We analyse the motion of a flagellated bacterium in a two-fluid medium using slender body theory. The two-fluid model is useful for describing a body moving through a complex fluid with a microstructure whose length scale is comparable to the characteristic scale of the body. This is true for bacterial motion in biological fluids (entangled polymer solutions), where the entanglement results in a porous microstructure with typical pore diameters comparable to or larger than the flagellar bundle diameter, but smaller than the diameter of the bacterial head. Thus, the polymer and solvent satisfy different boundary conditions on the flagellar bundle and move with different velocities close to it. This gives rise to a screening length $L_B$ within which the fluids exchange momentum and the relative velocity between the two fluids decays. In this work, both the solvent and polymer of the two-fluid medium are modelled as Newtonian fluids with different viscosities $\mu _s$ and $\mu _p$ (viscosity ratio $\lambda = \mu _p/\mu _s$), thereby capturing the effects solely introduced by the microstructure of the complex fluid. From our calculations, we observe an increased drag anisotropy for a rigid, slender flagellar bundle moving through this two-fluid medium, resulting in an enhanced swimming velocity of the organism. The results are sensitive to the interaction between the bundle and the polymer, and we discuss two physical scenarios corresponding to two types of interaction. Our model provides an explanation for the experimentally observed enhancement of swimming velocity of bacteria in entangled polymer solutions and motivates further experimental investigations.

Empirical evidence is provided that within the inertial sublayer (i.e. logarithmic region) of adiabatic turbulent flows over smooth walls, the skewness of the vertical-velocity component $S_w$ displays universal behaviour, being a positive constant and constrained within the range $S_w \approx 0.1\unicode{x2013}0.16$, regardless of flow configuration and Reynolds number. A theoretical model is then proposed to explain this behaviour, including the observed range of variations of $S_w$. The proposed model clarifies why $S_w$ cannot be predicted from down-gradient closure approximations routinely employed in large-scale meteorological and climate models. The proposed model also offers an alternative and implementable approach for such large-scale models.

The spherical Couette system consists of two differentially rotating concentric spheres with the space in between filled with fluid. We study a regime where the outer sphere is rotating rapidly enough so that the Coriolis force is important and the inner sphere is rotating either slower or in the opposite direction with respect to the outer sphere. We numerically study the sudden transition to turbulence at a critical differential rotation seen in experiments at BTU Cottbus-Senftenberg, Germany, and investigate its cause. We find that the source of turbulence is the boundary layer on the inner sphere, which becomes centrifugally unstable. We show that this instability leads to generation of small-scale structures which lead to turbulence in the bulk, dominated by inertial waves, a change in the force balance near the inner boundary, the formation of a mean flow through Reynolds stresses and, consequently, to an efficient angular momentum transport. We compare our findings with axisymmetric simulations and show that there are significant similarities in the nature of the flow in the turbulent regimes of full three-dimensional and axisymmetric simulations but differences in the evolution of the instability that leads to this transition. We find that a heuristic argument based on a Reynolds number defined using the thickness of the boundary layer as a length scale helps explain the scaling law of the variation of critical differential rotation for transition to turbulence with rotation rate observed in the experiments.

