Sound wave propagation in rarefied flows of molecular gases confined in micro-channels is investigated numerically. We first validate the employed kinetic model against the experimental results and then systematically study the gas damping and surface force on the transducer as well as the resonance/anti-resonance in confined space. To quantify the impact of the finite relaxation rates of the translational and internal energies on wave propagation, we examine the roles of bulk viscosity and thermal conductivity in depth over a wide range of rarefactions and oscillation frequencies. It is found that the bulk viscosity only exerts influence on the pressure amplitude and its resonance frequency in the slip regime in high oscillations. In addition, the internal degree of freedom is frozen when the bulk viscosity of a molecular gas is large, resulting in the pressure amplitude of sound waves in the molecular gas being the same as in a monatomic gas. Meanwhile, the thermal conductivity has a limited influence on the pressure amplitude in all the simulated flows. In the case of the thermoacoustic wave, we prove that the Onsager–Casimir reciprocal relation also holds for molecular gases, i.e. the pressure deviation induced by the temperature variation is equal to the heat flux induced by the plate oscillation. Our findings enable an enhanced understanding of sound wave propagation in molecular gases, which may facilitate the design of nano-/micro-scale devices.

Linear models, based on stochastically forced linearized equations, are deployed for spectral linear stochastic estimation (SLSE) of the velocity and temperature fluctuations in compressible turbulent channel flows with a bulk Mach number of 1.5. Through comparison with the direct numerical simulation (DNS) data, an eddy-viscosity-enhanced model (eLNS) outperforms the one not enhanced (LNS) in computing the coherence and amplitude ratio of streamwise velocity at different wall-normal heights, but they both largely deviate from DNS regarding the temperature prediction. For further investigation, the eigenspectra and pseudospectra of the linear operators are scrutinized. The eddy viscosity is shown to stabilize the eigenmodes and decrease the non-normality of the vortical modes. Consequently, the relative importance of acoustic and entropy modes increases, and they can contribute 20 % to 55 % of the response growth, which is not supported by DNS. Hence, it is an intrinsic defect of the eLNS model introduced by turbulence modelling. After a procedure of cospectrum decomposition, the contributions of acoustic and entropy components are filtered out. The resulting SLSE quantities for velocity, temperature and their coupling are basically agreeable with DNS, demonstrating that the coherent temperature fluctuation is dominated by advection and other vortical motions, instead of the compressibility effects. Moreover, a parameter study of Reynolds and Mach numbers (from 0.3 to 4) is conducted. The semi-local units are shown to well collapse the velocity SLSE quantities to the incompressible case for streamwise-elongated structures of high coherence.

The present work aims at clarifying the effects of a solid boundary on the salt fingers in the wall-bounded double diffusive convection turbulence driven by the salinity and temperature differences between the top and bottom plates. The fluid properties are the same as the seawater, and two-dimensional direct numerical simulations are conducted over a wide range of the thermal and salinity Rayleigh numbers which measure the strength of driving salinity difference and stabilising temperature difference. We find that the bulk density ratio $\varLambda _b$, defined by the mean temperature and salinity gradients at the bulk, controls the flow morphology. As $\varLambda _b$ exceeds unity, the bulk flow shifts from wide convection rolls to slender salt fingers. Two different regimes are further identified for the cases of salt-finger type. One is the confined salt-finger regime where the characteristic height of salt fingers is comparable to the bulk height and the influences of the solid boundary are noticeable. The other is the free salt-finger regime where the salt fingers are much shorter than the bulk height. In this latter regime, the transport properties versus $\varLambda _b$ are in quantitative agreement with those obtained in the fully periodic domain (e.g. Traxler et al., J. Fluid Mech., vol. 677, 2011, pp. 530–553). For a limited range of density ratio at the highest salinity Rayleigh number considered here, multiple states can be obtained from different initial conditions. The large-scale secondary instability and spontaneous formation of staircase from finger layers are not observed in the current study.

