An exact solution of a stable vortex tripole in two-dimensional (2-D) Euler flows is provided. The stable tripole is composed of an inner elliptical vortex and two small-amplitude lateral vortices. The non-vanishing vorticity field of this tripole, referred to as here as an embedded tripole because of the closeness of its vortices, is given in elliptical coordinates $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D708})$ by the even radial and angular order-0 Mathieu functions $\text{Je}_{0}(\unicode[STIX]{x1D707})\text{ce}_{0}(\unicode[STIX]{x1D708})$ truncated at the external branch of the vorticity isoline passing through the two critical points closest to the vortex centre. This tripole mode has a rigid vorticity field which rotates with constant angular velocity equal to $\unicode[STIX]{x1D701}_{0}\text{Je}_{0}(\unicode[STIX]{x1D707}_{1})\text{ce}_{0}(0)/2$, where $\unicode[STIX]{x1D707}_{1}$ is the first zero of $\text{Je}_{0}^{\prime }(\unicode[STIX]{x1D707})$ and $\unicode[STIX]{x1D701}_{0}$ is a constant modal amplitude. It is argued that embedded 2-D tripoles may be conceptually regarded as the superposition of two asymmetric Chaplygin–Lamb dipoles, separated a distance equal to $2R$, as long as their individual trajectory curvature radius $R$ is much shorter than their dipole extent radius.

In elastic macromolecules, the value of the short-time diffusion coefficient depends on the choice of the point the displacement of which is tracked. On the other hand, the experimentally more relevant long-time diffusion coefficient is independent of the reference point, but its estimation usually requires computationally expensive Brownian dynamics simulations. Here we show how to obtain a precise estimate of the long-time diffusion coefficient of elastic macromolecules in a fast and robust manner, without invoking Brownian dynamics.

In this paper we contrast the early impact stage of a highly viscous drop onto a liquid versus a solid substrate. Water drops impacting at low velocities can rebound from a solid surface without contact. This dynamic is mediated through lubrication of a thin air layer between the liquid and solid. Drops can also rebound from a liquid surface, but only for low Weber numbers. Impacts at higher velocities in both cases lead to circular contacts which entrap an air disc under the centre of the drop. Increasing the drop viscosity produces extended air films for impacts on a smooth solid surface even for much larger velocities. These air films eventually break through random wetting contacts with the solid. Herein we use high-speed interferometry to study the extent and thickness profile of the air film for a large-viscosity drop impacting onto a viscous film of the same liquid. We demonstrate a unified scaling of the centreline height of the air film for impacts on both solid and liquid, when using the effective impact velocity. On the other hand, we show that the large-viscosity liquid film promotes air films of larger extent. Furthermore, the rupture behaviour becomes fundamentally different, with the air film between the two compliant surfaces being more stable, lacking the random wetting patches seen on the solid. We map the parameter range where these air films occur and explore the transition from gliding to ring contact at the edge of the drop dimple. After the air film ruptures, the initial contraction occurs very rapidly and for viscosities greater than 100 cSt the retraction velocity of the air film is ${\sim}0.3~\text{m}~\text{s}^{-1}$, independent of the liquid viscosity and impact velocity, in sharp contrast with theoretical predictions.

