The linear stability of the two-dimensional Taylor–Green vortices, which is a spatially periodic array of vortices, in rotating stratified fluids is investigated by local and modal stability analysis. Five types of instability appear in general: the pure hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate, which are useful in interpreting numerical results, are obtained in the framework of local stability analysis. Realizability of an instability is introduced to predict whether an unstable mode corresponding to an unstable region found in the local stability analysis exists at finite Reynolds numbers. In the absence of stratification, the pure hyperbolic instability is dominant for weak rotation; it is stabilized for strong rotation. For strong anti-cyclonic rotation, the elliptic instability or the centrifugal instability becomes dominant depending on the parameter values; further stronger rotation stabilizes both instabilities. For strong cyclonic rotation, the rotational-hyperbolic instability or the elliptic instability becomes dominant, although the growth rate is smaller than the anti-cyclonic cases. Strong stratification changes the stability properties. The strato-hyperbolic instability occurs for weak rotation. The rotational-hyperbolic instability and the elliptic instability are weakened under cyclonic rotation, while the latter survives and extends the unstable range under anti-cyclonic rotation. The pure hyperbolic instability and the centrifugal instability are less affected by stratification. The mode structures of each instability are in good agreement with the corresponding solution to local stability equations, confirming the physical mechanism of the instability.

In this study, a novel analytical solution is derived rigorously for the mean wall-normal velocity in a turbulent boundary layer (TBL) under zero pressure gradient (ZPG), without relying on any a priori assumptions regarding the mean streamwise velocity. By neglecting the higher-order terms in the exact solution, an approximate formulation for the mean wall-normal velocity is obtained. The accuracy of this approximation is then validated through a comparison with data from direct numerical simulations (DNS). Drawing upon the distinct behaviours of force balance in the inner and outer regions of the ZPG TBL, simplified approximate mean momentum equations are derived by decomposing the Reynolds shear stress into inner and outer parts. The inner and outer Reynolds shear stresses are then obtained through integration of the corresponding approximate mean momentum equations. The outer Reynolds shear stress is revealed to exhibit a strong connection to the mean wall-normal velocity. Specifically, it is observed that $R_{uv-{out}}/u^2_\tau \approx 1 - UV/(U_e V_e)$, where $u_\tau$ represents the friction velocity, and $U_e$ and $V_e$ denote the mean streamwise and wall-normal velocities at the boundary layer edge, respectively. Finally, an approximate formulation for the Reynolds shear stress across the entire TBL is developed by synthesizing the inner and outer parts. Extensive validation against both DNS and experimental data, spanning a wide range of Reynolds numbers, demonstrates the excellent agreement achieved.

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

The Richtmyer–Meshkov instability (Richtmyer, Commun. Pure Appl. Maths, vol. 13, issue 2, 1960, pp. 297–319; Meshkov, Fluid Dyn., vol. 4, issue 5, 1972, pp. 101–104) of a twice-shocked gas interface is studied using both high spatial resolution single-shot (SS) and lower spatial resolution, time-resolved, high-speed (HS) simultaneous planar laser-induced fluorescence and particle image velocimetry in the Wisconsin Shock Tube Laboratory's vertical shock tube. The initial condition (IC) is a shear layer with broadband diffuse perturbations at the interface between a helium–acetone mixture and argon. This IC is accelerated by a shock of nominal strength Mach number $M = 1.75$, and then accelerated again by the transmitted shock that reflects off the end wall of the tube. An ensemble of experiments is analysed after reshock while the interface mixing width grows linearly with time. The kinetic and scalar energy spectra and the terms of their evolution equation are calculated and compared between SS and HS experiments. The inertial range scaling of the scalar power spectrum is found to follow Gibson's relation (Gibson, Phys. Fluids, vol. 11, issue 11, 1968, pp. 2316–2327) as a function of Schmidt number when the effective turbulent Schmidt number is used in place of the material Schmidt number that controls equilibrium scaling. Further, the spatially integrated scalar flux follows similar behaviour observed for the kinetic energy in large eddy simulation studies by Zeng et al. (Phys. Fluids, vol. 30, issue 6, 2018, 064106) while the spatially varying scalar flux exhibits back scatter along the centre of the mixing layer and forward energy transfer in the spike and bubble regions.

