We propose the first machine-learned control-oriented flow estimation for multiple-input, multiple-output plants. The starting point is constant actuation with open-loop actuation commands leading to a database with simultaneously recorded actuation commands, sensor signals and flow fields. A key enabler is an estimator input vector comprising sensor signals and actuation commands. The mapping from the sensor signals and actuation commands to the flow fields is realized in an analytically simple, data-centric and general nonlinear approach. The analytically simple estimator generalizes linear stochastic estimation for actuation commands. The data-centric approach yields flow fields from estimator inputs by interpolating from the database – similar to Loiseau, Noack & Brunton (J. Fluid Mech., vol. 844, 2018, pp. 459–490) for unforced flow. The interpolation is performed with $k$ nearest neighbours ($k$NN). The general global nonlinear mapping from inputs to flow fields is obtained from a deep neural network (DNN) via an iterative training approach. The estimator comparison is performed for the fluidic pinball plant, which is a multiple-input, multiple-output wake control benchmark (Deng et al., J. Fluid Mech., vol. 884, 2020, A37) featuring rich dynamics under steady controls. We conclude that the machine learning methods clearly outperform the linear model. The performance of $k$NN and DNN estimators are comparable for periodic dynamics. Yet the DNN performs consistently better when the flow is chaotic. Moreover, a thorough comparison regarding the complexity, computational cost and prediction accuracy is presented to demonstrate the relative merits of each estimator. The proposed method can be generalized for closed-loop flow control plants.

This paper investigates the steady coherent solutions, bifurcated from the linear stability of a stationary flow, in two-dimensional (2-D) penetrative convection. The results show that the thickness of the upper stably stratified layer, which is measured by $\theta _M$ (the dimensionless temperature at which the density is maximal, with $0\le \theta _M\le 1$), plays an important role in the linear and nonlinear dynamics. First, we investigate steady solutions of fixed aspect ratio $L=2{\rm \pi} /\alpha _c$ (where $\alpha _c$ is the critical wavenumber). The results show that the instability is supercritical when $\theta _M<0.4$ and is subcritical when $\theta _M>0.4$. When $\theta _M>0.4$, the results show that the type of solution depends on the Prandtl number ($Pr$). For instance, when $Pr\lesssim 2.4$ at $\theta _M=0.5$, the solution in one type of pair of convection cells does not exist, as the Rayleigh number $Ra$ exceeds a critical value due to a saddle-node bifurcation. When $Pr>2.4$, steady solutions can be found up to $Ra=10^{8}$ for all $\theta _M$, which exhibit the scaling of heat transfer (characterized by the Nusselt number $Nu$): $Nu\sim Ra^{1/4}$. Then, the optimal 2-D steady solutions are tracked up to $Ra=10^{9}$ by varying the aspect ratio $L$, which shows that heat transfer roughly follows the $Nu\sim Ra^{\gamma }$ ($\gamma \approx 1/3$) scaling in the regime of $10^{7}< Ra<10^{9}$. It is interesting that the optimal temperature field has an arm-like horizontal structure when $Pr<10$, while it has no significant horizontal structures when $Pr>10$. Thus, the mean temperature in the mixing region is higher at large $Pr$. The steady solutions show that $Nu\sim Pr^{-1/12}$ for $\theta _M=0$ in a certain range of $Pr$ by fixing the Rayleigh numbers, e.g. $1< Pr<10$ for 2-D optimal steady solutions at $Ra=10^{8}$ and $2< Pr<30$ for 2-D steady solutions of fixed aspect ratio at $Ra=10^{7}$. But when the Prandtl number is large or the upper stably stratified layer is thick, both the steady solutions of fixed aspect ratio and the 2-D optimal steady solutions are very weakly dependent on $Pr$.

Natural oscillations of sessile drops with a free or pinned contact line in different gravity environments are studied based on a linear inviscid irrotational theory. The inviscid Navier–Stokes equations and boundary conditions are reduced to a functional eigenvalue problem by the normal-mode decomposition. We develop a boundary element method model to numerically solve the eigenvalue problem for predicting the natural frequencies. Emphasis is placed on the frequency shifts of modes due to gravity for a wide range of contact angles $\alpha$ and Bond numbers $Bo$. Three types of $\alpha$–$Bo$ diagrams reflecting how gravity shifts the frequency are identified. Specifically, the frequency of zonal modes shifts downwards (upwards) when $\alpha$ is smaller (larger) than a critical value, while the frequencies of most sectoral modes are shifted downwards regardless of $\alpha$. As a result, gravity can transform the lowest mode from a zonal mode to a sectoral mode. The spectral degeneracy of hemispherical drops inherited from the Rayleigh–Lamb spectrum is also broken by gravity. However, we discover that gravity has no effect on the mode associated with the horizontal motion of the centre of mass, whose frequency is always zero regardless of $\alpha$ and $Bo$. This implies that the ‘walking’ drop instability reported in previous literature does not exist.

