Liquid spreading through a randomly packed particle-resolved bed influenced by capillary or inertial ($AB_s \sim 1$), and gravitational force (moderately ($AB_s \sim 0.1$) and strongly ($AB_s \sim 0.01$)) is investigated using the volume-of-fluid simulations. The relative contribution of governing forces at different stages of spreading is analysed using the time evolution of Weber ($We_I$) and $AB_I$ numbers. We show that the dynamics of liquid spreading at $AB_s \sim 1$ is primarily governed by the inertial force in the beginning ($AB_I > 1$, $We_I > 1$) followed by the capillary force at $t/t^* \sim 1$. This interplay of governing forces leads to inertia- and capillary-induced bubble entrapments at the void scale and promote lateral liquid spreading. When the $AB_s \sim 0.1$, the $t/t^*$ for which the flow is governed by inertial ($AB_I > 1$, $We_I > 1$) and capillary forces ($AB_I > 1$, $We_I < 1$) decreases and the relative contribution of gravitational force is substantial at large $t/t^*$ ($AB_I < 1$). This force balance leads to unified-void filling characterised by negligible bubble trapping and results in a decrease in the lateral spreading. Further decrease in the $AB_s$ to ${\sim } 0.01$ results in liquid spreading primarily governed by gravitational force ($AB_I < 1$) with small contribution of inertial and capillary forces at the very beginning leading to trickling flow and a further decrease in lateral spreading. Finally, a regime map is proposed, which provides the relationship between different forces, void-scale events, and the resultant liquid spreading at $t/t^* \sim 1$.

Resolvent analysis is a powerful tool for modelling and analysing transitional and turbulent flows and, in particular, for approximating coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this paper we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction, i.e. one in which the mean flow changes gradually. First, we derive a well-posed and convergent one-way equation describing the downstream-travelling waves supported by the linearized Navier–Stokes equations. The method is based on a projection operator that isolates downstream-travelling waves. Integrating these one-way Navier–Stokes (OWNS) equations in the slowly varying direction, which requires significantly less CPU and memory resources than a direct solution of the linearized Navier–Stokes equations, approximates the action of the resolvent operator on a forcing vector. Second, this capability is leveraged to compute approximate resolvent modes using an adjoint-based optimization framework in which the forward and adjoint OWNS equations are marched in the downstream and upstream directions, respectively. This avoids the need to solve direct and adjoint globally discretized equations, therefore bypassing the main computational bottleneck of a typical global resolvent calculation. The method is demonstrated using the examples of a simple acoustics problem, a Mach 1.5 turbulent jet and a Mach 4.5 transitional zero-pressure-gradient flat-plate boundary layer. The optimal OWNS results are validated against corresponding global calculations, and the close agreement demonstrates the near-parabolic nature of these flows.

We employ a recently introduced structured input–output analysis (SIOA) approach to analyse streamwise and spanwise wavelengths of flow structures in stably stratified plane Couette flow. In the low-Reynolds-number ($Re$) low-bulk Richardson number ($Ri_b$) spatially intermittent regime, we demonstrate that SIOA predicts high amplification associated with wavelengths corresponding to the characteristic oblique turbulent bands in this regime. SIOA also identifies quasi-horizontal flow structures resembling the turbulent–laminar layers commonly observed in the high-$Re$ high-$Ri_b$ intermittent regime. An SIOA across a range of $Ri_b$ and $Re$ values suggests that the classical Miles–Howard stability criterion ($Ri_b\leq 1/4$) is associated with a change in the most amplified flow structures when the Prandtl number is close to one ($Pr\approx 1$). However, for $Pr\ll 1$, the most amplified flow structures are determined by the product $PrRi_b$. For $Pr\gg 1$, SIOA identifies another quasi-horizontal flow structure that we show is principally associated with density perturbations. We further demonstrate the dominance of this density-associated flow structure in the high $Pr$ limit by constructing analytical scaling arguments for the amplification in terms of $Re$ and $Pr$ under the assumptions of unstratified flow (with $Ri_b=0$) and streamwise invariance.