The collision of binary droplets plays a key role in several industrial, chemical and biological processes. In these processes, the quality of the desired outcome is strongly dependent on the mixing of the liquid droplets as they collide in mid-air. In this work, multiphase direct numerical simulations based on the volume-of-fluid method have been used to investigate the process of mixing and analyse the effects of parameters such as injection velocity, timing and collision angles. The evolution of mixing due to convection and irreversible diffusive processes has been quantified by means of the segregation parameter. To synthesise the outcome of a collision, the impact parameter has been redefined to account for the collision of non-spherical droplets. It has been found that the optimal mixing does not occur for symmetric head-on collisions, but rather at moderately asymmetrical configurations. This behaviour has been explained by analysing the velocity gradient tensor. It has been demonstrated that by breaking the symmetry, the local topology of the flow is altered and the resulting convective flows increase the contact area between the liquids, thereby augmenting the mixing process. However, it was also observed that lateral misalignment transforms the initial kinetic energy into the spinning of the merged droplets, thus preventing an enhanced mixing.

The linear instability of viscoelastic film with insoluble surfactants on an oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of viscoelastic and insoluble surfactants on the instability are described using Floquet theory. For long-wavelength instability, the solution in the limit of long wave perturbations is obtained by the asymptotic expansion method. The results show that the presence of viscoelastic film shifts the stability boundaries to the low-frequency region in the absence of gravity when the imposed frequency is less than 6. The U-shaped neutral curves with separation bandwidth appear in the presence of gravity. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. Different from the previous results, a new branch point with special structure of a neutral curve is detected for clean-surface film. Results show that the presence of the surfactants will decrease the unstable frequency bandwidth and increase the critical Reynolds number. Both the travelling-wave mode and standing-wave mode are found due to the existence of surface surfactants. For high-frequency oscillation, the viscoelastic parameter may significantly destabilize the flow and the instability is determined by the finite-wavelength mode over a relatively large frequency range.

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

This paper presents a vortex breakdown study due to the interaction between a Batchelor vortex and the crossing of oblique shock waves with Mach numbers of 3.5 and 5.0, favourable for supersonic mixing and combustion, respectively. Numerical simulations were conducted to investigate the effects of the circulation intensity and shock angle on vortex breakdown. The results indicate that a breakdown occurs at the shock angle $\beta \geqslant 45^\circ$ or the vortex circulation $q = 0.32$, and the configuration is a bubble structure with a recirculation region; most of the breakdowns possess a stagnation point. Furthermore, the structure differs from that of a normal shock wave and vortex interaction because the bubble region is subsonic and does not comprise a normal shock wave on the inside. Additionally, this vortex breakdown shows that the momentum flux on the centreline decreases once at the tip of the bubble owing to a sudden drop in velocity in the subsonic region. In addition, the enstrophy production resulting from vortex stretching and tilting is found to have a significant advantage in the interaction region. Based on these results, the threshold required for a bubble vortex breakdown was theoretically derived as an inequality. The numerical simulation results support the theoretical criterion obtained from the proposed inequality. Therefore, a streamwise vortex breakdown resulting from the interaction between the vortex and intersecting oblique-shocks should be reasonably predicted.

Because the Navier–Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation. On this manifold, the state of the system can be exactly described in a coordinate system parameterising the manifold. Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis. We show, for turbulent Couette flow, that this description of the dynamics is possible using a data-driven manifold dynamics modelling method. This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at $Re=400$, where a fully resolved solutions requires ${O}(10^5)$ degrees of freedom. Using only data from this simulation we build models with fewer than $20$ degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. At short times, the models track the true trajectory for multiple Lyapunov times and, at long times, the models capture the Reynolds stress and the energy balance. For comparison, we show that the models outperform POD-Galerkin models with $\sim$2000 degrees of freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely resemble previously computed orbits for the full system; in addition, we find nine orbits that correspond to previously unknown solutions in the full system.