We present an investigation of the root-mean-square (r.m.s.) temperature $\unicode[STIX]{x1D70E}_{T}$ and the r.m.s. velocity $\unicode[STIX]{x1D70E}_{w}$ in the bulk of Rayleigh–Bénard turbulence, using new experimental data from the current study and experimental and numerical data from previous studies. We find that, once scaled by the convective temperature $\unicode[STIX]{x1D703}_{\ast }$, the value of $\unicode[STIX]{x1D70E}_{T}$ at the cell centre is a constant ($\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }\approx 0.85$) over a wide range of the Rayleigh number ($10^{8}\leqslant Ra\leqslant 10^{15}$) and the Prandtl number ($0.7\leqslant Pr\leqslant 23.34$), and is independent of the surface topographies of the top and bottom plates of the convection cell. A constant close to unity suggests that $\unicode[STIX]{x1D703}_{\ast }$ is a proper measure of the temperature fluctuation in the core region. On the other hand, $\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$, the vertical r.m.s. velocity at the cell centre scaled by the convective velocity $w_{\ast }$, shows a weak $Ra$-dependence (${\sim}Ra^{0.07\pm 0.02}$) over $10^{8}\leqslant Ra\leqslant 10^{10}$ at $Pr\sim 4.3$ and is independent of plate topography. Similar to a previous finding by He & Xia (Phys. Rev. Lett., vol. 122, 2019, 014503), we find that the r.m.s. temperature profile $\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$ in the region of the mixing zone with a mean horizontal shear exhibits a power-law dependence on the distance $z$ from the plate, but now the universal profile applies to both smooth and rough surface topographies and over a wider range of $Ra$. The vertical r.m.s. velocity profile $\unicode[STIX]{x1D70E}_{w}(z)/w_{\ast }$ obeys a logarithmic dependence on $z$. The study thus demonstrates that the typical scales for the temperature and the velocity are the convective temperature $\unicode[STIX]{x1D703}_{\ast }$ and the convective velocity $w_{\ast }$, respectively. Finally, we note that $\unicode[STIX]{x1D703}_{\ast }$ may be utilised to study the flow regime transitions in ultrahigh-$Ra$-number turbulent convection.

The streamwise inclination angle of large wall-attached structures, in the log region of a canonical turbulent boundary layer, is estimated via spectral coherence analysis, and is found to be approximately $45^{\circ }$. This is consistent with assumptions used in prior attached eddy model-based simulations. Given that the inclination angle obtained via standard two-point correlations is influenced by the range of scales in the turbulent flow (Marusic, Phys. Fluids, vol. 13 (3), 2001, pp. 735–743), the present result is obtained by isolating the large wall-attached structures from the rest of the turbulence. This is achieved by introducing a spanwise offset between two hot-wire probes, synchronously measuring the streamwise velocity at a near-wall and log-region reference location, to assess the wall coherence. The methodology is shown to be effective by applying it to data sets across Reynolds numbers, $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3})$–$O(10^{6})$.

We derive and test a new heuristic theory for third-order structure functions that resolves the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally, for example, in rotating stratified turbulence or magnetohydrodynamical (MHD) turbulence. The theory has three parameters – namely the upscale/downscale energy transfer rates and the forcing scale – and it includes the classic inertial-range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full range of data, therefore it has broader applications compared with the local theories, especially in situations where the data is imperfect. In addition, because of the resolution of forcing scales, the new theory can detect the scales of energy input, which was impossible before. We test our new theory with a two-dimensional simulation of MHD turbulence.

The string shooter experiment uses counter-rotating pulleys to propel a closed string forward. Its steady state exhibits a transition from a gravity-dominated regime at low velocity towards a high-velocity regime where the string takes the form of a self-supporting loop. Here we show that this loop of light string is not suspended in the air due to inertia, but through the hydrodynamic drag exerted by the surrounding fluid, namely air. We investigate this drag experimentally and theoretically for a smooth long cylinder moving along its axis. We then derive the equations describing the shape of the string loop in the limit of vanishing string radius. The solutions present a critical point, analogous to a hydraulic jump, separating a supercritical zone where the wave velocity is smaller than the rope velocity, from a subcritical zone where waves propagate faster than the rope velocity. This property could be leveraged to create a white hole analogue similar to what has been demonstrated using surface waves on a flowing fluid. Loop solutions that are regular at the critical point are derived, discussed and compared to the experiment. In the general case, however, the critical point turns out to be the locus of a sharp turn of the string, which is modelled theoretically as a discontinuity. The hydrodynamic regularisation of this geometrical singularity, which involves non-local and added mass effects, is discussed on the basis of dimensional analysis.

We present an experimental study of the reversal of the large-scale circulation (LSC) in quasi-two-dimensional turbulent Rayleigh–Bénard convection. It is found that there exists a transition in the Rayleigh number ($Ra$) dependence of the reversal rate $f$ with two distinct scalings: for $Ra$ less than a transitional value $Ra_{t}$, the non-dimensionalized reversal rate $ft_{E}\sim Ra^{-1.09}$; however, for higher $Ra$ the scaling changes to $ft_{E}\sim Ra^{-3.06}$, where $t_{E}$ is the turnover time of the LSC. Flow visualization shows that this regime transition originates from a change in flow topology from a single-roll state to a new, less stable, abnormal single-roll state with substructures inside the single roll. The emergence of the substructures inside the LSC lowers the energy barrier for the flow reversals to occur and leads to a slower decay of $f$ with $Ra$. Detailed analysis reveals that, although it is the corner rolls that trigger the reversal event, the probability for the occurrence of reversals mainly depends on the stability of the LSC. This is supported by a model we proposed to predict the critical condition for the transition, which agrees well with the experimental results.