This paper is concerned with the optimal upper bound on mean quantities (torque, dissipation and the Nusselt number) obtained in the framework of the background method for the Taylor–Couette flow with a stationary outer cylinder. Along the way, we perform the energy stability analysis of the laminar flow, and demonstrate that below radius ratio $0.0556$, the marginally stable perturbations are not the axisymmetric Taylor vortices but rather a fully three-dimensional flow. The main result of the paper is an analytical expression of the optimal bound as a function of the radius ratio. To obtain this bound, we begin by deriving a suboptimal analytical bound using analysis techniques. We use a definition of the background flow with two boundary layers, whose relative thicknesses are optimized to obtain the bound. In the limit of high Reynolds number, the dependence of this suboptimal bound on the radius ratio (the geometrical scaling) turns out to be the same as that of numerically computed optimal bounds in three different cases: (1) the perturbed flow only satisfies the homogeneous boundary conditions but need not be incompressible; (2) the perturbed flow is three-dimensional and incompressible; (3) the perturbed flow is two-dimensional and incompressible. We compare the geometrical scaling with the observations from the turbulent Taylor–Couette flow, and find that the analytical result indeed agrees well with the available direct numerical simulations data. In this paper, we also dismiss the applicability of the background method to certain flow problems and therefore establish the limitation of this method.

We discuss (i) the measurement, (ii) interpretation and (iii) technical impact of temperature-dependent wall slip and its activation energy for polar and non-polar hydrocarbon Newtonian fluids moving relative to machined metal surfaces. A newly developed apparatus, the slip length tribometer (SLT), overcomes the drawbacks of existing measurement devices in terms of characterising relevant rough surfaces by measuring the slip length at different temperatures over a sufficiently large wetted area. The experimental data show that the bulk viscosity and slip length at the fluid–metal interface is independent of the shear rate up to $10^5\,\mathrm {s}^{-1}$, being consistent with recent results from molecular dynamics simulations by Mehrnia & Pelz (J. Mol. Liq., vol. 336, 2021, 116589). Furthermore, the activation energies for wall slip and bulk shear determined by means of the SLT differ by a factor of two, i.e. $E_{\lambda }\approx 0.5 E_\mu$ for non-polar hydrocarbon molecules sliding relative to metal walls. This difference is explained by a generalised Eyring model applied to wall slip. The paper closes with the impact of wall slip on Sommerfeld's similarity theory of tribology and the resulting Stribeck curve (Sommerfeld, Z. Math. Phys., vol. 50, 1904, pp. 97–155; Z. Tech. Phys., vol. 2. Jahrg., 1921; Mechanik der deformierbaren Medien, 1944, Akademische Verlagsgesellschaft Becker & Erler). For this purpose, Reynolds’ equation generalised for wall slip is solved in combination with the experimentally determined constitutive relations for bulk shear and wall slip to predict typical characteristics of journal bearings. The results show that, for a typical journal bearing, where the ratio of slip length to average bearing clearance is of the order of $10^{-2}$, the influence of wall slip on both load-carrying capacity and dissipation is not negligible. This work combines nanofluidics and tribology in order to provide methods and knowledge for e.g. tailor-made fluids and interfaces.

The flow of polymer solutions past solid obstacles or through porous media gives rise to rich physical phenomena over a wide range of spatial and temporal scales. Viscoelasticity, in particular, can induce a strong nonlinear response with an increase of flow resistance even for a solution whose viscosity decreases in simple shear flow. Various hypotheses have been proposed to explain this phenomenon but a clear picture of the pore-scale mechanisms involved and their impact upon larger scales is still lacking. Here, we show that localized zones of large polymeric stress, known as birefringent strands, drive the flow of an Oldroyd-B fluid through two-dimensional arrays of cylinders. Combining a recently developed numerical scheme with high performance computing, we find that these strands generate a complete reorganization of the flow with an increase of stagnation zones, a reinforcement of preferential paths and a splitting of flow channels. Furthermore, we show that this reorganization is the source of an increase in the viscous dissipation of the solvent and also that the stretching of polymer molecules in the strands is associated with entropy production. Both these phenomena yield a global increase in dissipation that can be directly linked to the increase of flow resistance. Our results demonstrate that the birefringent strands – not the elongational viscosity – drive the flow of viscoelastic fluids through porous media and that the increase of flow resistance can occur even at steady state, before the transition to elastic turbulence.

We investigate the large-scale influence of numerical noises as tiny artificial stochastic disturbances on a sustained turbulence. Using two-dimensional (2-D) turbulent Rayleigh–Bénard convection (RBC) as an example, we solve numerically the Navier–Stokes equations, separately, by means of a traditional algorithm with double precision (denoted RKwD) and the so-called clean numerical simulation (CNS). The numerical simulation given by RKwD is a mixture of the ‘true’ physical solution and the ‘false’ numerical noises that are random and can be regarded as a kind of artificial stochastic disturbances; unfortunately, the ‘true’ physical solution is mostly at the same level as the ‘false’ numerical noises. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the ‘false’ numerical noises are negligible compared with the ‘true’ physical solution, thus the CNS solution can be used as a ‘clean’ benchmark solution for comparison. It is found that the numerical noises as tiny artificial stochastic disturbances could indeed lead to large-scale deviations of simulations not only in spatio-temporal trajectories but also even in statistics. In particular, these numerical noises (as artificial stochastic disturbances) even lead to different types of flows. The shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter; however, the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Thus we provide rigorous evidence that numerical noises as a kind of small-scale artificial stochastic disturbances have quantitatively and qualitatively large-scale influences on a sustained turbulence, i.e. the 2-D turbulent RBC considered in this paper.

