Energy transfer in turbulent fluids is non-Gaussian. We quantify non-Gaussian energy transfer between the atmosphere and bodies of water using a turbulent diffusion operator coupled with temporally self-affine velocity distributions and a recursive integration method that produce multifractal measures. The measures serve as input to a system of moment field equations (derived from Navier–Stokes) that generate and track high-frequency gravity waves that propagate through the water surface (as a result of the air–water interactions). The dimension of the support of the air–water turbulence produced by our methods falls within the range of theory and observation, and correspondingly, hindcast statistical measures of the water-wave surface such as significant water-wave height and wave period are well correlated to observational buoy data. Further, our recursive integration method can be used by spectral resolving phase-averaged models to interpolate temporal wind data to smaller scales to capture the non-Gaussian behaviour of the air–water interaction.

A series of laboratory experiments are reported in which a continuous stream of bubbles rise from a small source at the base of a tank of water. Using different nozzles, bubble sizes $d$ ranging from 1.2 to 11.6 mm were produced for a number of gas volume fluxes, $Q_b$, ranging between 1.1 and $21.1\times 10^{-6}\ {\rm m}^3\ {\rm s}^{-1}$. Within a small distance from the source, the slip speed of these bubbles exceeds the speed of the equivalent single-phase plume with the same buoyancy flux, leading to formation of what we refer to as the ‘slip plume’ regime. Through a combination of high-speed photography, coupled with flow visualisation in the plume and the ambient fluid using dye, we find that the bubble speed and the fluid speed remain nearly constant with height, with the maximum fluid speed being of order $0.30\pm 0.03$ of the bubble speed. Using the filling box method, we also find that the net fluid volume flux in the slip plume increases linearly with distance from the source at a rate $Q_l = \lambda Bz/v_s^2$, where $B$ is the buoyancy flux of the gas, $v_s$ the rise speed of the gas bubbles, $z$ the distance above the source and $\lambda$ is a constant related to the dimensionless volume of fluid in the wake of each bubble. This slip-dominated flow regime can be understood in terms of kinetic energy imparted to the fluid as the bubbles rise and release potential energy. Further experiments with particle-laden plumes illustrate similar scalings for the volume flux in a particle-driven slip plume once the slip speed of the particles exceeds the bulk speed of the equivalent single-phase buoyant plume with the same buoyancy flux. Near the source the slip speed may be smaller than the plume speed, and the flow follows the classical model for a turbulent buoyant plume, with the transition to the slip regime occurring at a distance $z^*\approx (32\pm 5)\lambda ^{3/2} B/v_s^3$ from the source, where the dimensionless parameter $\lambda$ relates to the dimensionless volume of the fluid wake, which we find varies with the Reynolds number of the particles.

We investigate the three-dimensional (3-D) flow around and through a porous screen for various porosities at high Reynolds number $Re = {O}(10^4)$. Historically, the study of this problem has been focused on two-dimensional cases and for screens spanning completely or partially a channel. Since many recent problems have involved a porous object in a 3-D free flow, we present a 3-D model initially based on Koo & James (J. Fluid Mech., vol. 60, 1973, pp. 513–538) and Steiros & Hultmark (J. Fluid Mech., vol. 853, 2018 pp. 1–11) for screens of arbitrary shapes. In addition, we include an empirical viscous correction factor accounting for viscous effects in the vicinity of the screen. We characterize experimentally the aerodynamic drag coefficient for a porous square screen composed of fibres, immersed in a laminar air flow with various solidities and different angles of attack. We test various fibre diameters to explore the effect of the space between the pores on the drag force. Using PIV and hot wire probe measurements, we visualize the flow around and through the screen, and in particular measure the proportion of fluid that is deviated around the screen. The predictions from the model for drag coefficient, flow velocities and streamlines are in good agreement with our experimental results. In particular, we show that local viscous effects are important: at the same solidity and with the same air flow, the drag coefficient and the flow deviations strongly depend on the Reynolds number based on the fibre diameter. The model, taking into account 3-D effects and the shape of the porous screen, and including an empirical viscous correction factor that is valid for fibrous screens may have many applications including the prediction of water collection efficiency for fog harvesters.

