Direct numerical simulations of a uniform flow past a fixed spherical droplet are performed to determine the parameter range within which the axisymmetric flow becomes unstable. The problem is governed by three dimensionless parameters: the drop-to-fluid dynamic viscosity ratio, $\mu ^\ast$, and the external and internal Reynolds numbers, ${\textit{Re}}^e$ and ${\textit{Re}}^i$, which are defined using the kinematic viscosities of the external and internal fluids, respectively. The present study confirms the existence of a regime at low-to-moderate viscosity ratio where the axisymmetric flow breaks down due to an internal flow instability. In the initial stages of this bifurcation, the external flow remains axisymmetric, while the asymmetry is generated and grows only inside the droplet. As the disturbance propagates outward, the entire flow first transits to a biplanar-symmetric flow, characterised by two pairs of counter-rotating streamwise vortices in the wake. A detailed examination of the flow field reveals that the vorticity on the internal side of the droplet interface is driving the flow instability. Specifically, the bifurcation sets in once the maximum internal vorticity exceeds a critical value that decreases with increasing ${\textit{Re}}^i$. For sufficiently large ${\textit{Re}}^i$, internal flow bifurcation may occur at viscosity ratios of $\mu ^\ast = {\mathcal{O}}(10)$, an order of magnitude higher than previously reported values. Finally, we demonstrate that the internal flow bifurcation in the configuration of a fixed droplet in a uniform fluid stream is closely related to the first path instability experienced by a buoyant, deformable droplet of low-to-moderate $\mu ^\ast$ freely rising in a stagnant liquid.

A literature review suggests that the flows past simply connected bodies with aspect ratio close to unity and symmetries aligned with the flow follow a consistent sequence of regimes (steady, periodic, quasiperiodic) as the Reynolds number increases. However, evidence is fragmented, and studies are rarely conducted using comparable numerical or experimental set-ups. This paper investigates the wake dynamics of two canonical bluff bodies with distinct symmetries: a cube (discrete) and a sphere (continuous). Employing three-dimensional (3-D) global linear stability analysis and nonlinear simulations within a unified numerical framework, we identify the bifurcation sequence driving these regime transitions. The sequence: a pitchfork bifurcation breaks spatial symmetry; a Hopf bifurcation introduces temporal periodicity ($St_1$); a Neimark–Sacker bifurcation destabilises the periodic orbit, leading to quasiperiodic dynamics with two incommensurate frequencies ($St_1, St_2$). A Newton–Krylov method computes the unstable steady and periodic base flows without imposing symmetry constraints. Linear stability reveals similarities between the cube and sphere in the spatial structure of the leading eigenvectors and in the eigenvalue trajectories approaching instability. This study provides the first confirmation of a Neimark–Sacker bifurcation to quasiperiodicity in these 3-D wakes, using Floquet stability analysis of computed unstable periodic orbits and their Floquet modes. The quasiperiodic regime is described in space and time by the Floquet modes’ effects on the base flow and a spectrum dominated by the two incommensurate frequencies and tones arising from nonlinear interactions. Although demonstrated for a cube and a sphere, this bifurcation sequence, leading from steady state to quasiperiodic dynamics, suggests broader applicability beyond these geometries.

Understanding the interplay between thermal, elastic and hydrodynamic effects is crucial for a variety of applications, including the design of soft materials and microfluidic systems. Motivated by these applications, we investigate the emergence of natural convection in a fluid layer that is supported from below by a rigid surface, and covered from above by a thin elastic sheet. The sheet is laterally compressed and is maintained at a constant temperature lower than that of the rigid surface. We show that for very stiff sheets, and below a certain magnitude of the lateral compression, the system behaves as if the fluid were confined between two rigid walls, where the emergent flow exhibits a periodic structure of vortices with a typical length scale proportional to the depth of the fluid, similar to patterns observed in Rayleigh–Bénard convection. However, for more compliant sheets, and above a certain threshold of the lateral compression, a new local minimum appears in the stability diagram, with a corresponding wavenumber that depends solely on the bending modulus of the sheet and the specific weight of the fluid, as in wrinkling instability of thin sheets. The emergent flow field in this region synchronises with the wrinkle pattern. We investigate the exchange of stabilities between these two solutions, and construct a stability diagram of the system.

