We formulate a model of the two-way interactions between surface gravity waves and ocean currents. The model couples the transport of wave action in the four-dimensional (horizontal) position–wavevector phase space with the Craik–Leibovich system for the currents. Coupling is via the Doppler shift in the dispersion relation governing action transport, and wave pseudomomentum in the Craik–Leibovich system. The velocity in the Doppler shift is a vertical integral of the Lagrangian mean velocity of the currents, with a weight that is consistent with the vertical structure of the pseudomomentum. This consistency ensures conservation of momentum and energy in the coupled wave–current system. The conservation properties of the wave–current model stem from an underlying variational structure. We derive this structure from that of the rotating Euler equations for an incompressible fluid with free surface by introducing a Lagrangian wave–mean decomposition, making simplifying approximations, and Whitham averaging. The wave–mean decomposition is constructed such that the Lagrangian mean velocity is non-divergent and satisfies a rigid-lid boundary condition. We apply the wave–current model to the problem of generation of inertial oscillations by surface waves originally considered by Hasselmann.

Convection on geophysical and astrophysical scales is subject to rapid rotation and strong heating from within the domain. In studying the long-time behaviour of the solutions for such a system, energy identities fail to capture the effects of rotation because the Coriolis force does no work, and rapid rotation can be prohibitive for direct numerical simulations. Instead, we derive an asymptotically reduced model for rapidly rotating convection driven by uniform internal heating between isothermal stress-free boundaries in a plane periodic layer. The main contribution is the proof of bounds on the mean temperature, and the mean vertical convective heat transport, in terms of the Rayleigh and Ekman numbers, in the limit of infinite Prandtl number. The first quantity measures thermal mixing, and the second the asymmetry in heat leaving the bottom and top boundaries due to convection, and unlike Rayleigh–Bénard convection, the two are not a priori related. We employ alternative estimation techniques to those used in previous studies (Grooms & Whitehead, 2014, Nonlinearity, vol. 28, 29), and identify two distinct scaling behaviours for both quantities. Finally, our bounds are optimised, within the methodology, and provide a rigorous constraint for future studies of rotation-dominated internally heated convection.

This study makes a twofold contribution regarding vorticity flux modulation by wall-pressure ‘footprints’. First, we present a continuous data assimilation (DA) framework for reconstructing flow fields from wall-pressure observations by embedding an adjoint-derived scalar source into the pressure Poisson equation. By leveraging elliptic pressure–velocity coupling, the method propagates wall-driven corrections across the domain, and avoids the costly global backward integration of four-dimensional variational DA while retaining phase-aware control of vorticity production and modal energy. Second, we validate the framework on a three-dimensional turbulent flow over a NACA0012 aerofoil at $12^{\circ}$, using synthetic wall-pressure data from high-fidelity large eddy simulations (LES). The DA strategy mitigates unresolved transition physics in coarse simulations by suppressing premature separation, restoring attachment, and recovering key flow metrics with near-LES fidelity. Spectral and wavelet analyses further demonstrate that the reconstruction captures both the scale-dependent energy distribution and the spatial coherence of the dominant structures. Two complementary mechanisms are identified: (i) an oscillator-like mechanism, where phase-locked tangential pressure gradients couple with normal vorticity gradients to modulate near-wall vorticity through linear and nonlinear interactions; and (ii) an amplifier-like mechanism, where trailing-edge inputs propagate upstream via scale-dependent elliptic coupling, exciting receptivity modes, and triggering transition cascades through Orr mechanisms and resolvent-matched instabilities. The sensor coverage study reveals scale-dependent sensitivity: streamwise-elongated motions remain robust under sparse arrays, whereas spanwise-dominated structures require near-wall sensing. These findings broaden the scope of continuous DA for wall-bounded flow reconstructions, offering a computationally efficient way to incorporate wall-pressure observations into existing solvers for enhanced accuracy.

