In this paper, we train turbulence models based on convolutional neural networks. These learned turbulence models improve under-resolved low-resolution solutions to the incompressible Navier–Stokes equations at simulation time. Our study involves the development of a differentiable numerical solver that supports the propagation of optimisation gradients through multiple solver steps. The significance of this property is demonstrated by the superior stability and accuracy of those models that unroll more solver steps during training. Furthermore, we introduce loss terms based on turbulence physics that further improve the model accuracy. This approach is applied to three two-dimensional turbulence flow scenarios, a homogeneous decaying turbulence case, a temporally evolving mixing layer and a spatially evolving mixing layer. Our models achieve significant improvements of long-term a posteriori statistics when compared with no-model simulations, without requiring these statistics to be directly included in the learning targets. At inference time, our proposed method also gains substantial performance improvements over similarly accurate, purely numerical methods.

The effect of surface vibrations on the propulsion augmentation and resistance in the relative movement of parallel plates has been studied. The analysis was focused on monochromatic waves and laminar flows. The effectiveness of the vibrations was gauged by determining the external force required to maintain the movement of one of the plates at a prescribed velocity. It is shown that waves propagating upstream always increase the resistance but the flow response to waves propagating downstream is more intricate and is a function of the flow Reynolds number. In general, waves must be sufficiently fast to reduce the flow resistance. This leads to a natural division between slow and fast waves; a characterization that is helpful for flows at a sufficiently small Reynolds number Re. An increase in Re brings into play the complication of possible resonances with the natural flow frequencies. Resonances are not possible with waves faster than the plate velocity and these supercritical waves generally decrease the flow resistance. More complex flow responses can occur with slower (subcritical) waves which tend to increase the flow resistance. A complete elimination of the resistance is possible if the waves are of sufficiently short wavelength and travel quickly. This suggests that our mechanism has great potential in the development of propulsion augmentation systems. None of the waves produced net energy savings.

Despite decades of investigations, there is still no consensus on whether inertial particles augment or dampen turbulence. Here, we perform the first experimental study in which the particle concentration is varied systematically across a broad range of volume fractions $\varPhi _{v}$, from nominally one-way coupled to heavily two-way coupled regimes, keeping all other parameters constant. We utilize a zero-mean flow chamber where steady, homogeneous and approximately isotropic air turbulence is realized, with a Taylor-microscale Reynolds number $Re_{\lambda } = 150\unicode{x2013}300$. We consider spherical solid particles of two sizes, both much smaller than the Kolmogorov length, and yielding Stokes numbers $St_{\eta } = 0.3$ and 2.6 based on the Kolmogorov time scale. By adjusting the turbulent intensity, the settling velocity parameter is kept constant for both cases, $Sv_{\eta } = V_{t}/u_{\eta }\approx 3$ (where $V_{t}$ is the still-air terminal velocity, and $u_{\eta }$ is the Kolmogorov velocity scale). Unlike previous studies focused on massively inertial particles, we find that the turbulent kinetic energy increases with particle loading, being more than doubled at $\varPhi _{v} =5\times 10^{-5}$. This is attributed to the energy input associated with gravitational settling: the particles release their potential energy into the fluid and increase its dissipation rate, while the time scale associated with the inter-scale energy transfer is not strongly changed. Two-point statistics indicate that the energy-containing eddies become vertically elongated in the presence of falling particles, and that the latter redistribute the energy more homogeneously across the scales compared to unladen turbulence. This is rooted in an enhanced cascade, as shown by the nonlinear inter-scale energy transfer rate.

In this paper, we determine the instability effects of a phase twist superposed on a quantum vortex defect governed by the Gross–Pitaevskii equation. For this, we consider the modified form of the equation in two cases: when a uniform phase twist is present everywhere in the condensate, and when the defect is subject to a localized phase twist confined to the defect healing region. In the first case, we show that a secondary, new defect is produced as manifestation of an Aharonov–Bohm type effect. In the second case, we prove that due to energy minimization, the defect changes its configurational energy by converting localized twist to writhe. This mechanism, typical of classical elastic systems, is shown to occur also in quantum defects, and it may find useful applications in science and technology.

