Closed-form expressions for aerodynamic force on an accelerating aerofoil were presented in the 1930s, relating instantaneous force to geometric and kinematic parameters under the following assumptions: a thin aerofoil, small-amplitude motions, planar wake development, and a flow that is inviscid, incompressible and two-dimensional. The present work is a step towards analogous closed-form expressions for large-amplitude motions of thick foils when the flow remains attached and boundary-layer thickness approaches (but does not equal) zero. A mathematical framework is derived from vortical flow theory to highlight the finite degrees of freedom that must be solved or predicted in order to yield a predictive aerodynamic model under the stated conditions. The special case of periodic motion is further considered, and an equation is derived to calculate mean forces from known or assumed time histories of circulation, vorticity-weighted mean wake convection velocity and trailing-edge velocity.

Jet penetration into soft gels is essential for optimising fluid delivery in medical therapies, biomedical engineering, and soft robotics. In this work, we visualise the jet flow of a Newtonian fluid into a soft viscoplastic gel using camera imaging and time-resolved tomographic particle image velocimetry (PIV) systems. The flow is primarily governed by the Reynolds number ($Re = 350-5000$) and the effective viscosity ratio ($m$ up to 22). We observe three flow regimes – mixing, jellyfish, and fingering – with transitions between them quantified in the $Re-m$ plane. An experimentally informed, systematic, practical, semi-analytical modelling framework is developed to estimate jet penetration depth over time, incorporating PIV results and an approximate functional decomposition approach to describe the velocity distribution and Reynolds stress contributions. The model provides reasonable estimations across all three regimes.

The effect of nucleation on cavitation inception in a high-Reynolds-number von Kármán wake from a bluff two-dimensional hydrofoil is studied experimentally in a variable pressure water tunnel. Nucleation effects are studied by seeding the flow with sparse monodisperse nuclei populations, with the critical pressure nominally equal to vapour pressure. The injected nuclei population and incipient cavitation events were imaged simultaneously using high-speed cameras to precisely quantify the number of activated nuclei of the total available. Three-dimensional spatial characterisation (orientation and location) of the incipient structures is obtained using two high-speed cameras mounted to the side and below the tunnel test section. Inception was observed predominantly in the stretched cores of secondary structures, with a negligible proportion of events occurring in the primary vortices. A broad peak in the vertical angle distribution is observed about the streamwise axis; however, events at all angles are seen. A symmetric distribution was observed for the horizontal angle, with a dominant orientation $45^{\circ }$ from the free-stream direction. The majority of events occur at approximately one hydrofoil thickness downstream of the hydrofoil trailing edge, with a bimodal symmetric distribution about the hydrofoil vertical centre plane. Nuclei activation rate is determined from the acoustic measurements, and was found to be proportional to the number of the injected nuclei. A power law increase in activation rate was observed following a decrease in cavitation number and an increase in Reynolds number. The nuclei activation rate was of the order of $0.1{-}10 \, \mathrm {s^{-1}}$, which combined with seeding rates of the orderof $100{-}1000 \, \mathrm {s^{-1}}$ reveals inception to be a rare occurrence (0.001 %–10 % of nuclei being activated), requiring the confluence of two unlikely events, the occurrence of a subvapour pressure vortex core with capture of a sufficiently weak nuclei. The presented study provides new insights into the physics of cavitation nucleation and inception and provides a comprehensive dataset for development of computational models.

We solve a Bayesian inverse Navier–Stokes (N–S) problem that assimilates velocimetry data by jointly reconstructing a flow field and learning its unknown N–S parameters. We devise an algorithm that learns the most likely parameters of a Carreau shear-thinning viscosity model, and estimates their uncertainties, from velocimetry data of a shear-thinning fluid. We conduct a magnetic resonance velocimetry experiment to obtain velocimetry data of an axisymmetric laminar jet in an idealised medical device (US Food and Drug Administration’s benchmark nozzle) for a blood analogue fluid. The algorithm successfully reconstructs the flow field and learns the most likely Carreau parameters. Predictions from the learned model agree well with rheometry measurements. The algorithm accepts any differentiable algebraic viscosity model, and can be extended to more complicated non-Newtonian fluids (e.g. Oldroyd-B fluid if a viscoelastic model is incorporated).

