyramidal roughness elements then act locally as a sloping boundary on which an Ekman layer can be formed.

Wavepackets obtained as solutions of the flow equations linearised around the mean flow have been shown in recent work to yield good agreement, in terms of amplitude and phase, with those educed from turbulent jets. Compelling agreement has been demonstrated, for the axisymmetric and first helical mode, up to Strouhal numbers close to unity. We here extend the range of validity of wavepacket models to Strouhal number $St=4.0$ and azimuthal wavenumber $m=4$ by comparing solutions of the parabolised stability equations with a well-validated large-eddy simulation of a Mach 0.9 turbulent jet. The results show that the near-nozzle dynamics can be correctly described by the homogeneous linear model, the initial growth rates being accurately predicted for the entire range of frequencies and azimuthal wavenumbers considered. Similarly to the lower-frequency wavepackets reported prior to this work, the high-frequency linear waves deviate from the data downstream of their stabilisation locations, which move progressively upstream as the frequency increases.

Wetting of a planar solid substrate is investigated in the presence of a macroscopic particle in the complete wetting regime. A drop of silicone oil spreads on the substrate and its macroscopic edge is incident on the particle at the late stage of spreading. The drop–particle interaction is observed in detail by shadowgraph and interferometry. Although the spreading drop edge is pinned by the particle for a short time, a sharp acceleration occurs when the liquid starts wetting the extra surface area offered by the particle and forming a meniscus. This process yields a net gain in spreading speed. A theoretical model based on the classical wetting dynamics dictated by Cox’s law is developed. It predicts that the capillary energy of the meniscus gives rise to a rapid motion of the liquid edge, showing good agreement with the dynamics observed in the experiments.

We present detailed experiments on transient growth of turbulent spots induced by external forcing in plane Couette–Poiseuille flow, which are studied in the framework of linear transient growth. The experimental investigation is supplemented with full theoretical analysis. We compare quantitatively the experimental and theoretical results, including maximal gain and the time at which it occurs. We also present the limits of validity for the application of the linear theory at high amplitude perturbation and Reynolds number, showing experiments with self-sustained states.

A generalized reciprocal theorem is formulated for the motion and hydrodynamic force moments of an active particle in an arbitrary background flow of a (weakly nonlinear) complex fluid. This formalism includes as special cases a number of previous calculations of the motion of both passive and active particles in Newtonian and non-Newtonian fluids.

Turbulent wakes past bluff bodies commonly present asymmetric flow states reminiscent of bifurcations in the laminar regime. Understanding the sensitivity of these states to flow forcing is crucial to the modelling and control of flow symmetry properties. In this study, the near wake of a rectangular bluff body in proximity to a wall is disturbed by the use of passive devices located between the model and the wall, upstream of the massive flow separation occurring at the blunt trailing edges. Due to the proximity to the boundary, the wake initially presents wall-normal asymmetry and a negative wall-normal pressure gradient along the base. The application of disturbances with variable size, however, sets flow symmetry along the wall-normal plane, leading to the intermittent spanwise wake reversals reported recently in the literature. A further increase in the size of perturbation suppresses wake switching, and wall-normal asymmetry is recovered, but with a positive wall-normal pressure gradient. The dynamical features of this bifurcation scenario can be retrieved using two coupled symmetry-breaking models for spanwise and wall-normal pressure gradients. This confirms the high sensitivity of the separated flow to external perturbations. More importantly, the results unify observations of the bluff-body wake topologies covered in previous investigations.

The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are $2\times 2$ systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.

A one-dimensional, unsteady nozzle flow is modelled to identify the sources of indirect noise in multicomponent gases. First, from non-equilibrium thermodynamics relations, it is shown that a compositional inhomogeneity advected in an accelerating flow is a source of sound induced by inhomogeneities in the mixture (i) chemical potentials and (ii) specific heat capacities. Second, it is shown that the acoustic, entropy and compositional linear perturbations evolve independently from each other and they become coupled through mean-flow gradients and/or at the boundaries. Third, the equations are cast in invariant formulation and a mathematical solution is found by asymptotic expansion of path-ordered integrals with an infinite radius of convergence. Finally, the transfer functions are calculated for a supersonic nozzle with finite spatial extent perturbed by a methane–air compositional inhomogeneity. The proposed framework will help identify and quantify the sources of sound in nozzles with relevance, for example, to aeronautical gas turbines.