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a ‘pure’ point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to \infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a ‘Liouville link’. In some cases, the emergent point vortex equilibrium as $A\to \infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a ‘Liouville chain’. We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717) and by Krishnamurthy et al. (J. Fluid Mech., vol. 874, 2019, R1) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (Proc. R. Soc. A, vol. 476, 2020, 20200310) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the two-dimensional Euler equation.

The linear and nonlinear dynamics of two-phase swirling flows produced by the Grabowski–Berger profile under the influence of a viscosity stratification are investigated. We perform axisymmetric nonlinear simulations and fully three-dimensional linear global stability analysis of the flow for several swirl numbers $S$ and viscosity ratios $\tilde {\mu }$. They are accompanied by nonlinear three-dimensional simulations and subsequent modal analysis using the bispectral mode decomposition, recently introduced by Schmidt (Nonlinear Dyn., vol. 102, issue 4, 2020, pp. 2479–2501). We find a pronounced destabilising effect of the viscosity stratification on both the onset of axisymmetric vortex breakdown and helical instability that is linked to the required shear stress continuity across the interface. Consequently, destabilisation is shifted to lower $S$ as compared with an equivalent flow with uniform viscosity. Further, the stability analysis reveals the simultaneous destabilisation of two global modes with wavenumbers $m=1$ and $m=2$ that have harmonic frequencies. The analysis of the nonlinear flow reveals a strong triadic resonance between these modes that governs the nonlinear dynamics and leads to a rapid departure from the linear dynamics. At larger swirl, the bifurcation of additional modes initiates an interaction cascade by means of triadic resonance which is elucidated by the bispectral analysis. It leads to the emergence of a variety of additional modes in the nonlinear flow. This study contributes to an improved understanding of the influence of viscosity stratification on the onset of vortex breakdown and the destabilisation of global modes. Further, it provides a clear picture of the dynamics of swirling flows with a codimension-two point and related triadic interaction of two global modes at harmonic frequencies and wavenumbers.

The flow-induced snap-through dynamics of a buckled flexible filament was explored using the penalty immersed boundary method. The effects of the filament length, bending rigidity and Reynolds number on the mode transition were systematically examined. Three different modes were observed when the aforementioned parameters were varied: an equilibrium mode, a streamwise oscillation mode and a snap-through oscillation mode. Two mode transitions occurred when the bending rigidity was lowered and the length and Reynolds number were increased: a direct transition from the equilibrium mode to the snap-through oscillation mode and the successive appearance of the three modes. An increase in transverse fluid force induced the snap-through oscillation mode. A vortex-induced vibration and a self-excited vibration occurred in the streamwise oscillation mode and the snap-through oscillation mode, respectively. A wake pattern of 2S appeared in the streamwise oscillation mode, and a pattern of 2S + 2P appeared in the snap-through oscillation mode. A hysteresis was observed near the critical Reynolds number. The hysteresis loop increased in magnitude with increasing bending rigidity. The greater energy harvesting was achieved by the larger deflection and the higher strain energy. We found that most of the strain energy was concentrated in the last half of the filament.

Resonant sloshing in circular cylinders was studied by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645), whose theory was used to describe steady-state resonant waves due to a time-harmonic container's elliptic orbits. In the limit of longitudinal container motions, a symmetry breaking of the planar wave solution occurs, with clockwise and anti-clockwise swirling equally likely. In addition to this primary harmonic dynamics, previous experiments have unveiled that diverse super-harmonic dynamics are observable far from primary resonances. Among these, the so-called double-crest (DC) dynamics, first observed by Reclari et al. (Phys. Fluids, vol. 26, no. 5, 2014, 052104) for circular sloshing, is particularly relevant, as its manifestation is the most favoured by the spatial structure of the external driving. Following Bongarzone et al. (J. Fluid Mech., vol. 943, 2022, A28), in this work we develop a weakly nonlinear analysis to describe the system response to super-harmonic longitudinal forcing. The resulting system of amplitude equations predicts that a planar wave symmetry breaking via stable swirling may also occur under super-harmonic excitation. This finding is confirmed by our experimental observations, which identify three possible super-harmonic regimes, i.e. (i) stable planar DC waves, (ii) irregular motion and (iii) stable swirling DC waves, whose corresponding stability boundaries in the forcing frequency-amplitude plane quantitatively match the present theoretical estimates.

Thin-film flow down a fibre exhibits rich dynamics and is relevant to applications such as desalination, fibre coating and fog harvesting. These flows are subject to instabilities that result in dynamic bead-on-fibre patterns. We perform an experimental study of shear-thinning flow down fibres using 20 different xanthan gum solutions as our working liquid. The bead-on-fibre morphology can be oriented either symmetrically or asymmetrically on the fibre, and this depends upon the surface tension, fibre diameter and liquid rheology, as defined by the Ostwald power-law index. For highly shear-thinning liquids, it is possible for the pattern to be complex and exhibit simultaneously both asymmetric large beads and symmetric small beads in the isolated and convective flow regimes. We quantify the transition between flow regimes and bead dynamics for the asymmetric morphology, and compare with Newtonian flow, as it depends upon the experimental parameters. Finally, the dimensionless bead frequency is shown to scale with the Bond number for all of our experimental data (symmetric and asymmetric).