Invariant maps are a useful tool for turbulence modelling, and the rapid growth of machine learning-based turbulence modelling research has led to renewed interest in them. They allow different turbulent states to be visualised in an interpretable manner and provide a mathematical framework to analyse or enforce realisability. Current invariant maps, however, are limited in machine learning models by the need for costly coordinate transformations and eigendecomposition at each point in the flow field. This paper introduces a new polar invariant map based on an angle that parametrises the relationship of the principal anisotropic stresses, and a scalar that describes the anisotropy magnitude relative to a maximum value. The polar invariant map reframes realisability in terms of a limiting anisotropy magnitude, allowing for new and simplified approaches to enforcing realisability that do not require coordinate transformations or explicit eigendecomposition. Potential applications to machine learning-based turbulence modelling include post-processing corrections for realisability, realisability-informed training, turbulence models with adaptive coefficients and general tensor basis models. The relationships to other invariant maps are illustrated through examples of plane channel flow and square duct flow. Sample calculations are provided for a comparison with a typical barycentric map-based method for enforcing realisability, showing an average 62 % reduction in calculation time using the equivalent polar formulation. The results provide a foundation for new approaches to enforcing realisability constraints in Reynolds-averaged turbulence modelling.

We study the interaction between a pair of particles suspended in a uniform oscillatory flow. The time-averaged behaviour of particles under these conditions, which arises from an interplay of inertial and viscous forces, is explored through a theoretical framework relying on small oscillation amplitude. We approximate the oscillatory flow in terms of dual multipole expansions, with which we compute time-averaged interaction forces using the Lorentz reciprocal theorem. We then develop analytic approximations for the force in the limit where Stokes layers surrounding the particles do not overlap. Finally, we show how the same formalism can be generalised to the situation where the particles are free to oscillate and drift in response to the applied flow. The results are shown to be in agreement with existing numerical data for forces and particle velocities. The theory thus provides an efficient means to quantify nonlinear particle interactions in oscillatory flows.

The dynamics of thin viscous liquid films flowing down an inclined wall under gravity in the presence of an upward flowing high-speed air stream is considered. The air stream induces nonlinear waves on the interface and asymptotic solutions are developed to derive a non-local evolution equation forced by the air pressure which is obtained analytically, and incorporating a constant tangential stress. Benney equations in the capillary (strong surface tension) and inertio-capillary regimes are derived and studied. The air stream produces Turing-type short wave instabilities in sub-critical Reynolds number regimes that would be stable in the absence of the outer flow. Extensive numerical experiments are carried out to elucidate the rich dynamics in the above-mentioned short-wave regime. The stability of different branches of solutions of non-uniform steady states is carried out, along with time-dependent nonlinear computations that are used to track the large-time behaviour of attractors. A fairly complete picture of different solution types are categorised in parameter space. The effect of the Reynolds number on the wave characteristics in the inertio-capillary regime is also investigated. It is observed that, for each value of the slenderness parameter $\delta$, there exists a critical Reynolds number $R_c$ above which the solutions become unbounded by encountering finite-time singularities. Increasing the air speed significantly decreases $R_c$, making the system more prone to large amplitude singular events even at low Reynolds numbers when the system would have been stable in the absence of the air stream.

An analytical theory is presented for linear, local, short-wavelength instabilities in swirling flows, in which axial shear, differential rotation, radial thermal stratification, viscosity and thermal diffusivity are all taken into account. A geometrical optics approach is applied to the Navier–Stokes equations, coupled with the energy equation, leading to a set of amplitude transport equations. From these, a dispersion relation is derived, capturing two distinct types of instability: a stationary centrifugal instability and an oscillatory, visco-diffusive McIntyre instability. Instability regions corresponding to different axial or azimuthal wavenumbers are found to possess envelopes in the plane of physical parameters, which are explicitly determined using the discriminants of polynomials. As these envelopes are shown to bound the union of instability regions associated with particular wavenumbers, it is concluded that the envelopes correspond to curves of critical values of physical parameters, thereby providing compact, closed-form criteria for the onset of instability. The derived analytical criteria are validated for swirling flows modelled by a cylindrical, differentially rotating annulus with axial flow induced by either a sliding inner cylinder, an axial pressure gradient or a radial temperature gradient combined with vertical gravity. These criteria unify and extend, to viscous and thermodiffusive differentially heated swirling flows, the Rayleigh criterion for centrifugally driven instabilities, the Ludwieg–Eckhoff–Leibovich–Stewartson criterion for isothermal swirling flows and the Goldreich–Schubert–Fricke criterion for non-isothermal azimuthal flows. Additionally, they predict oscillatory modes in swirling, differentially heated, visco-diffusive flows, thereby generalising the McIntyre instability criterion to these systems.