We numerically investigate the feasibility and limits of jointly estimating flow fields and unknown particle properties (e.g. position, size and density) from Lagrangian particle tracking data. Lagrangian particle tracking offers time-resolved, volumetric measurements of particle trajectories, which are markers of the carrier fluid motion. However, experimental tracks are spatially sparse and potentially noisy, and the problem of reconstructing flow fields may be further complicated by inertial particle transport, such that particle slip velocities must be determined to access the velocity field of the carrier fluid. To address this problem, we develop a data assimilation framework that couples an Eulerian representation of the flow with Lagrangian particle models, enabling the simultaneous inference of carrier fields and particle properties under the governing equations of disperse multiphase flow. We show that flow fields and particle properties can be jointly estimated in three representative regimes: (i) in a turbulent boundary layer with noisy tracer tracks ($\textit{St} \to 0$), flow fields and true particle positions are jointly estimated, which amounts to a physics-informed particle tracking problem; (ii) in homogeneous isotropic turbulence seeded with inertial particles ($\textit{St} \sim 1{-}5$), we demonstrate simultaneous recovery of flow states and particle diameters, showing the feasibility of implicit particle characterisation; and (iii) in a compressible, shock-dominated flow, we report the first joint reconstructions of velocity, pressure, density and inertial particle properties (diameter and density), highlighting both the potential and certain limits of joint estimation in supersonic regimes. A systematic sensitivity study reveals how the seeding density, noise level and Stokes number govern reconstruction accuracy for our method.

Estimating past events in wall turbulence based solely on surface measurements and first principles is an ill-posed problem that is complicated by chaos. The sensitivity of a measurement to the earlier flow state is described by the adjoint Navier–Stokes equations, which are solved in reverse time starting from the measurement kernel at the sensing position and time. The resulting adjoint field is the spatio-temporal domain of dependence (DOD) of the sensor, which is a dual to the concept of the domain of influence (DOI) of an actuator in the linearised forward equations. In channel turbulence, the energy of each adjoint realisation grows exponentially in backward time according to the Lyapunov exponent, even though the energy of the ensemble average should decay. We introduce a linear eddy-viscosity closure model in the ensemble-averaged adjoint equations, and directly compute the mean DOD and compare our prediction with the ensemble average. Furthermore, we demonstrate that the DOD of a wall-stress measurement and the DOI resulting from a wall-stress perturbation exhibit respective universal behaviours across Reynolds numbers. However, their spatio-temporal structures differ qualitatively, due to the time asymmetry of the governing equations. The DOD field has a two-part structure: one component is associated with the Orr mechanism, characterised by rapid reorientation under mean shear, and the other is related to self-similar expanding streaky structures. These two components jointly define the sensitivity of the wall-stress measurement to past flow events.

Absorbing sets are a central organising concept in the long-time analysis of dissipative partial differential equations, especially those arising in fluid mechanics. Roughly, an absorbing set is a bounded region in a suitable function space that eventually contains every trajectory that starts from any bounded set of initial data. Once the dynamics enter this region, they remain controlled in norm, and this ultimate boundedness becomes the gateway to more refined statements: existence of global attractors, finite-dimensional long-time behaviour. A variational framework, based on the background method, is developed to determine an ‘absorbing ball’ in the state space of incompressible shear flows described by the Navier–Stokes equations. This region is defined by the Reynolds–Orr energy identity and is guaranteed to contain all long-term dynamics, including chaotic and non-chaotic attractors. We employ a gradient-based optimisation to find a background flow corresponding to minimal absorbing radius for plane Poiseuille and Couette flows over a range of Reynolds numbers. The optimised background profiles are compared with the turbulent mean flow and they exhibit significantly steeper near-wall gradients than turbulent mean profiles obtained from direct numerical simulations and do not reproduce the universal law of the wall. Despite this quantitative discrepancy, the methodology provides rigorous, provable bounds on the region of state space accessible to the flow dynamics. This offers a novel and promising foundation for improving the global stability limit of the laminar state.

