In this work, we investigate the mixing of active scalars in two dimensions by the stirring action of stochastically generated weak shock waves. We use Fourier pseudospectral direct numerical simulations of the interaction of shock waves with two non-reacting species to analyse the mixing dynamics for different Atwood numbers (At). Unlike passive scalars, the presence of density gradients in active scalars alters the molecular diffusion term and makes the species diffusion nonlinear, introducing a concentration gradient-driven term and a density gradient-driven nonlinear dissipation term in the concentration evolution equation. We show that the direction of concentration gradient causes the interface across which molecular diffusion occurs to expand outwards or inwards, even without any stirring action. Shock waves enhance the mixing process by increasing the perimeter of the interface and by sustaining concentration gradients. Negative Atwood number mixtures sustain concentration gradients for a longer time than positive Atwood number mixtures due to the so-called nonlinear dissipation terms. We estimate the time until that when the action of stirring is dominant over molecular mixing. We also highlight the role of baroclinicity in increasing the interface perimeter in the stirring dominant regime. We compare the stirring effect of shock waves on mixing of passive scalars with active scalars and show that the vorticity generated by baroclinicity is responsible for the folding and stretching of the interface in the case of active scalars. We conclude by showing that lighter mixtures with denser inhomogeneities ($At\lt 0$) take a longer time to homogenise than the denser mixtures with lighter inhomogeneities ($At\gt 0$).

Elastoinertial turbulence (EIT) is a chaotic state that emerges in the flows of dilute polymer solutions. Direct numerical simulation (DNS) of EIT is highly computationally expensive due to the need to resolve the multiscale nature of the system. While DNS of two-dimensional (2-D) EIT typically requires $O(10^6)$ degrees of freedom, we demonstrate here that a data-driven modelling framework allows for the construction of an accurate model with 50 degrees of freedom. We achieve a low-dimensional representation of the full state by first applying a viscoelastic variant of proper orthogonal decomposition to DNS results, and then using an autoencoder. The dynamics of this low-dimensional representation is learned using the neural ordinary differential equation (NODE) method, which approximates the vector field for the reduced dynamics as a neural network. The resulting low-dimensional data-driven model effectively captures short-time dynamics over the span of one correlation time, as well as long-time dynamics, particularly the self-similar, nested travelling wave structure of 2-D EIT in the parameter range considered.

When a droplet impacts onto a superheated liquid pool, vapour generation and drainage within the gas cushion play a crucial role in postponing or even preventing contact between the droplet and the pool surface. Through direct numerical simulations, we closely examine the transient dynamics of vapour flow confined within the thin film, with a particular focus on the minimum thickness of this film under a range of impact conditions. Our numerical findings manifest the significant influence of evaporation on the vertical motion of the liquid–vapour interface, revealing how the minimum film thickness evolves in response to variations in impact velocity and degree of superheat. In our numerical simulations, we have identified two distinct evolution laws for the minimum film thickness, corresponding to moderate and high superheat regimes, respectively. These regimes are differentiated by the dominance of evaporation effects within the vapour film during the early falling stage. Subsequently, we establish scaling relations to characterize these regimes by carefully balancing inertial, pressure and evaporation effects within the thin vapour film. Furthermore, we observe that the vapour pressure eventually reaches equilibrium with the rapid increase in capillary pressure at the spreading front, thereby controlling the minimum thickness of the vapour layer in both moderate and high superheat regimes. We derive self-similar solutions based on this equilibrium, and the predicted minimum film thickness aligns remarkably well with our numerical results. This provides compelling evidence that evaporation alone is insufficient to prevent droplet–pool coalescence.

