The stability characteristics of a Mach $5.35$ boundary-layer flow over a flat plate with parametrised two-dimensional sinusoidal surface roughness are investigated. The investigation involves varying the roughness height from $10\,\%$ to $44\,\%$ of the boundary-layer thickness and exploring wavelengths ranging between $0.44$ and $3.56$ times the dominant second-mode wavelength in the region. The introduction of surface roughness leads to notable variations in the mean flow, resulting in separation behind the roughness elements and the propagation of local compression and expansion waves into the free stream. Stability investigations involved the utilisation of wave packet tracking in a linear disturbance simulation (LDS) framework and linear stability theory. The findings revealed significant effects on Mack modes including a reduction in frequency corresponding to maximum amplification with increased roughness height. Proper scaling of the dominant wavelength facilitates a collapse of the growth rate data. In contrast to the commonly reported stabilisation effects for roughness wavelengths significantly larger than the instability mode’s wavelength, the findings primarily revealed destabilisation compared with the smooth-wall case, except for cases with very small roughness wavelengths and large amplitudes approaching the threshold of being classified as porous media. The LDS findings depicted lobed wall pressure amplitude plots, indicating potential undiscovered instability mechanisms or differences compared with the smooth wall. A detailed stability analysis elucidates these LDS findings, establishing a connection between the lobed amplitude structures and substantial changes in local stability characteristics, along with the emergence of Mack’s first, second and third modes.

A typical dielectric-barrier-discharge plasma actuator operating in burst mode generates periodic vortices resembling the starting vortex. This paper presents the three-dimensional (3-D) characteristics and instability mechanism of these vortices. The experimental investigation is carried out using smoke visualisation and time-resolved particle image velocimetry techniques in three orthogonal measurement planes. The size of the vortices decreases with an increase in burst signal frequency, $ f_{b}$, at a constant duty cycle, $ \alpha$. At higher burst frequencies, dipole vortices are formed due to the roll-up of the wall boundary layer. The angle of travel also decreases with an increase in $ f_{b}$. The evolution of $ \lambda _{2}$-criterion clearly demonstrates the vortex merging of co-rotating vortices. The vortex merging occurs at a critical ratio $ a_{c}/l_{c}$ of core size, $ a_{c}$, and separation distance, $ l_{c}$, equal to $ 0.22\pm 0.01$ which is close to $ a_{c}/l_{c} = 0.24\pm 0.01$ reported by Meunier et al. (Phys. Fluids,vol.14, 2002, pp. 2757–2766) for merging of a pair of equal two-dimensional co-rotating vortices. The periodic vortices are self-similar in nature and the vorticity distribution inside their core region follows the Lamb–Oseen vortex model. Cell structures form in the spanwise direction, which develops wave-like behaviour with an increase in burst frequency. Subsequently, these cell-like structures separate from each other, whose size and spacing correlate well with that of vorticity patches. The alternating sign of vorticity indicates that the circular cells have rotational motion in opposite sense with respect to each other. These cells grow downstream and appear in pairs of counter-rotating vortices (vortex dipole) akin to mushroom-like structures. At low values of $ \alpha$ and $ f_{b}$, the periodic vortex is subjected to a very weak strain and centrifugal instability dominates. The vortices are subjected to a higher strain at elevated burst frequencies, leading to the elliptic instability phenomenon similar to that observed in counter-rotating (Leweke & Williamson, J. Fluid Mech. 1998, vol. 360, pp. 85–119) and co-rotating (Meunier & Leweke, J. Fluid Mech.2005, vol. 533, pp. 125–159) vortex pair generated in water. The present experimental results based on the cross-cut visualisation, Galilean streamlines and vorticity decomposition confirm the role of the instability mechanism on the 3-D vortical structures generated by the dielectric-barrier-discharge plasma actuator.

