Direct numerical solutions are described for flow past a body placed in an otherwise uniform shear layer adjoining a wall. The study is associated with potential impact of the body onto the wall. Steady two-dimensional flow solutions are calculated for an inclined flat plate in particular, covering cases of zero wall velocity, positive wall velocity and negative wall velocity, with the plate being at varying orientations and distances from the wall. Substantial flow separation is found with reduced proximity to the wall or increased plate incidence, caused partly by the cutting off of the mass flux in the gap between the body and the wall as impact is neared. Other distinct flow characteristics that emerge with increased local Reynolds number are the extent of the enhanced wake responses, greatly condensed upstream influence near the leading edge, increased sensitivity to body orientation, the pressure dominance in the total lift and moment on the body, new insight into the complex flow structure and quantitative agreement with a recent viscous–inviscid interaction analysis on scales.

This paper analyses the turbulent energy cascade from the perspective of statistical mechanics, and relates interscale energy fluxes to statistical irreversibility and information entropy production. The microscopical reversibility of the energy cascade is tested by constructing a reversible three-dimensional turbulent system using a dynamic model for the sub-grid stresses. This system, when reversed in time, develops a sustained inverse cascade towards the large scales, evidencing that the characterisation of the inertial energy cascade must consider the possibility of an inverse regime. This experiment is used to study the origin of statistical irreversibility and the prevalence of direct over inverse energy cascades in isotropic turbulence. Statistical irreversibility, a property of statistical ensembles in phase space related to entropy production, is connected to the dynamics of the energy cascade in physical space by considering the space locality of the energy fluxes and their relation to the local structure of the flow. A mechanism to explain the probabilistic prevalence of direct energy transfer is proposed based on the dynamics of the rate-of-strain tensor, which is identified as the most important source of statistical irreversibility in the energy cascade.

It is known that the pitchfork bifurcation of Kelvin–Helmholtz instability occurring at minimum gradient Richardson number $Ri_m \simeq 1/4$ in viscous stratified shear flows can be subcritical or supercritical depending on the value of the Prandtl number, $Pr$. Here, we study stratified shear flow restricted to two dimensions at finite Reynolds number, continuously forced to have a constant background density gradient and a hyperbolic tangent shear profile, corresponding to the ‘Drazin model’ base flow. Bifurcation diagrams are produced for fluids with $Pr=0.7$ (typical for air), 3 and $7$ (typical for water). For $Pr=3$ and $7$, steady billow-like solutions are found to exist for strongly stable stratification of $Ri_m$ beyond $1/2$. Interestingly, these solutions are not a direct product of a Kelvin–Helmholtz instability, having half the wavelength of the linear instability, and arising through a superharmonic bifurcation. These short-wavelength states can be tracked down to at least $Pr \approx 2.3$ and act as instigators of complex dynamics, even in strongly stratified flows. Direct numerical simulations of forced and unforced two-dimensional flows are performed, which support the results of the bifurcation analyses. Perturbations are observed to grow approximately exponentially from random initial conditions where no modal instability is predicted by a linear stability analysis.

This paper presents closed analytical solutions for the pressure and velocity fields of flows in two-dimensional porous media. The flow field is modelled through a potential function which allows the use of the Laplace equation to describe the pressure field. The boundary conditions of the porous medium are tailored to represent general cases encountered in transpiration cooling applications. These include mixed Neumann and Dirichlet boundary conditions to represent a pressurised plenum driving a coolant mass flux, and impermeable sections where the plenum is attached to a non-porous substructure. The external pressure boundary is modelled as an arbitrary function representing a flow around the porous domain, and the wall thickness of the porous domain can take any arbitrary distribution. General solutions in Cartesian coordinates and cylindrical coordinates are provided describing the entire porous domain of a flat plate or curved geometry, respectively. In addition, special simplified solutions are provided for regions of particular interest, such as the interface of external flow and porous medium. The obtained solutions are verified through a comparison to a numerical simulation of two test cases, a rectangular flat plate geometry and 90$^\circ$ section of a cylindrical case.