Exact solutions of the time-dependent three-dimensional nonlinear vorticity equation for Euler flows with spherical geometry are provided. The velocity solution is the sum of a multipolar oscillatory function and a rigid cylindrical motion with swirl. The multipolar oscillation is a velocity mode whose radial and angular dependencies are given by the spherical Bessel functions and vector spherical harmonics, respectively. The local frequency of the velocity oscillations equals the angular speed of the rigid flow times the angular azimuthal wavenumber of the oscillating flow. The unsteady motion corresponds to inertial oscillations in multipolar flows with spatial azimuthal waves (non-vanishing azimuthal wavenumber) in the presence of a background flow with constant axial vorticity. In these nonlinear solutions, the curl of the Lamb vector has a linear dependence with the oscillation velocity, a property that makes it possible for the oscillating motion to satisfy different linear wave equations. Based on these inviscid time-dependent velocity modes, new exact solutions to the time-dependent Navier–Stokes equation are also provided.

Surface distortions to an otherwise planar channel flow introduce vorticity perturbations. In Newtonian fluids, the vorticity induced by small surface undulations on the lower wall is advected by the background flow and diffuses into the fluid. When the fluid is viscoelastic, we identify new mechanisms by which significant vorticity perturbations can be generated in both inertialess and elasto-inertial channel flows. We focus on the case where the lengthscale of the surface distortion is much longer than the channel depth, where we find significant departure from plane shear (Page & Zaki, J. Fluid Mech., vol. 901, 2016, pp. 392–429) due to the non-monotonic base-flow streamwise-normal elastic stress. In inertialess flows, a purely elastic response results in streamlines deforming to match the bottom topography in the lower half the channel. However, the vanishing stress at the centreline introduces a blocking effect, and the associated $O(1)$ jump in normal velocity is balanced by a large-amplitude streamwise-oscillating ‘jet’ in a boundary layer, resulting in a localised, chevron-shaped vorticity perturbation field. In elasto-inertial flows, resonance between the frequency of elasto-inertial ‘Alfvén’ waves and the frequency apparent to an observer moving with the fluid results in vorticity amplification in a pair of critical layers on either side of the channel. The vorticity in both layers is equal in magnitude, to leading order in Weissenberg number, and as such the perturbation vorticity field penetrates the full channel depth even when inertia is dominant. The results demonstrate that long-wave distortions, which are relatively innocuous in Newtonian fluids, can drive a significant flow distortion in viscoelastic fluids for a wide range of parameter values.

The linear stability of the axisymmetric steady thermocapillary flow in a liquid bridge made from 2 cSt silicone oil (Prandtl number 28) is investigated numerically in the framework of the Boussinesq approximation. The flow and temperature fields in the surrounding gas phase (air) are taken into account for a generic cylindrical container hosting the liquid bridge. The flows in the liquid and in the gas are fully coupled across the hydrostatically deformed liquid–gas interface, neglecting dynamic interface deformations. Originating from a common reference case, the linear stability boundary is computed varying the length of the liquid bridge (aspect ratio), its volume and the gravity level, providing accurate critical data. The qualitative dependence of the critical threshold on these parameters is explained in terms of the characteristics of the critical mode. The heat exchange between the ambient gas and the liquid bridge that is fully resolved has an important influence on the critical conditions.

In this study, we develop an analytical model to predict the turbulent boundary layer downstream of a step-change in the surface roughness where upstream flow conditions are given. We first revisit the classical model of Elliott (Trans. Am. Geophys. Union, vol. 39, 1958, pp. 1048–1054), who modelled the velocity distribution within and above the internal layer with a simple piecewise logarithmic profile, and evolved the velocity profile using the streamwise momentum equation. Elliott's model was originally developed for an atmospheric surface layer, and to make the model applicable to a spatially developing turbulent boundary layer with finite thickness, we propose a number of more physical refinements, including adding a wake function to the velocity profile, considering the growth of the entire boundary layer in the streamwise direction, and using a more realistic shear stress profile in the momentum equation. In particular, we implement the blending model (Li et al., J. Fluid Mech., vol. 923, 2021, p. A18) to account for the deviation of the mean flow within the internal layer from a canonical velocity profile based on the local wall condition. These refinements lead to improved agreement between the prediction and the measurement, especially in the vicinity of the rough-to-smooth change.