Spatial optimal responses to both inlet disturbances and harmonic external forcing for hypersonic flows over a blunt cone at non-zero angles of attack are obtained by efficiently solving the direct–adjoint equations with a parabolic approach. In either case, the most amplified disturbances initially take the form of localised streamwise vortices on the windward side and will undergo a two-stage evolution process when propagating downstream: they first experience a substantial algebraic growth by exploiting the Orr and lift-up mechanisms, and then smoothly transition to a quasi-exponential growth stage driven by the crossflow-instability mechanism, accompanied by an azimuthal advection of the disturbance structure towards the leeward side. The algebraic growth phase is most receptive to the external forcing, whereas the exponential growth stage relies on the disturbance frequency and can be significantly strengthened by increasing the angle of attack. The wavemaker delineating the structural sensitivity region for the optimal gain is shown to lie on the windward side immediately downstream of the inlet, implying a potent control strategy. Additionally, considerable non-modal growth is also observed for broadband high-frequency disturbances residing in the entropy layer.

A complete analytical solution procedure is proposed here to work out the mixed boundary value problems associated with the oblique wave scattering problem involving either a complete elastic porous plate or a permeable membrane in both the cases of finite and infinite depth water in a two-layer fluid. Problems for two different velocity potentials with a phase difference are described in the upper half-planes. They are redefined as the solution potentials for the problems in the quarter-plane. A couple of novel integro-differential relations are constructed to connect the solution potentials of the redefined problems with auxiliary wave potentials. The subsequent potentials are solutions to relatively simpler boundary value problems for the modified Helmholtz equation, with structural boundary conditions of the Neumann type. The generalised orthogonal relation is then used to address the auxiliary wave potential problems analytically. The solution wave potentials are then derived in terms of these auxiliary wave potentials with the aid of the integro-differential relations. Further, explicit analytical expressions are derived for the significant hydrodynamic quantities such as energy reflection and transmission coefficients corresponding to the surface mode (SM) and interface mode (IM), respectively. Moreover, the deflection of the flexible porous structures is derived analytically. The scattering quantities in both SM and IM are presented graphically against the wavenumber and angle of incidence for various values of non-dimensional parameters involved in the structures.

Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity – this is known as ‘contour dynamics’. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like mergers) or ‘filamentation’. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly– and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a re-scaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full contour dynamics equations, this corresponds to the onset of filamentation.

We consider the dynamics of a liquid film with a pinned contact line (for example, a drop), as described by the one-dimensional, surface-tension-driven thin-film equation $h_t + (h^n h_{xxx})_x = 0$, where $h(x,t)$ is the thickness of the film. The case $n=3$ corresponds to a film on a solid substrate. We derive an evolution equation for the contact angle $\theta (t)$, which couples to the shape of the film. Starting from a regular initial condition $h_0(x)$, we investigate the dynamics of the drop both analytically and numerically, focusing on the contact angle. For short times $t\ll 1$, and if $n\ne 3$, the contact angle changes according to a power law $\displaystyle t^{\frac {n-2}{4-n}}$. In the critical case $n=3$, the dynamics become non-local, and $\dot {\theta }$ is now of order $\displaystyle {\rm{e}}^{-3/(2t^{1/3})}$. This implies that, for $n=3$, the standard contact line problem with prescribed contact angle is ill posed. In the long time limit, the solution relaxes exponentially towards equilibrium.

We present numerical analysis of the lateral movement of a spherical capsule in the steady and pulsatile channel flow of a Newtonian fluid for a wide range of oscillatory frequencies. Each capsule membrane satisfying strain-hardening characteristics is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$. Our numerical results showed that capsules with high $Ca$ exhibit axial focusing at finite $Re$ similarly to the inertialess case. We observe that the speed of the axial focusing can be substantially accelerated by making the driving pressure gradient oscillate in time. We also confirm the existence of an optimal frequency that maximises the speed of axial focusing, which remains the same found in the absence of inertia. For relatively low $Ca$, however, the capsule exhibits off-centre focusing, resulting in various equilibrium radial positions depending on $Re$. Our numerical results further clarify the existence of a specific $Re$ for which the effect of the flow pulsation to the equilibrium radial position is maximum. The roles of channel size on the lateral movements of the capsule are also addressed. Throughout our analyses, we have quantified the radial position of the capsule in a tube based on an empirical expression. Given that the speed of inertial focusing can be controlled by the oscillatory frequency, the results obtained here can be used for label-free cell alignment/sorting/separation techniques, e.g. for circulating tumour cells in cancer patients or precious haematopoietic cells such as colony-forming cells.