We present a robust optimisation framework for computing invariant solutions of wall-bounded flows by recasting the Navier–Stokes equations as a variational problem as established in Ashtari & Schneider (J. Fluid Mech., vol. 977, 2023, A7). The approach minimises the residual of the governing equations over a finite-time horizon, seeking periodic or equilibrium solutions. A novel contribution is made by including a Galerkin projection onto a basis of divergence-free modes that satisfy the no-slip boundary conditions. This projection not only makes the variational framework applicable to wall-bounded flows but it also yields a low-order representation of the dynamics. The basis is derived from resolvent analysis, which provides an orthonormal set. We demonstrate the method on a streamwise invariant formulation of rotating plane Couette flow, obtaining exact equilibrium and periodic solutions consistent with direct numerical simulations. The conditioning of the optimisation problem is analysed in detail, showing that convergence rates depend on the stability properties of the targeted solutions. Finally, we highlight a direct link between the conditioning of the optimisation and the structure of the resolvent operator, suggesting a unifying perspective on both the efficiency of the optimisation and the dynamical significance of resolvent modes.

Following recent work (Gonzalez & Taha 2022 J. Fluid Mech., vol. 941, A58; Peters & Ormiston 2025 AIAA J., vol. 63, pp. 6–20), this manuscript clarifies what the Gauss–Appell principle determines in incompressible, inviscid flow and how it connects to classical projection methods. At a fixed time, freezing the velocity and varying only the material acceleration leads to minimisation of a quadratic subject to acceleration-level constraints. First-order conditions yield a Poisson–Neumann problem for a reaction pressure whose gradient removes the non-solenoidal and wall-normal content of the provisional residual, precisely the well-known Leray–Hodge projection. Thus, Gauss–Appell enforces the instantaneous kinematic constraints and recovers Euler at the instant. Once the impressed physics is specified, for instance via external body forces, the reaction pressure is uniquely determined (up to an additive constant) as the Lagrange multiplier enforcing incompressibility and wall impermeability; it does no work on divergence-free, wall-tangent motions. This is the well-established interpretation of pressure in incompressible flow. The direct, fixed-time application of this principle determines the reaction pressure for an already-specified velocity field and does not, by itself, select circulation or stagnation points, because these are properties of the velocity state, not the instantaneous acceleration correction. The formal decomposition of the pressure into impressed and reaction components admits representational freedom that does not imply physical non-uniqueness of the constraint force. Orthogonality conventions such as Dirichlet orthogonality can fix the representational freedom as an additional modelling choice. This variational viewpoint also yields a simple computational diagnostic: the minimised Appellian equals a $L^2$ norm of the reaction-pressure gradient which vanishes for constraint-compatible updates and grows with the magnitude of divergence and wall-flux mismatch.

We investigate the ability of two data assimilation (DA) methods – ensemble Kalman filter (EnKF) and variational smoother (4D-Var) – to reconstruct the Kuramoto–Sivashinsky and complex Ginzburg–Landau systems from sparse observations. While turbulent systems can reconstruct small scales below a critical resolution by substituting larger scales into governing equations, we examine whether DA methods can accurately recover the full field with observations sparser than this threshold, and consider which method performs better. Our findings show that both methods can accurately reconstruct the full field even with observations much sparser than the substitution-based critical resolution. However, likelihood of successful reconstruction within a fixed assimilation time decreases as sparsity increases. The EnKF method needs smaller assimilation time and lower temporal sampling rate than 4D-Var, but needs ad hoc stabilisation (e.g. inflation) and higher memory for ensemble storage. We validate these results by applying EnKF to turbulent two-dimensional Kolmogorov flows at Reynolds numbers from 200 to 2000, and forcing at wavenumbers 4 and 5. For these flows, we achieve full field reconstruction from observations as sparse as $4\times 4$, outperforming existing 4D-Var and machine learning results where denser observations are required for reconstruction. These findings highlight the strengths and trade-offs of DA methods, offer guidance for reconstructing turbulent flows, and establish benchmarks for evaluating alternative methods.