High resolution direct numerical simulation (DNS) data are used to investigate properties of similarity solutions for the mean velocity ($U$) and Reynolds shear stress ($T$) in turbulent channel flow. The analysis uses the method of Fife et al. (Multiscale Model. Simul., vol. 4, 2005, p. 936) for determining the asymptotic scaling properties of indeterminate equations. Subject to its primary assumption, the analysis yields an invariant form of the mean momentum equation that is valid over a significant portion of the flow domain. Because the requisite coordinate transformations yield a self-similar relationship between scaled wall-normal derivatives of $U$ and $T$, this procedure inherently closes the mean equation. These transformations incorporate use of the so-called Fife similarity parameter, $\phi$. The asymptotic constancy of $\phi$ ($\to \phi _c$) on the inertial sublayer is analytically related to the von Kármán constant by $\kappa = \phi _c^{-2}$. Integrations of the transformed mean equation are shown to independently yield similarity solutions for both $U$ and $T$, and the veracity of these solutions is examined in detail. DNS data at $\delta ^+ \simeq 8000$ reveal that the primary assumption of the analysis is satisfied to within a deviation of less than $0.1\,\%$. The present solutions and their associated similarity structure are further used to generate a number of new results. These include a cogent specification for the both the inner and outer boundaries of the inertial sublayer and a variety of well-founded ways to estimate $\phi _c$ at finite Reynolds number. Extensions of the analytical arguments by Klewicki et al. (Phys. Rev. E, vol. 90, 2014, 063015) support the conjecture that $\phi _c \to \varPhi = (1 + \sqrt {5})/2$ (or equivalently, $\kappa = 2/(3 + \sqrt {5})$) at large Reynolds number. Connections between the channel solutions and those in the other canonical wall flows are briefly discussed, as are potential implications of the present results relative to wall-flow model development.

Vibration-based flow energy harvesting enables robust, in situ energy extraction for low-power applications, such as distributed sensor networks. Fluid–structure instabilities dictate a harvester's viability since the structural response to the flow determines its power output. Previous work on a flextensional based flow energy harvester demonstrated that an elastic member within a converging-diverging channel is susceptible to the aeroelastic flutter. This work explores the mechanism driving flutter through experiments and simulations. A model is then developed based on channel flow rate modulation and considering the effects of both normal and spanwise flow confinement on the instability. Linear stability analysis of the model replicates flutter onset, critical frequency and mode shapes observed in experiments. The model suggests that flow modulation through the channel throat is the principal mechanism for the fluid-induced vibration. The generalized model presented can serve as the foundation of design parameter exploration for energy harvesters, perhaps leading to more powerful devices in the future, but also to other similar flow geometries where the flutter instability arises in an elastic member within a narrow flow passage.

Geometric focusing of monochromatic internal waves is a well-known linear mechanism that provides a pathway to smaller length scales and hence energy dissipation, especially when repeated reflection from inclined boundaries focuses these waves onto internal wave attractors (IWAs). While simple attractors have been well documented both theoretically and in idealized laboratory settings, the small aspect ratio and complex shape of the ocean has made it difficult to detect the narrow frequency bands where attractors may form. The greatest obstruction however is the restricted spatiotemporal resolution of measurements of the ocean's internal wave field. Moored instruments provide a detailed temporal but poor spatial resolution, except along line-segments spanning part of the water column only. The ability to measure a fully resolved two-dimensional section through a three-dimensional field, as in present-day laboratory experiments, is missing in the ocean. This prohibits the search for IWAs in ocean data, especially if this concerns a multi-frequency wave field. Mimicking an ocean observation set-up, our experimental study aims at detecting IWA signatures from laboratory observations along a vertical line-segment. We perform small and large amplitude forcing experiments at a fixed wavenumber with both monochromatic and broadband frequency components. We examine energy intensification at dissipative length scales at frequencies known to support attractors. Importantly, our laboratory experiments indicate that even in the presence of multiple forcing frequencies, or for large-amplitude monochromatic forcing (where beams become unstable to Triadic Resonant Instability), focusing reflections due to attractors are the predominant mechanism for transferring energy to dissipative scales.

Stereoscopic particle image velocimetry measurements of open-channel flows over streamwise-orientated triangular-shaped ridges were used to explore interactions between ridge-induced secondary currents (SCs) and turbulence. Terms in the double-averaged (in space and in time) momentum and energy conservation equations were analysed for a range of ridge spacings ($s$) between 0.4 and 4.0 flow depths ($H$). The double-averaged equations neatly partition momentum and energy fluxes into turbulence and SC contributions, making them well suited to this study. The obtained data indicate that for a range of $s/H$ between 0.4 and 2.0, the normalised momentum and energy fluxes due to SCs approximately collapse when plotted as functions of $(z-d)/s$, where $z$ is the vertical coordinate and $d$ a constant that aligns the elevations of SC cell centres. The SCs controlled the shape of the mean velocity distribution with the vertical gradient of the double-averaged streamwise velocity found to be inversely proportional to $s$ near the elevations of SC cell centres. Partitioning the total kinetic energy into double-mean (DMKE), dispersive (DKE) and turbulent (SATKE) components and considering the balance equation for each component indicated that at the elevation of SC cell centres the production rate of SATKE via exchange with DMKE was comparable in magnitude to the production rate via exchange with DKE (due to SCs). For all ridge spacings, SATKE was reduced compared to a no-ridge benchmark case due to suppression of very-large-scale turbulent motions by the SCs. Finally, it is demonstrated that energy is supplied to SCs by turbulence.