Planar linear flows are a one-parameter family, with the parameter $\hat {\alpha }\in [-1,1]$ being a measure of the relative magnitudes of extension and vorticity; $\hat {\alpha } = -1$, $0$ and $1$ correspond to solid-body rotation, simple shear flow and planar extension, respectively. For a neutrally buoyant spherical drop in a hyperbolic planar linear flow with $\hat {\alpha }\in (0,1]$, the near-field streamlines are closed for $\lambda \gt \lambda _c = 2 \hat {\alpha } / (1 - \hat {\alpha })$, $\lambda$ being the drop-to-medium viscosity ratio; all streamlines are closed for an ambient elliptic linear flow with $\hat {\alpha }\in [-1,0)$. We use both analytical and numerical tools to show that drop deformation, as characterized by a non-zero capillary number ($Ca$), destroys the aforementioned closed-streamline topology. While inertia has previously been shown to transform closed Stokesian streamlines into open spiralling ones that run from upstream to downstream infinity, the streamline topology around a deformed drop, for small but finite $Ca$, is more complicated. Only a subset of the original closed streamlines transforms to open spiralling ones, while the remaining ones densely wind around a configuration of nested invariant tori. Our results contradict previous efforts pointing to the persistence of the closed streamline topology exterior to a deformed drop, and have important implications for transport and mixing.

The wave kinetic equation has become an important tool in different fields of physics. In particular, for surface gravity waves, it is the backbone of wave forecasting models. Its derivation is based on the Hamiltonian dynamics of surface gravity waves. Only at the end of the derivation are the non-conservative effects, such as forcing and dissipation, included as additional terms to the collision integral. In this paper, we present a first attempt to derive the wave kinetic equation when the dissipation/forcing is included in the deterministic dynamics. If, in the dynamical equations, the dissipation/forcing is one order of magnitude smaller than the nonlinear effect, then the classical wave action balance equation is obtained and the kinetic time scale corresponds to the dissipation/forcing time scale. However, if we assume that the nonlinearity and the dissipation/forcing act on the same dynamical time scale, we find that the dissipation/forcing dominates the dynamics and the resulting collision integral appears in a modified form, at a higher order.

Debris flows are a growing natural hazard as a result of climate change and population density. To effectively assess this hazard, simulating field-scale debris flows at a reasonable computational cost is crucial. We enhance existing debris flow models by rigorously deriving a series of depth-averaged shallow models with varying complexities describing the behaviour of grain–fluid flows, considering granular mass dilatancy and pore fluid pressure feedback. The most complete model includes a mixture layer with an upper fluid layer, and solves for solid and fluid velocity in the mixture and for the upper fluid velocity. Simpler models are obtained by assuming velocity equality in the mixture or single-layer descriptions with a virtual thickness. Simulations in a uniform configuration mimicking submarine landslides and debris flows reveal that these models are extremely sensitive to the rheology, the permeability (grain diameter) and initial volume fraction, parameters that are hard to measure in the field. Notably, velocity equality assumptions in the mixture hold true only for low permeability (corresponding to grain diameter $d=10^{-3}$ m). The one-layer models’ results can strongly differ from those of the complete model, for example, the mass can stop much earlier. One-layer models, however, provide a rough estimate of two-layer models when permeability is low, initial volume fraction is far from critical and the upper fluid layer is very thin. Our work with uniform settings highlights the need of developing two-layer models accounting for dilatancy and for an upper layer made either of fluid or grains.