We introduce a novel unsteady shear protocol, which we name rotary shear (RS), where the flow and vorticity directions are continuously rotated around the velocity-gradient direction by imposing two out-of-phase oscillatory shears (OSs) in orthogonal directions. We perform numerical simulations of dense suspensions of rigid non-Brownian spherical particles at volume fractions ($\phi$) between 0.40 and 0.55, subject to this new RS protocol, and compare with the classical OS protocol. We find that the suspension viscosity displays a similar non-monotonic response as the strain amplitude ($\gamma _0$) is increased: a minimum viscosity is found at an intermediate, volume-fraction-dependent strain amplitude. However, the suspension dynamics is different in the new protocol. Unlike the OS protocol, suspensions under RS do not show absorbing states at any $\gamma _0$ and do not undergo the reversible–irreversible transition: the stroboscopic particle dynamics is always diffusive, which we attribute to the fact that the RS protocol is inherently irreversible due to its design. To validate this hypothesis, we introduce a reversible-RS (RRS) protocol, a combination of RS and OS, where we rotate the shear direction (as in RS) until it is instantaneously reversed (as in OS), and find the resulting rheology and dynamics to be closer to OS. Detailed microstructure analysis shows that both the OS and RRS protocols result in a contact-free, isotropic to an in-contact, anisotropic microstructure at the dynamically reversible-to-irreversible transition. The RS protocol does not render such a transition, and the dynamics remains diffusive with an in-contact, anisotropic microstructure for all strain amplitudes.

Direct numerical simulations are conducted to investigate the transition flow over a flat plate featuring pressure gradients and a three-dimensional rough surface. The rough surface is categorised into nine types based on the effective slope ratio ${E{{S}_{z}}}/{E{{S}_{x}}}$ ($ES_{z}$: spanwise effective slope, $ES_{x}$: streamwise effective slope) and skewness $Sk$, with the embedded boundary method employed for resolving the solid wall. Findings indicate that the influence of ${E{{S}_{z}}}/{E{{S}_{x}}}$ on the streamwise vortex pair counters the effects on the wall-normal shear and the two-dimensional spanwise vortex sheet. Negative skewness alone can stimulate all three components of the hairpin vortex simultaneously. The new formula for predicting the sheltering angle, which incorporates the up-ejecting segment, demonstrates enhanced accuracy in predicting the sheltering area across the entire rough surface, outperforming the previous formulation. The forward displacement relative to the drag peak of the pressure stagnation point along the streamwise direction remains unaffected by the spanwise effective slope and the skewness. In the upper transition region, negative skewness significantly intensifies both the production and dissipation terms of the fluctuating kinetic energy, which correlate with the inviscid instability of the separation flow and the viscous instability induced by the lift-up mechanism. During the early phase of transition, negative skewness is capable of producing linear modes that match the intensity of nonlinear coherent structures at intermediate to high frequencies, exhibiting quasi-orthogonality. During the late transition phase, zero skewness can give rise to linear modes featuring robust quasi-orthogonality at low frequencies.

When a low Mach flow is imposed through an orifice at the end of a cavity, intense whistling can occur. It results from the constructive feedback loop between the acoustic field of the cavity and coherent vortex shedding at the edges of the orifice with bias flow. Whistling is often a source of unwanted noise, demanding passive control strategies. In this study, it is shown that whistling can be suppressed by utilising the slow-sound effect. This periodic arrangement of small cavities detunes the cavity from the frequency range where the orifice flow exhibits a potential for acoustic energy amplification, by reducing the effective speed of sound inside the cavity. Acoustic and optical measurement techniques are employed, including scattering matrix and impedance measurements, and particle image velocimetry to reconstruct the velocity field downstream of the orifice. The production and dissipation of acoustic energy is investigated using Howe’s energy corollary. The spatio-temporal patterns of the vortex sound downstream of the orifice are revealed. They are deduced from phase-averaged acoustic and Lamb vector fields and give qualitative insight into the physical mechanisms of the whistling phenomenon.