This work aims to perform a parametric study on a round supersonic jet with a design Mach number Md = 1.8, which is manipulated using a single steady radial minijet with a view to enhancing its mixing. Four control parameters are examined, i.e. the mass flow rate ratio Cm and diameter ratio d/D of the minijet to main jet, and exit pressure ratio Pe/Pa and fully expanded jet Mach number Mj, where Pe and Pa are the nozzle exit and atmospheric pressures, respectively. Extensive pressure and schlieren flow visualization measurements are conducted on the natural and manipulated jets. The supersonic jet core length Lc/D exhibits a strong dependence on the four control parameters. Careful scaling analysis of experimental data reveals that Lc/D = f1(Cm, d/D, Pe/Pa, Mj) may be reduced to Lc/D = f2(ξ), where f1 and f2 are different functions. The scaling factor $\xi = J({d_i}/{D_j})/(\gamma M_j^2{P_e}/{P_a})$ is physically the penetration depth of the minijet into the main jet, where $J({d_i}/{D_j})$ is the square root of the momentum ratio of the minijet to main jet (di and Dj are the fully expanded diameters of d and D, respectively), γ is the specific heat ratio and $\gamma M_j^2{P_e}/{P_a}$ is the non-dimensional exit pressure ratio. Important physical insight may be gained from this scaling law into the optimal choice of control parameters such as d/D and Pe/Pa for practical applications. It has been found for the first time that the minijet may induce a street of quasi-periodical coherent structures once Cm exceeds a certain level for a given ${P_e}/{P_a}$. Its predominant dimensionless frequency Ste (≡ feDj/Uj) scales with a factor $\zeta = J({d_i}/{D_j})\; \sqrt {\gamma M_j^2{P_e}/{P_a}} $, which is physically the ratio of the minijet momentum thrust to the ambient pressure thrust. The formation mechanism of the street and its role in enhancing jet mixing are also discussed.

The simulation of rarefied gas flow based on the Boltzmann equation is challenging, especially when the gas mixtures have disparate molecular masses. In this paper, a computationally tractable kinetic model is proposed for monatomic gas mixtures, to mimic the Boltzmann collision operator as closely as possible. The intra- and inter-collisions are modelled separately using relaxation approximations, to correctly recover the relaxation time scales that could span several orders of magnitude. The proposed kinetic model preserves the accuracy of the Boltzmann equation in the continuum regime by recovering four critical transport properties of a gas mixture: the shear viscosity, the thermal conductivity, the coefficients of diffusion and the thermal diffusion. While in the rarefied flow regimes, the kinetic model is found to be accurate when comparing its solutions with those from the direct simulation Monte Carlo method in several representative cases (e.g. one-dimensional normal shock wave, Fourier flow and Couette flow, two-dimensional supersonic flow passing a cylinder and nozzle flow into a vacuum), for binary mixtures with a wide range of mass ratios, species concentrations and different intermolecular potentials. Pronounced separations in species properties have been observed, and the flow characteristics of gas mixtures in shock waves are found to change as the molecular mass ratio increases from 10 to 1000.

We present a theory that quantifies the interplay between intrapore and interpore flow variabilities and their impact on hydrodynamic dispersion. The theory reveals that porous media with varying levels of structural disorder exhibit notable differences in interpore flow variability, characterised by the flux-weighted probability density function (PDF), $\hat {\psi }_\tau (\tau ) \sim \tau ^{-\theta -2}$, for advection times $\tau$ through conduits. These differences result in varying relative strengths of interpore and intrapore flow variabilities, leading to distinct scaling behaviours of the hydrodynamic dispersion coefficient $D_L$, normalised by the molecular diffusion coefficient $D_m$, with respect to the Péclet number $Pe$. Specifically, when $\hat {\psi }_\tau (\tau )$ exhibits a broad distribution of $\tau$ with $\theta$ in the range of $(0, 1)$, the dispersion undergoes a transition from power-law scaling, $D_L/D_m \sim Pe^{2-\theta }$, to linear scaling, $D_L/D_m \sim Pe$, and eventually to logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$, as $Pe$ increases. Conversely, when $\tau$ is narrowly distributed or when $\theta$ exceeds 1, dispersion consistently follows a logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$. The power-law and linear scaling occur when interpore variability predominates over intrapore variability, while logarithmic scaling arises under the opposite condition. These theoretical predictions are supported by experimental data and network simulations across a broad spectrum of porous media.

