The deformation of fluid elements governs many processes in porous media, including the transport, dispersion and mixing of solutes, chemical reactions and colloidal particles. Recently, it has been shown that even in strongly heterogeneous porous media, steady three-dimensional (3-D) Darcy flow with isotropic hydraulic conductivity admits only constrained kinematics. This is due to its inherently helicity-free nature, as the velocity is everywhere orthogonal to the vorticity. This property precludes braided streamlines and instead admits a pair of coherent 3-D streamfunctions, and the streamlines cannot wander freely throughout the flow domain. In this study, we consider the impact of these kinematic constraints upon fluid deformation at the Darcy scale. We show that the helicity-free condition corresponds to an orthogonal 3-D streamline coordinate system, which we use to derive an ab initio continuous time random walk framework for fluid deformation. We find that the helicity-free condition combined with the intermittent nature of shear events leads to fluid deformation that is limited to algebraic growth, with stretching ranging from sublinear to superlinear behaviour. Fluid deformation in 3-D isotropic Darcy flow is remarkably similar to that of two-dimensional (2-D) Darcy flow, and the structure of 3-D Darcy flow is fundamentally the same as two superposed 2-D Darcy flows. These results have implications for understanding flow and transport in heterogeneous porous media, and provide a basis for quantification of mixing and dispersion in isotropic 3-D Darcy flow.

A numerical insight into the radial displacement of two viscously stable reactants undergoing an infinitely fast chemical reaction is obtained. This work broadens the numerical knowledge about the interaction between chemical reaction and hydrodynamic instability. A suitable transformation is utilised to deal with an infinite dimensionless parameter and reduce computational cost. Viscous fingering instability originates when the product has a different viscosity than the reactants. We calculate the onset time $t_{on}$ when the instability begins to appear for different reactants by varying the log mobility ratio $R_c$ and the Péclet number $Pe$. Based on $t_{on}$, we divide the time domain into stable and unstable zones with respect to instability. This helps to characterise reactants so as to have a flow with/without instability. For a fixed $Pe$ and $\vert R_c \vert$, it is found that $t_{on}$ is early for $R_c > 0$ in contrast to the result for rectilinear displacement. This results in a wider stable zone for $R_c < 0$. Interestingly, it is found that the stable zone can be made independent of $Pe$ by using a careful rescaling and we obtain the dependence of the onset time of instability on $R_c$ and $Pe$ as $t_{on} \propto (\vert R_c\vert Pe^{\beta _1}-\beta _2)^{\beta _3}$ where the constant of proportionality and $\beta _i$, $i=1,2,3$, depend upon the sign of $R_c$. In the unstable zone, we find that the length of the fingers depends upon the sign of $R_c$, which is not observed for any radial displacement of reactants undergoing a slow or moderately fast chemical reaction.

The transition to turbulence induced by counter-rotating wall-normal rotating cylindrical roughness pairs immersed within a laminar boundary layer on a flat plate is investigated with direct numerical simulations, dynamic mode decomposition (DMD) and perturbation kinetic energy (PKE) analysis. As long as the cylinder stub is rotating, the wake contains a steady dominating inner vortex (DIV) surrounded by a secondary inner vortex. Its circumferential velocity accelerates the fluid on one side of the cylinder and decelerates it on the other side. With low rotation speed, the perturbation is initiated by a combination of elliptical and centrifugal instabilities in the near wake. At medium rotation speeds, Taylor–Couette-like streamwise vortices are generated on the decelerated side, resulting in a protruding reverse-flow zone. Results from DMD analysis and corresponding PKE analysis reveal the unstable nature of the deceleration region and the wake. At the largest rotation speed investigated, the onset of perturbations is directly located on the decelerated side of the cylinder stubs, where a deceleration mechanism feeds the instability. In the near wake, the mechanism gradually changes to a pure centrifugal instability when the rotation speed increases. In the far wake, both elliptical and centrifugal instabilities fade away, and the streaky flow featuring a vigorous DIV is then only subject to inviscid inflectional instability.

