We present a non-perturbative mesh-free spectral method for the axisymmetric scenario of a radially highly deformed sphere in Stokes flow. Spectra of harmonic and biharmonic Stokes flow modes are used to provide a general algebraic spectral solution to axisymmetric Stokes flow. A solution for the flow field around a sphere with prescribed surface velocity field is presented, and is used to obtain the velocity field around a deformed sphere. The method is demonstrated on two problems: hydrodynamic radii of radially deformed spheres for a range of strength and angular dependence of the deformation, with results in good agreement with the boundary element method; and self-phoretic velocity of spheroids with surface flux of the driving field in a source/sink or source/inert configuration.

A scaling law to predict the onset of the primary Hopf instability in the steady flow past two-dimensional symmetric bluff bodies is proposed. It uses a measure of the spatial extent of the separation bubble as the length scale and the largest reverse-flow speed within it as the velocity scale. The ensuing Reynolds number, evaluated at the onset of the primary Hopf bifurcation, collapses quite nicely for bodies of different shape and aspect ratio even when a small angle of attack perturbs the symmetry; its relative variation is one order of magnitude smaller than that of the usual Reynolds number defined with the free-stream velocity and the cross-stream body size. With the new scaling, it can be roughly assessed whether the steady flow past a two-dimensional bluff body is absolutely and globally unstable to two-dimensional perturbations without a computationally expensive stability analysis: only the inspection of the base flow is required. More importantly, the scaling provides an insight into the flow mechanism that produces the instability.

The unsteady response of nozzles with steady heat transfer forced by acoustic and/or entropy waves is modelled. The approach is based on the quasi-one-dimensional linearised Euler equations. The equations are cast in terms of three variables, namely the dimensionless mass, stagnation temperature and entropy fluctuations, which are invariants of the system at zero frequency and with no heat transfer. The resulting first-order system of differential equations is then solved using the Magnus expansion method, where the perturbation parameters are the normalised frequency and the volumetric heat transfer. In this work, a measure of the flow non-isentropicity (in this case the steady heat transfer) is used for the first time as an expansion parameter. The solution method was applied to a converging–diverging nozzle with constant heat transfer for both subcritical and supercritical flow cases, showing good agreement with numerical predictions. It was observed that the acoustic and entropy transfer functions of the nozzle strongly depend on the frequency and heat transfer.

Can morphodynamic problems be solved using a first-principles approach in multiphase fluid mechanics? This is the holy grail for many sediment transport researchers but has yet to be achieved in practice. Using a fully resolved direct numerical simulation for turbulent flow over a bed of spheres, the study of Scherer et al. (J. Fluid Mech., vol. 930, 2022, A11) investigates the onset of morphodynamics from a statistically flat bed. The study shows that the formation of streamwise-aligned sediment ridges is due to large-scale turbulent streaks in the logarithmic layer, which drives local sediment sweeps and bursts. The study provides a solid physical justification for introducing initial perturbations in other reduced-complexity models and opens up new perspectives for simulating sediment transport and morphodynamic problems using high-fidelity models.

Theoretical predictions and numerical simulations are used to determine the transition to bubble and conical vortex breakdown in low-Mach-number laminar axisymmetric variable-density swirling jets. A critical value of the swirl number $S$ for the onset of the bubble ($S^*_B$) and the cone ($S^*_C$) is determined as the jet-to-ambient density ratio $\varLambda$ is varied, with the temperature dependence of the gas density and viscosity appropriate to that of air. The criterion of failure of the slender quasi-cylindrical approximation predicts $S^*_B$ that decreases with increasing values of $\varLambda$ for a jet in solid-body rotation emerging sharply into a quiescent atmosphere. In addition, a new criterion for the onset of conical breakdown is derived from divergence of the initial value of the radial spreading rate of the jet occurring at $S^*_C$, found to be independent of $\varLambda$, in an asymptotic analysis for small distances from the inlet plane. To maintain stable flow in the unsteady numerical simulations, an effective Reynolds number $Re_{eff}$, defined employing the geometric mean of the viscosity in the jet and ambient atmosphere, is fixed at $Re_{eff}=200$ for all $\varLambda$. Similar to the theoretical predictions, numerical calculations of $S^*_B$ decrease monotonically as $\varLambda$ is increased. The critical swirl numbers for the cone, $S^*_C$, are found to depend strongly on viscous effects; for $\varLambda =1/5$, the low jet Reynolds number (51) at $Re_{eff}=200$ delays the transition to the cone, while for $\varLambda =5$ at $Re_{eff}=200$, the large increase in kinematic viscosity in the external fluid produces a similar trend, significantly increasing $S^*_C$.