To aid in prediction of turbulent boundary layer flows over rough surfaces, a new model is proposed to estimate hydrodynamic roughness based solely on geometric surface information. The model is based on a fluid-mechanics motivated geometric parameter called the wind-shade factor. Sheltering is included using a rapid algorithm adapted from the landscape shadow literature, while local pressure drag is estimated using a piecewise potential flow approximation. Similarly to evaluating traditional surface parameters such as skewness or average slope magnitude, the wind-shade factor is purely geometric and can be evaluated efficiently from knowing the surface elevation map and the mean flow direction. The wind-shade roughness model is applied to over 100 different surfaces available in a public roughness database and some others, and the predicted sandgrain-roughness heights are compared with measured values. Effects of various model ingredients are analysed, and transitionally rough surfaces are treated by adding a term representing the viscous stress component.

A model of imbibition dynamics in a channel of flattened triangular cross-section is presented, taking into account the liquid film flow in the corners of the channel. The quasi-analytical solutions are derived on the basis of a lubrication approximation. The analysis encompasses two imbibition scenarios corresponding to a constant flow rate or constant pressure imposed in the wetting fluid at the inlet of the channel. In the former case, the process starts with a liquid film flow regime in the corners that is followed by a bulk and corner film flow regime characterised by a triple point advancing (far) ahead of the bulk meniscus after its entrance in the channel. In the latter case, the occurrence of the bulk and corner film flow regime is conditioned by an imposed pressure yielding a capillary pressure at the inlet smaller than a threshold capillary pressure. Above this threshold, the liquid film regime remains. For both imbibition scenarios under concern, important features are highlighted, including (i) the time scalings of the dynamics of both the triple point and apex of the bulk meniscus (when it exists), (ii) the contrast in the positions of these two points showing that the classical Washburn approach, which neglects the effect of the corner films, overpredicts the dynamics of the bulk meniscus. The important consequence is an early wetting fluid breakthrough at the channel outlet much before the bulk meniscus arrival. Comparisons with experimental data available in the literature are provided, validating the approach proposed in this work.

This paper presents a numerical study on the flow around two tandem circular cylinders beneath a free surface at a Reynolds number of $180$. The free-surface effects on the wake dynamics and hydrodynamic forces are investigated through a parametric study, covering a parameter space of gap ratios from $0.20$ to $2.00$, spacing ratios from $1.50$ to $4.00$ and Froude numbers from $0.2$ to $0.8$. A jet-like flow accompanied by a shear layer of positive vorticity separating from the free surface is formed in the wake at small gap ratios, which significantly alters the wake pattern through its dynamic behaviours. At shallow submergence depths, the three-dimensional wake transitions from mode B to mode A as the distance between the cylinders increases. As submergence depth increases, the wavy deformation of the primary vortex cores disappears in the wake, and the flow transitions to a two-dimensional state. Higher Froude numbers can extend the effect of the free surface to deeper submergence depths. The critical spacing ratio tends to be larger at higher Froude numbers. Furthermore, the free-surface deformation is examined. The free-surface profile typically comprises a hydraulic jump immediately ahead of the upstream cylinder, trapped waves in the vicinity of the two tandem cylinders and well-defined travelling waves on the downstream side. The frequencies of the waves cluster around the vortex shedding frequency, indicating a close association between the generation of waves and the vortex shedding process.

Metastructures composed of a closely spaced plate array have been widely used in bespoke manipulation of waves in contexts of acoustics, electromagnetics, elasticity and water waves. This paper focuses on wave scattering by discrete plate array metastructures of arbitrary cross-sections, including isolated vertical metacylinders, periodic arrays and horizontal surface-piercing metacylinders. A suitable transform-based method has been applied to each problem to reduce the influence of barriers in a two-dimensional problem to a set of points in a one-dimensional wave equation wherein the solution is constructed using a corresponding Green's function. A key difference from the existing work is the use of an exact description of the plate array rather than an effective medium approximation, enabling the exploration of wave frequencies above resonance where homogenisation models fail but where the most intriguing physical findings are unravelled. The new findings are particularly notable for graded plate array metastructures that produce a dense spectrum of resonant frequencies, leading to broadband ‘rainbow reflection’ effects. This study provides new ideas for the design of structures for the bespoke control of waves with the potential for innovative solutions to coastal protection schemes or wave energy converters.