A modal stability analysis shows that plane Poiseuille flow of an Oldroyd-B fluid becomes unstable to a ‘centre mode’ with phase speed close to the maximum base-flow velocity, $U_{max}$. The governing dimensionless groups are the Reynolds number $Re = \rho U_{max} H/\eta$, the elasticity number $E = \lambda \eta /(H^2 \rho )$ and the ratio of solvent to solution viscosity $\beta = \eta _s/\eta$; here, $\lambda$ is the polymer relaxation time, $H$ is the channel half-width and $\rho$ is the fluid density. For experimentally relevant values (e.g. $E \sim 0.1$ and $\beta \sim 0.9$), the critical Reynolds number, $Re_c$, is around $200$, with the associated eigenmodes being spread out across the channel. For $E(1-\beta ) \ll 1$, with $E$ fixed, corresponding to strongly elastic dilute polymer solutions, $Re_c \propto (E(1-\beta ))^{-3/2}$ and the critical wavenumber $k_c \propto (E(1-\beta ))^{-1/2}$. The unstable eigenmode in this limit is confined in a thin layer near the channel centreline. These features are largely analogous to the centre-mode instability in viscoelastic pipe flow (Garg et al., Phys. Rev. Lett., vol. 121, 2018, 024502), and suggest a universal linear mechanism underlying the onset of turbulence in both channel and pipe flows of sufficiently elastic dilute polymer solutions. Although the centre-mode instability continues down to $\beta \sim 10^{-2}$ for pipe flow, it ceases to exist for $\beta < 0.5$ in channels. Whereas inertia, elasticity and solvent viscous effects are simultaneously required for this instability, a higher viscous threshold is required for channel flow. Further, in the opposite limit of $\beta \rightarrow 1$, the centre-mode instability in channel flow continues to exist at $Re \approx 5$, again in contrast to pipe flow where the instability ceases to exist below $Re \approx 63$, regardless of $E$ or $\beta$. Our predictions are in reasonable agreement with experimental observations for the onset of turbulence in the flow of polymer solutions through microchannels.

This study aims to shed light on hypersonic attachment-line instabilities with large sweep Mach numbers. Highly swept flows over a cold cylinder that give rise to large sweep Mach numbers are studied. High-fidelity basic flows are obtained by solving full Navier–Stokes equations with a high-order shock-fitting method. Using local and global stability theories, an attachment-line mode is found to be dominant for the laminar–turbulent transition along the leading edge that agrees qualitatively with the experimental observations (Gaillard et al., Exp. Fluids, vol. 26, 1999, pp. 169–176). The behaviour of this mode explains the reason for the transition occurring earlier as the sweep Mach number is above 5. In addition, this attachment-line mode is absent if the basic flow is calculated with boundary layer assumptions, indicating that the influence of inviscid flow outside the boundary layer cannot be ignored as is normally done. It is clearly demonstrated that the global modes display the features of both attachment-line modes, as in sweep Hiemenz flow, and the second Mack modes further downstream along the surface. In the large sweep Mach number regime, the attachment-line mode is inviscid in nature and its growth rate increases with the sweep angle. In contrast, in the lower sweep Mach number regime, the attachment-line instability exhibits the features of viscous Tollmien–Schlichting waves, and the sweep angle first increases but then decreases the maximum growth rate.

We examine the axisymmetric and non-axisymmetric flows of thin fluid films over a spherical glass dome. A thin film is formed by raising a submerged dome through a silicone oil mixture composed of a volatile, low surface tension species (1 cSt, solvent) and a non-volatile species at a higher surface tension (5 cSt, initial solute volume fraction $\phi _0$). Evaporation of the 1 cSt silicone oil establishes a concentration gradient and, thus, a surface tension gradient that drives a Marangoni flow that leads to the formation of an initially axisymmetric mound. Experimentally, when $\phi _0 \leqslant 0.3\,\%$, the mound grows axisymmetrically for long times (Rodríguez-Hakim et al., Phys. Rev. Fluids, vol. 4, 2019, pp. 1–22), whereas when $\phi _0 \geqslant 0.35\,\%$, the mound discharges in a preferred direction, thereby breaking symmetry. Using lubrication theory and numerical solutions, we demonstrate that, under the right conditions, external disturbances can cause an imbalance between the Marangoni flow and the capillary flow, leading to symmetry breaking. In both experiments and simulations, we observe that (i) the apparent, most amplified disturbance has an azimuthal wavenumber of unity, and (ii) an enhanced Marangoni driving force (larger $\phi _0$) leads to an earlier onset of the instability. The linear stability analysis shows that capillarity and diffusion stabilize the system, while Marangoni driving forces contribute to the growth in the disturbances.