Recent microfluidic experiments have evidenced complex spatio-temporal fluctuations in low-Reynolds-number flows of polymer solutions through lattices of obstacles. However, understanding the nonlinear physics of such systems remains a challenge. Here, we use high performance simulations to study viscoelastic flows through a hexagonal lattice of cylindrical obstacles. We find that structures of localized polymer stress – in particular birefringent strands – control the stability and the dynamics. We first show that, at steady state, strands act as a web of sticky flow barriers that induce channelization, multistability and hysteresis. We then demonstrate that a spontaneous destabilization of the strands drives the transition to unsteady flow with regimes of self-sustained oscillations, travelling waves and strand pulsations. We further show that these pulsations, which result from the destabilization of envelope patterns of stress with strands wrapped around multiple obstacles, are integral to the transition towards elastic turbulence in our two-dimensional simulations. Our study provides a new perspective on the role of birefringent strands and a framework for understanding experimental observations. We anticipate that it is an important step towards unifying existing interpretations of the nonlinear physics of viscoelastic flows through complex structures.

The behaviour of turbulent, buoyant, planar plumes is fundamentally coupled to the environment within which they develop. The effect of a background stratification directly influences a plumes buoyancy and has been the subject of numerous studies. Conversely, the effect of an ambient co-flow, which directly influences the vertical momentum of a plume, has not previously been the subject of theoretical investigation. The governing conservation equations for the case of a uniform co-flow are derived and the local dynamical behaviour of the plume is shown to be characterised by the scaled source Richardson number and the relative magnitude of the co-flow and plume source velocities. For forced, pure and lazy plume release conditions the co-flow acts to narrow the plume and reduce both the dilution and the asymptotic Richardson number relative to the classic zero co-flow case. Analytical solutions are developed for pure plumes from line sources, and for highly forced and highly lazy releases from sources of finite width in a weak co-flow. Contrary to releases in quiescent surroundings, our solutions show that all classes of release can exhibit plume contraction and the associated necking. For entraining plumes, a dynamical invariance spatially only occurs for pure and forced releases and we derive the co-flow strengths that lead to this invariance.

We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of $10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of $0.25 \leq Pr \leq 20$. The transition is characterized by the emergence of multiple satellite eddies with increasing $Ra$, which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as $Ra_t \sim Pr^{1.41}$ where $Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).

This study investigates the scenario of flow past a circular cylinder in proximity to a moving wall (or equally a body translating in still fluid parallel to a stationary wall). Fifty high-fidelity three-dimensional direct numerical simulations are performed over a parameter space of turbulent Reynolds numbers (Re) of 300 to 1000 combined with gap-to-diameter ratios (G/D) of 0.2 to 3. The flow, hydrodynamic and turbulence characteristics over the (Re, G/D) parameter space are examined in detail. Small-scale rib-like mode B structures and alternate vortex shedding are observed for all cases. The streamwise location for the vortex shedding (quantified by the wake recirculation length Lr) varies strongly with both Re and G/D. The variation of Lr with Re and G/D can be explained by the spanwise circulation $\varGamma_z$ fed into the wake, where the variation trends of Lr and $\varGamma_z$ are inversely correlated. The variations of the mean drag and fluctuating lift coefficients are also governed by the variations in Lr and $\varGamma_z$. The total kinetic energy in the wake region reduces drastically as G/D reduces below 0.8, which is contributed collectively by (i) reduction in the strength of the shed vortices, (ii) downstream movement of the location of vortex shedding and (iii) associated delayed generation of streamwise vortices. The present results on a moving wall also help to explain several flow and hydrodynamic characteristics reported in the literature for a stationary wall, because the moving wall eliminates the complex wall boundary layer and retains a ‘clean’ near-wall effect.