A gas bubble sitting at a liquid–gas interface can burst following the rupture of the thin liquid film separating it from the ambient, owing to the large surface energy of the resultant cavity. This bursting bubble forms capillary waves, a Worthington jet and subsequent droplets for a Newtonian liquid medium. However, rheological properties of the liquid medium like elastoviscoplasticity can greatly affect these dynamics. Using direct numerical simulations, this study exemplifies how the complex interplay between elasticity (in terms of elastic stress relaxation) and yield stress influences the transient interfacial phenomenon of bursting bubbles. We investigate how bursting dynamics depends on capillary, elastic and yield stresses by exploring the parameter space of the Deborah number ${{\textit {De}}}$ (dimensionless relaxation time of elastic stresses) and the plastocapillary number $\mathcal {J}$ (dimensionless yield-stress of the medium), delineating four distinct characteristic behaviours. Overall, we observe a non-monotonic effect of elastic stress relaxation on the jet development while plasticity of the elastoviscoplastic (EVP) medium is shown to affect primarily the jet evolution only at faster relaxation times (low ${{\textit {De}}}$). The role of elastic stresses on jet development is elucidated with the support of energy budgets identifying different modes of energy transfer within the EVP medium. The effects of elasticity on the initial progression of capillary waves and droplet formation are also studied. In passing, we study the effects of solvent–polymer viscosity ratio on bursting dynamics and show that polymer viscosity can increase the jet thickness apart from reducing the maximum height of the jet.

This study is dedicated to achieving efficient active noise control in a supersonic underexpanded planar jet, utilizing control parameters informed by resolvent analysis. The baseline supersonic underexpanded jet exhibits complex wave structures and substantial high-amplitude noise radiations. To perform the active control, unsteady blowing and suction are applied along the nozzle inner wall close to the exit. Employing both standard and acoustic resolvent analyses, a suitable frequency and spanwise wavenumber range for the blowing and suction is identified. Within this range, the control forcing can be significantly amplified in the near field, effectively altering the original sound-producing energetic structure while minimizing far-field amplification to prevent excessive noise. A series of large-eddy simulations are further conducted to validate the control efficiency, demonstrating an over 10 dB reduction in upstream-propagated screech noise. It is identified that the present unsteady control proves more effective than steady control at the same momentum coefficient. The controlled jet flow indicates that the shock structures become more stable, and the stronger the streamwise amplification of the forcing, the more likely it is to modify the mean flow characteristics, which is beneficial for reducing far-field noise radiation. Spectral proper orthogonal decomposition analysis of the controlled flow confirms that the control redistributes energy to higher forcing frequencies and suppresses large-scale antisymmetric and symmetric modes related to screech and its harmonics. The findings of this study highlight the potential of resolvent-guided control techniques in reducing noise in supersonic underexpanded jets and provide a detailed understanding of the inherent mechanisms for effective noise reduction through active control strategies.

In this study, direct numerical simulation of the particle dispersion and turbulence modulation in a sonic transverse jet injected into a supersonic cross-flow with a Mach number of 2 was carried out with the Eulerian–Lagrangian point-particle method. One single-phase case and two particle-laden cases with different particle diameters were simulated. The jet and particle trajectories, the dispersion characteristics of particles, and the modulation effect of particles on the flow were investigated systematically. It was found that large particles primarily accumulate around shear layer structures situated on the windward side of the jet trajectory. In contrast, small particles exhibit radial transport, accessing both upstream and downstream recirculation zones. Moreover, small particles disperse extensively within the boundary layer and large-scale shear layers, evidently influenced by the streamwise vortices. The particles increase the mean wall-normal velocity near the wall in the wake region of the transverse jet, while reducing the mean streamwise and wall-normal velocities in outer regions. Particles significantly alter the flow velocity adjacent to shock fronts. In particular, the turbulent fluctuations near the windward barrel shock and bow shock are reduced, while those around the leeward barrel shock are increased. An upward displacement of the bow shock in the wall-normal direction is also observed due to particles. In the regions away from the shocks, small particles tend to amplify the Reynolds stress, while large particles attenuate the turbulent kinetic energy.