In a seminal paper, Pironneau (1973 J. Fluid Mech., vol. 59, pp. 117–128) showed that the lowest-drag shape of fixed volume in Stokes flow has a surface vorticity with constant magnitude over the entire body. In this paper, the viscoplastic version of the problem is analysed. The first result is that the surface vorticity on the optimal body in a Herschel–Bulkley fluid cannot vanish or become singular (in both two- and three-dimensional geometries). For the special cases of power-law fluids and high-yield-strength fluids, the change in drag following a small change in body shape is directly related to the surface vorticity, which is then shown to be constant on the optimal body. These results inform a local analysis of the flow near the sharp tips at either end of the optimal body, which determines the tip angle in different non-Newtonian fluids. In shear-thinning and viscoplastic fluids, the viscosity decreases with strain rate and so the fluid effectively self-lubricates in regions of high shear allowing for a sharper optimal body. Indeed, in a high-yield-strength fluid, the optimal body is entirely surrounded by a thin viscoplastic boundary layer and in a planar geometry, the interior tip angles converge to $90^\circ$ in the plastic limit (the tip angles are $102.6^\circ$ in a Newtonian fluid). In the other limit of a perfect shear-thickening fluid, any regions of high strain rate are heavily penalised and so the optimal body is much blunter with the two tip angles converging to $150^\circ$.

This study explores partial synchronisation in turbulent channel flows using sequential variational data assimilation with sparse observations, emphasising the roles of model and observation uncertainties. Unlike previous work that focused on synchronisation using direct numerical simulation, this study considers synchronisation under imperfect models and noisy data. In the first part, a synchronisation map is constructed, revealing invariance with respect to variations in the predictive model, Reynolds number and mesh resolution. Full synchronisation emerges above a critical level of equivalent observation density. At lower observation densities, modal synchronisation is observed, where the energies of dominant modes evolve independently of initial conditions. As data become sparser, the system transitions to a non-synchronisation regime, with assimilated flows exhibiting minimal correlation with the observations. The second part of this study uses the master flow interpolated from down-sampled sparse observations. The delay-coordinate strategy is introduced to enhance the modal synchronisation. Results indicate that the optimal $\sigma$ lies near the threshold between modal synchronisation and non-synchronisation. This demonstrates that the modal synchronisation serves as a critical prerequisite for leveraging historical information in data assimilation to improve the accuracy of turbulence reconstruction. These findings extend the scope of synchronisation theory and provide valuable guidance for advancing data assimilation methodologies.

This paper is concerned with the existence of normalized solutions for the following class of Hamiltonian elliptic systems:

\begin{align*}\left\{\begin{array}{ll}- \Delta u = \lambda u + |v|^{q-2}v \quad \text{in } \mathbb{R}^{N}, \\ - \Delta v = \lambda v + |u|^{p-2}u \quad \text{in } \mathbb{R}^{N}, \\ \displaystyle\int_{\mathbb{R}^N}(|u|^2 + |v|^2) = m,\end{array}\right.\end{align*}

$m \gt 0$

$2 \lt p,q \lt 2^{*}=2N/(N-2)$

$p,q$

$H^{1}(\mathbb{R}^N) \hookrightarrow L^{2}(\mathbb{R}^N)$

Slow viscous flow around a fixed body generates a shape-dependent drag. We explore the drag-minimising shapes of bodies centred between two parallel plates in two-dimensional viscous flow. The channel width introduces a length scale so that the optimal profile is area-dependent. We solve the shape optimisation problem numerically over a wide range of areas. We also compute the optimal elliptical shapes and this identifies how these shapes should be slightly altered to reduce the drag with reductions of up to $3.8\,\%$ attained at high areas. More broadly, we derive two properties of general optimal shapes within the confined flow: the magnitude of the surface vorticity is approximately (but not exactly) constant and the noses have sharp angles that are independent of area. For relatively small bodies, the optimal shape becomes identical to that in an unconfined geometry, but the drag is qualitatively different owing to the influence of confinement; within a channel, it is proportional to the inverse of the logarithm of the body area. At relatively large areas, the optimal body becomes long and its surface is approximately parallel to the channel boundaries, except in the vicinity of the noses. Using a lubrication approximation, we recast the optimisation problem as an Euler–Lagrange equation that is solved to determine the drag-minimising shape, finding that the drag is proportional to the body area in this regime.