We consider the flow of a thermoviscous fluid within a vertical tube which is heated from below, modelling a scenario where a fluid melts, flows and eventually drips due to a temperature-dependent viscosity. To do so, we develop a two-dimensional axisymmetric model comprising three regions, a solid granular upper region (modelled as a region of hydrostatic pressure), a middle highly viscous ‘crust’ region which flows and a lower cavity region within which the material can drip. New material is continuously added to the top, yet the highly viscous middle region can slow mass transfer from the top region to the cavity if it becomes too thick or does not drip fast enough. In the limit of a tall, thin geometry, akin to what is often seen in industrial applications, the resulting model comprises a moving boundary problem governed by an energy equation with a Stefan condition, both subject to a non-local radially averaged convective term. We carry out numerical simulations and an asymptotic analysis of the model in this tall, thin limit, for a variety of physically relevant parameter regimes. Our results reveal a variety of qualitatively different behaviours, and enable us to explore how various parameter regimes influence the salient features of the flow, including the ‘crust’ thickness and the flux of material through the lower moving boundary.

A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Green's function is constructed using Frobenius series. The critical layer gives a non-modal contribution from the continuous spectrum branch cut, and dominates the downstream pressure perturbation in certain cases, particularly for thicker boundary layers. The continuous spectrum branch cut is also found to stabilize what are otherwise convectively unstable modes by hiding them behind the branch cut. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave when the source is located within the sheared flow region, and to decay algebraically along the duct as $O(x^{-{5}/{2}})$ for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods lose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find; this includes the stabilized hydrodynamic instability. Matlab code is provided to compute the Green's function.

The stability and sensitivity of two- and three-dimensional global modes developing on steady spanwise-homogeneous laminar separated flows around NACA 4412 swept wings are numerically investigated for different Reynolds numbers ${\textit {Re}}$ and angles of attack $\alpha$. The wake dynamics is driven by the two-dimensional von Kármán mode whose emergence threshold in the $\alpha \unicode{x2013}{\textit {Re}}$ plane is computed with that of the three-dimensional centrifugal mode. At the critical Reynolds number, the Strouhal number, the streamwise wavenumber of the von Kármán mode and the spanwise wavenumber of the leading three-dimensional centrifugal mode scale as a power law of $\alpha$. The introduction of a sweep angle attenuates the growth of all unstable modes and entails a Doppler effect in the leading modes’ dynamics and a shift towards non-zero frequencies of the three-dimensional centrifugal modes. These are found to be non-dispersive as opposed to the von Kármán modes. The sensitivity of the leading global modes is investigated in the vicinity of the critical conditions through adjoint-based methods. The growth-rate sensitivity map displays a region on the suction side of the wing, wherein a streamwise-oriented force has a net stabilising effect, comparable to what could have been obtained inside the recirculation bubble. In agreement with the predictions of the sensitivity analysis, a spanwise-homogeneous force suppresses the Hopf bifurcation and stabilises the entire branch of von Kármán modes. In the limit of small amplitudes, passive control via spanwise-wavy forcing produces a stabilising effect similar to that of a spanwise-homogeneous control and is more effective than localised spherical forces.

The impact of a drop train, a series of identical liquid drops separated by a constant distance, on a liquid pool initially generates a long slender cavity. However, the cavity soon collapses and turns into a shallow funnel. Here we theoretically model the dynamic profile of the elongated cavity and the steady shape of the funnel, which are then shown to agree well with experiment. When the liquid inertia plays a dominant role, the cavity assumes a parabolic profile that depends only on the drop diameter and the centre-to-centre spacing of adjacent drops. We consider the capillary forces as well as the drop impact force to obtain the shape of the funnel that persists once the elongated cavity collapses. Our study allows for predicting the interfacial morphology by the impact of a drop train and designing impact conditions useful for semiconductor cleaning processes.

Direct numerical simulations were performed to characterize fully developed supersonic turbulent channel flows over isothermal rough walls. The effect of roughness was incorporated using a level-set/volume-of-fluid immersed boundary method. Turbulence statistics of five channel flows are compared, including one reference case with both walls smooth and four cases with smooth top walls and bottom walls with two-dimensional (2-D) and three-dimensional (3-D) sinusoidal roughnesses. Results reveal a strong dependence of the turbulence on the roughness topography and the associated shock patterns. Specifically, the 2-D geometries generate strong oblique shock waves that propagate across the channel and are reflected back to the rough-wall side. These strong shocks are absent in the smooth-wall channel and are significantly weaker in cases with 3-D roughness geometries, replaced by weak shocklets. At the impingement locations of the shocks on the top wall in the 2-D roughness cases, localized augmentations of turbulence shear production are observed. Such regions of augmented production also exist for the 3-D cases, at a much weaker level. The oblique shock waves are thought to be responsible for a more significant entropy generation for cases with 2-D surfaces than those with 3-D ones, leading to a higher irreversible heat generation and consequently higher temperature values in 2-D roughness cases. In the present supersonic channels, the effects of roughness extend beyond the near-wall layer due to the shocks. This suggests that outer layer similarity may not fully apply to a rough-wall supersonic turbulent flow.