Ice-crystal icing (ICI) in aircraft engines is a major threat to flight safety. Due to the complex thermodynamic and phase-change conditions involved in ICI, rigorous modelling of the accretion process remains limited. The present study proposes a novel modelling approach based on the physically observed mixed-phase nature of the accretion layers. The mathematical model, which is derived from the enthalpy change after accretion (the enthalpy model), is compared with an existing pure-phase layer model (the three-layer model). Scaling laws and asymptotic solutions are developed for both models. The onset of ice accretion, the icing layer thickness and solid ice fraction within the layer are determined by a set of non-dimensional parameters including the Péclet number, the Stefan number, the Biot number, the melt ratio and the evaporative rate. Thresholds for freezing and non-freezing conditions are developed. The asymptotic solutions present good agreement with numerical solutions at low Péclet numbers. Both the asymptotic and numerical solutions show that, when compared with the three-layer model, the enthalpy model presents a thicker icing layer and a thicker water layer above the substrate due to mixed-phased features and modified Stefan conditions. Modelling in terms of the enthalpy poses significant advantages in the development of numerical methods to complex three-dimensional geometrical and flow configurations. These results improve understanding of the accretion process and provide a novel, rigorous mathematical framework for accurate modelling of ICI.

Describing the evolution of a wind turbine's wake from a top-hat profile near the turbine to a Gaussian profile in the far wake is a central feature of many engineering wake models. Existing approaches, such as super-Gaussian wake models, rely on a set of tuning parameters that are typically obtained from fitting high-fidelity data. In the current study, we present a new engineering wake model that leverages the similarity between the shape of a turbine's wake normal to the streamwise direction and the diffusion of a passive scalar from a disk source. This new wake model provides an analytical expression for a streamwise scaling function that ensures the conservation of linear momentum in the wake region downstream of a turbine. The model also considers the different rates of wake expansion that are known to occur in the near- and far-wake regions. Validation is presented against high-fidelity numerical data and experimental measurements from the literature, confirming a consistent good agreement across a wide range of turbine operating conditions. A comparison is also drawn with several existing engineering wake models, indicating that the diffusion-based model consistently provides more accurate wake predictions. This new unified framework allows for extensions to more complex wake profiles by making adjustments to the diffusion equation. The derivation of the proposed model included the evaluation of analytical solutions to several mathematical integrals that can be useful for other physical applications.

A long-wave asymptotic model is developed for a viscoelastic falling film along the inside of a tube; viscoelasticity is incorporated using an upper convected Maxwell model. The dynamics of the resulting model in the inertialess limit is determined by three parameters: Bond number Bo, Weissenberg number We and a film thickness parameter $a$. The free surface is unstable to long waves due to the Plateau–Rayleigh instability; linear stability analysis of the model equation quantifies the degree to which viscoelasticity increases both the rate and wavenumber of maximum growth of instability. Elasticity also affects the classification of instabilities as absolute or convective, with elasticity promoting absolute instability. Numerical solutions of the nonlinear evolution equation demonstrate that elasticity promotes plug formation by reducing the critical film thickness required for plugs to form. Turning points in travelling wave solution families may be used as a proxy for this critical thickness in the model. By continuation of these turning points, it is demonstrated that in contrast to Newtonian films in the inertialess limit, in which plug formation may be suppressed for a film of any thickness so long as the base flow is strong enough relative to surface tension, elasticity introduces a maximum critical thickness past which plug formation occurs regardless of the base flow strength. Attention is also paid to the trade-off of the competing effects introduced by increasing We (which increases growth rate and promotes plug formation) and increasing Bo (which decreases growth rate and inhibits plug formation) simultaneously.