A new flamelet model is developed to be used for subgrid modelling and coupled with the resolved flow description for turbulent combustion. The model differs from current models in several critical ways. (i) Non-premixed flames, premixed flames or multi-branched flame structures are determined rather than prescribed. (ii) The effects of shear strain and vorticity on the flames are determined. (iii) The strain rates and vorticity applied at the subgrid level are directly determined from the resolved-scale strain rates and vorticity without the use of a contrived progress variable. (iv) The flamelet model is three-dimensional without need for assuming axisymmetry or planar geometry. (v) The effect of variable density is addressed in the flamelet model. Solutions to the Navier–Stokes equations and the associated scalar equations governing the flamelet model are obtained without the boundary-layer approximation. By appropriate coordinate transformation, a similar solution is found for the rotational flamelet model, reducing it to a system of ordinary differential equations. Vorticity is shown to create a centrifugal force on the subgrid counterflow that modifies the molecular transport rates and burning rate. Sample computations of the rotational flamelet model without coupling to the resolved flow are presented first to demonstrate the importance of the new features of the model. Scaling laws are presented for relating strain rates and vorticity at the subgrid level to those quantities at the resolved-flow level for coupling with large-eddy simulations or the time-averaged mean-flow level for Reynolds-averaged flows. The time-averaged behaviour of a simple turbulent flow is resolved with coupling to the rotational flamelet model. Specifically, a two-dimensional, multicomponent, time-averaged planar shear layer with variable density and energy release is employed using a mixing-length description for the eddy viscosity. Needs for future study are identified.

Turbulent superstructures in horizontally extended three-dimensional Rayleigh–Bénard convection flows are investigated in controlled laboratory experiments in water at Prandtl number ${Pr}=7$. A Rayleigh–Bénard cell with square cross-section, aspect ratio $\varGamma =l/h=25$, side length $l$ and height $h$ is used. Three different Rayleigh numbers in the range $10^{5} < {Ra} < 10^{6}$ are considered. The cell is accessible optically, such that thermochromic liquid crystals can be seeded as tracer particles to monitor simultaneously temperature and velocity fields in a large section of the horizontal mid-plane for long time periods of up to 6 h, corresponding to approximately $10^{4}$ convective free-fall time units. The joint application of stereoscopic particle image velocimetry and thermometry opens the possibility to assess the local convective heat flux fields in the bulk of the convection cell and thus to analyse the characteristic large-scale transport patterns in the flow. A direct comparison with existing direct numerical simulation data in the same parameter range of $Pr$, ${Ra}$ and $\varGamma$ reveals the same superstructure patterns and global turbulent heat transfer scaling ${Nu}({Ra})$. Slight quantitative differences can be traced back to violations of the isothermal boundary condition at the extended water-cooled glass plate at the top. The characteristic scales of the patterns fall into the same size range, but are systematically larger. It is confirmed experimentally that the superstructure patterns are an important backbone of the heat transfer. The present experiments enable, furthermore, the study of the gradual evolution of the large-scale patterns in time, which is challenging in simulations of large-aspect-ratio turbulent convection.

This paper describes a study of spatially accelerating turbulent flow based on the direct numerical simulation of a flow with longitudinally accelerating moving walls to create a relative acceleration between the fluid and the wall without inducing streamline curvature. The results show a broad similarity to those of previous investigations of spatial acceleration, albeit with some differences. A new interpretation has been proposed considering this spatially accelerating flow to be characterised by the formation of a new boundary layer superimposed on the pre-existing turbulent flow. This is followed by the transition of the flow in response to the development of this new boundary layer. This can be seen as an extension of the transition theory for temporally accelerating turbulent flows (He & Seddighi, J. Fluid Mech., vol. 715, 2013, pp. 60–102). The existing turbulent structures act as disturbances for the new boundary layer similar to the role of free-stream turbulence in bypass transition. This boundary layer modulates the pre-existing near-wall structures, amplifying and elongating the streaks. Some streaks eventually become unstable in a sinuous mechanism reminiscent of streak breakdown in near-wall turbulence, resulting in the formation localised turbulent spots which spread until the entire wall is covered in new turbulence. This interpretation naturally splits the flow into a new boundary layer region and a core (or free-stream) flow with interactions between the two dominating a significant length of the flow development and potentially offers a new explanation for the slow evolution of the turbulent stresses observed previously.