The three-dimensional near wake of a square-back bluff body in ground proximity is experimentally perturbed by placing a pair of D-shaped obstacles under the body. Five obstacle widths $d$, from 12 % to 26 % of the height of the body, are used to perform a sensitivity study of the body's pressure drag by varying the relative distance $l$ between the obstacle pair and the base. Two successive drag-sensitive regimes are identified for obstacle-to-base distances $l/d < 2.5$, where the pressure drag of the body is increased up to 22 %. In different regimes, the flow dynamics measured downstream of the obstacles are found to be very different. When the obstacles are the closest to the base, $l/d<1.5$, the pressure drag changes of the main body are driven by mean merging between the wakes of the obstacles and of the main body, and scale with $d$. Contrarily, when the obstacles are located farther from the base, $1.5< l/d<2.5$, the wakes of the obstacles are isolated from the main body wake. Here the dynamics of the obstacle wake drive the pressure drag changes of the main body, which scale with $d^{2}$. In both regimes, we measure a mean mass transfer from the wake of the main body to the wakes of the obstacles. This is the main mechanism responsible for the pressure drag changes. Using our results and reference studies describing the effects of base suction on the pressure drag of bluff bodies, a physical model is proposed to explain the contrasting scalings of the pressure drag increase in the different regimes observed in this study.

To generate aerodynamic forces required for flight, two-winged insects (Diptera) move their wings back and forth at high wing-beat frequencies. This results in exceptionally high wing-stroke accelerations, and consequently relatively high acceleration-dependent fluid forces. Quasi-steady fluid force models have reasonable success in relating the generated aerodynamic forces to the instantaneous wing motion kinematics. However, existing approaches model the stroke-rate and stroke-acceleration effects independently from each other, which might be too simplified for capturing the complex unsteady aerodynamics of accelerating wings. Here, we use computational-fluid-dynamics simulations to systematically explore how aerodynamic forces and flow dynamics depend on wing-stroke rate, wing-stroke acceleration and wing-planform geometry. Based on this, we developed and calibrated a novel unsteady aerodynamic force model for insect wings with stroke accelerations. This includes improved versions of the translational-force model and the added-mass force model, and we identify a third novel component generated by the interaction of the two. This term reflects the delay in bound-circulation build-up as the wing accelerates. The physical interpretation of this effect is analogous to the Wagner effect experienced by a wing starting from rest. Here, we show that this effect can be modelled in the context of flapping wings as a stroke-acceleration-dependent correction on the translational-force model. Our revised added-mass model includes a viscous force component, which is relatively small but not negligible. We subsequently applied our new model to realistic wing-beat kinematics of hovering Dipteran insects, in a quasi-steady approach. This revealed that stroke-acceleration-related aerodynamic forces contribute substantially to lift and drag production, particularly for high-frequency flapping mosquito wings.

The turbulent flow in the wake of a discontinuous cylinder (DC) was investigated. The DC geometry consisted of cylinder segments $5D$ long (with $D$ being the diameter of the cylinder) separated by gaps of width $2.5D$. Particle image velocimetry and hot-wire anemometry were used to analyse the flow at two Reynolds numbers, $Re = 4000$ and $10\,000$, for $x/D \le 180$. Large eddy simulations for both the DC and the infinite continuous cylinder (CC) wakes were also carried out at $Re = 10\,000$. The DC configuration was devised to trigger the shedding of horseshoe vortices (HSV) in the very-near-wake region with the intent of illustrating the role that these three-dimensional HSVs, previously identified in the far-wake region of CC, play in the entrainment process in turbulent wakes. The DC geometry produced HSVs by the interaction between the high momentum flow through the gaps and the main spanwise vortex shed behind each cylinder segment, while in the CC wake they evolve from near-wake instabilities straddled with hairpin vortices that detach spanwise vorticity from the shed Kármán vortices. The DC wake was found to grow and spread in the transverse direction with a much faster rate than for the CC wake, up until approximately $x/D \approx 50$. Prior to this location, the enhanced growth rate caused by the shear-aligned HSV led to a wake width of approximately 3 times that of the CC wake, with a maximum mean velocity deficit that was approximately half.