The unsteadiness of a turbulent separation bubble (TSB) formed close to the trailing edge of a two-dimensional wing was investigated using time-resolved particle image velocimetry. The angle of attack was set to 9.7° and the chord-based Reynolds number was 720 000. The TSB consisted of two shear layers and formed a triangular shape in the streamwise–wall-normal plane. The vertices of this triangle consisted of an intermittent detachment point, a fixed corner close to the airfoil trailing edge and an intermittent endpoint in the wake region. The velocity field had three energetic regions each with different Strouhal numbers (Stl): (a) an upstream turbulent boundary layer (TBL) with Stl = 0.1 to 4, (b) a TSB with Stl = 0.03 to 0.08 and (c) two shear layers with Stl = 0.4 to 0.8. The low-frequency motions in the TSB consisted of large zones of positive and negative streamwise velocity fluctuation that were several times wider than the large-scale structures of the upstream TBL. These zones forced an undulation of the separation line and were attributed to Görtler structures. They were also correlated with the velocity fluctuations between the two shear layers. The breathing motion of the TSB occurred at Stl = 0.05. This breathing correlated with the location of the TSB endpoint and the flapping of the upper shear layer. The detachment point of the TSB featured broad fluctuations and did not demonstrate a strong correlation with the breathing motion.

Production and inter-component redistribution of turbulence in adverse pressure gradient (APG) turbulent boundary layers (TBLs) with small and large velocity defects are investigated, along with the structures playing a role in these energy transfer mechanisms. We examine the wall-normal and spectral distributions of energy, production and pressure-strain in APG TBLs, and compare these distributions with those in canonical flows. It is found that the spectral distributions of production and pressure-strain are not affected profoundly by an increase of the velocity defect, although the energy spectra change drastically in the inner layer of the large-defect APG TBL. In the latter, the signature of the inner-layer streaks is absent from the energy spectra. In the outer layer, energetic, production and pressure-strain structures appear to change from wall-attached to wall-detached structures with increasing velocity defect. Despite this, the two-dimensional spectral distributions have similar shapes and wavelength aspect ratios of the peaks in all these flows. Therefore, the conclusion is that the mechanisms responsible for turbulence production and inter-component energy transfer may remain the same within each layer in all these flows. It is the intensity of these mechanisms within one layer that changes with velocity defect, because of the local mean shear variation.

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of non-normal dynamical systems, when they experience transient growth or respond to harmonic forcing. This approach reconciles the non-modal nature of these growth mechanisms and the need for a centre manifold to project the leading-order dynamics. Under the hypothesis of strong non-normality, we take advantage of the fact that small operator perturbations suffice to make the inverse resolvent and the inverse propagator singular, which we encompass in a multiple-scale asymptotic expansion. The methodology is outlined for a generic nonlinear dynamical system, and four application cases highlight common non-normal mechanisms in hydrodynamics: the streamwise convective non-normal amplification in the flow past a backward-facing step, and the Orr and lift-up mechanisms in the plane Poiseuille flow.

A nonlinear ensemble-variational data assimilation is performed in order to estimate the unknown flow field over a slender cone at Mach 6, from isolated wall-pressure measurements. The cost functional accounts for discrepancies in wall-pressure spectra and total intensity between the experiment and the prediction using direct numerical simulations, as well as our relative confidence in the measurements and the estimated state. We demonstrate the robustness of the predicted flow by direct propagation of posterior statistics. The approach provides a unique first look at the flow beyond the sensor data, and rigorously accounts for the role of nonlinearity, unlike previous efforts that adopted ad hoc inflow syntheses. Away from the wall, two- and three-dimensional assimilated states both show rope-like structures, qualitatively similar to independent schlieren visualizations. Despite this resemblance, and even though the planar second modes are the most unstable upstream, three-dimensional waves must be included in the assimilation in order to accurately reproduce the wall-pressure measurements recorded in the AFRL Ludwieg Tube facility. The results highlight the importance of three-dimensionality of the field and of the base-state distortion on the instability waves in this experiment, and motivate future measurements that probe the three-dimensional nature of the flow field.