The interaction of opposite-signed twist waves on vortex tubes can lead to vortex bursting, a process where the core expands into a double ring-like structure with strong swirling flows. Previous works have studied vortex bursting on rectilinear vortices by axially perturbing the initial core size to generate the twist waves, and observed largely axisymmetric bursting dynamics. In this work, we numerically study bursting on vortical structures with curved centrelines, analysing the interaction between the centreline dynamics, twist wave generation and propagation, and vortex bursting. We focus on axially perturbed helical vortex tubes with small radius-to-pitch ratios up to $0.0625$, as well as vortex rings with a large radius-to-core size ratio $10$, both at a circulation-based Reynolds number $5000$. The results show that though the initial twist wave propagation speeds are relatively unaffected by the curvature and torsion of the centreline, the bursting process is altered significantly compared with rectilinear vortices. The self-induced rotation of the centreline of the helical tube induces a non-axisymmetric distortion of the bursting structure, which rapidly breaks up the vortex core into small-scale helical structures. A similar destabilization of the bursting structure also occurs on vortex rings. The enstrophy increase and accelerated energy decay associated with bursting are predominantly determined by the twist wave strength, rather than the curvature and torsion of the centreline. Combined, our findings imply that bursting could play an important role in transferring and dissipating energy of vortical structures in wakes, and turbulent flows in general.

We study the behaviour of the streamwise velocity variance in turbulent wall-bounded flows using a direct numerical simulation (DNS) database of pipe flow up to friction Reynolds number ${{Re}}_{\tau } \approx 12000$. The analysis of the spanwise spectra in the viscous near-wall region strongly hints to the presence of an overlap layer between the inner- and the outer-scaled spectral ranges, featuring a $k_{\theta }^{-1+\alpha }$ decay (with $k_{\theta }$ the wavenumber in the azimuthal direction, and $\alpha \approx 0.18$), hence shallower than suggested by the classical formulation of the attached-eddy model. The key implication is that the contribution to the streamwise velocity variance $(\langle{u}^2\rangle)$ from the largest scales of motion (superstructures) slowly declines as ${{Re}}_{\tau }^{-\alpha }$, and the integrated inner-scaled variance follows a defect power law of the type $\langle u^2 \rangle ^+ = A - B \, {{Re}}_{\tau }^{-\alpha }$, with constants $A$ and $B$ depending on $y^+$. The DNS data very well support this behaviour, which implies that strict wall scaling is restored in the infinite-Reynolds-number limit. The extrapolated limit distribution of the streamwise velocity variance features a buffer-layer peak value of $\langle u^2 \rangle ^+ \approx 12.1$, and an additional outer peak with larger magnitude. The analysis of the velocity spectra also suggests a similar behaviour of the dissipation rate of the streamwise velocity variance at the wall, which is expected to attain a limiting value of approximately $0.28$, hence slightly exceeding the value $0.25$ which was assumed in previous analyses (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3). We have found evidence suggesting that the reduced near-wall influence of wall-attached eddies is likely linked to the formation of underlying turbulent Stokes layers.

The entrainment of ambient fluid into a variable-density jet is typically quantified using an entrainment coefficient $\alpha$. Here, we investigate the dependence of $\alpha$ on the ratio of the jet's density $\rho _m$ and that of the ambient fluid $\rho _0$. Current parametrisations of $\alpha$ rely on a scaling inferred from early laboratory experiments (Ricou & Spalding, J. Fluid Mech., vol. 11, 1961, pp. 21–32). We demonstrate analytically that the experiments preclude definitive conclusions regarding the dependence of $\alpha$ on $\rho _m / \rho _0$ and that the underlying physical processes therefore warrant closer attention. To investigate the physics behind the dependence of entrainment on the density ratio we use a Favre-averaged entrainment decomposition. The decomposition is applied to data from large-eddy simulations of jets characterised by density ratios $\rho _m / \rho _0$ spanning over two orders of magnitude that have been verified against experimental data. Changes in the shape of the velocity profile are a significant contributor to entrainment in the near field due to the breakdown of the potential core, and persist over larger streamwise distances in heavy releases than in light releases. Therefore, to focus exclusively on the effects of density ratio, we study the region where the shape changes have become small but the density ratio is still significant. We show that the dimensionless turbulent kinetic energy production and mean kinetic energy flux depend strongly on the density ratio, both for our large-eddy simulation data and for recent experiments. Despite this, the entrainment coefficient is practically constant in this region and has value $\alpha \approx 0.07$ for all simulations.