Dust storms are a unique form of high-Reynolds-number particle-laden turbulence associated with intense electrical activity. Using a wavelet-based analysis method on field measurement data, Zhang et al. (2023 J. Fluid Mech. 963, A15) found that wind velocity intermittency intensifies during dust storms, but it is weaker than both dust concentration and electric field. However, the linear and nonlinear multifield coupling characteristics, which significantly influence particle transport and turbulence modulation, remain poorly understood. To address this issue, we obtained high-fidelity datasets of wind velocity, dust concentration, and electric field at the Qingtu Lake Observation Array. By extending the wavelet-based data analysis method, we investigated localised linear and quadratic nonlinear coupling characteristics in strong turbulence–particle–electrostatics coupling regimes. Our findings reveal that linear coupling behaviour is largely dominated by the multifield intermittent components. At small scales, due to very high intermittency, no strong phase synchronisation can be formed, and the interphase linear coupling is weak and notably intermittent. At larger scales, however, perfect phase synchronisation emerges, and dust concentration and electric field exhibit strong, non-intermittent linear coupling, suggesting that large-scale coherent structures play a dominant role in driving the coupling. Importantly, the multifield spectra show well-developed $-1$ and $-5/3$ power-law regions, but the spectral breakpoints for dust concentration and electric field are two decades lower than that for streamwise wind velocity. This difference is due to the broader range and stronger intensity of quadratic nonlinear coupling in dust concentration and electric field, which leads to the broadening of Kolmogorov’s $-5/3$ power-law spectrum.

The dynamics of small-scale structures in free-surface turbulence is crucial to large-scale phenomena in natural and industrial environments. Here, we conduct experiments on the quasi-flat free surface of a zero-mean-flow turbulent water tank over the Reynolds number range $Re_{\lambda } = 207$–312. By seeding microscopic floating particles at high concentrations, the fine scales of the flow and the velocity-gradient tensor are resolved. A kinematic relation is derived expressing the contribution of surface divergence and vorticity to the dissipation rate. The probability density functions of divergence, vorticity and strain rate collapse once normalised by the Kolmogorov scales. Their magnitude displays strong intermittency and follows chi-square distributions with power-law tails at small values. The topology of high-intensity events and two-point statistics indicate that the surface divergence is characterised by dissipative spatial and temporal scales, while the high-vorticity and high-strain-rate regions are larger, long-lived, concurrent and elongated. The second-order velocity structure functions obey the classic Kolmogorov scaling in the inertial range when the dissipation rate on the surface is considered, with a different numerical constant than in three-dimensional turbulence. The cross-correlation among divergence, vorticity and strain rate indicates that the surface-attached vortices are strengthened during downwellings and diffuse when those dissipate. Sources (sinks) in the surface velocity fields are associated with strong (weak) surface-parallel stretching and compression along perpendicular directions. The floating particles cluster over spatial and temporal scales larger than those of the sinks. These results demonstrate that, compared with three-dimensional turbulence, in free-surface turbulence the energetic scales leave a stronger imprint on the small-scale quantities.

The Liebau effect generates a net flow without the need for valves. For the Liebau effect pumping phenomenon to occur, the pump must have specific characteristics. It needs tubes with different elastic properties and an actuator to provide energy to the fluid. The actuator periodically compresses the more flexible element. Furthermore, asymmetry is a crucial factor that differentiates between two pumping mechanisms: impedance pumping and asymmetric pumping. In this work, a model based on the fluid dynamics of an asymmetric valveless pump under resonant conditions is proposed to determine which parameters influence the pumped flow rate. Experimental work is used to validate the model, after which each of the parameters involved in the pump performance is dimensionlessly analysed. This highlights the most significant parameters influencing the pump performance such as the actuator period, length tube ratio and tube diameters. The results point out ways to increase a valveless asymmetric pump’s net-propelled flow rate, which has exciting applications in fields such as biomedicine. The model also allows for predicting the resonance period, a fundamental operating parameter for asymmetric pumping.