The migration of polydisperse particles and the formation of self-organized particle chains in a square channel flow of non-Newtonian fluids is studied. The effects of rheological behaviour of the fluid, solution concentration and flow rate are explored experimentally. The direct forcing/fictitious domain method is adopted to qualitatively verify the experiments and further analyse the mechanisms of particle migration and particle chain self-organization. The results show that only particles in viscoelastic fluids with negligible shear-thinning effect will remain at the channel centreline as the flow rate increases. The monodisperse particles reach the same velocity when migrating to the equilibrium position. However, in polydisperse suspensions, the smaller the particle diameter, the greater the velocity when the particle migrates to the equilibrium position. In a viscoelastic fluid, the polydisperse particles are more likely to self-organize into long particle chains along the channel centreline than the monodisperse particles, where the large and small particles are at the front and end of the chain. The dimensionless alignment factor (Af) is adopted to quantify the formation of particle chains, which is the largest in viscoelastic fluids and rapidly increases before decreasing to a stable value as the flow rate increases. For larger particle diameter ratios and stronger shear-thinning effect, the long particle chain self-organization is less obvious. The self-organizing particle chains at the channel centreline are strongly influenced by the fluid elastic properties and weakly by the inertial effect; however, the shear-thinning effect disperses the particles and prevents the formation of long straight particle chains.

The instability characteristic of a viscoelastic jet in a co-flowing gas stream is studied comprehensively. The important role of the non-uniform basic velocity in the instability analysis of viscoelastic jets is clarified, which first induces an unrelaxed elastic tension, and then produces a coupling term between the elastic tension and perturbation velocity. The elastic tension promotes the instability of the jet, while the coupling term exhibits a stabilizing effect, which is essentially related to the nonlinear constitutive relation of viscoelastic fluids and the non-uniform basic velocity. The competition between these two factors leads to the non-monotonic effect of fluid elasticity on the disturbance growth rate, which can be divided into two different regimes characterized by the Weissenberg number with values smaller or larger than unity. In different regimes, the structure of the eigenspectrum is also significantly different. Furthermore, three instability mechanisms are identified using the energy budget analysis, corresponding to the predominance of the surface tension, elastic tension and shear and pressure of the external gas, respectively. By analysing the variations of the growth rate and phase speed of the disturbances, the general features of viscoelastic jet instability are obtained. Finally, the transitions of instability modes in parameter spaces are investigated theoretically and the transition boundaries among them are provided. This study provides guidance for understanding the underlying mechanism of instability of a viscoelastic jet surrounded by a co-flowing gas stream and the transition criterion of different instability modes.

This study investigates the stability features of spatially spreading heated jets in the viscous regime with real gas effects, using both unsteady two-dimensional axisymmetric simulations and linear analyses of the steady state and time-averaged states. At a Reynolds number of $400$, the heated jets are found to undergo a subcritical Hopf bifurcation, marking the start of self-sustained oscillations when decreasing the temperature ratio $S=T_{\infty }/T_{{c}}$, which highly depends on the thermodynamic and transport property assumptions imposed in the simulations. Once the flow enters a limit cycle past the Hopf bifurcation, the linear analyses over the steady base state are unable to capture the oscillation's frequency. Nevertheless, this study confirms that including real gas effects in the stability equations has a strong effect on the growth rate of the global mode once the centreline temperature of the jet reaches that of the dissociation reaction onset, which is $T=2800$ K for air at $p_s=100$ mbar. The linear global analyses over the time-averaged states lead to a satisfying prediction of the oscillation's frequency for the cases studied, and the baroclinic torque obtained from the resulting global mode matches well with that of the simulations.

We investigate the effect of inertial particles on the stability and decay of a prototypical vortex tube, represented by a two-dimensional Lamb–Oseen vortex. In the absence of particles, the strong stability of this flow makes it resilient to perturbations, whereby vorticity and enstrophy decay at a slow rate controlled by viscosity. Using Eulerian–Lagrangian simulations, we show that the dispersion of semidilute inertial particles accelerates the decay of the vortex tube by orders of magnitude. In this work, mass loading is unity, ensuring that the fluid and particle phases are tightly coupled. Particle inertia and vortex strength are varied to yield Stokes numbers 0.1–0.4 and circulation Reynolds numbers 800–5000. Preferential concentration causes these inertial particles to be ejected from the vortex core forming a ring-shaped cluster and a void fraction bubble that expand outwards. The outward migration of the particles causes a flattening of the vorticity distribution, which enhances the decay of the vortex. The latter is further accelerated by small-scale clustering that causes enstrophy to grow, in contrast with the monotonic decay of enstrophy in single-phase two-dimensional vortices. These dynamics unfold on a time scale that is set by preferential concentration and is two orders of magnitude lower than the viscous time scale. Increasing particle inertia causes a faster decay of the vortex. This work shows that the injection of inertial particles could provide an effective strategy for the control and suppression of resilient vortex tubes.