Developing a consistent near-wall turbulence model remains an unsolved problem. The machine learning method has the potential to become the workhorse for turbulence modelling. However, the learned model suffers from limited generalisability, especially for flows without similarity laws (e.g. separated flows). In this work, we propose a knowledge-integrated additive (KIA) learning approach for learning wall models in large-eddy simulations. The proposed approach integrates the knowledge in the simplified thin-boundary-layer equation with a data-driven forcing term for the non-equilibrium effects induced by pressure gradients and flow separations. The capability learned from each flow dataset is encapsulated using basis functions with the corresponding weights approximated using neural networks. The fusion of capabilities learned from various datasets is enabled using a distance function, in a way that the learned capability is preserved and the generalisability to other cases is allowed. The additive learning capability is demonstrated via training the model sequentially using the data of the flow with pressure gradient but no separation, and the separated flow data. The capability of the learned model to preserve previously learned capabilities is tested using turbulent channel flow cases. The periodic hill and the 2-D Gaussian bump cases showcase the generalisability of the model to flows with different surface curvatures and different Reynolds numbers. Good agreements with the references are obtained for all the test cases.

The transition to chaos in the subcritical regime of counter-rotating Taylor–Couette flow is investigated using a minimal periodic domain capable of sustaining coherent structures. Following a Feigenbaum cascade, the dynamics is found to be remarkably well approximated by a simple discrete map that admits rigorous proof of its chaotic nature. The chaotic set that arises for the map features densely distributed periodic points that are in one-to-one correspondence with unstable periodic orbits (UPOs) of the Navier–Stokes system. This supports the increasingly accepted view that UPOs may serve as the backbone of turbulence and, indeed, we demonstrate that it is possible to reconstruct every statistical property of chaotic fluid flow from UPOs.

Planar entropy waves are commonly assumed for predicting indirect combustion noise. However, the non-planar and turbulent nature of flows found in most practical combustors challenges this assumption. In the present paper, we examine the indirect noise generated by non-planar and turbulent entropy fields through subsonic nozzles. Firstly, we introduce a new transfer function framework that accounts for the contribution of non-planar Fourier modes of the entropy field to the indirect noise spectra. When applied to a turbulent flow field, this method demonstrates a significant improvement in spectral predictions compared with a conventional approach that only considers the planar mode. Secondly, simulations show that non-planar Fourier modes become significant above a threshold frequency $f_{thresh}$, found in the mid- to high-frequency range. This contribution of non-planar modes is explained by two-dimensional shear effects that distort the entropy waves. A scaling relation that uses residence times along streamlines is developed for $f_{thresh}$, showing good agreement with simulation results. Finally, we show that the indirect noise from non-planar entropy modes found in aviation combustors can be significant at frequencies below 1 kHz, which might be relevant in situations of thermo-acoustic instabilities coupled to indirect noise.

This study seeks a low-rank representation of turbulent flow data obtained from multiple sources. To uncover such a representation, we consider finding a finite-dimensional manifold that captures underlying turbulent flow structures and characteristics. While nonlinear machine-learning techniques can be considered to seek a low-order manifold from flow field data, there exists an infinite number of transformations between data-driven low-order representations, causing difficulty in understanding turbulent flows on a manifold. Finding a manifold that captures turbulence characteristics becomes further challenging when considering multi-source data together due to the presence of inherent noise or uncertainties and the difference in the spatiotemporal length scale resolved in flow snapshots, which depends on approaches in collecting data. With an example of numerical and experimental data sets of transitional turbulent boundary layers, this study considers an observable-augmented nonlinear autoencoder-based compression, enabling data-driven feature extraction with prior knowledge of turbulence. We show that it is possible to find a low-rank subspace that not only captures structural features of flows across the Reynolds number but also distinguishes the data source. Along with machine-learning-based super-resolution, we further argue that the present manifold can be used to validate the outcome of modern data-driven techniques when training and evaluating across data sets collected through different techniques. The current approach could serve as a foundation for a range of analyses including reduced-complexity modelling and state estimation with multi-source turbulent flow data.