We consider the flow of a volume $\mathcal {V} = q t^\alpha$ of viscous fluid injected into a gap $H$ between two horizontal plates ($q$ and $\alpha$ are positive constants, $t$ is time). When the viscosity of the displaced fluid is negligible, the injected fluid forms a slug in contact with both plates connected (at a moving grounding line) to a gravity current (GC) with a downward-inclined interface. Hutchinson et al. (J. Fluid Mech., 598, 2023, pp. A4–1–13) considered a constant source ($\alpha = 1$) of Newtonian fluid at the center of an axisymmetric gap; the flow, governed by the parameter $J$ (the height ratio of the unconfined GC to $H$), admits a similarity solution. Here, the self-similar flow theory is (a) extended to rectangular geometry and power-law fluids, and (b) simplified. Similarity appears when $\alpha = n/(n+1)$ (two-dimensional) and $\alpha = 2n/(n+1)$ (axisymmetric), with propagation $\sim t^\beta$, where $\beta /\alpha = 1$ and $1/2$, respectively, and $n-1$ is the power of the shear in the viscosity law ($n=1$ for Newtonian fluid). The flow is governed by a single parameter $J$, representing the above-mentioned ratio. For small $J$, the GC is mostly unconfined; for large $J$, almost all the injected fluid is in contact with both boundaries of the gap. For given geometry and $n$, we solve one ordinary differential equation (ODE) for the reduced thickness over the reduced length $0\lt y \leqslant 1$, with a singular-regular condition at $y=1$. The details of the confined GC, functions of $J$, follow by simple formulae.

The present study aims to provide an understanding of the influence of an afterbody on the flow-induced vibration (FIV) of cylinders. This is achieved through experimental and numerical investigations into the FIV response of a reverse-D-cross-section cylinder of aspect ratio $AR=5$. By carefully monitoring the point of flow separation to show it always occurs at the sharp top and bottom edges and never further upstream, it is demonstrated that vortex-induced vibration (VIV) can occur without an afterbody. However, for other aspect ratios, an afterbody does play a crucial role in determining the type of fluid forces responsible for sustaining VIV from low to moderate Reynolds numbers in the range $100$–$4700$. For a cylinder without an afterbody, it is found that the viscous force originating from the presence of strong compact vortices forming close to the leeward side of the cylinder is responsible for sustaining strong transverse vibration. In contrast, for a cylinder with an afterbody, the dominant force component depends on the size of the afterbody. In cylinders with a small afterbody, such as a reverse-D semi-circular cylinder, the viscous force dominates, while in cylinders with a larger afterbody such as a circular cylinder, the pressure force dominates.

The intrinsic uncertainty of fluid properties, including the equation-of-state, viscosity and thermal conductivity, on boundary layer stability has scarcely been addressed. When a fluid is operating in the vicinity of the Widom line (defined as the maximum of isobaric specific heat) in supercritical state, its properties exhibit highly non-ideal behavior, which is an ongoing research field leading to refined and more accurate fluid property databases. Upon crossing the Widom line, new mechanisms of flow instability emerge, feasibly leading to changes in dominating modes that yield turbulence. The present work investigates the sensitivity of three-dimensional boundary layer modal instability to these intrinsic uncertainties in fluid properties. The uncertainty, regardless of its source and the fluid regimes, gives rise to distortions of all profiles that constitute the inputs of the stability operator. The effect of these distortions on flow stability is measured by sensitivity coefficients, which are formulated with the adjoint operator and validated against linear modal stability analysis. The results are presented for carbon dioxide at a representative supercritical pressure of approximately 80 bar. The sensitivity to different inputs of the stability operator across various thermodynamic regimes shows an immense range of sensitivity amplitude. A balancing relationship between the density gradient and its perturbation leads to a quadratic effect across the Widom line, provoking significant sensitivity to distortions of the second derivative of the pressure with respect to the density, $\partial ^2 p/\partial \rho ^2$. From an application-oriented point of view, one important question is whether the correct baseflow profiles can be meaningfully analysed by the simplified ideal-fluid model. The integrated modal disturbance growth – the N factor calculated with different partly idealised models – indicates that the answer depends strongly on the thermodynamic regime investigated.