In turbulence near a free surface, strong vortices attach to the surface, creating surface imprints visible as nearly circular ‘dimples’. By studying these imprints in direct numerical simulation (DNS) data we make two observations. First, the imprints of surface-attached vortices can be very effectively distinguished from other turbulent surface features using two physical features: they are nearly circular in shape, and persist for a long time compared with other pertinent time scales. Secondly, the instantaneous number of surface dimples from surface-attached vortices in an area, $N(t)$, is intimately related to its mean-square surface divergence, $\beta ^2(t)$. We develop a simple and physically transparent computer vision procedure which, using the properties of low eccentricity and longevity, detects and tracks vortices from their surface features only, with sensitivity and accuracy of $90\,\%$ or better. We compare $N(t)$ and $\beta ^2(t)$, finding a normalised cross-correlation of $0.90$, with changes in $N$ lagging around $0.8T_\infty$ behind those in $\beta ^2$ ($T_\infty$ is an integral time scale), confirming the common observation that vortices are spawned by strong upwelling events where $\beta ^2$ is large. These findings suggest that the rate of mass flux across the surface, being closely related to surface divergence, can be estimated remotely in some natural flows using visible free-surface dimples as proxy.

In this paper, we study the linear stability of a two-dimensional shear flow of an air layer overriding a water layer of finite depth. The air layer is considered to be of an infinite extent with an exponential velocity profile. Three different background conditions are considered in the finite-depth water layer: a quiescent background, a linear velocity profile and a quadratic velocity profile. It is known that the cases of the quiescent water layer and the linear velocity profile allow for analytical treatment. We further provide an analytical solution for the case of the quadratic velocity field: we specifically consider a flow-reversal profile, although the result could be generalized to other quadratic profiles as well. The role of water layer depth on the growth rate of the Miles and rippling instabilities is studied in each of the three cases. Using asymptotic analysis, with the air–water density ratio being a small parameter, we obtain an analytical expression for the growth rate of the Miles mode and discuss the condition for the existence of a long-wave cutoff for these profiles. We provide analytical expressions for the stability boundary in the parameter space of inverse squared Froude number and wavenumber. In scenarios where a long-wave cutoff does not exist, we have carried out a long-wave asymptotic study to obtain the growth rate behaviour in that regime.

When viscous fluid in a corner is disturbed, eddies can form in the absence of inertia. Examples of flow configurations in which this motion occurs include flow through an abrupt contraction and over a cavity. Six decades ago, Moffatt (J. Fluid Mech., vol. 18, 1964, pp. 1–18) calculated the slow viscous flow of Newtonian fluids in sharp corners, detailing his eponymous ‘Moffatt eddies’. In this study we examine corner flows of viscoplastic materials, a class of non-Newtonian fluids which exhibit solid-like behaviour for stresses below a yield stress. Specifically, we consider a Bingham fluid, for which the material is perfectly rigid at stresses below the yield stress. While a static unyielded plug forms at the tip of the corner, eddies analogous to those found by Moffatt can also form. We examine these viscoplastic eddies numerically, by computing finite element solutions using the augmented-Lagrangian method, and analytically, by employing a viscoplastic boundary layer formulation and scaling arguments. We measure the depth of the static plug as a function of the Bingham number (dimensionless yield stress), show that the process of a new eddy forming as the Bingham number is decreased is driven by the pressure in the yielded fluid adjacent to the static plug, and provide a heuristic argument for the critical Bingham number at which this occurs.

This study presents the findings of a wind tunnel experiment investigating the behaviour of micrometric inertial particles with Stokes numbers around unity in the turbulent wake of a stationary porous disk. Various concentrations $\varPhi _{v}\in ([6-19] \times 10^{-6})$ of poly-disperse water droplets (average diameter 40–50 $\mathrm {\mu }$m) are compared with sub-inertial tracer particles. Hot-wire anemometry, phase Doppler interferometry and particle image velocimetry were implemented in the near- and far-wake regions to study the complex dynamics of such particles. Quadrant analysis is used to explore the shear effects of the particle wake interaction. Turbulence statistics and particle size distributions reveal distinct differences in the structure of the wake when inertial particles are present in the flow. Additionally, there are different structures in the near and far wake regions and structures change with particle volume fraction.