The flow behind impulsively started circular and polygonal plates is investigated experimentally, using particle image velocimetry at several azimuthal angles. Observing plates accelerating up to a steady Reynolds number $Re=27\,000$, the three invariants of the motion, circulation $\Gamma$, hydrodynamic impulse $I$ and kinetic energy $E$, were scaled against four candidate lengths: the hydraulic diameter, perimeter, circumscribed diameter and the square root of the area. Of these, the square root of the area was found to best collapse all the data. Investigating the three-dimensionality of the flow, it is found that, while a single-plane measurement can provide a reasonable approximation for $\Gamma$ behind plates, multiple planes are necessary to accurately estimate $E$ and $I$.

Wall turbulence consists of various sizes of vortical structures that induce flow circulation around a wide range of closed Eulerian loops. Here we investigate the multiscale properties of circulation around such loops in statistically homogeneous planes parallel to the wall. Using a high-resolution direct numerical simulation database of turbulent channels at Reynolds numbers of $Re_\tau =180$, 550, 1000 and 5200, circulation statistics are obtained in planes at different wall-normal heights. Intermittency of circulation in the planes of the outer flow ($y^+ \gtrsim 0.1Re_\tau$) takes the form of universal bifractality as in homogeneous and isotropic turbulence. The bifractal character simplifies to space-filling character close to the wall, with scaling exponents that are linear in the moment order, and lower than those given by the Kolmogorov paradigm. The probability density functions of circulation are long-tailed in the outer bifractal region, with evidence showing their invariance with respect to the loop aspect ratio, while those in the inner region are closely Gaussian. The unifractality near the wall implies that the circulation there is not intermittent in character.

Droplet coalescence is an essential multiphase flow process in nature and industry. For the inviscid coalescence of two spherical droplets, our experiment shows that the classical 1/2 power-law scaling for equal-size droplets still holds for the unequal-size situation of small size ratios, but it diverges as the size ratio increases. Employing an energy balance analysis, we develop the first theory for asymmetric droplet coalescence, yielding a solution that collapses all experimental data of different size ratios. This confirms the physical relevance of the new set of length and time scales given by the theory. The functionality of the solution reveals an exponential dependence of the bridge’s radial growth on time, implying a scaling-free nature. Nevertheless, the small-time asymptote of the model is able to recover the classical power-law scaling, so that the actual bridge evolution still follows the scaling law asymptotically in a wide parameter space. Further analysis suggests that the scaling-free evolution behaviour emerges only at late coalescence time and large size ratios.

Experimental and numerical observations in turbulent shear flows point to the persistence of the anisotropy imprinted by the large-scale velocity gradient down to the smallest scales of turbulence. This is reminiscent of the strong anisotropy induced by a mean passive scalar gradient, which manifests itself by the ‘ramp–cliff’ structures. In the shear flow problem, the anisotropy can be characterised by the odd-order moments of $\partial _y u$, where $u$ is the fluctuating streamwise velocity component, and $y$ is the direction of mean shear. Here, we extend the approach proposed by Buaria et al. (Phys. Rev. Lett., 126, 034504, 2021) for the passive scalar fields, and postulate that fronts of width $\delta \sim \eta Re_\lambda ^{1/4}$, where $\eta$ is the Kolmogorov length scale, and $Re_\lambda$ is the Taylor-based Reynolds number, explain the observed small-scale anisotropy for shear flows. This model is supported by the collapse of the positive tails of the probability density functions (PDFs) of $(\partial _y u)/(u^{\prime }/\delta )$ in turbulent homogeneous shear flows (THSF) when the PDFs are normalised by $\delta /L$, where $u^{\prime }$ is the root-mean-square of $u$ and $L$ is the integral length scale. The predictions of this model for the odd-order moments of $\partial _y u$ in THSF agree well with direct numerical simulation (DNS) and experimental results. Moreover, the extension of our analysis to the log-layer of turbulent channel flows (TCF) leads to the prediction that the odd-order moments of order $p (p \gt 1)$ of $\partial _y u$ have power-law dependencies on the wall distance $y^{+}$: $\langle (\partial _y u)^p \rangle /\langle (\partial _y u)^2 \rangle ^{p/2} \sim (y^{+})^{(p-5)/8}$, which is consistent with DNS results.