Self-propulsive performances of the flexible plates undergoing pitching and heaving motions are investigated numerically. The effects of multiple key dimensionless parameters are considered, such as bending stiffness, heaving amplitude, pitching amplitude and flapping frequency. Despite so many influence factors, results indicate that the cruising speed $U$ (or the cruising Reynolds number $Re_c$), the thrust $T$ and the input power $P$ can be summarized as some simple scaling laws vs the flapping Reynolds number $Re_f$. In the heaving motion, the scaling laws may be not fully independent of bending stiffness because in the motion the role of bending stiffness is more complicated for the thrust generation. Our scaling laws are well supported by biological data on swimming aquatic animals.

This paper explores the screech closure mechanism for different axisymmetric modes in shock-containing jets. While many of the discontinuities in tonal frequency exhibited by screeching jets can be associated with a change in the azimuthal mode, there has to date been no satisfactory explanation for the existence of multiple axisymmetric modes at different frequencies. This paper provides just such an explanation. As shown in previous works, specific wavenumbers arise from the interaction of waves in the flow with the shocks. This provides new paths for driving upstream-travelling waves that can potentially close the resonance loop. Predictions using locally parallel and spatially periodic linear stability analyses and the wavenumber spectrum of the shock-cell structure suggest that the A1 mode resonance is closed by a wave generated when the Kelvin–Helmholtz mode interacts with the leading wavenumber of the shock-cell structure. The A2 mode is closed by a wave that arises owing to the interaction between the Kelvin–Helmholtz wave and a secondary wavenumber peak, which arises from the spatial variation of the shock-cell wavelength. The predictions are shown to closely match experimental data, and possible justifications for the dominance of each mode are provided based on the growth rates of the absolute instability.

Numerical simulation based on the discrete element method (DEM) is used to investigate the flow field generated when a cylindrical obstacle is placed in a supersonic granular stream. Robust validation of the simulation model is performed by comparing numerical results with experiments. Experiments are performed using a two-dimensional set-up generating rapid granular flow owing to gravity. DEM simulations demonstrate that a rapid gas-like stream of grains suddenly decelerates across the shock wave and finally collapses into a slow-moving heap at the cylinder. The volume fraction suddenly increases across the shock layer and remains constant thereafter. The flow physics of the shock wave and the granular heap is elucidated through fundamental fluid dynamic quantities such as the velocity, volume fraction, pressure and granular temperature. It is shown that the interaction of grains with a cylindrical obstacle results in the generation of pressure, which is responsible for sustaining static granular heaps on the cylinder. The total pressure is resolved into collisional and streaming components. A streaming pressure is generated owing to velocity fluctuations, and is found to be significant only in the shock wave region. The observations show that the rheological complexity offered by granular shock waves is a direct manifestation of the dissipative and frictional nature of granular collisions. The new insight into the granular heaps could be relevant to a variety of applications involving granular-fluid–solid interactions.

The Leidenfrost effect, where a drop levitates on a vapour film above a hot solid, is simulated using an efficient computational model that captures the internal flow within the droplet, models the vapour flow in a lubrication framework and is capable of resolving the dynamics of the process. The initial focus is on quasi-static droplets and the associated geometry of the vapour film formed beneath the drop, where we are able to compare with experimental analyses and assess the range of validity of the theoretical model developed in Sobac et al. (Phys. Rev. E, vol. 103, 2021, 039901). The computational model also allows us to explore parameter space, varying both the drop size and viscosity of the liquid, with computational results in excellent agreement with the theoretical model for high-viscosity liquids. Interestingly, for large water drops, discrepancies between the computational model and experiments occur, and possible reasons for this observation are provided. Our predictions reveal features including a regime with a dimpleless bottom surface of the drop and a minimum in the vapour layer thickness as a function of the drop size. Finally, the capability to simulate dynamics is revealed by computations that predict and track the vapour ‘chimney’ instability for large drops.