We have studied the motion of drops on inclined liquid-impregnated surfaces (LISs) subject to vertical vibrations. The liquid drops comprise distilled water and different aqueous solutions of glycerol of increasing viscosity. The use of weak pinning LISs strongly affects the dynamical phase diagram. First of all, there is no trace of the dominant static region at low oscillating amplitudes reported for oscillating solid surfaces characterized by contact angle hysteresis. On the contrary, at sufficiently low oscillating amplitudes, the drops always move downwards with a velocity that depends only on the drop viscosity. Further increasing the oscillating amplitude may drive the drop upwards against gravity, as reported for dry surfaces. The use of more viscous drops widens this climbing region. Arguably, the main novelty of this work concerns the observation of two distinct descending regimes where the downhill speed differs by a factor of five or more. Fast-rate videos show that the evolution of the drop profile is diverse in the two regimes, likely because the vertical oscillations reduce the effect of the oil meniscus surrounding the drop at high accelerations.

In a recent paper (J. Fluid Mech., vol. 861, 2019, pp. 328–348), Benilov derived equations governing a laminar liquid sheet (a curtain) that emanates from a slot whose centreline is inclined to the vertical. The equations are valid for slender sheets whose characteristic length scale in the direction of flow is much larger than its cross-sectional thickness. For a liquid that leaves a slot with average speed, $u_{0}$, volumetric flow rate per unit width, $q$, surface tension, $\unicode[STIX]{x1D70E}$, and density, $\unicode[STIX]{x1D70C}$, Benilov obtains parametric equations that predict steady-state curtain shapes that bend upwards against gravity provided $\unicode[STIX]{x1D70C}qu_{0}/2\unicode[STIX]{x1D70E}<1$. Benilov’s parametric equations are shown to be identical to those derived by Finnicum, Weinstein, and Ruschak (J. Fluid Mech., vol. 255, 1993, pp. 647–665). In the latter form, it is straightforward to deduce an alternative solution of Benilov’s equations where a curtain falls vertically regardless of the slot’s orientation. This solution is consistent with prior experimental and theoretical results that show that a liquid curtain can emerge from a slot at an angle different from that of the slot centreline.

We show experimentally that the mechanisms of energy transport in turbulent flows of superfluid $^{4}\text{He}$ are strikingly different from those occurring in turbulent flows of viscous fluids. We argue that the result can be related to the role played by quantized vortices in this unique type of turbulence. The flow-induced motions of relatively small particles suspended in the liquid reveal that, for scales of the order of the mean distance between the vortices, the particles do not tend on average to decelerate faster than they accelerate, whereas, at larger scales, a classical-like asymmetry is recovered. It follows that, in the range of investigated parameters, flight-crash events are less apparent than in classical turbulence. We specifically link the outcome to the time symmetry of quantized vortex reconnections observed at scales comparable to the typical particle size.

While it has long been recognized that Lagrangian drift at the ocean surface plays a critical role in the kinematics and dynamics of upper ocean processes, only recently has the contribution of wave breaking to this drift begun to be investigated through direct numerical simulations (Deike et al., J. Fluid Mech., vol. 829, 2017, pp. 364–391; Pizzo et al., J. Phys. Oceanogr., vol. 49(4), 2019, pp. 983–992). In this work, laboratory measurements of the surface Lagrangian transport due to focusing deep-water non-breaking and breaking waves are presented. It is found that wave breaking greatly enhances mass transport, compared to non-breaking focusing wave packets. These results are in agreement with the direct numerical simulations of Deike et al. (J. Fluid Mech., vol. 829, 2017, pp. 364–391), and the increased transport due to breaking agrees with their scaling argument. In particular, the transport at the surface scales with $S$, the linear prediction of the maximum slope at focusing, while the surface transport due to non-breaking waves scales with $S^{2}$, in agreement with the classical Stokes prediction.