Global stability analysis and direct numerical simulation (DNS) are used to study boundary layer flows with an isolated roughness element. The aspect ratio of the element ($\eta$) is small, while the ratio of element height to displacement boundary layer thickness ($h/\delta ^{*}$) is large. Both steady base flows and time-averaged mean flows are able to capture the frequencies of the primary vortical structures and mode shapes. Global stability results highlight that although the varicose instability is dominant for large $h/\delta ^{*}$, sinuous instability becomes more pronounced as $Re_h$ increases for the thin geometry ($\eta =0.5$), due to increased spanwise shear in the near-wake region. Wavemaker results indicate that $\eta$ affects the convective nature of the shear layer more than the type of instability. DNS results show that different instability mechanisms lead to different development and evolution of vortical structures in the transition process. For $\eta =1$, the varicose instability is associated with the periodic shedding of hairpin vortices, and its stronger spatial transient growth indicated by wavemaker results aids the formation of hairpin vortices farther downstream. In contrast, for $\eta =0.5$, the interplay between varicose and sinuous instabilities results in a broader-banded energy spectrum and leads to the sinuous wiggling of hairpin vortices in the near wake when $Re_h$ is sufficiently high. A sinuous mode with a lower frequency captured by dynamic mode decomposition analysis, and associated with the ‘wiggling’ of streaks, persists far downstream and promotes transition to turbulence. A new regime map is developed to classify and predict instability mechanisms based on $Re_{hh}^{1/2}$ and $d/\delta ^{*}$ using a logistic regression model. Although the mean skin friction demonstrates different evolutions for the two geometries, both of them efficiently trip the flow to turbulence at $Re_h=1100$. An earlier location of a fully-developed turbulent state is established for $\eta =1$ at $x \approx 110h$.

We perform a three-dimensional direct numerical simulation of flow over the Tacoma Narrows Bridge to understand the vertical and torsional vibrations that occurred before its collapse in 1940. Real-scale structural parameters of the bridge are used for the simulation. The Reynolds number based on the free-stream velocity and height of the deck fence is lower (${Re}=10\ 000$) than the actual one on the day of its collapse (${Re}=3.06 \times 10^{6}$), but the magnitude of a fluid property is modified to provide the real-scale aerodynamic force and moment on the deck. The vertical and torsional vibrations are simulated through two-way coupling of the fluid flow and structural motion. The vertical vibration occurs from the frequency lock-in with the vortex shedding, and its wavelength and frequency agree well with the recorded data in 1940. After saturation of the vertical vibration, a torsional vibration resulting from the aeroelastic fluttering grows exponentially in time, with its wavelength and frequency in excellent agreement with the recorded data of the incident. The critical flutter wind speed for the growth of torsional vibration is obtained as $3.56 < U_c / (f_{nat} B) \le 4$, where $U_{c}$ is the critical flutter wind speed, $f_{nat}$ is the natural frequency of the torsional vibration and $B$ is the deck width. Finally, apart from the actual vibration process in 1940, we perform more numerical simulations to investigate the roles of the free-stream velocity and vertical vibration in the growth of the torsional vibration.

The flow over a forward-facing step (FFS) at $Ma_\infty =1.7$ and $Re_{\delta _0}=1.3718\times 10^{4}$ is investigated by well-resolved large-eddy simulation. To investigate effects of upstream flow structures and turbulence on the low-frequency dynamics of the shock wave/boundary layer interaction (SWBLI), two cases are considered: one with a laminar inflow and one with a turbulent inflow. The laminar inflow case shows signs of a rapid transition to turbulence upstream of the step, as inferred from the streamwise variation of $\langle C_f \rangle$ and the evolution of the coherent vortical structures. Nevertheless, the separation length is more than twice as large for the laminar inflow case, and the coalescence of compression waves into a separation shock is observed only for the fully turbulent inflow case. The dynamics at low and medium frequencies is characterized by a spectral analysis, where the lower frequency range is related to the unsteady separation region, and the intermediate one is associated with the shedding of shear layer vortices. For the turbulent inflow case, we furthermore use a three-dimensional dynamic mode decomposition to analyse the individual contributions of selected modes to the unsteadiness of the SWBLI. The separation shock and Görtler-like vortices, which are induced by the centrifugal forces in the separation region, are strongly correlated with the low-frequency unsteadiness in the current FFS case. Similarly as observed previously for the backward-facing steps, we observe a slightly higher non-dimensional frequency (based on the separation length) of the low-frequency mode than for SWBLI in flat plate and ramp configurations.

