In detonation engines and accidental explosions, a detonation may propagate in an inhomogeneous mixture with non-uniform reactant concentration. In this study, one- and two-dimensional simulations are conducted for detonation propagation in hydrogen/oxygen/nitrogen mixtures with periodic sinusoidal or square wave distribution of the reactant concentration. The objective is to assess the properties of detonation propagation in such inhomogeneous mixtures. Specifically, detonation quenching and reinitiation, cellular structure, cell size and detonation speed deficit are investigated. It is found that there exists a critical amplitude of the periodic mixture composition distribution, above which the detonation quenches. When the amplitude is below the critical value, detonation quenching and reinitiation occur alternately. A double cellular structure consisting of substructures and a large-scale structure is found for a two-dimensional detonation propagating in inhomogeneous mixtures with a periodic reactant concentration gradient. The detonation reinitiation process and the formation of the double cellular structure are interpreted. To quantify the properties of detonation propagation in different inhomogeneous mixtures, the large cell size, critical amplitude, transition distance and detonation speed deficit are compared for hydrogen/air without and with nitrogen dilution and for periodic sine wave and square wave distributions of the reactant concentration. The large-scale cell size is found to be linearly proportional to the wavelength, and both the critical amplitude and the transition distance decrease with the wavelength. The small detonation speed deficit is shown to be due to the incomplete combustion of the reactant. This work provides helpful understanding of the features of detonation propagation in inhomogeneous mixtures.

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

A mesoscopic lattice Boltzmann model is implemented for modelling isothermal two-component evaporation in porous media. The model is based on the pseudopotential multiphase model with two components to mimic the phase-change component (e.g. water and its vapour) and the non-condensible component (e.g. dry air), and the cascaded collision operator is used to enhance the numerical performance. The model is first analysed based on Chapman–Enskog expansion and then validated by the theoretical solution of an isothermal diffusive evaporation problem. We then discuss in detail the implementation of wettability based on a geometric function scheme and further validate the model with microfluidic evaporation experiments. We apply the method to simulate the convective evaporation of a dual-porosity medium and investigate the effects of inflow vapour concentration (${Y_{vapour,in}}$) and contact angle ($\theta$) on the evaporation. Simulation results reproduce the typical transition from the constant evaporation regime (CRP) at large liquid saturation (S) to the receding front period (RFP) at small S, with an intermediate falling rate period in between. The dependence of the average evaporation rates on ${Y_{vapour,in}}$ and $\theta$ during CRP and RFP is investigated. A universal scaling formulation for the evaporation rate during CRP is found with respect to the concentration-related mass transfer number $B_Y$, contact angle $\theta$ and inflow Reynolds number Re, i.e. $E{R_{CRP}} = {k_3}\ln \left ( {1 + {B_Y}} \right ) {\cdot } \left [ {\ln \left ( {1 + {Re}} \right ) + {k_2}} \right ]\left [ {\cos (\theta ) + {k_1}} \right ]$, where ${k_1}$, ${k_2}$ and ${k_3}$ are fitting parameters.

The motion of a finite-size particle in the cuboidal lid-driven cavity flow is investigated experimentally for Reynolds numbers $100$ and $200$ for which the flow is steady. These steady three-dimensional flows exhibit chaotic and regular streamlines, where the latter are confined to Kolmogorov–Arnold–Moser (KAM) tori. The interaction between the moving wall and the particle creates global particle attractors. For neutrally buoyant particles, these attractors are periodic or quasi-periodic, strongly attracting and located in or near KAM tori of the flow. As the density mismatch between particle and fluid increases, buoyancy and inertia become important, and the attractors evolve from those for neutrally buoyant particles, changing their shape, position and attraction rates.

In this paper, we analyse the instability of a thin-film suspension of micro-swimmers subjected to an attractant gradient. The imposed attractant gradient introduces a preferential swimming direction for the swimmers, resulting in an anisotropic orientation field. By modelling the swimmers as force dipoles, the study by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, 038101, J. Fluid Mech., vol. 741, 2014, pp. 619–657) found an instability arising from the coupling between the active stress associated with an anisotropic orientation field and perturbations to the swimmer density field. In this work we begin by presenting a stability analysis that calculates the modification of this instability in the presence of an interface. We then detail the presence of a new mode of instability that arises solely from the deformation of the interface mediated by the active stress in the suspension. The resulting hydrodynamic instabilities in the system are observed to be sensitive to the direction of the attractant gradient relative to the interface. Furthermore, we show that the coupling between the two modes involving a jump in interfacial viscous stresses and normal stress differences within the suspension allows for an instability to manifest in a suspension of chemotactic pullers, previously thought to be unconditionally stable. We then use a long-wave theory to write down a set of nonlinear equations governing the film evolution. The numerical solution of the system using the long-wave theory aids in explaining the mechanism associated with the instability in addition to validating the predictions from the linear stability theory.