We study the linear stability of a compressible sessile bubble in an ambient fluid that partially wets a planar solid support, where the gas is assumed to be an ideal gas that obeys the adiabatic law. The frequency spectrum is computed from an integrodifferential boundary value problem and depends upon the wetting conditions through the static contact angle $\alpha$, the dimensionless equilibrium bubble pressure $\varPi$, and the contact-line dynamics that we assume to be either (i) pinned or (ii) freely moving with fixed contact angle. Corresponding mode shapes are defined by the polar-azimuthal mode number pair $[k,\ell ]$ with $k+\ell =\mathbb {Z}^{+}_{even}$. We report instabilities to the (i) $[0,0]$ breathing mode associated with volume change, and (ii) $[1,1]$ mode that is linked to horizontal centre-of-mass motion of the bubble. Stability diagrams and instability growth rates are computed, and the respective instability mechanisms are revealed through an energy analysis. The zonal $\ell =0$ modes are associated with volume change, and we show that there is a complex dependence between the classical volume and shape change modes for wetting conditions that differ from neutral wetting $\alpha =90^\circ$. Finally, we show how the classical frequency degeneracy for the Rayleigh–Lamb modes of the free bubble splits for the azimuthal modes $\ell \neq 0,1$.

The steady dense granular flow in a vertical channel bounded by flat frictional walls in one horizontal direction and with periodic boundary conditions in the other horizontal and vertical directions is studied using the discrete element method. The shape of the scaled velocity profile is characterized quantitatively by a universal exponential function, and the ratio of the maximum and slip velocities is independent of the average volume fraction $\bar {\phi }$ and the channel width $W$. For sufficiently wide channels, the maximum and slip velocities increase proportional to $\sqrt {W}$, and the thickness of the shearing zones increases proportional to $W$. There are four zones in the flow, each with distinct dynamical properties. There is no shear in the plug zone at the centre, but there is particle agitation, and the volume fraction $\phi$ is lower than the random close packing volume fraction $\phi _{rcp}$. In the adjoining dense shearing zone, $\phi$ is greater than the volume fraction for arrested dynamics $\phi _{ad} = 0.587$, and the granular temperature and shear rate depend on the particle stiffness. Adjacent to the dense shearing zone is the loose shearing zone where $\phi < \phi _{ad}$. Here, the properties do not depend on the particle stiffness, and the constitutive relations are well described by hard-particle models. The rheology in the loose shearing zone is similar to the dense flow down an inclined plane. There is high shear and a sharp decrease in $\phi$ in the wall shearing zone of thickness about two particle diameters, where the particle angular velocity is different from the material rotation rate due to the presence of the wall.

This paper examines thermoacoustic effects on the propagation of non-planar sound in a circular duct subjected to an axial temperature gradient. Of particular concern are thermoviscous diffusive effects, which are taken into account by the boundary-layer approximation in a framework of the linear theory. For disturbances expanded into Fourier and Fourier–Bessel series in the azimuthal and radial directions, respectively, the pressure in each mode is described by a one-dimensional, dispersive wave equation, if non-diffusive propagation is assumed. When the diffusive effects are included, each radial mode is coupled to the other radial modes through the boundary layer. Focusing on a single azimuthal and radial mode only, the dispersion relation for the propagation along an infinite duct of a uniform gas is first derived. Effects of the temperature gradient are then examined by solving boundary-value problems for a duct of finite length in four typical cases. Assuming that the wall temperature increases exponentially along the duct, eigenfrequencies and decay rates in the lowest axial mode are obtained as well as axial distributions of the sound pressure and the axial velocity in the duct. The frequency and the decay rate increase as the temperature ratio at both ends becomes higher. It is found from the acoustic energy equation that the dispersion combined with the diffusion acts to reduce the damping and that the temperature gradient makes little contribution to the production of the energy. However, it is unveiled that the non-uniformity in temperature yields thermoacoustic sound confinement in the vicinity of the cold end.

This paper explores the effect of axial and radial confinement on the flow topology of laminar swirling jets. Its objective is to provide a unifying perspective toward swirling jet mechanics that connects earlier reports across a variety of confined and unconfined flow situations, and over a range of swirl ratio $S$ values. The analysis focuses separately on the influence of the jet's injection depth $L$ in a radially unconfined flow and of the chamber diameter $C$ in radially confined jets. In the former case, it shows that axial confinement influences strongly the jet's behaviour when $L$ is small, allowing bistable steady states: a central jet (CJ) solution with or without a small central recirculation zone (CRZ), and a wall jet (WJ) solution with a wide-open CRZ spreading along the reservoir's edge. Similar behaviour is identified for radially confined jets, where bistable CJ and WJ states appear over a range of moderate $C$ values, and the WJ state adopts a conical CRZ. In either case, the WJ solution appears or disappears via saddle–node bifurcations when the confinement is made sufficiently strong or weak, respectively. This dynamics is attributed to an exchange of dominance between central and outer low-pressure regions as the flow transitions from CJ to WJ, or vice versa. The findings demonstrate that the hysteresis associated widely with swirling jets is controlled not just by vortex breakdown, but also by confinement through the Coandă effect. Such confinement is found to alter significantly the state-space structure even when the walls are far from the nozzle.