In low-density axisymmetric jets, the onset of global instability is known to depend on three control parameters, namely the jet-to-ambient density ratio $S$, the initial momentum thickness $\unicode[STIX]{x1D703}_{0}$ and the Reynolds number $Re$. For sufficiently low values of $S$ and $\unicode[STIX]{x1D703}_{0}$, these jets bifurcate from a steady state (a fixed point) to a self-excited oscillatory state (a limit cycle) when $Re$ increases above a critical value corresponding to the Hopf point, $Re_{H}$. In the literature, this Hopf bifurcation is often regarded as supercritical. In this experimental study, however, we find that under some conditions, there exists a hysteretic bistable region at $Re_{SN}<Re<Re_{H}$, where $Re_{SN}$ denotes a saddle-node point. This shows that, contrary to expectations, the Hopf bifurcation can also be subcritical, which we explore by evaluating the coefficients of a truncated Landau model. The existence of subcritical bifurcations implies the potential for triggering and the need for weakly nonlinear analyses to be performed to at least fifth order if one is to be able to predict saturation and bistability. We conclude by proposing a universal scaling for $Re_{H}$ in terms of $S$ and $\unicode[STIX]{x1D703}_{0}$. This scaling, which is insensitive to the super/subcritical nature of the bifurcations, can be used to predict the onset of self-excited oscillations, providing further evidence to support Hallberg & Strykowski’s concept (J. Fluid Mech., vol. 569, 2006, pp. 493–507) of universal global modes in low-density jets.

Pressure- and volume-imposed rheology is used to study suspensions of non-colloidal, rigid fibres in the concentrated regime for aspect ratios ranging from 3 to 15. The suspensions exhibit yield stresses. Subtracting these apparent yield stresses reveals a viscous scaling for both the shear and normal stresses. The variation in aspect ratio does not affect the friction coefficient (ratio of shear and normal stresses), but increasing the aspect ratio lowers the maximum volume fraction at which the suspension flows. Constitutive laws are proposed for the viscosities and the friction coefficient close to the jamming transition.

In this study, we experimentally investigate the turbulent drag-reduction (DR) mechanism in flow through ducts of circular, rectangular and square cross-sections using two grades of polyacrylamide in aqueous solution having different molecular weights and various semidilute concentrations. Specifically, we explore the relationship between drag reduction and fluid elasticity, purposely exploiting the mechanical degradation of polymer molecules to vary their rheological properties. We also obtain time-resolved velocity data for various DR levels using particle image velocimetry and laser Doppler velocimetry. Elasticity is quantified via relaxation times determined from uniaxial extensional flow using a capillary breakup apparatus. A plot of DR against Weissenberg number ($Wi$) is found to approximately collapse the data, with the onset of DR occurring at $Wi\approx 0.5$ and the maximum drag-reduction asymptote being approached for $Wi\gtrsim 5$. Thus quantitative predictions of DR in a range of shear flows can be made from a single measurable material property of a polymer solution, at least for this particular flexible linear polymer.

New variational principles are given for the two-dimensional interactions between gravity-driven water waves and a rotating and translating rectangular vessel dynamically coupled to its interior potential flow with uniform vorticity. The complete set of equations of motion for the exterior water waves, the exact nonlinear hydrodynamic equations of motion for the vessel in the roll/pitch, sway/surge and heave directions, and also the full set of equations of motion for the interior fluid of the vessel, relative to the body coordinate system attached to the rotating–translating vessel, are derived from two Lagrangian functionals.

In shear flows at transitional Reynolds numbers, localized patches of turbulence, known as puffs, coexist with the laminar flow. Recently, Avila et al. (Phys. Rev. Lett., vol. 110, 2013, 224502) discovered two spatially localized relative periodic solutions for pipe flow, which appeared in a saddle-node bifurcation at low Reynolds number. Combining slicing methods for continuous symmetry reduction with Poincaré sections for the first time in a shear flow setting, we compute and visualize the unstable manifold of the lower-branch solution and show that it extends towards the neighbourhood of the upper-branch solution. Surprisingly, this connection even persists far above the bifurcation point and appears to mediate the first stage of the puff generation: amplification of streamwise localized fluctuations. When the state-space trajectories on the unstable manifold reach the vicinity of the upper branch, corresponding fluctuations expand in space and eventually take the usual shape of a puff.

The stability of an accelerating cylindrical blob of a time-varying radius is considered with the goals of understanding the effects of time dependence of the underlying base state on a Rayleigh–Plateau instability as well as of evaluating a contribution due to a lateral acceleration of the blob, treated as a perturbation here. All of the key processes contributing to instability development are dissected, with analytical analyses of the exact incompressible inviscid potential flow formulation. Herein, without invoking the ‘frozen’ base state assumption, the entire time interval of the evolution of a perturbation is explored, discerning physical mechanisms at each stage of development. It transpires that the stability picture proves to be cardinally different from Rayleigh’s standard analysis.