Unsteadiness lies at the heart of turbulent fluid dynamics, eddy formation and instabilities in flows, thus making it central to both understanding and controlling fluid systems. In this work, we present an objective measure for the unsteadiness of a time-dependent velocity field, the deformation unsteadiness, derived from a spatio-temporal variational principle, allowing for a frame-independent assessment of the unsteadiness of a given flow field. Additionally, as an application of our main result, we define an objective analogue of the classical $Q$-criterion based on extremisers of unsteadiness minimisation. We apply our results to several examples of analytical flows as well as simulated flow data sets in two and three dimensions. In particular, we apply our newly derived vortex criterion to several explicit, time-dependent solutions of the Navier–Stokes equation and compare the results with existing vortex criteria. We give a physical interpretation of the deformation unsteadiness and discuss future research directions.

We derive a self-consistent hydrodynamic theory of coupled binary fluid–surfactant systems from the underlying microscopic physics using Rayleigh’s variational principle. At the microscopic level, surfactant molecules are modelled as dumbbells that exert forces and torques on the fluid and interface while undergoing Brownian motion. We obtain the overdamped stochastic dynamics of these particles from a Rayleighian dissipation functional, which we then coarse-grain to derive a set of continuum equations governing the surfactant concentration, orientation, fluid density and velocity. This approach introduces a polarisation field $\boldsymbol{p}(\boldsymbol{r},t)$, representing the average orientation of surfactants, which plays a central role in suppressing droplet coalescence. The remaining hydrodynamic equations are consistently obtained from a mesoscopic free energy functional. The resulting model accurately captures key surfactant phenomena, including surface tension reduction and droplet stabilisation, as confirmed by both perturbation theory and numerical simulations, and is thermodynamically consistent with both the Gibbs adsorption isotherm and Henry’s law for adsorbed surfactant concentration.

This study presents a novel extension of the Onsager variational principle to incorporate inertial and thermal effects in fluid dynamics, thereby establishing a unified variational framework for modelling non-isothermal two-phase flows with liquid–vapour phase transitions and wetting effects on solid substrates. From this framework, we naturally derive a thermodynamically consistent model for the fluid system, comprising two-phase Navier–Stokes equations, an equation for the total energy, and dynamic boundary conditions that account for thermal and wetting effects. The derivation is independent of the equation of state, and generalises the dynamic van der Waals theory. To address the computational complexity of the resulting dynamic system, we propose a lattice Boltzmann method based on double distribution functions, which enables accurate and robust simulations of coupled fluid and thermal transport. Numerical experiments – including droplet evaporation, bubble nucleation and departure, and Leidenfrost droplet impact – demonstrate good agreement with theoretical predictions and experimental data, indicating that the proposed numerical method can effectively capture complex thermohydrodynamic phenomena.

Bounds on turbulent averages in shear flows can be derived from the Navier–Stokes equations by a mathematical approach called the background method. Bounds that are optimal within this method can be computed at each Reynolds number $ \textit{Re}$ by numerically optimising subject to a spectral constraint, which requires a quadratic integral to be non-negative for all possible velocity fields. Past authors have eased computations by enforcing the spectral constraint only for streamwise-invariant (2.5-D) velocity fields, assuming this gives the same result as enforcing it for three-dimensional (3-D) fields. Here, we compute optimal bounds over 2.5-D fields and then verify, without doing computations over 3-D fields, that the bounds indeed apply to 3-D flows. One way is to directly check that an optimiser computed using 2.5-D fields satisfies the spectral constraint for all 3-D fields. A second way uses a criterion we derive that is based on a theorem of Busse (1972 Arch. Ration. Mech. Anal., vol. 47, pp. 28–35) for energy stability analysis of models with certain symmetry. The advantage of checking this criterion, as opposed to directly checking the 3-D constraint, is lower computational cost and natural extrapolation of the criterion to large $ \textit{Re}$. We compute optimal upper bounds on friction coefficients for the wall-bounded Kolmogorov flow known as Waleffe flow and for plane Couette flow. This requires lower bounds on dissipation in the first model and upper bounds in the second. For Waleffe flow, all bounds computed using 2.5-D fields satisfy our criterion, so they hold for 3-D flows. For Couette flow, where bounds have been previously computed using 2.5-D fields by Plasting & Kerswell (2003 J. Fluid Mech., vol. 477, pp. 363–379), our criterion holds only up to moderate $ \textit{Re}$, so at larger $ \textit{Re}$ we directly verify the 3-D spectral constraint. Over the $ \textit{Re}$ range of our computations, this confirms the assumption by Plasting & Kerswell that their bounds hold for 3-D flows.