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re,\theta )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $\theta =\arctan (Re_M/Re_c)$, where $Re_c,Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at $Re=4000,6000$, wall-resolved large-eddy simulations at $Re = 10\,000, 20\,000$, and some wall-modelled large-eddy simulations (WMLES) up to $Re=10^{10}$. Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall $Re_{\tau,b}$ changes sign at $\theta =\theta _c(Re)$. The mean flow in the $(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of $Re_{\tau,b}(Re,\theta )$, $Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and $\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as $\theta$ is increased at fixed $Re$, $Re_{\tau,t}$ passes through a peak at approximately $\theta = 45^{\circ }$, while $Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as $\theta$ passes through $\theta _c$ and then re-transitions to turbulence. As $Re$ increases, $\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with $Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re\to \infty$ limit, giving $\theta _c \to 45^\circ$ as $Re\to \infty$.

The spatial structure and time evolution of tornado-like vortices in a three-dimensional cavity are studied by topological analysis and numerical simulation. The topology theory of the unsteady vortex in the rectangular coordinate system (Zhang, Zhang & Shu, J. Fluid Mech., vol. 639, 2009, pp. 343–372) is generalized to the curvilinear coordinate system. Two functions $\lambda (q_1,t)$ and $q(q_1,t)$ are obtained to determine the topology structure of the sectional streamline pattern in the cross-section perpendicular to the vortex axis and the meridional plane, respectively. The spiral direction of the sectional streamlines in the cross-section perpendicular to the vortex axis depends on the sign of $\lambda (q_1,t)$. The types of critical points in the meridional plane depend on the sign of $q(q_1,t)$. The relation between the critical points of the streamline pattern in the meridional plane and that in the cross-section perpendicular to the vortex axis is set up. The flow in a three-dimensional rectangular cavity is numerically simulated by solving the three-dimensional Navier–Stokes equations using high-order numerical methods. The spatial structures and the time evolutions of the tornado-like vortices in the cavity are analysed with our topology theory. Both the bubble type and spiral type of vortex breakdown are observed. They have a close relationship with the vortex structure in the cross-section perpendicular to the vortex axis. The bubble-type breakdown has a conical core and the core is non-axisymmetric in the sense of topology. A criterion for the bubble type and the spiral type based on the spatial structure characteristic of the two breakdown types is provided.

Numerically computed with high accuracy are periodic travelling waves at the free surface of a two-dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are the following. (i) There is a boundary layer where the angle rises sharply from $0^\circ$ at the crest to a local maximum, which converges to $30.3787\ldots ^\circ$, independently of the vorticity, as the amplitude increases towards that of the extreme wave, which displays a corner at the crest with a $30^\circ$ angle. (ii) There is an outer region where the angle descends to $0^\circ$ at the trough for negative vorticity, while it rises to a maximum, greater than $30^\circ$, and then falls sharply to $0^\circ$ at the trough for large positive vorticity. (iii) There is a transition region where the angle oscillates about $30^\circ$, resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.

Active matter exploits motion to induce changes in shape and conformation via external input. In this paper, we establish theoretically that viscous liquid droplets containing magnetic nanoparticles with frozen-in magnetic moments, sitting on a solid substrate and surrounded by an ambient gas phase, can deform and migrate under the influence of a magnetic torque. The effect arises because the collective rotation of the magnetic nanoparticles at the liquid–gas interface tilts the droplet away from a symmetric configuration, breaks the reflection symmetry with respect to the centre axis, and leads to a left–right asymmetry of the contact angles. A sufficiently strong magnetic torque leads the contact angles to overcome hysteresis effects leading the droplet to migrate. We develop a general framework to explain how symmetry-breaking affects droplet migration. Thus previous results of droplet spreading and migration can be recovered as special cases. Such droplets can be employed as agents in active surfaces and can move against gravity, chemical and thermal gradients, providing a mechanism that could be utilized by both industry and medicine.

When a fluid is injected into a porous medium saturated with an ambient fluid of a greater density, the injected fluid forms a plume that rises upwards due to buoyancy. In the near field of the injection point, the plume adjusts its speed to match the buoyancy velocity of the porous medium, either thinning or thickening to conserve mass. These adjustments are the dominant controls on the near-field plume shape, rather than mixing with the ambient fluid, which occurs over larger vertical distances. In this study, we focus on the plume behaviour in the near field, demonstrating that for moderate injection rates, the plume will reach a steady state, whereby it matches the buoyancy velocity over a few plume width scales from the injection point. However, for very small injection rates, an instability occurs in which the steady plume breaks apart due to the insurmountable density contrast with the surrounding fluid. The steady shape of the plume in the near field depends only on a single dimensionless parameter, which is the ratio between the inlet velocity and the buoyancy velocity. A linear stability analysis is performed, indicating that for small velocity ratios, an infinitesimal perturbation can be constructed that becomes unstable, whilst for moderate velocity ratios, the shape is shown to be stable. Finally, we comment on the application of such flows to the context of CO$_2$ sequestration in porous geological reservoirs.