In this paper, the interaction between an upper-layer vortex and a bottom topography is investigated using an $f$-plane two-layer quasi-geostrophic model with a point vortex and step-like topography. The contour dynamics method is used to formulate the model. A steadily propagating linear solution along the topography, known as the pseudoimage solution, is derived analytically for a weak point vortex, and the nonlinear solution is obtained numerically. Numerical experiments show that the nonlinear pseudoimage solution collapses with time. Saddle-node points in the velocity field are critical in this collapse. Even after the collapse, the point vortex propagates along the topography similarly to in the steadily propagating solution. Numerical experiments with various initial conditions show that the point vortex has two types of motion in this system: motion along the topography and motion away from the topography. In the latter case, the point vortex and lower-layer potential vorticity anomaly form a heton-like dipole structure. The motion classification results show that an anticyclonic (cyclonic) point vortex on the deeper (shallower) side is more likely to form a dipole structure than a cyclonic (anticyclonic) vortex on the deeper (shallower) side when their initial distance from the topography is the same.

The flow generated by the breaking of free-surface waves in a periodic domain is simulated numerically with a gas–liquid Navier–Stokes solver. The solver relies on the volume-of-fluid method to account for different phases, and the interface tracking is carried out by using novel schemes based on a tailored total-variation-diminishing limiter. The numerical solver is proved to be characterized by a low numerical dissipation, thanks to the use of a scheme that guarantees energy conservation in the discrete form. Both two- and three-dimensional simulations have been performed, and the analysis is presented in terms of energy dissipation, air entrainment, bubble fragmentation, statistics and distribution. Particular attention is paid to the analysis of the mechanisms of viscous dissipation. To this purpose, coherent vortical structures, such as vortex tubes and vortex sheets, are identified, and the different behaviours of the vortex sheets and tubes at various Reynolds numbers are highlighted. The correlation between vortical structures and energy dissipation demonstrates clearly their close link both in the mixing zone and in the pure water domain, where the coherent structures propagate as a consequence of the downward transport. Notably, it is found that the dissipation is identified primarily by the vortex sheets, whereas the vortex tubes govern mainly the intermittency.

We study the dynamics of a cavitation bubble near beds of sand of different grain sizes. We use high-speed imaging to observe the motion of the bubble and the sand for different values of the stand-off parameter $\gamma$ (dimensionless bubble-boundary distance) between 0.3 and 5.3. Compared with a rigid boundary, we find that a granular boundary leads to bubbles with shorter lifetimes and reduced centroid displacements. Above $\gamma \approx 1.3$, the behaviour of the bubble is almost independent of the granularity of the sand. When the stand-off parameter lies between $0.6$ and $1.3$, a mound of sand develops beneath the bubble, which can force the latter to assume a conical shape as it collapses. For $\gamma \lesssim 0.6$, the bubble develops a bell-shaped form, leading to the formation of thin and surprisingly fast micro-jets ($v_{jet} > 1000\,{\rm m}\,{\rm s}^{-1}$). Moreover, between $\gamma \approx 1.3$ and $\gamma \approx 0.3$, granular jets erupt from the sand surface following the bubble collapse. We additionally develop a simple numerical model, based on the boundary integral method, to predict the dynamics of a cavitation bubble near a bed of sand, which we replace by an equivalent liquid. The simulations are remarkably consistent with experimental observations for values of $\gamma$ down to $1.3$. We also show how the anisotropy parameter $\zeta$, a dimensionless version of the Kelvin impulse, can be adapted for the case of a nearby bed of sand and predict the displacement of the bubble centroid for $\zeta \lesssim 0.08$.

We derive from first principles analytic relations for the second- and third-order moments of $\boldsymbol{\mathsf{m}}$, the spatial gradient of fluid velocity $\boldsymbol{u}$, $\boldsymbol{\mathsf{m}} = \nabla \boldsymbol{u}$, in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second- and third-order moments of the velocity gradient experimentally for isotropic compressible turbulence.

The transient flow of a circular liquid jet impinging on a horizontal disk, and the hydraulic jump formation, are examined theoretically and numerically. The interplay between inertia and gravity is particularly emphasised. The flow is governed by the thin-film equations, which are solved along with a force balance across the jump. The unsteadiness of the flow is caused by a linearly accelerating jet from an initial to a final steady state. To validate the predicted boundary-layer flow evolution, an analytical development is conducted for small distance from impingement, and for small time. In addition, the predictions of the film profile and jump location are compared against numerical simulation for the transient flow, and are further validated against experiment for steady flow. The evolutions of the film thickness and the wall shear stress in the developing boundary-layer region are found to be similar to those reported for a fluid lying on a stretching surface. The flow responds to the jet acceleration quasi-steadily near impingement but exhibits a long-term transient behaviour near the jump. Analysis of the jump evolution is considered in the range 5 < Fr < 40 for the Froude number (based on the jet radius and velocity). For Fr < 10, the jump reaches the final state instantly when the jet acceleration ceases. At higher Froude number, the jump settles at a later time, exhibiting an overshoot in the thickness due to the dominance of inertia.