Explaining fast magnetic reconnection in electrically conducting plasmas has been a theoretical challenge in plasma physics since its first description by Eugene N. Parker. In recent years, the observed reconnection rate has been shown by numerical simulations to be explained by the plasmoid instability that appears in highly conductive plasmas. In this work, by studying numerically the Orszag–Tang vortex, we show that the plasmoid instability is very sensitive to the numerical resolution used. It is shown that well-resolved runs display no plasmoid instability even at Lundquist numbers as large as $5\times 10^5$ achieved at resolutions of $32\,768^2$ grid points. On the contrary, in simulations that are under-resolved below a threshold, the plasmoid instability manifests itself with the formation of larger plasmoids the larger the under-resolving is. The present results thus emphasize the importance of performing convergence tests in numerical simulations and suggest that further investigations on the nonlinear evolution of the plasmoid instability are required.

We investigate viscous dissipation in linear flows driven by small-amplitude longitudinal librations in rotating fluid spheres focusing on the rapid rotation regime applicable to planets. Viscous coupling can resonate with inertial modes in the bulk of the fluid when the frequency of the forcing is within the range $(0,2\Omega _0)$, where $\Omega _0$ is the mean angular velocity of the sphere. We solve the linearised equations of motion with a semi-spectral numerical method and with an asymptotic expansion exploiting the small Ekman number, $E$, which quantifies the strength of viscous forces relative to the Coriolis force. Our results confirm that the dominant contribution to the dissipation occurs in the Ekman boundary layer with leading-order scaling $E^{1/2}$. When the forcing frequency coincides with that of an inertial mode, dissipation is reduced by as much as 9 % compared with boundary layer theory alone. The percentage-wise reduction is independent of $E$ and the frequency width of the reduction envelope scales as $E^{1/2}$. At non-resonant frequencies conic shear layers develop in the bulk interior and, together with the Ekman layer bulge at critical latitude, slightly enhance dissipation. We confirm critical latitude bulge and shear layer contributions to the overall dissipation scale as $E^{4/5}$ and $E^{6/5}$ respectively, becoming negligible compared with dissipation in the main boundary layer as $E\rightarrow 0$. The frequencies at which the dissipation enhancement from critical latitude effects is maximised are displaced from the inviscid limit periodic orbit frequencies by a factor that scales with $E^{0.23}$.

We review some of the processes leading to dispersion and mixing in porous media, exploring the differences between the travel time distribution of fluid particles within a pore throat and between pore throats of different size within the porous layer. A recent paper of Liu et al. (2024) has combined a model of these travel time distributions with a continuous time random walk to quantify the dispersion as a function of the Peclet number. We describe some further problems relating to dispersive mixing of tracer which may be amenable to this approach, including dispersion caused by macroscopic lenses of different permeability, dispersion of tracer which partitions between the fluid and matrix and the effects of buoyancy on mixing.

In this paper, the reflection of shock waves with downstream expansion fan interference in two-dimensional, inviscid flow is investigated, including steady Mach reflection (MR) and the unsteady transition process from regular reflection (RR) to MR. A threshold for the configuration based on non-dimensional wedge length is proposed. The analytical model for the steady MR and RR$\rightarrow$MR transition process is established based on the classical shock and expansion wave relations, whose prediction agrees well with results obtained through inviscid numerical simulation. It is found that the expansion fan interference significantly influences the steady flow patterns, especially the height of the Mach stem and the shape of the slip line. The interaction accelerates the formation of the sonic throat, stabilizing the flow structure rapidly, and results in generally small Mach stem heights. The exposure of the triple point to the expansion fan eliminates the inflection point on the slip line, whose slope increases smoothly. The interaction further affects the time evolution of the Mach stem during the multiple-interaction stage of the RR$\rightarrow$MR transition process. It appears that the modifications come from the curvature of the incident shock brought by the wave interference. During the multiple-interaction stage, the triple point moves upstream along the curved incident shock, where the incident shock angle changes according to the curvature, resulting in the variation of the evolution velocity.