It is well known that the Poiseuille mass flow rate along microchannels shows a stationary point as the fluid density decreases, referred to as the Knudsen minimum. Surprisingly, if the flow characteristic length is comparable to the molecular size, the Knudsen minimum disappears, as reported for the first time by Wu et al. (J. Fluid Mech., vol. 794, 2016, pp. 252–266). However, there is still no fundamental understanding why the mass flow rate monotonically increases throughout the entire range of flow regimes. Although diffusion is believed to dominate the fluid transport at the nanoscale, here we show that the Fick's first law fails in capturing this behaviour, and so diffusion alone is insufficient to explain this confined flow phenomenon. Rather, we show that the Knudsen minimum disappears in tight confinements because the decay of the mass flow rate due to the decreasing density effects is overcome by the enhancing contribution to the flow provided by the fluid velocity slip at the wall.

We numerically investigate turbulent Rayleigh–Bénard convection with gas bubbles attached to the hot plate, mimicking a core feature in electrolysis, catalysis or boiling. The existence of bubbles on the plate reduces the global heat transfer due to the much lower thermal conductivity of gases as compared with liquids and changes the structure of the boundary layers. The numerical simulations are performed in three dimensions at Prandtl number $\mbox{Pr}=4.38$ (water) and Rayleigh number $10^7\leqslant \mbox{Ra}\leqslant 10^8$. For simplicity, we assume the bubbles to be equally sized and having pinned contact lines. We vary the total gas-covered area fraction $0.18 \leqslant S_0 \leqslant 0.62$, the relative bubble height $0.02\leqslant h/H \leqslant 0.05$ (where $H$ is the height of the Rayleigh–Bénard cell), the bubble number $40 \leqslant n \leqslant 144$ and their spatial distribution. In all cases, asymmetric temperature profiles are observed, which we quantitatively explain based on the heat flux conservation at each horizontal section. We further propose the idea of using an equivalent single-phase set-up to mimic the system with attached bubbles. Based on this equivalence, we can calculate the heat transfer. Without introducing any free parameter, the predictions for the Nusselt number, the upper and lower thermal boundary layer thicknesses and the mean centre temperature agree well with the numerical results. Finally, our predictions also work for the cases with much larger $\mbox{Pr}$ (e.g. $400$), which indicates that our results can also be applied to predict the mass transfer in water electrolysis with bubbles attached to the electrode surface or in catalysis.

We study the structure of high-speed zero-pressure-gradient turbulent boundary layers up to friction Reynolds number $Re_{\tau } \approx 2000$ using direct numerical simulation of the Navier–Stokes equations. Both supersonic and hypersonic conditions with nominal free-stream Mach numbers $M_{\infty }=2$, $M_{\infty }=5.86$ and heat transfer at the wall are considered. The present simulations extend the database currently available for wall-bounded flows, enabling us to explore high-Reynolds-number effects even in the hypersonic regime. We first analyse the instantaneous fields to characterize the structure of both velocity and temperature fluctuations. In all cases elongated strips of uniform velocity and temperature (superstructures) are observed in the outer portion of the boundary layer, characterized by a clear association between low-/high-speed momentum and high/low temperature streaks. The results highlight important deviations from the typical organization observed in the inner region of adiabatic boundary layers, revealing that the near-wall temperature streaks disappear in strongly non-adiabatic flow cases. We also focus on the structural properties of regions of uniform streamwise momentum (De Silva, Hutchins & Marusic, J. Fluid Mech., vol. 786, 2016, pp. 309–331) observed in turbulent boundary layers, confirming the presence of such zones in the high-speed regime at high Reynolds number and revealing the existence of similar regions for the temperature field. The accuracy of different compressibility transformations and temperature–velocity relations is assessed extending their range of validation to moderate/high Reynolds numbers. Spanwise spectral densities of the velocity and temperature fluctuations at various wall distances have been calculated revealing the energy content and the size of the turbulent eddies across the boundary layer. Finally, we propose a revised scaling for the characteristic length scales, that is based on the local mean shear computed according to the recent theory by Griffin, Fu & Moin [Proc. Natl Acad. Sci. USA, vol. 118 (34)].