In this study, we present a three-dimensional numerical model for the interaction of flow with submerged flexible vegetation, based on a large-eddy simulation and the immersed boundary method. The model innovatively realises the interaction between the flow and highly flexible vegetation with clustered leaves. Besides being a three-dimensional model of motion with full degrees of freedom, this study improves the consideration of the motion of the vegetation in all directions, and in addition the energy and momentum transfer in the spanwise direction. Furthermore, we perform a flume experiment for the flow with submerged flexible vegetation, the results of which are used to validate the simulation effects of the numerical model. It is found that the numerical model can effectively simulate the velocity profiles and the movement of vegetation induced by the flow. Using the model to analyse the flow–vegetation interaction, we find that the movement of vegetation is closely related to the flow velocity. As the flow velocity increases, both the offset angle and the vegetation swaying amplitude increase. Compared to vertical rigid vegetation, the tilting of flexible vegetation does not significantly change the velocity difference and the magnitude of the turbulent kinetic energy between the inside and the outside of the vegetation canopy, but it does weaken the disturbance to flow, thus reducing the resistance to flow. However, the swaying of vegetation dose significantly increase the velocity difference between the inside and the outside of the canopy. It forms Kelvin–Helmholtz hairpin vortices intensifying the turbulence production, and enhancing the disturbance and resistance to flow.

Laminar miscible displacement flows in narrow, vertical, eccentric annuli show a wide range of interesting physical phenomena, e.g. static layers and channels, dispersive spikes/fronts and various instabilities. These are of relevance to the primary cementing of wells, where gas leakage and greenhouse gas emissions can result from ineffective cementing. The current popular way to model this process is via a Hele-Shaw approach, reducing the Navier–Stokes equations via scaling arguments and then averaging across the annular gap: the two-dimensional (2-D) gap averaged (2DGA) model. This leads to a reduced model that is computationally efficient and represents some important features of cementing flows, but is also deficient in capturing the effects of dispersion. While three-dimensional (3-D) simulations are able to capture dispersion and most other physical phenomena, they suffer from excessive computational times due to the extreme aspect ratios of cementing geometries and non-Newtonian properties of the fluids. Here we extend the 2DGA approach by modelling the high Péclet number limit of miscible duct displacement flows, to take into account the leading-order effects of dispersion. Our modified dispersive 2DGA model is well able to approximate dispersive effects that are observed by gap-averaging the results of 3-D computations, including geometric effects of eccentricity and instabilities. This represents a marked improvement over the 2DGA approach, while keeping the computational advantages of a 2-D model.

We study the morphology of Eulerian vortical structures and their interaction with density interfaces in increasingly turbulent stably stratified shear layers. We analyse the three-dimensional, simultaneous velocity and density fields obtained in the stratified inclined duct laboratory experiment (SID). We track, across 15 datasets, the evolution of coherent structures from pre-turbulent Holmboe waves, through intermittent turbulence, to full turbulence and mixing. We use the rortex–shear decomposition of the local vorticity vectors into a rortex vector capturing rigid-body rotation and a shear vector. We describe the morphology of ubiquitous hairpin-like vortical structures (revealed by the rortex), similar to those commonly observed in boundary-layer turbulence. These are born as relatively weak vortices around the strong three-dimensional shearing structures of confined Holmboe waves, and gradually strengthen and deform under increasing turbulence, transforming into pairs of upward- and downward-pointing hairpins propagating in opposite directions on the top and bottom edge of the shear layer. The pair of legs for each hairpin are counter-rotating and entrain fluid laterally and vertically, whereas their arched-up ‘heads’, which are transverse vortices, entrain fluid vertically. We then elucidate how this large-scale vortex morphology stirs and mixes the density field. Essentially, vortices located at the sharp density interface on either edge of the mixing layer (mostly hairpin heads) engulf blobs of unmixed fluid into the mixing layer, whereas vortices inside the mixing layer (mostly hairpin legs) further stir it, generating strong, small-scale shear, enhancing mixing. These findings provide new insights into the role of turbulent coherent structures in shear-driven stratified mixing.