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 analyse a 36-year hydrodynamic and morphological dataset from the Hasaki coast, Japan, comprising 501 wave storm events (405 individual and 96 clustered events) to investigate the impact of storm dynamics and clustering on beach erosion. Focusing on the wave component of storms, events are identified using wave height thresholds. Daily and weekly beach profile measurements from the Hasaki Oceanographic Research Station are used to quantify erosion. The study examines the seasonal influences on Hasaki beach, the characteristics and temporal evolution of storms, and their associated erosional impacts. Moreover, we test two supervised machine learning (ML) algorithms, support vector regression (SVR), and deep neural network (DNN), in predicting shoreline change using 16 wave, storm, and morphological features. SVR showed reasonable accuracy on the training dataset but underperformed on testing, while DNN failed to produce reliable predictions on both. With SVR yielding an R2 of 0.18 and DNN 0.27 on the testing dataset, we conclude that, given the limited data and available features, such ML models may not generalise well. However, separate analyses using observed data reveal clear seasonal variations in wave storm dynamics and distinct behaviours of clustered events associated with beach erosion, highlighting important insights beyond the ML results.