We take advantage of the extremely small kinematic viscosity of superfluid $^4$He to investigate the propagation of macroscopic vortex rings at Reynolds numbers between $2 \times 10^4$ and $4 \times 10^6$. These inhomogeneous flow structures are thermally generated by releasing short power pulses into a small volume of liquid, open to the surrounding bath through a vertical tube $2$ mm in diameter. We study specifically the ring behaviour between $1.30$ and $1.80$ K using the flow visualization and second sound attenuation techniques. From the obtained data sets, containing more than $2600$ realizations, we find that the rings remain well-defined in space and time for distances up to at least $40$ tube diameters, and that their circulation depends significantly on the travelled distance, in a way similar to that observed for turbulent vortex rings propagating in Newtonian fluids. Additionally, the ring velocity and circulation appear to be influenced solely by a single, experimentally accessible parameter, combining the liquid temperature with the magnitude and duration of the power pulse. Overall, our results support the view that macroscopic vortex rings moving in superfluid $^4$He closely resemble their Newtonian analogues, at least in the absence of significant thermal effects and at sufficiently large flow scales.

We propose a semi-analytical solution for the advection–diffusion equation in cylindrical domains, with an aim towards extracting blood flow rates from contrast variations in a coronary computed tomography angiography image. The solution proposed in this work, in contrast with existing methods, which only consider advection, incorporates both radial velocity variation and diffusion. By means of a Galerkin approach using Bessel functions, a solution for a three-dimensional concentration field at a single time point is obtained after a Laplace transformation. This semi-analytical solution forms the basis for a novel advection–diffusion flow estimation (ADFE) method. The ADFE is derived, validated through numerical spectral-element method computations, and shown to exhibit improved accuracy against the state-of-the-art method for image-based blood flow extraction.

The dynamics of turbulent kinetic energy (TKE), turbulence dissipation rate (TDR) and turbulence production rate (TPR) are explored in fully developed turbulent channel flow using direct numerical simulations up to $\textit {Re}_\tau \approx 2000$ with minimal computational box for large-scale structures. Time correlation analysis based on volume-averaged TKE and TDR shows a well-defined average time lag, as in periodic/homogeneous turbulence, which, unlike periodic/homogeneous turbulence, appears to be Reynolds-number-dependent. On the basis of a spatio-temporal correlation analysis, we show that plane-averaged TKE fluctuations in the near-equilibrium region are transported towards both the core and near-wall regions, and are positively correlated with plane-averaged TDR fluctuations there with combined wall-distance and time lags. In the path towards the core region, the wall-distance lag is very close to the time lag multiplied by the friction velocity. The path towards the near-wall region has a wide spread of time lags, which increases with Reynolds number. The spatio-temporal correlation paths both towards the core and towards the wall are reproduced when the reference plane TKE is conditionally averaged on either ejections or sweeps, and are in fact stronger in correlation values in the case of ejections, which are better organised than sweeps. While volume-averaged TPR evidently precedes volume-averaged TKE, a more complex picture of non-local space–time correlations between reference plane TKE and TPR is revealed. A mechanistic model is proposed to elucidate these correlations between TKE and TPR through the interaction between the mean shear and the Reynolds shear stress.

A model is formulated of a two-dimensional migrating, or swimming, inviscid bubble in a viscous fluid whose unsteady displacement is caused by the spreading over its surface of an initial distribution of insoluble surfactant. Assuming small capillary and Reynolds numbers, and a linear equation of state giving the surface tension as a function of surfactant concentration, the quasi-steady Stokes flow around the bubble is found analytically and explicit formulas are determined for the time-dependent bubble speed and its final overall displacement. At infinite surface Péclet number this is done using a complex version of the method of characteristics to solve a complex partial differential equation of Burgers type. For a finite non-zero surface Péclet number, the problem is shown to be linearizable by a complex variant of the classical Cole–Hopf transformation. The formulation allows general statements to be made on the bubble speed and its total net displacement in terms of the initial surfactant distribution. A weak finite-time singularity in the surface activity associated with an isolated clean point on the bubble surface is also identified and studied in detail.

