We carry out a linear stability analysis of the flow of a thin layer of Newtonian fluid with a deformable free surface bounded at the bottom by a horizontal wall subjected to quasi-periodic oscillation in its own plane. Or's model (J. Fluid Mech., vol. 335, 1997, pp. 213–232), using a periodic oscillation, is extended to the configuration where oscillation has two incommensurate frequencies, $\omega _1$ and $\omega _2$, with an irrational ratio $\omega ={\omega _2}/{\omega _1}$. Using the long-wave expansion, we derive the asymptotic function involved in the long-wave instability criterion while taking into account the frequency ratio. It turns out that the maximum of this asymptotic function, as well as the frequency parameter at which long-wave instabilities occur, depend strongly on the frequency ratio. For arbitrary wavenumbers, the equations governing the problem under consideration are solved in space using Chebyshev's spectral collocation method, while the temporal resolution is performed using Floquet theory, knowing that an irrational number can be approximated by a rational number. For a large frequency ratio and for a velocity amplitude ratio equal to unity, we obtain, as in Or's work (J. Fluid Mech., vol. 335, 1997, pp. 213–232) considering the same frequency parameter interval, an alternation between the U shape and oblique shape referring respectively to instabilities of long wavelength and finite wavelength appearing in the diagram representing Reynolds number as a function of frequency parameter. By decreasing the frequency ratio towards $1/\sqrt {37}$, the three initial U-shaped and three oblique instabilities merge into a single U-shaped and a single oblique instability. This merging phenomenon also occurs when the ratio of the amplitudes of the superimposed velocities, linked to the introduction of the second frequency, increases from small values to unity. For a fixed frequency parameter, the effect of frequency ratio and velocity amplitude ratio on the marginal stability curves in terms of Reynolds number versus wavenumber is also investigated, focusing on the appearance of long wavelength instability and finite wavelength instability.

When one fluid is injected into a confined geometry such as a porous medium filled with another immiscible fluid, even at an extremely low injection speed, rapid filling of several pore spaces accompanied by retraction of multiple fluid–fluid interfaces can be observed. Such processes with fast liquid redistribution within the solid structure, called Haines jumps, are ubiquitous in many multiphase flow systems, which can impact fluid trapping, energy dissipation and hysteretic saturation in various engineering applications. Inspired by this mechanism, here, we propose a dual-channel structure to realise controlled Haines jumps during fluid displacement processes. Via theoretical analysis and numerical simulations, we show that the dynamics of fluid interfaces during Haines jumps can be quantitatively correlated with the driving capillary pressure and dissipating viscous stress, which enables simultaneous determination of the fluid viscosity and interfacial tension in the dual-channel multiphase system.

We investigate nonlinear energy transfer for channel flows at friction Reynolds numbers $Re_{\tau }=180$ and $590$. The key feature of the analysis is that we quantify the energy transferred from a source mode to a recipient mode, with each mode characterised by a streamwise wavenumber and a spanwise wavenumber. This is achieved through an explicit examination of the triadic interactions of the nonlinear energy transfer term in the spectral turbulent kinetic energy equation. First, we quantify the nonlinear energy transfer gain and loss for individual Fourier modes. The gain and loss cannot be obtained without expanding the nonlinear triadic interactions. Second, we quantify the nonlinear energy transfer budgets for three types of modes. Each type of mode is characterised by a specific region in streamwise–spanwise wavenumber space. We find that a transverse cascade from streamwise-elongated modes to spanwise-elongated modes exists for all three types of modes. Third, we quantify the forward and inverse cascades between resolved scales and subgrid scales in the spirit of large-eddy simulations. For the cutoff wavelength range that we consider, the forward and inverse cascades between the resolved scales and subgrid scales result in a net forward cascade from the resolved scales to the subgrid scales. The shape of the net forward cascade curve with respect to the cutoff wavelength resembles the net forward cascade predicted by the Smagorinsky eddy viscosity.