We investigate the effects of thermal boundary conditions and Mach number on turbulence close to walls. In particular, we study the near-wall asymptotic behaviour for adiabatic and pseudo-adiabatic walls, and compare to the asymptotic behaviour recently found near isothermal cold walls (Baranwal et al. 2022. J. Fluid Mech. 933, A28). This is done by analysing a new large database of highly-resolved direct numerical simulations of turbulent channels with different wall thermal conditions and centreline Mach numbers. We observe that the asymptotic power-law behaviour of Reynolds stresses as well as heat fluxes does change with both centreline Mach number and thermal condition at the wall. Power-law exponents transition from their analytical expansion for solenoidal fields to those for non-solenoidal field as the Mach number is increased, though this transition is found to be dependent on the thermal boundary conditions. The correlation coefficients between velocity and temperature are also found to be affected by these factors. Consistent with recent proposals on universal behaviour of compressible turbulence, we find that dilatation at the wall is the key scaling parameter for these power-law exponents, providing a universal functional law that can provide a basis for general models of near-wall behaviour.

It is generally accepted that the evolution of the deep-water surface gravity wave spectrum is governed by quartet resonant and quasi-resonant interactions. However, it has also been reported in both experimental and computational studies that non-resonant triad interactions can play a role, e.g. generation of bound waves. In this study, we investigate the effects of triad and quartet interactions on the spectral evolution, by numerically tracking the contributions from quadratic and cubic terms in the dynamical equation. In a finite time interval, we find that the contribution from triad interactions follows the trend of that from quartet resonances (with comparable magnitude) for most wavenumbers, except that it peaks at low wavenumbers with very low initial energy. This result reveals two effects of triad interactions. (1) The non-resonant triad interactions can be connected to form quartet resonant interactions (hence exhibiting the comparable trend), which is a reflection of the normal form transformation applied in wave turbulence theory of surface gravity waves. (2) The triad interactions can fill energy into the low-energy portion of the spectrum (low wavenumber part in this case) on a very fast time scale, with energy distributed in both bound and free modes at the same wavenumber. We further analyse the latter mechanism using a simple model with two initially active modes in the wavenumber domain. Analytical formulae describing the distribution of energy in free and bound modes are provided, along with numerical validations.

In a quiescent medium, chemically active particles propel themselves by emitting or absorbing solutes, creating concentration gradients that induce a slip at the particle surface. This self-propulsion occurs when solute advection overcomes diffusion. However, an imposed flow field can alter these dynamics. This study explores the propulsion characteristics and the related rheological consequences of chemically active particles in an imposed uniaxial extensional flow analytically and numerically. An asymptotic solution is obtained for weak imposed flow relative to self-induced diffusiophoretic slip. Meanwhile, finite element simulations are carried out over a wide range of imposed flow strength and Péclet number. The results reveal that the interplay between solute advection, imposed flow and diffusiophoretic slip significantly affects particle propulsion and suspension rheology. While solute advection and diffusiophoretic slip tend to create asymmetric solute distributions, promoting self-propulsion, imposed extensional flow promotes symmetric distributions, hindering self-propulsion. This not only delays the start of self-propulsion but also results in an early transition from a propulsion state to a stationary state characterised by an abrupt halt at relatively lower Péclet number compared to a quiescent medium. Post the abrupt halt, a stirring effect induced by particle activity and imposed extensional flow results in an increased magnitude of stresslet, thus a sudden change in the effective viscosity of the active suspension. The effect of imposed extensional flow on active particle dynamics and suspension rheology can be described succinctly by categorising the overall dynamics into three separate regimes, determined by the Péclet number and the intensity of the extensional flow.

Thermal Marangoni effects play important roles in bubble dynamics such as bubbles generated by water electrolysis or boiling. As macroscopic bubbles often originate from nucleated nanobubbles, it is crucial to understand how thermocapillarity operates at the nanoscale. In this study, the motion of transient bulk gas nanobubbles in water driven by a vertical temperature gradient between two solid plates is investigated using molecular dynamics simulations and analytical theory. The simulation results show that due to the thermal Marangoni force, nanobubbles move towards the hot plate at a constant velocity, similar to the behaviour of macroscale gas bubbles. However, unlike macroscale gas bubbles whose thermal conductivity and viscosity are negligible compared to those of water, the thermal conductivity and viscosity of nanoscale gas bubbles are significantly increased due to their large gas density. The thermal resistance and the slip length are also found to matter at the liquid–gas interface, though they decrease with increasing gas densities. The previous thermocapillary theory for macroscale bubbles is extended by considering all these nanoscopic effects. Expressions of the Marangoni force and the drag force are derived. By balancing the Marangoni force and the drag force, the theoretical velocity of the nanobubble migration in a thermal gradient is obtained. When using the measured transport properties of liquid, gas, and their interfaces, the theoretically obtained velocity is consistent with the result of the molecular simulations. We find that the slip length is too small to have considerable effects on nanobubble motions in the current liquid–gas system.