We develop a novel ‘moving-capacitor’ dynamic network model to simulate immiscible fluid–fluid displacement in porous media. Traditional network models approximate the pore geometry as a network of fixed resistors, directly analogous to an electrical circuit. Our model additionally captures the motion of individual fluid–fluid interfaces through the pore geometry by completing this analogy, representing interfaces as a set of moving capacitors. By incorporating pore-scale invasion events, the model reproduces, for the first time, both the displacement pattern and the injection-pressure signal under a wide range of capillary numbers and substrate wettabilities. We show that at high capillary numbers the invading patterns advance symmetrically through viscous fingers. In contrast, at low capillary numbers the flow is governed by the wettability-dependent fluid–fluid interactions with the pore structure. The signature of the transition between the two regimes manifests itself in the fluctuations of the injection-pressure signal.

The dynamics of ideally polarizable spherical particles in concentrated suspensions under the effects of nonlinear electrokinetic phenomena is analysed using large-scale numerical simulations. Particles are assumed to carry no net charge and considered to undergo the combination of dielectrophoresis and induced-charge electrophoresis termed dipolophoresis. Chaotic motion and resulting hydrodynamic diffusion are known to be driven by the induced-charge electrophoresis, which dominates the dielectrophoresis. Up to a volume fraction $\unicode[STIX]{x1D719}\approx 35\,\%$, the particle dynamics seems to be hindered by the increase in the magnitude of excluded volume interactions with concentration. However, a non-trivial suspension behaviour is observed in concentrated regimes, where the hydrodynamic diffusivity starts to increase with the volume fraction at $\unicode[STIX]{x1D719}\approx 35\,\%$, before reaching a local maximum, and then drastically decreases on approaching random close packing. Similar non-trivial behaviours are observed in the particle velocity and number-density fluctuations around volume fractions at which the non-trivial behaviour of the hydrodynamic diffusion is observed. We explain these non-trivial behaviours as a consequence of particle contacts, which are related to the dominant mechanism of particle pairings. The particle contacts are classified into attractive and repulsive classes by the nature of contacts, and in particular, the strong repulsive contact becomes predominant at $\unicode[STIX]{x1D719}>20\,\%$. Moreover, this transition is visible in the pair distribution functions, which also reveal the change in the suspension microstructure in concentrated regimes. It appears that strong and massive repulsive contacts along the direction perpendicular to an electric field promote the non-trivial suspension behaviours observed in concentrated regimes.

We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $(\unicode[STIX]{x1D716}d)^{1/3}$ and the $p$th-order structure function scales as $(\unicode[STIX]{x1D716}d)^{p/3}r/d$, $\unicode[STIX]{x1D716}$ being the mean energy dissipation per unit mass, $d$ the mean distance between the shocks and $r$ the separation distance. However, for the acoustic field, $\unicode[STIX]{x1D716}$ should be replaced by $\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712}$, where $\unicode[STIX]{x1D712}$ is associated with entropy production due to heat conduction. In particular, the third-order longitudinal structure function scales as $\langle \unicode[STIX]{x1D6FF}u_{r}^{3}\rangle =-C(\unicode[STIX]{x1D716}+\unicode[STIX]{x1D712})r$, where $C$ takes the value $12/5(\unicode[STIX]{x1D6FE}+1)$ in the weak shock limit, $\unicode[STIX]{x1D6FE}=c_{p}/c_{v}$ being the ratio between the specific heats at constant pressure and constant volume.

Online in situ prediction of 3-D flame evolution has been long desired and is considered to be the Holy Grail for the combustion community. Recent advances in computational power have facilitated the development of computational fluid dynamics (CFD), which can be used to predict flame behaviours. However, the most advanced CFD techniques are still incapable of realizing online in situ prediction of practical flames due to the enormous computational costs involved. In this work, we aim to combine the state-of-the-art experimental technique (that is, time-resolved volumetric tomography) with deep learning algorithms for rapid prediction of 3-D flame evolution. Proof-of-concept experiments conducted suggest that the evolution of both a laminar diffusion flame and a typical non-premixed turbulent swirl-stabilized flame can be predicted faithfully in a time scale on the order of milliseconds, which can be further reduced by simply using a few more GPUs. We believe this is the first time that online in situ prediction of 3-D flame evolution has become feasible, and we expect this method to be extremely useful, as for most application scenarios the online in situ prediction of even the large-scale flame features are already useful for an effective flame control.