In this paper, we study the disturbance velocity and density fields induced by a sphere translating vertically in a viscous density-stratified ambient. Specifically, we consider the limit of a vanishingly small Reynolds number $(Re = \rho U a/\mu \ll 1)$, a small but finite viscous Richardson number $(Ri_v = \gamma a^3g/\mu U\ll 1)$ and large Péclet number $(Pe = Ua/D\gg 1)$. Here, $a$ is the sphere's radius, $U$ its translational velocity, $\rho$ an appropriate reference density within the framework of the Boussinesq approximation, $\mu$ the ambient viscosity, $\gamma$ the absolute value of the background density gradient, g is acceleration due to gravity and $D$ the diffusivity of the stratifying agent. For the scenario where buoyancy forces first become comparable to viscous forces at large distances, corresponding to the Stokes-stratification regime defined by $Re \ll Ri_v^{1/3} \ll 1$ for $Pe \gg 1$, important flow features have been identified by Varanasi & Subramanian (J. Fluid Mech., vol. 949, 2022, A29) – these include a vertically oriented reverse jet, and a horizontal axisymmetric wake, on scales larger than the primary (stratification) screening length of ${O}(aRi_v^{-1/3})$. Here, we study the reverse-jet region in more detail, and show that it is only the central portion of a columnar structure with multiple annular cells concentric about the rear stagnation streamline. In the absence of diffusion, corresponding to $Pe = \infty$ $( \beta _\infty = Ri_v^{1/3}Pe^{-1} = 0)$, this columnar structure extends to downstream infinity with the number of annular cells diverging in this limit. We provide expressions for the boundary of the structure, and the number of cells within, as a function of the downstream distance. For small but finite $\beta _\infty$, two length scales emerge in addition to the primary screening length – a secondary screening length of ${O}(aRi_v^{-1/2}Pe^{1/2})$ where diffusion starts to smear out density variations across cells, leading to exponentially decaying density and velocity fields; and a tertiary screening length, $l_t \sim {O}(aRi_v^{-1/2}Pe^{1/2}[\zeta + \frac {13}{4}\ln {\zeta } + ({13^2}/{4^2})({\ln \zeta }/{\zeta })])$ with $\zeta = \frac {1}{2}\ln ({\sqrt {{\rm \pi} }Ri_v^{-1}Pe^3}/{2160})$, beyond which the columnar structure ceases to exist. The latter causes a transition from a vertical to a predominantly horizontal flow, with the downstream disturbance fields reverting from an exponential to an eventual algebraic decay, analogous to that prevalent at large distances upstream.

Flow field in the near wake of a short-finite circular cylinder at $L/D=1.0$ with an angle of attack between 0$^\circ$–15$^\circ$, where the transition from the non-reattaching flow to the reattaching flow appears, is investigated in wind tunnel tests with a supportless condition. Stereo particle image velocimetry measurements were applied to the experiments at the Reynolds number of $3.46\times 10^4$, and velocity fields in the near wake were obtained. The data was mainly analysed using spectral proper orthogonal decomposition. Characteristic large-scale wake structures of recirculation bubble pumping and large-scale vortex shedding were identified in the near wake of the cylinder regardless of the angle of attack. The phase difference of expansion and contraction of the recirculation flow appears in the recirculation bubble pumping at $\alpha \neq 0^\circ$. On the other hand, the eigenfunctions of velocity fluctuations at the vortex shedding frequency show a similar spatial pattern regardless of $\alpha$. Frequency analyses of wake position calculated from the reconstructed velocity field clarified that peak frequency is different between two in-plane directions when $\alpha \neq 0^\circ$. In addition, three vortex shedding patterns (anticlockwise/clockwise circular and flapping) are identified not only at $\alpha =0^\circ$ but also $\alpha \neq 0^\circ$. The feature of wake position in the radial direction for each pattern is observed regardless of the angle of attack. The relationship between the recirculation bubble pumping and the wake position in the radial direction is apparent in the non-reattaching flow but is weaker with $\alpha$ in the reattaching flow.

We investigate experimentally the planar paths displayed by cylinders falling freely in a thin-gap cell containing liquid at rest, by varying the elongation ratio and the Archimedes number of the cylinders, and the solid-to-fluid density ratio. In the investigated conditions, the oscillatory falling motion features two main characteristics: the mean fall velocity $\overline {u_v}$ does not scale with the gravitational velocity, which overestimates $\overline {u_v}$ and is unable to capture the influence of the density ratio on it; and high-amplitude oscillations of the order of $\overline {u_v}$ are observed for both translational and rotational velocities. To model the body behaviour, we propose a force balance, including proper and added inertia terms, the buoyancy force and vortical contributions accounting for the production of vorticity at the body surface and its interaction with the cell walls. Averaging the equations over a temporal period provides a mean force balance that governs the mean fall velocity of the cylinder, revealing that the coupling between the translational and rotational velocity components induces a mean upward inertial force responsible for the decrease of $\overline {u_v}$. This mean force balance also provides a normalization for the frequency of oscillation of the cylinder in agreement with experimental measurements. We then consider the instantaneous force balance experienced by the body, and propose three contributions for the modelling of the vortical force. These can be interpreted as drag, lift and history forces, and their dependence on the control parameters is adjusted on the basis of the experimental measurements.