In this paper, we study the spreading of droplets of density-matched granular suspensions on the surface of a solid. Bidispersity of the particle size distribution enriches the conclusions drawn from monodisperse experiments by highlighting key elements of the wetting dynamics. In all cases, the relation between the dynamic contact angle and the velocity of the contact line is similar to that for a simple fluid, despite the complexity introduced by the presence of particles. We extract from this relation an apparent wetting viscosity of the suspensions that differs from that measured in the bulk. Dimensional analysis supported by experimental measurements yields an estimate of the size of the region inside the droplet where the value of the dynamic contact angle depends on a balance of viscous dissipation and capillary stresses. How particle size compares with this viscous cut-off length seems crucial in determining the value of the apparent wetting viscosity. With bimodal blends, the particle size ratio can be used to show the effects of the local structure and volume fraction at the contact line, both impacting the value of the corresponding wetting viscosity.

A dilute magnetic emulsion under the combined action of a uniform external magnetic field and a small amplitude oscillatory shear is studied using numerical simulations. We consider a three-dimensional domain with a single ferrofluid droplet suspended in a non-magnetizable Newtonian fluid. We present results of droplet shape and orientation, viscoelastic functions and bulk emulsion magnetization as functions of the shear oscillation frequency, magnetic field intensity and orientation. We also investigate how the magnetic field induces mechanical anisotropy by producing internal torques in oscillatory conditions. We found that, when the magnetic field is parallel to the shear plane, the droplet shape is mostly independent of the shear oscillation frequency. Regarding the viscometric functions, we show how the external magnetic field modifies the storage and loss moduli, especially for a field aligned to the main velocity gradient. The bulk emulsion magnetization is studied in the same fashion as the viscoelastic functions of the oscillatory shear. We show that the in-phase component of the magnetization with respect to the shear rate reaches a saturation magnetization, at the high frequencies limit, dependent on the magnetic field intensity and orientation. On the other hand, we found a non-zero out-of-phase response, which indicates a finite emulsion magnetization relaxation time. Our results indicate that the magnetization relaxation is closely related to the mechanical relaxation for dilute magnetic emulsions under oscillatory shear.

Recent findings on the Reynolds-number-dependent behaviour of near-wall turbulence in terms of the ‘foot-printing’ of outer large-scale structures call for a new modelling development. A two-scale framework was proposed to couple a local fine-mesh solution with a global coarse-mesh solution by He (Intl J. Numer. Meth. Fluids, vol. 86, 2018, pp. 655–677). The methodology was implemented and demonstrated by Chen & He (J. Fluid Mech, vol. 933, 2022, p. A47) for a canonical turbulent channel flow, where the mesh-count scaling with Reynolds number is potentially reduced from $O(R{e^2})$ for a conventional wall-resolved large-eddy simulation (WRLES) to $O(R{e^1})$. The present work extends the two-scale method to turbulent boundary layers. A two-dimensional roughness element is used to trip a turbulent boundary layer. It is observed that large-scale disturbances originating at the trip have a much shorter lifetime and weaker foot-printing signatures on near-wall flow compared to those long streaky coherent structures in well-developed wall-bounded turbulent flows. Modal analyses show that the impact of trip-induced large scales can be adequately captured by a locally embedded fine-mesh block. For the tripped turbulent boundary layer, a Chebyshev block-spectral mapping is adopted to propagate source terms from the local fine-mesh blocks to the global coarse-mesh domain, driving to a target solution for the upscaled equations. The computed mean statistics and energy spectra are in good agreement with corresponding experimental data, WRLES and direct numerical simulation (DNS) results. The overall mesh count–$Re$ scaling is estimated to reduce from $O(R{e^{1.8}})$ for the full wall-resolved LES to $O(R{e^{0.9}})$ for the present two-scale solution.

We propose an interpretable deep learning (DL) model that extracts physical features from turbulence data. Based on a conditional generative adversarial network combined with a new decomposition algorithm for the Prandtl number effect, we developed a DL model that is capable of predicting the local surface heat flux very accurately using only the wall-shear stress information and Prandtl number as inputs in channel turbulence. The considered range of Prandtl number is $Pr = 0.001 \sim 7$, with a focus on the subrange of $Pr = 0.1 \sim 7$. Through an investigation of the gradient maps of the trained prediction model, we were able to identify the nonlinear physical relationship between the wall-shear stresses and heat flux, which is quite diverse depending on the Prandtl number. Furthermore, the decomposition algorithm, which is used to separate the Prandtl number dependent field from the common field of the surface heat flux, helps not only in learning for good prediction of an arbitrary Prandtl number but also in analysing the effect of the Prandtl number on the determination of the heat flux for the given turbulent flow fields. We demonstrate that a physical interpretation of a trained network is possible.