Experiments and computations are presented for a foil pitching about its leading edge near a planar, solid boundary. The foil is examined when it is constrained in space and when it is unconstrained or freely swimming in the cross-stream direction. It was found that the foil has stable equilibrium altitudes: the time-averaged lift is zero at certain altitudes and acts to return the foil to these equilibria. These stable equilibrium altitudes exist for both constrained and freely swimming foils and are independent of the initial conditions of the foil. In all cases, the equilibrium altitudes move farther from the ground when the Strouhal number is increased or the reduced frequency is decreased. Potential flow simulations predict the equilibrium altitudes to within 3 %–11 %, indicating that the equilibrium altitudes are primarily due to inviscid mechanisms. In fact, it is determined that stable equilibrium altitudes arise from an interplay among three time-averaged forces: a negative jet deflection circulatory force, a positive quasistatic circulatory force and a negative added mass force. At equilibrium, the foil exhibits a deflected wake and experiences a thrust enhancement of 4 %–17 % with no penalty in efficiency as compared to a pitching foil far from the ground. These newfound lateral stability characteristics suggest that unsteady ground effect may play a role in the control strategies of near-boundary fish and fish-inspired robots.

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Reversed von Kármán streets are responsible for a velocity surplus in the wake of flapping foils, indicating the onset of thrust generation. However, the wake pattern cannot be predicted based solely on the flapping peak-to-peak amplitude $A$ and frequency $f$ because the transition also depends sensitively on other details of the kinematics. In this work we replace $A$ with the cycle-averaged swept trajectory ${\mathcal{T}}$ of the foil chordline. Two-dimensional simulations are performed for pure heave, pure pitch and a variety of heave-to-pitch coupling. In a phase space of dimensionless ${\mathcal{T}}-f$ we show that the drag-to-thrust wake transition of all tested modes occurs for a modified Strouhal $St_{{\mathcal{T}}}\rightarrow 1$. Physically, the product ${\mathcal{T}}f$ expresses the induced velocity of the foil and indicates that propulsive jets occur when this velocity exceeds $U_{\infty }$. The new metric offers a unique insight into the thrust-producing strategies of biological swimmers and flyers alike, as it directly connects the wake development to the chosen kinematics, enabling a self-similar characterisation of flapping foil propulsion.

As a counterpart of energy cascade, turbulent momentum cascade (TMC) in the wall-normal direction is important for understanding wall turbulence. Here, we report an analytic prediction of non-universal Reynolds number ($Re_{\unicode[STIX]{x1D70F}}$) scaling transition of the maximum TMC located at $y_{p}$. We show that in viscous units, $y_{p}^{+}$ (and $1+\overline{u^{\prime }v^{\prime }}_{p}^{+}$) displays a scaling transition from $Re_{\unicode[STIX]{x1D70F}}^{3/7}$ ($Re_{\unicode[STIX]{x1D70F}}^{-6/7}$) to $Re_{\unicode[STIX]{x1D70F}}^{3/5}$ ($Re_{\unicode[STIX]{x1D70F}}^{-3/5}$) in turbulent boundary layer, in sharp contrast to that from $Re_{\unicode[STIX]{x1D70F}}^{1/3}$ ($Re_{\unicode[STIX]{x1D70F}}^{-2/3}$) to $Re_{\unicode[STIX]{x1D70F}}^{1/2}$ ($Re_{\unicode[STIX]{x1D70F}}^{-1/2}$) in a channel/pipe, countering the prevailing view of a single universal near-wall scaling. This scaling transition reflects different near-wall motions in the buffer layer for small $Re_{\unicode[STIX]{x1D70F}}$ and log layer for large $Re_{\unicode[STIX]{x1D70F}}$, with the non-universality being ascribed to the presence/absence of mean wall-normal velocity $V$. Our predictions are validated by a large set of data, and a probable flow state with a full coupling between momentum and energy cascades beyond a critical $Re_{\unicode[STIX]{x1D70F}}$ is envisaged.