Many fluid flow configurations nominally contain symmetries, which are always imperfect in real systems. In this study, we reduce the degree of rotational symmetry and break the mirror symmetry of an annular combustor’s thermoacoustic model by using non-uniform flame response distributions. It is known that, in the linear regime, asymmetries lift the degeneracy of some azimuthal thermoacoustic eigenvalues, which are nominally degenerate in the symmetric case. In this work, we prove that a second asymmetric perturbation, which does not restore any trivial symmetry, can be exploited to create an exceptional point (EP). If the only source of asymmetry is the non-uniform distribution of flame responses, at this symmetry-breaking induced EP the single remaining eigenvector is a perfectly spinning mode. We demonstrate that symmetry-breaking induced EPs may be linearly unstable. For an EP obtained for vanishingly small asymmetric perturbations, the linearly stable/unstable nature of the EP follows that of the degenerate eigenvalue of the perfectly symmetric system. Our results are derived theoretically with a low-order model, and validated on a state-space model extracted from experimental data.

The helicity is a topological conserved quantity of the Euler equations which imposes significant constraints on the dynamics of vortex lines. In the compressible setting, the conservation law holds only under the assumption that the pressure is barotropic. Let us consider a volume $V$ containing a compressible fluid with density $\rho$, velocity field $\textbf{u}$ and vorticity $\boldsymbol{\omega}$. We show that by introducing a new definition of helicity density $h_{\rho }=(\rho {\boldsymbol {u}})\cdot \mbox {curl}\,(\rho {\boldsymbol {u}})$ the barotropic assumption on the pressure can be removed, although ${\int _{V}} h_{\rho }{\rm d}V$ is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density $h_{\rho }$ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux ${\boldsymbol {J}}_{\rho }$ contains all the pressure terms and whose source involves the potential vorticity $q = \boldsymbol{\omega} \cdot \nabla \rho$. Therefore, the rate of change of ${\int _{V}} h_{\rho }{\rm d}V$ no longer depends on the pressure and is easier to analyse, as it depends only on the potential vorticity and kinetic energy as well as $\mbox {div}\,{\boldsymbol {u}}$. This result also carries over to the inhomogeneous incompressible Euler equations for which the potential vorticity $q$ is a material constant. Therefore, $q$ is bounded by its initial value $q_{0}=q({\boldsymbol {x}},\,0)$, which enables us to define an inverse resolution length scale $\lambda _{H}^{-1}$ whose upper bound is found to be proportional to $\|q_{0}\|_{\infty }^{2/7}$. In a similar manner, we also introduce a new cross-helicity density for the ideal non-barotropic magnetohydrodynamic (MHD) equations.

Double-diffusive convection can arise when the fluid density is set by multiple species which diffuse at different rates. Different flow regimes are possible depending on the distribution of the diffusing species, including salt fingering and diffusive convection. Flows arising from diffusive convection commonly exhibit step-like density profiles with sharp density interfaces separated by well-mixed layers. The formation of density layers is also seen in stratified turbulence, where a framework based on sorted density coordinates (Winters & D’Asaro 1996 J. Fluid Mech. 317, 179–193) has been used to diagnose layer formation (Zhou et al. 2017 J. Fluid Mech. 823, 198–229; Taylor & Zhou 2017 J. Fluid Mech. 823, R5). In this framework, the evolution of the sorted density profile can be expressed solely in terms of the eddy diffusivity, $\kappa _e$. Here, we use the same framework to diagnose layer formation in two-dimensional simulations of double-diffusive convection. We show that $\kappa _e$ is negative everywhere, with the antidiffusive effect strongest in ‘well-mixed’ layers where a positive diffusion coefficient may be expected. By considering a decomposition of the budget of the square of the Brunt-Väisälä frequency $\partial N^2_*/\partial t$, we demonstrate that the density layers are maintained by fundamentally different processes than in single-component stratified turbulence. In more complicated flows where stratified turbulence and double-diffusive convection can coexist, this framework could provide a method to distinguish between the mechanisms responsible for generating density layers.