Multipolar spherical solutions to the three-dimensional steady vorticity equation are provided. These solutions are based on the separation of radial and angular contributions in terms of the spherical Bessel functions and vector spherical harmonics, respectively. In this set of multipolar vortex solutions, the Hicks–Moffatt swirling vortex is categorized as a vortex of degree ${\ell }=1$ and therefore as a vortex dipole. This swirling vortex is the three-dimensional dipole in spherical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry. The three-dimensional dipole solution admits two linearly superposable solutions. The first one is a Trkalian flow and the second one is a cylindrical solid-body rotation with swirl. The higher ${\ell }>1$ multipolar vortices found are either vanishing-helicity vortices or Trkalian flow vortices. The multipolar Trkalian flows admit two circular polarizations given by the sign of the wavenumber $k$. It is also found that piecewise vortex solutions, consisting of interior rotational and exterior potential flow domains, satisfying velocity continuity conditions at the vortex boundary, are possible in the general multipolar Trkalian spherical vortex. A particular polarized dipole solution in three-dimensional cylindrical geometry, consisting as well in the superposition of a Trkalian flow and a rigid motion, is also analysed. This swirling vortex may be interpreted as the three-dimensional dipole in cylindrical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry.

The pseudo-incompressible approximation, which assumes small pressure perturbations from a one-dimensional reference state, has long been used to model large-scale dynamics in stellar and planetary atmospheres. However, existing implementations do not conserve energy when the reference state is time-dependent. We use a variational formulation to derive an energy-conserving pseudo-incompressible model in which the reference state evolves while remaining hydrostatic. We present an algorithm for solving these equations in the case of closed boundaries, for which the pseudo-incompressible velocity constraint is degenerate. We implement the model within the low-Mach-number code MAESTROeX, and validate it against a fully compressible model in several test cases, finding that our hybrid pseudo-incompressible–hydrostatic model generally shows better agreement with the compressible results than the existing MAESTROeX implementation.

Secondary instabilities growing over a time-evolving variable-density round jet subject to the primary Kelvin–Helmholtz (KH) instability at Atwood number $\vert \textit {At} \vert = 0.25$ are investigated with a non-modal linear stability analysis. Despite local modifications of the base flow vorticity induced by the baroclinic torque, these disturbances experience a short-term universal growth due to a combination of the Orr and lift-up mechanisms, whatever the azimuthal wavenumber $m$. At $\textit {Re}=1000$, the secondary energy growth stems from the development of elliptical and hyperbolic instabilities, with an E-type-to-H-type transition as $m$ and $\textit {Re}$ increase, as in the homogeneous case (Nastro et al., J. Fluid Mech., vol. 900, 2020, A13). In the light jet at $\textit {Re} = 1000$, after the KH mode saturation, the high-$m$ H-type instability is replaced by a perturbation organised as counter-rotating streamwise vortices located in the base flow region of promoted strain rate. Increasing the Reynolds number up to $\textit {Re} = 10\,000$ yields larger energy growths and a strong anisotropy among energy and enstrophy components with a preferential increase of axial velocity and azimuthal vorticity. Both come from the linearised baroclinic source that drives the optimal response towards folded sheets of axial velocity that differ from those observed in the variable-density plane shear layers. When the perturbation is injected around the KH saturation time for $\textit {Re}=10\,000$, the response to optimal perturbation takes the form of fast growing secondary KH instabilities whatever $m$. We find these three-dimensional secondary KH instabilities to be good candidates for the transition to turbulence in variable-density jet flows.

The generation of acoustic tones in four round jets at a Mach number of 0.9 impinging on a plate at a distance $L=6r_0$ from the nozzle exit, where $r_0$ is the nozzle radius, has been investigated by large-eddy simulation. Three plates are perforated by holes of diameters $h=2r_0$, $3r_0$ and $4.4r_0$, centred on the jet axis, whereas the fourth plate has no hole, in order to study the effects of the hole on the jet flow and acoustic fields. In all cases, acoustic tones emerge in the jet near field, upstream but also downstream of the plate for the perforated plates. Their frequencies are similar for all jets and close to those predicted for aeroacoustic feedback loops establishing between the nozzle and the plate, involving flow disturbances convected downstream and waves propagating upstream at the ambient speed of sound. Their levels, however, decrease with the hole diameter, by a few dB for $h\leqslant 3r_0$ but approximately by 40 dB for $h=4.4r_0$. The features of the feedback loops are identified by computing two-dimensional space–time correlations and frequency–wavenumber spectra of the pressure fluctuations in the jet mixing layers. These loops are found to be closed by free-stream upstream-propagating guided jet waves, produced by the impingement of vortical structures on the plate for $h\leqslant 3r_0$ and by the scattering of the jet aerodynamic pressure fluctuations by the hole edges for $h=4.4r_0$. Finally, an acoustic analogy is used to show that the contributions of the pressure fluctuations on the plate to the radiated noise are dominant.