We present a back-in-time analysis for the origin of vorticity in viscous separated flows over immersed bodies, using the adjoint-vorticity framework recently introduced by Xiang et al. (2025 J. Fluid Mech. vol. 1011, A33. The solution of the adjoint-vorticity equations yields the volume density of mean deformation, which captures the stretching and tilting of the earlier vorticity that leads to the terminal value. The analysis also takes into account the boundary contributions of vorticity and its flux. Three examples are considered. Steady, axisymmetric separation in the flow over a sphere at Reynolds number $Re=200$ is shown to be established due to wall flux from both upstream and downstream of separation, the latter contribution being absent from the classical description by Lighthill. For unsteady separation at higher $Re=300$, the streamwise vorticity within the wake hairpin vortex is traced back, quantitatively, to the azimuthal vorticity on the sphere. The third configuration is the flow over a prolate spheroid at $Re=3000$. The null vorticity at three-dimensional separation originates from the cancellation of opposite interior contributions adjacent to the separation surface. The contribution from the downstream side migrates across the separation surface into the upstream region due to a tilting effect – a fundamental distinction between two- and three-dimensional separation. We also examine the detached vortical structures. The streamwise vorticity in the primary vortex originates from tilting of near-wall azimuthal vorticity, differing from Lighthill’s conjecture that the origin is streamwise near-wall vorticity that arises due to the reduced Coriolis force. Finally, a necklace vortex in the turbulent wake is traced back in time, and is shown to have contributions from the spheroid trailing-edge shed shear layer and the large-scale counter-rotating primary vortices.

In this paper, we consider the flow of a nematic liquid crystal in the domain exterior to a small spherical particle. We work within the framework of the $\unicode{x1D64C}$-tensor model, taking into account the orientational elasticity of the medium. Under a suitable regime of physical parameters, the governing equations can be reduced to a system of linear partial differential equations. Our focus is on precise far-field asymptotics of the flow velocity with an emphasis on its anisotropic behaviour. We are able to analytically characterize the flow pattern and compare it with that of the classical isotropic Stokes flow. The expression for velocity away from the particle can be computed numerically or symbolically.

Nearly fifty years ago, Roberts (1978) postulated that the Earth’s magnetic field, which is generated by turbulent motions of liquid metal in its outer core, likely results from a subcritical dynamo instability characterised by a dominant balance between Coriolis, pressure and Lorentz forces (requiring a finite-amplitude magnetic field). Here, we numerically explore subcritical convective dynamo action in a spherical shell, using techniques from optimal control and dynamical systems theory to uncover the nonlinear dynamics of magnetic field generation. Through nonlinear optimisation, via direct-adjoint looping, we identify the minimal seed – the smallest magnetic field that attracts to a nonlinear dynamo solution. Additionally, using the Newton-hookstep algorithm, we converge stable and unstable travelling wave solutions to the governing equations. By combining these two techniques, complex nonlinear pathways between attracting states are revealed, providing insight into a potential subcritical origin of the geodynamo. This paper showcases these methods on the widely studied benchmark of Christensen et al. (2001, Phys. Earth Planet. Inter., vol. 128, pp. 25–34), laying the foundations for future studies in more extreme and realistic parameter regimes. We show that the minimal seed reaches a nonlinear dynamo solution by first approaching an unstable travelling wave solution, which acts as an edge state separating a hydrodynamic solution from a magnetohydrodynamic one. Furthermore, by carefully examining the choice of cost functional, we establish a robust optimisation procedure that can systematically locate dynamo solutions on short time horizons with no prior knowledge of its structure.