A rotating disk is the canonical experiment for measuring surface reaction rates in geochemical and electrochemical systems. Using the similarity solution for laminar flow around an infinite disk, the mass transfer coefficient can be simply related to the intrinsic reaction rate at the surface. However, measurements of mass transfer rates use a finite-size disk within a larger container of solution; here the flow is no longer strictly laminar, but there must always be some recirculation. Our interest was initially in the assumption of a uniform radial concentration field, how this breaks down near the perimeter of the disk, and what effect that might have on the measured mass transfer rates. However, our numerical simulations suggest that the flow around a finite-size disk becomes time dependent at Reynolds number ($Re$) below 1000, which is much smaller than the typical values in mass-transfer measurements (${Re} \sim 10^4$). We observe the formation of coherent structures in the flow, which suggest a non-uniform mass transfer at the disk surface. The rotating-disk flow follows a similar sequence of instabilities to the Taylor–Couette flow: a centrifugal instability leading an axisymmetric, time-invariant flow, followed by a Hopf bifurcation to a time-periodic flow. To minimise the possibility that our results are a numerical artefact, we have also simulated the instability in the stationary boundary layer of a rotor–stator flow, comparing with self-similar solutions at low ${Re}$ and with spectral methods near the critical Reynolds number.

The natural and forced dynamic response of a gravitational plane liquid sheet (curtain) of finite length interacting with an unconfined gaseous ambient is numerically and experimentally investigated. The global eigenvalue spectrum obtained by means of a linear inviscid one-dimensional model, accounting for the coupling between the curtain motion and the ambient pressure disturbances, clearly shows an abrupt increase (jump) in the characteristic natural frequency of the flow when the supercritical ($We>1$) to subcritical ($We<1$) transition occurs, with the Weber number $We$ defined as the ratio between inertia and capillary forces. On the other hand, the numerical simulation of the forced sheet response does not show any discontinuity between supercritical and subcritical conditions, as recently found by Torsey et al. (J. Fluid Mech., vol. 910, 2021, pp. 1–14) in the case of an infinite liquid sheet subjected to imposed ambient pressure disturbances not coupled with the curtain motion. It is argued that the forced liquid sheet behaviour varies continuously in shape and amplitude between the two regimes, not depending on the specific liquid–gas interaction model considered, whilst the natural frequency of the finite flow system does undergo a discontinuity, which can be theoretically predicted by the model of sheet–ambient interaction employed here. As a major result, the experimental evidence of the natural frequency jump is for the first time provided as well.

Large-scale coherent rolls are observed frequently in unstably stratified turbulent wall-bounded flows where they influence strongly the turbulent transport and the mean flow properties. Here we address the question of their genesis by means of a linear stability analysis of the turbulent mean flow where a model of turbulent Reynolds stresses is embedded in the linear stability operator. We use the unstably stratified turbulent channel flow as a testbed for the analysis. We show that the onset of large-scale convection is associated to the linear instability of the mean flow to large-scale streamwise-uniform coherent rolls of aspect ratio $A \approx 3\unicode{x2013}3.3$ when the friction Richardson number exceeds $|Ri_{\tau,c}|=0.86$. This corresponds to critical Rayleigh numbers that increase with the Reynolds number approximately as $Ra_c \approx 0.04\,Re_b^{1.8}$. These results are consistent with those obtained in recent direct numerical simulations performed in the same setting. It is also found that if turbulent Reynolds stresses are not modelled in the linear operator used in the stability analysis, then predictions are not consistent with direct numerical simulations results.

The moving contact line between a fluid, liquid and solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models, previous studies have shown that the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number, $Ca_{{crit}}$, above which no steady-state solution can be found. Below $Ca_{{crit}}$, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against $Ca$, a fold bifurcation appears where the stable and unstable branches meet. Interestingly, the significance of this bifurcation structure to the transient dynamics has yet to be explored. This article develops a computational model and uses ideas from dynamical systems theory to show the profound importance of the unstable solutions on the transient behaviour. By perturbing the stable state by the eigenmodes calculated from a linear stability analysis it is shown that the unstable branch is an ‘edge’ state that is responsible for the eventual dynamical outcomes and that the system can become transient when $Ca< Ca_{{crit}}$ due to finite-amplitude perturbations. Furthermore, when $Ca>Ca_{{crit}}$, we show that the trajectories in phase space closely follow the unstable branch.