Granular column collapse is a simple but important problem to the granular material community, due to its links to dynamics of natural hazards, such as landslides and pyroclastic flows, and many industrial situations, as well as its potential of analysing transient and non-local rheology of granular flows. This article proposes a new dimensionless number to describe the run-out behaviour of granular columns on inclined planes based on both previous experimental data and dimensional analysis. With the assistance of the sphero-polyhedral discrete element method (DEM), we simulate inclined granular column collapses with different initial aspect ratios, particle contact properties and initial solid fractions on inclined planes with different inclination angles ($2.5^{\circ }\unicode{x2013}20.0^{\circ }$) to verify the proposed dimensional analysis. Detailed analyses are further provided for better understanding of the influence of different initial conditions and boundary conditions, and to help unify the description of the run-out scaling of systems with different inclination angles. This work determines the similarity and unity between granular column collapses on inclined planes and those on horizontal planes, and helps investigate the transient rheological behaviour of granular flows, which has direct relevance to various natural and engineering systems.

In freely decaying stably stratified turbulent flows, numerical evidence shows that the horizontal displacement of Lagrangian tracers is diffusive while the vertical displacement converges towards a stationary distribution, as shown numerically by Kimura & Herring (J. Fluid Mech., vol. 328, 1996, pp. 253–269). Here, we develop a stochastic model for the vertical dispersion of Lagrangian tracers in stably stratified turbulent flows that aims to replicate and explain the emergence of a stationary probability distribution for the vertical displacement of such tracers. More precisely, our model is based on the assumption that the dynamical evolution of the tracers results from the competing effects of buoyancy forces that tend to bring a vertically perturbed fluid parcel (carrying tracers) to its equilibrium position and turbulent fluctuations that tend to disperse tracers. When the density of a fluid parcel is allowed to change due to molecular diffusion, a third effect needs to be taken into account: irreversible mixing. Indeed, ‘mixing’ dynamically and irreversibly changes the equilibrium position of the parcel and affects the buoyancy force that ‘stirs’ it on larger scales. These intricate couplings are modelled using a stochastic resetting process (Evans & Majumdar, Phys. Rev. Lett., vol. 106, issue 16, 2011, 160601) with memory. More precisely, Lagrangian tracers in stratified turbulent flows are assumed to follow random trajectories that obey a Brownian process. In addition, their stochastic paths can be reset to a given position (corresponding to the dynamically changing equilibrium position of a density structure containing the tracers) at a given rate. Scalings for the model parameters as functions of the molecular properties of the fluid and the turbulent characteristics of the flow are obtained by analysing the dynamics of an idealised density structure. Even though highly idealised, the model has the advantage of being analytically solvable. In particular, we show the emergence of a stationary distribution for the vertical displacement of Lagrangian tracers. We compare the predictions of this model with direct numerical simulation data at various Prandtl numbers $Pr$, the ratio of kinematic viscosity to molecular diffusion.

The transformation of internal waves on a stepwise underwater obstacle is studied in the linear approximation. The transmission and reflection coefficients are derived for a two-layer fluid. The results are obtained and presented as functions of incident wave wavenumber, density ratio of layers, pycnocline position, and height of the bottom step. Excitation coefficients of evanescent modes are also calculated, and their importance is demonstrated. This allows one to estimate the number of evanescent modes necessary to take into account to attain the required accuracy for the transformation coefficients.

This work presents an experimental investigation of the effects of vortex shedding suppression on the properties and recovery of turbulent wakes. Four plates, properly modified so that they produce different vortex shedding strengths, are tested using high speed particle image velocimetry and hot-wire anemometry, and analysed using spectral proper orthogonal decomposition, mean-flow linear stability analysis and various turbulence statistics. When present, vortex shedding is found to exhibit a characteristic frequency that scales with the mean shear, providing a link between the mean flow and the main turbulent motion. To achieve full suppression of shedding, we combine the effects of porosity and fractal perimeter. The mean shear is then decreased to the point where the flow becomes convectively unstable and shedding vanishes. In that case, the onset of self-similarity is delayed, compared with the case with vortex shedding, and appears after another large-scale structure, the secondary vortex street, emerges. It is also found that both large- and small-scale intermittency are starkly reduced when shedding is absent. A simple theoretical representation of the wake dynamics explains the evolution of the wake properties and its connection to the coherent structures in the flow.

Mangroves are a natural defence of the coastal strip against extreme waves. Furthermore, innovative techniques of naturally based coast defence are used increasingly, according to the canons of eco-hydraulics. Therefore, it is important to correctly evaluate the transmission of waves through cylinder arrays. In the present paper, the attenuation of solitary waves propagating through an array of rigid emergent and submerged cylindrical stems on a horizontal bottom is investigated theoretically, numerically and experimentally. The results of the theoretical model are compared with the numerical simulations obtained with the smoothed particle hydrodynamics meshless Lagrangian numerical code and with experimental laboratory data. In the latter case, solitary waves were tested on a background current, in order to reproduce more realistic sea conditions, since the absence of circulation currents is very rare in the sea. The comparison confirmed the validity of the theoretical model, allowing its use for the purposes indicated above. Furthermore, the present study allowed for an evaluation of the bulk drag coefficient of the rigid stem arrays used, as a function of their density, the stem diameter, and their submergence ratio.