We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M>0$. Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $m$ becomes unstable. This critical mass loading drops to zero as $m$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure.

Statistically homogeneous flows obey exact kinematic relations. The Betchov homogeneity constraints (Betchov, J. Fluid Mech., vol. 1, 1956, pp. 497–504) for the average principal invariants of the velocity gradient are among the most well-known and extensively employed homogeneity relations. These homogeneity relations have far-reaching implications for the coupled dynamics of strain and vorticity, as well as for the turbulent energy cascade. Whether the Betchov homogeneity constraints are the only possible ones or whether additional homogeneity relations exist has not been proven yet. Here we show that the Betchov homogeneity constraints are the only homogeneity constraints for incompressible and statistically isotropic velocity gradient fields. Our analysis also applies to compressible/perceived velocity gradients, and it allows the derivation of homogeneity relations involving the velocity gradient and other dynamically relevant quantities, such as the pressure Hessian and viscous stresses.

During solid surface impact, a falling drop's energy is transformed into oscillations of its liquid/gas interface. We consider drop deposition during oblique impact in the capillary-ballistic regime characterized by high Reynolds number and moderate Weber number. We treat this as an inverse problem showing that post-impact observations of the frequency spectrum and modal partition of energy allow one to determine a drop's pre-impact characteristics and wetting properties. Our analysis is useful for quantifying contact-line dissipation during inertial spreading and can be used as a diagnostic technique for determining substrate wetting properties.

Our understanding of homogeneous vapour bubble growth is currently restricted to asymptotic descriptions of their limiting behaviour. While attempts have been made to incorporate both the inertial and thermal limits into bubble growth models, the early stages of bubble growth have not been captured. By accounting for both the changing inertial driving force and the thermal restriction to growth, we present an inertio-thermal model of homogeneous vapour bubble growth, capable of accurately capturing the evolution of a bubble from the nano- to the macro-scale. We compare our model predictions with: (a) published experimental and numerical data, and (b) our own molecular simulations, showing significant improvement over previous models. This has potential application in improving the performance of engineering devices, such as ultrasonic cleaning and microprocessor cooling, as well as in understanding of natural phenomena involving vapour bubble growth.

We propose an efficient model for the description of the three-dimensional (3-D) evolution of breaking water waves in the nearshore region. A fundamental property of the model is its intrinsic ability to account for the 3-D dynamics of vorticity and the energy dissipation induced by wave breaking. In particular, the vorticity evolution is achieved through the use of mollified operators, an approach similar in spirit to that adopted in smoothed particle hydrodynamics. Further, since the model is based on depth-semi-averaged equations with a core structure similar to that of nonlinear shallow-water equations, it takes advantage of well-known numerical methods for hyperbolic equations, while permitting computation of local flows. Finally, the model relies on a limited number of tunable parameters and a very simple breaking criterion. All the above aspects allow for a simple and reliable representation of the main features of wave breaking at the time and spatial scales typical of the nearshore wave dynamics. A number of benchmarks are used to explore the properties of the model, which is tuned only once for all cases. Wave height decay rates are well described for both sloshing (thin) and shoaling (thick) spillers, and a good description is also provided of the vorticity field. A final run of an impulsive wave over a submerged breakwater is used to illustrate the representation of the 3-D vorticity dynamics.