Leading-edge noise is a complex phenomenon that occurs when a turbulent fluid encounters a solid object, and is a notable concern in various engineering applications. This study enhances a mathematical leading-edge noise model (Hales et al., J. Fluid Mech., vol. 970, 2023, A29) for anisotropic flow and porous boundaries. The model has two key components. First, we adjust the velocity spectrum to account for the possibility of anisotropy in the flow. This paper rigorously introduces a third dimension for the turbulence spectrum that preserves the turbulence kinetic energy and mathematical definitions for integral length scales. Second, we adapt the fully analytical acoustic transfer function to account for different boundaries by implementing convective impedance boundary conditions when formulating the gust-diffraction problem. This problem is then solved using the Wiener–Hopf technique. We discuss important aspects of this method, including the factorisation of a non-trivial scalar kernel function and the application of suitable edge conditions for the problem. Each modification is inspired by experimental leading-edge noise data using a series of different porous leading edges and anisotropic turbulence generated by a cylinder upstream of the edge. Experimental data demonstrate the interplay between anisotropy and leading-edge modifications while achieving the characteristic mid-frequency noise reduction expected from porous leading edges. Our model is adapted to best fit the trends of the data via a tailored impedance function, leading to good agreement with all datasets across an extended frequency range. This tailored function is used to successfully validate the model against other datasets from a different set of experiments.

Liquid plug formation in thin channels due to the Plateau–Rayleigh instability of a liquid film is observed in a variety of fields. In this paper, complementarity between theoretical solutions and direct numerical simulations (DNS) based on a front-tracking algorithm is explored to evaluate the importance of inertia for the case of a cylindrical capillary. A linear stability analysis is first performed and DNS results are then used to investigate the spatial distributions of inertial, convective and viscous terms of the Navier–Stokes equation. The existence of both viscous and inertial regimes is evidenced with a threshold given by the film thickness. The presence of the core fluid slows down the instability. In the viscous regime, predictions of the lubrication theory are verified. An example of liquid water as the outer fluid film and water vapour as the inner core fluid is simulated with application to the fuel cells.

We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.

The motion of small non-spherical particles is often studied using the unsteady Stokes equations. Zhang & Stone (J. Fluid Mech., vol. 367, 1998, pp. 329–358) reported an asymptotic treatment for nearly spherical particles, to first order in particle non-sphericity, i.e. $O(\epsilon )$, where $\epsilon$ quantifies the shape deviation from a sphere. Importantly, key physical phenomena are absent at $O(\epsilon )$, including (1) coupling between the torque experienced by the particle and its linear translation, (2) coupling between the force the particle experiences and its rotation and (3) the effect of non-sphericity on the orientation averages of these forces and torques. We present an explicit asymptotic theory to second order in particle non-sphericity, i.e. $O(\epsilon ^2)$, for the force and torque acting on a particle in a general unsteady Stokes flow. The derived analytical formulae apply to particles of arbitrary shape, providing the leading-order asymptotic theory for the three above-mentioned phenomena. The theory is demonstrated for several example nearly spherical particles including a spheroid, a ‘pear-shaped’ particle and a simple model for a SARS-CoV-2 virion. This includes formulae for force and torque as a function of particle orientation and their corresponding orientation averages. Our study reveals that the orientation-averaged forces and torques experienced by a nearly spherical particle cannot be generally represented by a perfect sphere. The reported formulae are validated using finite-amplitude three-dimensional direct numerical simulations of the Navier–Stokes equations. A Mathematica notebook is also provided, facilitating implementation of the theory for particle shapes of the user's choosing.