The three-dimensional (3-D) transition of the leading-edge vortex (LEV) and the force characteristics of the plunging airfoil are investigated in the chord-based Strouhal number $St_c$ range of 0.10 to 1.0 by means of experimental measurements, numerical simulations and linear stability analysis in order to understand the spanwise instabilities and the effects on the force. We find that the interaction pattern of the LEV, the LEV from a previous cycle (pLEV) and the trailing-edge vortex (TEV) is the primary mechanism that affects the 3-D transition and associated force characteristics. For $St_c \leq 0.16$, the 3-D transition is dominated by the LEV–TEV interaction. For $0.16 < St_c \leq 0.44$, the TEV lies in the middle of the LEV and the pLEV and therefore vortex interaction between them is relatively weak; as a result, the LEV remains two-dimensional up to a relatively high Reynolds number of $Re = 4000$ at $St_c = 0.32$. For $0.44 < St_{c} \leq 0.54$, and at relatively low Reynolds numbers, the pLEV and the TEV tend to form a clockwise vortex pair, which is beneficial for the high lift and stability of the LEV. For $0.49 \leq St_c$, the pLEV and TEV tend to form an anticlockwise vortex pair, which is detrimental to the lift and flow stability. In the last $St_c$ range, vortex interaction involving the LEV, the TEV and the pLEV results in an unstable period-doubling mode which has a wavelength of about two chord-lengths and the 3-D transition enhances the lift.

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$, where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

Characterizing the haemodynamics in intracranial aneurysms is of high interest as it impacts aneurysm growth, rupture and treatment, especially with flow-diverting stents (FDS). Flow in these geometries is known to depend on the Dean, Reynolds and Womersley numbers, $De$, $Re$, $Wo$, but is also influenced by geometrical parameters such as the sac shape or the size of the opening. Via particle image velocimetry, this parametric study aimed at evaluating the combined effects of $Re$, $De$, $Wo$ and the geometry of the aneurysmal sac on the haemodynamics before and after treatment with FDS. Eight ellipsoidal idealized aneurysm models were created with two curvatures of the parent vessel, two aspect ratios of the sac and two neck sizes. Before treatment, a single counter-rotating vortex, whose strength increases with $Re$ and $De$, as well as with the neck size and the aspect ratio, was observed in the sac for all but one geometry. After treatment with FDS, four different flow topologies were observed, depending on the geometry: no separation, separation for part of the cycle, two opposing vortices or a single counter-rotating vortex. A linear model with interaction revealed the predominant effect of $De$ and the curvature of the parent vessel on the haemodynamics before and after treatment. This work once more demonstrated the primary role of haemodynamics in the treatment of intracranial aneurysms with FDS. Future work will consider the complexity of patient-specific geometries, and their effects on both the haemodynamics in the sac and the porosity of the FDS.

This study investigates the linear instability of a thin-film coating inside a rigid tube. The flow is assumed to be inertialess and driven by an axial body force (e.g. gravity), an interfacial shearing force, or their combinations. The interface and the bulk of the film are laden with soluble surfactant. The properties of the soluble surfactant, i.e. solubility, sorption kinetics and bulk diffusivity, modulate the interfacial dynamics of the film. The influence of these properties on the linear instability of the film is comprehensively investigated via long-wave approximation analysis and numerical calculation. Two modes, namely the interface mode and the surfactant mode, are identified to dominate the instability. For a quiescent film, it is found that solubility, sorption kinetics and bulk diffusivity act to improve the uniformity of the surface surfactant and mitigate the stabilizing effect of the Marangoni force. For the film driven by the axial body/interfacial shearing force, the results reveal that solubility plays contrasting roles in the interface mode and the surfactant mode. A window with intermediate solubility is detected where the film can be linearly stabilized. Moreover, sorption kinetics is found to destabilize the perturbations with long wavelength whereas it stabilizes the perturbations with finite wavelength. The bulk diffusivity of the surfactant has a non-monotonic influence on the flow instability, and the film can be relatively stable at both strong and weak diffusivity.

This study theoretically establishes a flow rule for a granular flow down a rough inclined plane, capable of determining granular rheology in the presence of strong non-local effects resulting from grain cooperativity. To describe the non-local rheology, a Landau–Ginzburg model is formulated in terms of the fluidisation parameter represented by the granular inertial number. The exact solutions of the inertial-number field are solved and provide physical insights into the evolution of the internal rheology and the flow arrest process controlled by the flow height. Through asymptotic analysis in the regime dominated by strong non-locality, the exact solutions are further reduced to yield an analytical flow rule for the mean flow velocity. A comparison between the prediction of the flow rule and experimental data from the literature for sand grains determines the underlying rheology law and the relevant rheological parameters. Thus, the proposed flow rule serves as an effective tool for inferring granular rheology from strongly non-local inclined flow data, surpassing the limitations of the classical flow rule deduced from the local rheology framework.