The tangential strain rate in premixed flames impacts significantly the flame surface area generation and thus the combustion process. Studies on incompressible isotropic turbulence have revealed that the mean tangential strain rate at material and iso-scalar surfaces is positive and exhibits a universal value when normalized by the Kolmogorov time. This is associated with the preferential alignment of the surface normal with the most compressive principal strain rate. The present study investigates such effects in premixed hydrogen and iso-octane flame kernels using direct numerical simulations. It is shown that the normalized mean tangential strain rate of the investigated flames has a very similar value compared with the incompressible flows. However, in the reaction zone, the flame surface normal aligns preferentially with the most extensive principal strain rate. Furthermore, this alignment depends on the reaction progress variable and the Lewis number, while the tangential strain rate remains independent of these parameters. Such counter-intuitive behaviour is systematically investigated by decomposing the effects of dilatation and residual solenoidal turbulence. It is found that the solenoidal turbulence influences significantly the tangential strain rate. A general effect of turbulence on the tangential strain rate is identified, which is consistent with incompressible flows and independent of the Lewis number and the reaction progress variable. This is a remarkable finding indicating that models of the tangential strain rate developed based on incompressible flows apply also to premixed flames with different Lewis numbers, and, for the modelling, only the solenoidal turbulence should be considered.

Drainage and spreading processes in thin liquid films have received considerable attention in the past decades. Yet, our understanding of three-dimensional cases remains sparse, with only a few studies focusing on flat and axisymmetric substrates. Here, we exploit differential geometry to understand the drainage and spreading of thin films on curved substrates, under the assumption of negligible surface tension and hydrostatic gravity effects. We develop a solution for the drainage on a local maximum of a generic substrate. We then investigate the role of geometry in defining the spatial thickness distribution via an asymptotic expansion in the vicinity of the maximum. Spheroids with a much larger (respectively smaller) height than the equatorial radius are characterized by an increasing (respectively decreasing) coating thickness when moving away from the pole. These thickness variations result from a competition between the variations of the substrate's slope and mean curvature. The coating of a torus presents larger thicknesses and a faster spreading on the inner region than on the outer region, owing to the different curvatures in these two regions. In the case of an ellipsoid with three different axes, spatial modulations in the drainage solution are observed as a consequence of a faster drainage along the short principal axis, faithfully reproduced by a three-dimensional asymptotic solution. Leveraging the conservation of mass, an analytical solution for the average spreading front is obtained. The solutions are in agreement with numerical simulations and experimental measurements obtained from the coating of a curing polymer on diverse substrates.

This study discusses turbulent suspension flows of non-Brownian, non-colloidal, neutrally buoyant and rigid spherical particles in a Newtonian fluid over porous media with particles too large to penetrate and move through the porous layer. We consider suspension flows with the solid volume fraction ${{\varPhi _b}}$ ranging from 0 to 0.2, and different wall permeabilities, while porosity is constant at 0.6. Direct numerical simulations with an immersed boundary method are employed to resolve the particles and flow phase, with the volume-averaged Navier–Stokes equations modelling the flow within the porous layer. The results show that in the presence of particles in the free-flow region, the mean velocity and the concentration profiles are altered with increasing porous layer permeability because of the variations in the slip velocity and wall-normal fluctuations at the suspension-porous interface. Furthermore, we show that variations in the stress condition at the interface significantly affect the particle near-wall dynamics and migration toward the channel core, thereby inducing large modulations of the overall flow drag. At the highest volume fraction investigated here, ${{\varPhi _b}}= 0.2$, the velocity fluctuations and the Reynolds shear stress are found to decrease, and the overall drag increases due to the increase in the particle-induced stresses.

In this paper, we conduct a systematic study of the instability of a boundary layer over a rotating cone that is inserting into a supersonic stream with zero angle of attack. The base flow is obtained by solving the compressible boundary-layer equations using a marching scheme, whose accuracy is confirmed by comparing with the full Navier–Stokes solution. Setting the oncoming Mach number and the semi-apex angle to be 3 and 7$^\circ$, respectively, the instability characteristics for different rotating rates ($\bar \varOmega$, defined as the ratio of the rotating speed of the cone to the axial velocity) and Reynolds numbers ($R$) are revealed. For a rather weak rotation, $\bar \varOmega \ll 1$, only the modified Mack mode (MMM) exists, which is an extension of the supersonic Mack mode in a quasi-two-dimensional boundary layer to a rotation configuration. Further increase of $\bar \varOmega$ leads to the appearance of a cross-flow mode (CFM), coexisting with the MMM but in the quasi-zero frequency band. The unstable zones of the MMM and CFM merge together, and so they are referred to as the type-I instability. When $\bar \varOmega$ is increased to an $O(1)$ level, an additional unstable zone emerges, which is referred to as the type-II instability to be distinguished from the aforementioned type-I instability. The type-II instability appears as a centrifugal mode (CM) when $R$ is less than a certain value, but appears as a new CFM for higher Reynolds numbers. The unstable zone of the type-II CM enlarges as $\bar \varOmega$ increases. The vortex structures of these types of instability modes are compared, and their large-$R$ behaviours are also discussed.