The turbulent external flow around a three-dimensional stepped cylinder is studied by means of direct numerical simulations with the adaptive mesh refinement technique. We give a broad perspective of the flow regimes from laminar to turbulent wake at $Re_D=5000$, which is the highest ever considered for this flow case. In particular, we focus on the intermediate Reynolds number $Re_D=1000$ that reveals a turbulent wake coupled with a stable cylinder shear layer (subcritical regime). This flow shows a junction dynamics similar to the laminar $Re_D=150$, where no hairpin vortex appears around the edges, and just two horseshoe vortices are visible. A new stable vortex in the form of a ring, which coils around the rear area, is also identified. In the turbulent wake, the presence of three wake cells is pointed out: the large and small cylinder cells together with the modulation region. However, the modulation dynamics varies between the subcritical and turbulent regimes. A time-averaged, three-dimensional set of statistics is computed, and spatially coherent structures are extracted via proper orthogonal decomposition (POD). The POD identifies the (long-debated) connection between the N-cell and the downwash behind the junction. Furthermore, as the Reynolds number increases, the downwash phenomenon becomes less prominent. Eventually, a reduced-order reconstruction with the most energetically relevant modes is defined to explain the wake vortex interactions. This also serves as a valuable starting point for simulating the stepped cylinder wake behaviour within complex frameworks, e.g. fluid–structure interaction.

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.

Trip-resolved large-eddy simulations of the DARPA SUBOFF are performed to investigate the development of turbulent boundary layers (TBLs) in model-scale studies. The primary consideration of the study is the extent to which the details of tripping affect statistics in large-eddy simulations of complex geometries, which are presently limited to moderate Reynolds number TBLs. Two trip wire configurations are considered, along with a simple numerical trip (wall-normal blowing), which serves as an exemplar of artificial computational tripping methods often used in practice. When the trip wire height exceeds the laminar boundary layer thickness, shedding from the trip wire initiates transition, and the near field is characterized by an elevation of the wall-normal Reynolds stress and a modification of the turbulence anisotropy and mean momentum balance. This trip wire also induces a large jump in the boundary layer thickness, which affects the way in which the TBL responds to the pressure gradients and streamwise curvature of the hull. The trip-induced turbulence decays along the edge of the TBL as a wake component that sits on top of the underlying TBL structure, which dictates the evolution of the momentum and displacement thicknesses. In contrast, for a trip wire height shorter than the laminar boundary layer thickness, transition is initiated at the reattachment point of the trip-induced recirculation bubble, and the artificial trip reasonably replicates the resolved trip wire behaviour relatively shortly downstream of the trip location. For each case, the inner layer collapses rapidly in terms of the mean profile, Reynolds stresses and mean momentum balance, which is followed by the collapse of the Reynolds stresses in coordinates normalized by the local momentum thickness, and finally against the 99 % thickness. By this point, the lasting impact of the trip is the offset in boundary layer thickness due to the trip itself, which becomes a diminishing fraction of the total boundary layer thickness as the TBL grows. The importance of tripping the model appendages is also highlighted due to their lower Reynolds numbers and susceptibility to laminar separations.

Three-dimensional (3-D) transition in the wake of two tandem co-rotating cylinders is studied numerically for Reynolds numbers $180\le Re\le 500$. Infinite cylinders of equal diameter ($D$) and span ($8D$) are placed at streamwise gaps $Lx=2.5D$ and $7.5D$, and rotated at rotation rates $\alpha =0$, $0.5$, $1$ and $2$. For $\alpha =0$ and $Lx=7.5D$, 3-D transition in the wake of the upstream cylinder resembles that for an isolated cylinder with the formation of Mode-A and Mode-B instabilities, whereas for the downstream cylinder, only Mode-A instability occurs in both the $Lx=2.5D$ and $7.5D$ cases with $\alpha =0$. When the cylinders rotate at $\alpha =1$ for $Lx=7.5D$, staggered transition takes place with the appearance of only Mode-B and only Mode-A instabilities in the near-wake regions of the upstream and downstream bodies, respectively. In the case $Lx=2.5D$, the wake of the rotating cylinders transitions via Mode-C and Mode-D instabilities for $\alpha =0.5$, Mode-D and Mode-D$^\prime$ for $\alpha =1$, and Mode-C and Mode-A for $\alpha =2$. A sudden decrease in near-wake fluctuations, accompanied by a downward jump in the force coefficients, is observed as the 3-D instability shifts from one mode to another, for $\alpha \ge 1$ and $Lx=2.5D$. This discontinuity is caused by the formation of a pair of opposite-signed recirculation zones that mitigate each other's effects in the near-wake region.