The formation of vortex rings emanating from orifices with different orifice-to-tube diameter ratios $D_0/D_p$ is studied using time-resolved particle image velocimetry. The invariants of the motion in their non-dimensional form are computed and presented in the non-dimensional time space $t^*=U_0 t/D_0$, where the subscript 0 refers to the exhaust quantities. The classic slug-flow model is revisited and extended to account for the contraction of the flow when fluid is being pushed out through the orifice. Accordingly, a new time scale in terms of the contracted quantities (subscript $\star$) is defined as $T^*=U_\star t/D_\star$. Results show that the modified slug-flow model unifies the formation number of orifices and straight nozzles with a value of approximately 4.

In this study, the response of a supercritical round jet to various excitation modes including varicose, helical, flapping, dual varicose/helical and dual varicose/flapping is studied using large eddy simulations. A translation method is proposed to enhance the accuracy of the equation-of-state and transport correlations. Results show that the excitations, especially the dual modes and the varicose mode (when the forcing frequency matches the preferred mode in the potential core), considerably increase the turbulent mixing, the pitch distance and the penetration depth of the coherent structures as compared with the unexcited case. However, the excitations, especially the dual modes, de-energize the coherent structures and reduce the degree of three-dimensionality of the coherent structures. The excitations reduce the potential core length drastically, especially under the flapping and the dual mode excitations. Analyses show that the dual varicose/flapping mode excitations have the highest impacts on the jet development and the cross-section shape as compared with the other modes. Moreover, the dual varicose/flapping excitations have the highest impact on the large-scale turbulent mixing. However, the small-scale turbulent mixing is at the maximum value, when the supercritical jet is stimulated by the dual varicose/flapping mode excitations with the varicose-to-flapping frequency ratio of 2. The cross-correlations between the density fluctuations and the imposed perturbations indicate that the impact of the excitations on the turbulent diffusion is at the maximum value at the potential core breakdown location, while the correlation diminishes at the other locations.

This paper presents an analytical study of the Riemann problem for dilute granular gas using initial conditions that result in a shock–contact–shock wave structure. The Euler equations for molecular gas were perturbed with small granular energy dissipation, resulting in an approximate analytical solution that is valid for early evolution time. This approximate analytical solution shows good agreement with the numerical solution of the full problem obtained using a shock-capturing scheme. It is shown that the wave structure of the Riemann problem for dilute granular gas follows that of molecular gas. However, the variables in regions between the discontinuities are functions of both space and time. Our solution shows that the ‘density overshoot’ – reported by Reddy & Alam (J. Fluid Mech., vol. 779, 2015, R2) – is not part of the shock layer but a signature of the density variation in the star region between the left and right shocks, with the maximum density occurring at the contact discontinuity.

Elastohydrodynamic lubrication, or simply soft lubrication, refers to the motion of deformable objects near a boundary lubricated by a fluid, and is one of the key physical mechanisms to minimise friction and wear in natural and engineered systems. Hence, it is of particular interest to relate the thickness of the lubricant layer to the entrainment (sliding/rolling) velocity, the mechanical loading exerted onto the contacting elements and the properties of the elastic boundary. In this work, we provide an overview of the various regimes of soft lubrication for two-dimensional cylinders in lubricated contact with compliant walls. We discuss the limits of small and large entrainment velocity, which are equivalent to large and small elastic deformations, as the cylinder moves near thick or thin elastic layers. The analysis focusses on thin elastic coatings, both compressible and incompressible, for which analytical scaling laws are not yet available in the regime of large deformations. By analysing the elastohydrodynamic boundary layers that appear at the edge of the contact, we establish the missing scaling laws – including prefactors. As such, we offer a rather complete overview of the physically relevant limits of soft lubrication.

The evolution of the boundary layer vortex sheet on a rotating and translating accelerating circular cylinder at Reynolds numbers of 10 000 and 20 000 is investigated using planar particle image velocimetry. The vortex sheet is decomposed into contributions resulting from translation and rotation as well as from local and far-field vorticity. Their individual development is explored to understand the overall time history of the boundary layer as well as its evolution at the unsteady separation point. The boundary layer vortex sheet distribution changes considerably throughout the motion as well as between different kinematic cases. The same is observed for the vortex sheet strength at the unsteady separation point. A non-dimensional parameter is proposed which removes the effect of rotation rate, instantaneous velocity and shed vorticity accumulating in the far field. It was found that this was successful at collapsing the vortex sheet strength at the unsteady separation point during cylinder motion as well as for the individual kinematic test cases investigated. This confirms that cylinder kinematics and far-field vorticity are driving factors contributing to the development of the unsteady boundary layer and its strength at the separation point.