We find the optimally time-dependent (OTD) orthogonal modes about a time-varying flow generated by a strong gust vortex impacting a NACA 0012 airfoil. This OTD analysis reveals the amplification characteristics of perturbations about the unsteady base flow and their amplified spatiotemporal structures that evolve over time. We consider four time-varying laminar base flows in which a vortex with a strength corresponding to the gust ratio $G$ of $\{-1,-0.5,0.5,1\}$ impinges on the leading edge of the airfoil at an angle of attack of $12^\circ$. In these cases, the impingement of the strong gust vortex causes massive separation and the generation of large-scale vortices around the airfoil within two convective time units. As these flow structures develop around the airfoil on a short time scale, the airfoil experiences large transient vortical lift variations in the positive and negative directions that are approximately five to ten times larger than the baseline lift. The highly unsteady nature of these vortex–airfoil interactions necessitates an advanced analytical technique capable of capturing the transient perturbation dynamics. For each of the considered gust ratios, the OTD analysis identifies the most amplified region to perturbations, the location of which changes as the wake evolves differently. For interactions between a moderate positive vortex gust ($G=0.5$) and the airfoil, the area where perturbations are amplified transitions from the leading-edge vortex (LEV) sheet to the forming LEV. Later, this most amplified structure becomes supported in the airfoil wake directly behind the trailing edge. In contrast, a strong vortex gust ($G=\pm 1$) encountered by the airfoil shows the most amplified OTD mode to appear around the core of the shed vortices. This study provides an analysis technique and fundamental insights into the broader family of unsteady aerodynamic problems.

We derive leading-order governing equations and boundary conditions for a sheet of viscous fluid retracting freely under surface tension. We show that small thickness perturbations about a flat base state can lead to regions of compression, where one or both of the principal tensions in the sheet becomes negative, and thus drive transient buckling of the sheet centre-surface. The general theory is applied to the simple model problem of a retracting viscous disc with small axisymmetric thickness variations. Transient growth in the centre-surface is found to be possible generically, with the dominant mode selected depending on the imposed initial thickness and centre-surface perturbations. An asymptotic reduction of the boundary conditions at the edge of the disc, valid in the limit of large normalised thickness perturbations, reduces the centre-surface evolution equation to an ordinary differentional equation (ODE) eigenvalue problem. Analysis of this eigenvalue problem leads to insights such as how the degree of transient buckling depends on the imposed thickness perturbation and which thickness perturbation gives rise to the largest transient buckling.

Electro-osmotic flow (EOF) in nanochannels exhibits a puzzling non-monotonic dependence on salt concentration, which contrasts with observations in microchannels and remains not fully understood. In this work, we address this phenomenon through a theoretical investigation of EOF in $\mathrm{pH}$-regulated channels. New analytical approximations for electrostatic potential, EOF profile and electro-osmotic mobility beyond the Debye–Hückel limit are derived through asymptotic analysis. Our findings reveal that the surface electrostatic potential is independent of the channel size only when the half-channel size exceeds the Gouy–Chapman length. In contrast, surface ionization and net charge distribution play more crucial roles in EOF at the nanoscale, as they govern both the magnitude and the spatial distribution of the Coulomb driving force. As salt concentration increases, EOF velocity initially rises due to enhanced surface ionization, followed by a decline attributed to increased wall shear stress. This work provides key insights for EOF applications in nanofluidics and biomedical devices, and deepens the understanding of electrokinetic phenomena influenced by $\mathrm{pH}$-regulation effects.

The flow-induced oscillation of a transversely clamped buckled flexible filament in a uniform flow was explored using the penalty immersed boundary method. Both inverted and conventional configurations were analysed. The effects of bending rigidity, filament length and Reynolds number were examined. As these parameters were varied, four distinct modes were identified: conventional transverse oscillation mode, deflected oscillation mode, inverted transverse oscillation mode and structurally steady mode. The filament exhibited a 2S wake pattern under the conventional transverse oscillation mode and the small-amplitude inverted transverse oscillation mode, a P wake pattern under the deflected oscillation mode and a 2S + 2P wake pattern for the large-amplitude inverted transverse oscillation mode. Irrespective of their initial conditions, all of the filaments converged to the conventional transverse oscillation mode under low bending rigidity. Multistability was observed in the transversely clamped buckled flexible filament under moderate bending rigidity. The deflection in the oscillation mode increased with increasing filament length. The inverted buckled filament was sensitive to the Reynolds number, unlike the conventional buckled filament. The transverse oscillation mode demonstrated superior energy-harvesting performance.