The temporal characteristics of fully localised turbulent bands in transitional channel flow remains unclear due to the difficulty in resolving the large length and time scales involved. Here, we tackle this problem by performing statistical lifetime studies in sufficiently large computational domains. The results show signs of stochastic memoryless decay of a fully localised band, suggesting a chaotic-saddle behaviour of the entire band as a coherent entity. Although the mean lifetime of a turbulent band was reported to increase with the band length, our data suggest that it saturates at a certain length. This saturation results in a characteristic lifetime for a fully developed band with a changing length due to the intermittent chipping and decay of turbulence at the upstream end. This memoryless behaviour is observed down to Reynolds number $Re=630$ in our study and we propose that the onset of the memoryless behaviour is in the range of $Re\simeq 620{-}630$. Our data also show that the time it takes for a perturbed flow to enter the saddle, i.e. to start behaving memorylessly, can be thousands of convective time units, which is comparable to the maximum achievable observation time in existing channel set-ups and may pose difficulties for experiments.

Parametric oscillations of an interface separating two fluid phases create nonlinear surface waves, called Faraday waves, which organise into simple patterns, such as squares and hexagons, as well as complex structures, such as double hexagonal and superlattice patterns. In this work, we study the influence of surfactant-induced Marangoni stresses on the formation and transition of Faraday-wave patterns. We use a control parameter, $B$, that assesses the relative importance of Marangoni stresses as compared with the surface-wave dynamics. Our results show that the threshold acceleration required to destabilise a surfactant-covered interface through vibration increases with increasing $B$. For a surfactant-free interface, a square-wave pattern is observed. As $B$ is incremented, we report transitions from squares to asymmetric squares, weakly wavy stripes and ultimately to ridges and hills. These hills are a consequence of the bidirectional Marangoni stresses at the neck of the ridges. The mechanisms underlying the pattern transitions and the formation of exotic ridges and hills are discussed.

An optimal microswimmer with a given geometry has a surface velocity profile that minimises energy dissipation for a given swimming speed. An axisymmetric swimmer can be puller-, pusher- or neutral-type depending on the sign of the stresslet strength. We numerically investigate the type of optimal surface-driven active microswimmers using a minimum dissipation theorem for optimum microswimmers. We examine the hydrodynamic resistance and stresslet strength with nonlinear dependence on various deformation modes. Optimum microswimmers with fore-and-aft symmetry exhibit neutral-type behaviour. Asymmetrical geometries exhibit pusher-, puller- or neutral-type behaviour, depending on the dominant deformation mode and the nonlinear dependence of the stresslet for an optimum microswimmer on deformation mode and amplitude.

Using clean numerical simulation (CNS) in which artificial numerical noise is negligible over a finite, sufficiently long interval of time, we provide evidence, for the first time, that artificial numerical noise in direct numerical simulation (DNS) of turbulence is approximately equivalent to thermal fluctuation and/or stochastic environmental noise. This confers physical significance on the artificial numerical noise of DNS of the Navier–Stokes equations. As a result, DNS on a fine mesh should correspond to turbulence under small internal/external physical disturbance, whereas DNS on a sparse mesh corresponds to turbulent flow under large physical disturbance. The key point is that all of them have physical meanings and so are correct in terms of their deterministic physics, even if their statistics are quite different. This is illustrated herein. Our paper provides a positive viewpoint regarding the presence of artificial numerical noise in DNS.

In Rayleigh–Bénard convection and Taylor–Couette flow cellular patterns emerge at the onset of instability and persist as large-scale coherent structures in the turbulent regime. Their long-term dynamics has been thoroughly characterised and modelled for the case of turbulent convection, whereas turbulent Taylor rolls have received much less attention. Here we present direct numerical simulations of axisymmetric Taylor–Couette flow in the corotating regime and reveal a transition to spatio–temporal chaos as the system size increases. Beyond this transition, Taylor rolls suddenly undergo erratic drifts evolving on a very slow time scale. We estimate an effective diffusion coefficient for the drift and compare the dynamics with analogous motions in Rayleigh–Bénard convection and Poiseuille flow, suggesting that this spontaneous diffusive displacement of large coherent structures is common among different types of wall-bounded turbulent flows.