The elastohydrodynamics of slender bodies in a viscous fluid have long been the source of theoretical investigation, being pertinent to the microscale world of ciliates and flagellates as well as to biological and engineered active matter more generally. Although recent works have overcome the severe numerical stiffness typically associated with slender elastohydrodynamics, employing both local and non-local couplings to the surrounding fluid, there is no framework of comparable efficiency that rigorously justifies its hydrodynamic accuracy. In this study, we combine developments in filament elastohydrodynamics with a recent regularised slender-body theory, affording algebraic asymptotic accuracy to the commonly imposed no-slip condition on the surface of a slender filament of potentially non-uniform cross-sectional radius. Further, we do this whilst retaining the remarkable practical efficiency of contemporary elastohydrodynamic approaches, having drawn inspiration from the method of regularised Stokeslet segments to yield an efficient and flexible slender-body theory of regularised non-uniform segments.

This study experimentally investigates the effect of a surface pattern applied to a flat disk on the impact force during water entry. A macroscale mesh-like pattern with square holes is applied to the bottom surface of a flat disk, and the shape of the pattern and the falling speed of the disk are varied to find a universal parameter that characterizes the impact force. When a surface-patterned disk impacts a free surface, air is trapped inside the holes, and the subsequent compression and depressurization of the trapped air is accompanied by a rise and fall in the impact force. Under the cushioning effect of the trapped air, the peak of the impact force decreases and its period extends as the total volume of holes in the surface pattern increases. These changes are independent of the specific shape of the pattern. In contrast, the impulse exerted on the disk remains similar, regardless of the total volume of holes and the pattern. We conduct a simple theoretical analysis based on the added mass of the disk to estimate the impact force, and confirm the trends observed in our experiments.

In an earlier paper, Oruba et al. (J. Fluid Mech., vol. 818, 2017, pp. 205–240) considered the primary quasi-steady geostrophic (QG) motion of a constant density fluid of viscosity $\nu$ that occurs during linear spin-down in a cylindrical container of radius $L$ and height $H$, rotating rapidly (angular velocity $\varOmega$) about its axis of symmetry subject to mixed rigid and stress-free boundary conditions for the case $L=H$. Direct numerical simulation (DNS) of the linear system at large $L= 10 H$ and Ekman number $E\leqslant \nu /H^2\varOmega =10^{-3}$ by Oruba et al. (J. Fluid Mech., vol. 888, 2020, p. 44) reveals significant inertial wave activity on the spin-down time scale. That analytic study, for $E\ll 1$, builds on the results of Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404) for an infinite plane layer $L\to \infty$. At large but finite distance from the symmetry axis, the meridional (QG-)flow, that causes the QG-spin-down, is blocked by the lateral boundary, which provides the primary QG-trigger for inertial wave generation. For the laterally unbounded layer, Greenspan and Howard identified, in addition to the QG-flow, inertial waves of maximum frequency (MF) $2\varOmega$, which are a manifestation of the transient Ekman layer. The blocking of these additional MF-waves by the lateral boundary provides an extra trigger that complements the QG-triggered inertial waves. Here we obtain analytic results for the full wave activity caused by the combined trigger ($\text {QG}+\text {MF}$) that faithfully capture their true character.

Based on the linear shallow-water equations, new analytical solutions are derived for trapped waves over a ridge with a hyperbolic-cosine squared cross-sectional profile which may be used to idealize many real-world ocean ridges. In the new analytical formulation, the free surface of the trapped waves is described using the combination of the first and second kinds of the associated Legendre functions, which is further analysed to reveal the existence of both symmetrical and anti-symmetrical trapped waves on the ridge under consideration. New algebraic equations are also derived to depict the wave dispersion relationships, allowing explicit quantification of their sensitivity to the topographic profile. Furthermore, a ray-tracing method is applied to interpret the propagation paths of trapped waves over the ridge and better understand the excitation mechanisms. Finally, an extensively validated Boussinesq wave model is used to conduct numerical experiments for trapped waves induced by tsunamis. The numerical predictions are consistent with the new analytical solutions, which effectively confirms the validity of the new analytical framework for trapped waves over a more general type of oceanic ridges.