Although poroelastic clusters in nature, such as bristled wings and plumed seeds, exhibit remarkable flight performances by virtue of their porous structure, the effects of another key feature, elasticity, on aerodynamic loading remain elusive. For a poroelastic cluster, we investigate the aerodynamic effects of elastic deformation that occurs through the collective rearrangement of many elastic components and the fluid-dynamic interactions between them. As a simple two-dimensional model, an array of multiple cylinders which are individually and elastically mounted is employed with diverse values of porosity and elasticity. Under a uniform free stream, the poroelastic cluster enlarges its frontal area and augments the total drag force in the quasi-steady state; this is in contrast to the general reconfiguration of fixed elastic structures, which tends to reduce the frontal area and drag. The rearrangement of the poroelastic cluster is dominated by the virtual fluid barrier that develops in a gap between the elastic components, interrupting the flow penetrating between them. The effects of this hydrodynamic blockage on changes in the frontal area and drag force are analysed in terms of porosity and elasticity, revealing the fluid-dynamic mechanism underlying the appearance of peak drag at an intermediate porosity. Moreover, to represent the coupled effects of porosity and elasticity on the rearrangement, a scaled elastic energy is derived through a consideration of the energy balance.

The growth and characteristics of linear, oblique instabilities on a highly swept fin on a straight cone in Mach 6 flow are examined. Large streamwise pressure gradients cause doubly inflected cross-flow profiles and reversed flow near the wall, which necessitates using the harmonic linearized Navier–Stokes equations. The cross-flow instability is responsible for the most-amplified disturbances, however, not all disturbances show typical cross-flow characteristics. Distinct differences in perturbation structure are shown between small ($\sim$3–5 mm) and large ($\sim$10 mm) wavelength disturbances at the unit Reynolds number $Re' = 11 \times 10^6$ m$^{-1}$. As a result, amplification measurements based solely on wall quantities bias a most-amplified disturbance assessment towards larger wavelengths and lower frequencies than would otherwise be determined by an off-wall total-energy approach. A spatial-amplification energy-budget analysis demonstrates (i) that wall-normal Reynolds-flux terms dictate the local growth rate, despite other terms having a locally larger magnitude and (ii) that the Reynolds-stress terms are responsible for large-wavelength disturbances propagating closer to the wall compared with small-wavelength disturbances. Additionally, the effect of free-stream unit Reynolds number and small yaw angles on the perturbation amplification and energy budget is considered. At a higher Reynolds number ($Re' = 22 \times 10^6$ m$^{-1}$), the most-amplified wavelength shrinks. Perturbations do not behave self-similarly in the thinner boundary layer, and the shift in most-amplified wavelength is due to decreased dissipation relative to the lower-Reynolds-number case. Small yaw angles produce a streamwise shift in the boundary layer and disturbance amplification. The yaw results quantify a potential uncertainty source in experiments and flight.

We examine the gravity-driven flow of a thin film of viscous fluid spreading over a rigid plate that is lubricated by another viscous fluid. We model the flow over such a ‘soft’ substrate by applying the principles of lubrication theory, assuming that vertical shear provides the dominant resistance to the flow. We do so in axisymmetric and two-dimensional geometries in settings in which the flow is self-similar. Different flow regimes arise, depending on the values of four key dimensionless parameters. As the viscosity ratio varies, the behaviour of the intruding layer ranges from that of a thin coating film, which exerts negligible traction on the underlying layer, to a very viscous gravity current spreading over a low-viscosity, near-rigid layer. As the density difference between the two layers approaches zero, the nose of the intruding layer steepens, approaching a shock front in the equal-density limit. We characterise a frontal stress singularity, which forms near the nose of the intruding layer, by performing an asymptotic analysis in a small neighbourhood of the front. We find from our asymptotic analysis that unlike single-layer viscous gravity currents, which exhibit a cube-root frontal singularity, the nose of a viscous gravity current propagating over another viscous fluid instead exhibits a square-root singularity, to leading order. We also find that large differences in the densities between the two fluids give rise to flows similar to that of thin films of a single viscous fluid spreading over a rigid, yet mobile, substrate.