An ‘inverted flag’ – a flexible plate clamped at its trailing edge – undergoes large-amplitude flow-induced flapping when immersed in a uniform and steady flow. Here, we report direct numerical simulations of a related single degree-of-freedom mechanical system: a rigid plate attached at its trailing edge to a torsional spring. This system is termed a ‘rigid inverted flag’ and exhibits the dynamical states reported for the (flexible) inverted flag, with additional behaviour. This finding shows that the flapping dynamics of inverted flags is not reliant on their continuous flexibility, i.e. many degrees of freedom. The rigid inverted flag exhibits additional, novel states including a heteroclinic-type orbit that results in small-amplitude flapping, and a number of chaotic large-amplitude flapping regimes. We show that the various routes to chaos are driven by a series of periodic states, including at least two which are subharmonic. The instability and competition between these periodic states lead to chaos via type-I intermittency, mode competition and mode locking. The rigid inverted flag allows these periodic states and their subsequent interaction to be explained simply: they arise from an interaction between a preferred vortex shedding frequency and a single natural frequency of the structure. The dynamics of rigid inverted flags is yet to be studied experimentally, and this numerical study provides impetus for such future work.

The processing of dense suspensions is a crucial step in many industries including mining; the production of ceramics; and the manufacture of pharmaceuticals. It is widely reported that these suspensions exhibit nonlinear behaviours such as shear thinning and thickening, with particle surface contacts recently being accepted as a primary culprit in the latter. In light of this, the modelling community have started to explore the role of particle surface tribology, predominantly by incorporating Coulombic friction models borrowed from the field of dry granular matter. Full details of the interactions between particle surfaces remain unclear, however, and it is suggested that physical interlocking of particle asperities may be key. Here, we use particle-based simulations to explore explicitly the effect of interlocking on the rheology of dense suspensions of micron-sized solids in a Newtonian fluid. Our simplified model recovers shear thinning, thickening and jamming phenomena commonly seen in experiments.

This work reveals that the dynamic response of a spherical cap bubble in contact with a rigid wall depends on the effective contact angle at the instant prior to collapse. This parameter allows us to discriminate between two regimes in which the mechanisms of interaction between the collapsing bubble and its surrounding medium differ significantly: when the contact angle is smaller than $90^{\circ }$, a classical jet directed towards the wall is observed, whereas if the initial contact angle is larger than $90^{\circ }$, an annular re-entrant jet parallel to the wall appears. We show that this change of behaviour can be explained using the impulse potential flow theory for small times, which shows the presence of a singularity on the initial acceleration of the contact line when the contact angle is larger than $90^{\circ }$. Direct numerical simulations show that although viscosity regularises the solution at $t > 0$, the solution remains singular at $t=0$. In these circumstances, numerical and experimental results show that the collapse of flat bubbles can eventually lead to the formation of a vortex ring that unexpectedly induces long-range effects. The role of the bubble geometry at the instant of maximum expansion on the overall collapse process is shown to be well captured by the impulse potential flow theory, which can be generalised easily to other bubble shapes. These results may find direct application in the interpretation of geophysical flows as well as the control and design of biomedical, naval, manufacturing and sonochemistry applications.

This work focuses on the dynamics of a train of solid particles, separated by a distance $L$, flowing near a deformable interface formed by two co-flowing immiscible fluids in a microchannel of height $h$. Our study includes a systematic analysis of the influence of the governing parameters (fluids viscosity ratio, interface and particle positions, Reynolds $Re$ and capillary $Ca$ numbers and the inter-particle distance $L$) on the hydrodynamic force $f$ exerted on the particle. In the pure inertial regime with non-deformable interfaces $Ca=0$, the particle is driven towards the wall (interface) when the particle is close to the interface (wall). Up to three neutral equilibrium positions $f=0$, two of them stable, are found in this limit. The contrary is obtained in the pure capillary regime $Re=0$. In this limit, we also carried out an asymptotic analysis in the distinguished limits of very large and very small surface tension. In the latter case, the amplitude of the interface deformation induced by the particle is large, comparable to its diameter, but its influence is limited to a small region upstream and downstream of the particle. In the limit of very large surface tension, the amplitude of the interface deformation is small but the presence of the particle modifies the shape of the interface in a region of length $2\lambda$, much larger than the particle diameter $d$. The parameter $\lambda$, introduces an additional characteristic length that determines the asymptotic behaviour of the flow properties in the limit of large surface tension.