Granular surface flows are frequently encountered in nature as well as during handling of powders in different industries. An experimental study of granular surface flow on a heap is carried out. The heap is formed by pouring nearly monodisperse spherical particles from the rectangular slit orifice of a hopper on a rough rectangular plate. A flow of particles is developed on the heap surface, which is planar in the central region, with particles flowing over the edge of the plate into a collection chamber. The geometry considered in this study is an example of a fully three-dimensional heap without side walls. The surface velocities of the particles are measured using high-speed videography and particle tracking velocimetry for different mass flow rates with steel balls and glass beads of two different sizes, for heaps of different aspect ratios. The flow is uni-directional and fully developed in a central zone on the heap surface. The flowing layer thickness is measured in this zone by immersing a soot-coated blade into the flow. The angle of inclination of the free surface of the heap is found to be nearly constant for a ten-fold increase of the mass flow rate. The scaled flowing layer thickness is found to vary linearly with the scaled flow rate and the data for all the particles collapse to a single line over a ten-fold increase in the scaled flow rate and an increase in the aspect ratio of the heap by a factor of 1.75. The predicted scaled surface velocity and scaled shear rate using this correlation match the measured values.

The evolution of turbulent spots in a flat plate boundary layer is examined using time-resolved tomographic particle image velocimetry (Tomo-PIV) experiments and direct numerical simulation (DNS). The characteristics of flow structures are examined using timelines and material surfaces. Both the numerical and experimental results reveal a notable behaviour in the developmental process of turbulent spots: the development of low-speed streaks at the spanwise edges of turbulent spots, followed by the subsequent formation of hairpin vortices. The behaviour of these low-speed streaks is further investigated using timelines and material surfaces generated for a series of regions and development times. The results indicate that these low-speed streaks exhibit characteristic wave behaviour. The low-speed streaks are observed to lift up as three-dimensional (3-D) waves, with high-shear layers forming at the interface of these waves. These induced high-shear layers become unstable and evolve into vortices, which contribute to the expansion of the turbulent spot. These findings show the significant role of 3-D waves in the development of turbulent spots, supporting the hypothesis that 3-D waves serve as initiators of vortices at the bounding surface of a turbulent spot.

This paper presents detailed analyses of the Reynolds stresses and their budgets in temporally evolving stratified wakes using direct numerical simulation. Ensemble averaging is employed to mitigate statistical errors in the data, and the results are presented as functions of both the transverse and vertical coordinates – at time instants across the near-wake, non-equilibrium, and quasi-two-dimensional regimes for wakes in weakly and strongly stratified environments. Key findings include the identification of dominant terms in the Reynolds stress transport equations and their spatial structures, the generation and destruction processes of the Reynolds stresses, and the energy transfer between the Reynolds stress and the mean flow. The study also clarifies the effects of the Reynolds number and the Froude number. Additionally, we assess the validity of the eddy-viscosity type models and some existing closures for the Reynolds stress model, highlighting the limitations of isotropy and return-to-isotropy hypotheses in stratified flows.

Steady-state Bloch wave systems at resonance with fixed frequencies and amplitudes are investigated using the homotopy analysis method. Nonlinear waves propagate over a stationary undulating bottom topography of infinite extent, modelled as a superposition of two waveforms. The wave systems are classified as type 1 if the primary transmitted and resonant wave components have equal energies, and type 2 if the energy distribution is unequal. Two subtypes of type 2 are identified, distinguished by their responses to frequency detuning and bottom topography: the wave steepness in subtype 1 shows monotonic variations with detuning, while in subtype 2 it exhibits a peak at a particular detuning value, indicating downward resonance that intensifies with greater wave steepness. A pair of peaks in wave steepness arises in each subtype at certain values of the angle $\theta$ between the waveforms of the bottom topography. In both subtypes, the peaks are slightly affected by the ratio $k_{{b}1}/k_{{b}2}$ of the two bottom wave vectors, and significantly affected by the propagation angle $\alpha$ of the primary transmitted wave, but remain stable under changes to other topographic parameters. As the topography amplitude and $\theta$ vary, significant additional contributions to the total energy of the wave system appear from components other than resonant and primary transmitted waves. The most pronounced effects occur with changes in $\theta$, with the additional components accounting for up to 12 % of the total energy. This study provides an enriched understanding of resonant Bloch wave systems and a basis for improving the effectiveness of wave energy converters.

Turbulent flames in practical devices are subject to a superposition of broadband turbulence and narrowband harmonic flow oscillations. In such cases, flames have a superposition of space–time correlated wrinkles, superposed with broadband turbulent disturbances that interact nonlinearly. This paper extends our prior experimental work to characterise and quantify these flame dynamics. We extract ensemble-averaged flame edge and velocity by ensemble-averaging the instantaneous data at the same phase with respect to the forcing cycle. This paper shows that the ensemble-averaged spatio-temporal dynamics of the flame changes significantly with turbulence intensity. From a spatial viewpoint, the ensemble-averaged flame at weak turbulence intensities exhibits clear cusps and a large ratio between curvature in concave and convex regions. In contrast, at high turbulence intensities, the concave and convex parts of the ensemble-averaged flame are nearly symmetric. From a temporal viewpoint, increasing turbulence intensity monotonically suppresses higher harmonics of the forcing frequency that are manifestations of flame nonlinearities. Taken together, these both point to the interesting observation that the ensemble-averaged flame exhibits increasingly linear dynamics with increasing turbulence intensities, in contrast to its very strong nonlinear behaviours at weak turbulence intensities and juxtaposed with the increasingly nonlinear nature of its instantaneous dynamics with increasing turbulence intensity. In addition, prior studies have shown clear coherent modulation of turbulent flame speed correlated with coherent curvature modulation and that this relationship could be quantified via a ‘turbulent Markstein number’, $M_{T}$. We develop correlations for $M_{T}$ showing how it scales with turbulent and narrowband disturbance quantities, such as turbulent flame brush thickness and convective length scale.

The laminar flow past rectangular prisms is studied in the space of length-to-height ratio ($1 \leqslant L/H \leqslant 5$), width-to-height ratio ($1.2 \leqslant W/H \leqslant 5$) and Reynolds number ($Re \lessapprox 700$); $L$ and $W$ are the streamwise and cross-flow dimensions of the prisms. The primary bifurcation is investigated with linear stability analysis. For large $W/L$, an oscillating mode breaks the top/bottom planar symmetry. For smaller $W/L$, the flow becomes unstable to stationary perturbations and the wake experiences a static deflection, vertical for intermediate $W/L$ and horizontal for small $W/L$. Weakly nonlinear analysis and nonlinear direct numerical simulations are used for $L/H = 5$ and larger $Re$. For $W/H = 1.2$ and 2.25, the flow recovers the top/bottom planar symmetry but loses the left/right one, via supercritical and subcritical pitchfork bifurcations, respectively. For even larger $Re$, the flow becomes unsteady and oscillates around either the deflected (small $W/H$) or the non-deflected (intermediate $W/H$) wake. For intermediate $W/H$ and $Re$, a fully symmetric periodic regime is detected, with hairpin vortices shed from the top and bottom leading-edge (LE) shear layers; its triggering mechanism is discussed. At large $Re$ and for all $W/H$, the flow approaches a chaotic state characterised by the superposition of different modes: shedding of hairpin vortices from the LE shear layers, and wake oscillations in the horizontal and vertical directions. In some portions of the parameter space the different modes synchronise, giving rise to periodic regimes also at relatively large $Re$.