In separation control with air-jet vortex generators in supersonic flow, the spacing of the jets in a row array has a crucial effect on the control effectiveness. Previous experimental studies have revealed the overall influence of jet spacing on air-jet vortex generator controlled shock-induced separation zones, focusing mostly on the separation regions. There is, however, a gap in knowledge regarding the mechanisms leading to these changes in control effectiveness, particularly on the interaction between multiple jets and the details of downstream flow evolution. Therefore, the objective of the current study is to provide detailed information on the underlying flow dynamics associated with the injection of a row of spanwise-inclined jets into a supersonic turbulent boundary layer – and in particular the effects of different jet spacings. Four different spacings were studied with large-eddy simulations. In addition, we performed oil-flow and schlieren visualizations of separation control in a 24$^\circ$ compression–ramp interaction with different jet-spacing configurations to validate and discuss our conclusions regarding the effects of jet/jet interactions on the separation-control effectiveness. We analyse the influence of jet spacing on the flow topology, induced vortical structures, and boundary-layer statistics. The jet-induced major vortex pair is the dominant flow structure energizing the near-wall boundary layer. The paper details how interactions amongst adjacent vortex pairs differ for varying jet spacings, thus influencing the momentum transfer and eventually the control efficiency. The dynamic behaviour of the flow was analysed using a three-dimensional dynamic mode decomposition. The resulting insights are key to the development of efficient control set-ups.

We study the dynamics of two air bubbles driven by the motion of a suspending viscous fluid in a Hele-Shaw channel with a small elevation along its centreline via physical experiment and numerical simulation of a depth-averaged model. For a single-bubble system we establish that, in general, the bubble propagation speed monotonically increases with bubble volume so that two bubbles of different sizes, in the absence of any hydrodynamic interactions, will either coalesce or separate in a finite time. However, our experiments indicate that the bubbles interact and that an unstable two-bubble state is responsible for the eventual dynamical outcome: coalescence or separation. These results motivate us to develop an edge-tracking routine and to calculate these weakly unstable two-bubble steady states from the governing equations. The steady states consist of pairs of ‘aligned’ bubbles that appear on the same side of the centreline with the larger bubble leading. We also discover, through time-dependent simulations and physical experiment, another class of two-bubble states that, surprisingly, are stable. In contrast to the ‘aligned’ steady states, these bubbles appear on either side of the centreline and are ‘offset’ from each other. We calculate the bifurcation structures of both classes of steady states as the flow rate and bubble volume ratio are varied. We find that they exhibit intriguing similarities to the single-bubble bifurcation structure, which has implications for the existence of $n$-bubble steady states.

We investigate scalings of turbulence dissipation and turbulence length/time scales in the fully developed turbulent channel flow region of wall distances $y$ where the ratio of turbulence production to turbulence dissipation oscillates close to 1. First, we study averages over both time and wall-parallel streamwise ($x$) and spanwise ($z$) planes at $y$. Turbulent channel flow data with friction velocity $u_{\tau }$, and global Reynolds number $Re_{\tau }$ ranging from $550$ to $5200$, suggest that the integral length scales of streamwise fluctuating velocities along the streamwise direction, and of wall-normal fluctuating velocities along the transverse direction, tend towards scaling with $y$, and that the respective turbulence dissipation coefficients tend towards being constant with increasing $Re_{\tau }$. However, the data for integral lengths of transverse fluctuating velocities in the transverse direction suggest that these lengths obey an asymptotic scaling $\sqrt {\delta y}$ (where $\delta$ is the channel half-width) with increasing $Re_{\tau }$. The corresponding turbulence dissipation's scaling seems to tend towards $\sqrt {Re_{\tau }}/Re_{\lambda }$, which is reminiscent of the non-equilibrium turbulence dissipation scaling found in boundary-free turbulent flows, $Re_{\lambda }$ being a $y$-local Taylor-length-based Reynolds number. The data do not exclude minor corrections from these asymptotic scalings, and in fact, suggest finite Reynolds number deviations to them. Second, we remove time averaging and study time-fluctuating averages over wall-parallel planes at $y$. We find that the time fluctuations of the turbulence dissipation coefficients and the Taylor-length-based Reynolds number are very strongly anti-correlated at all wall distances $y$ considered, reflecting a dominance of turbulent kinetic energy fluctuations at the lower frequencies, but a dominance of both turbulent kinetic energy and turbulence dissipation at the higher frequencies. In the case of the turbulence dissipation coefficient corresponding to the integral length of the wall-normal velocity along the transverse direction, it is possible to determine the cross-over frequency $f_c^*$ between these two behaviours, and we find $f_c^{*}\sim u_{\tau }/y$ for $Re_{\tau } = 950$, but $f_c^* \sim u_{\tau }/\delta$ for $Re_{\tau }=2000$, where there is evidence of very-large-scale motions.

A model for the third-order mixed velocity–scalar structure function $-\overline {(\delta u)( \delta \phi )^{2}}$ ($\delta a$ represents the spatial increment of the quantity $a$; $u$ is the velocity and $\phi$ is a scalar) is proposed for closing the transport equation of the second-order moment of the scalar increment $\overline {( \delta \phi )^{2}}$. The closure model is based on a gradient-type hypothesis with an eddy-viscosity model which exploits the analogy between the turbulent kinetic energy and a passive scalar when the Prandtl number, $Pr$, is equal to 1. The solutions of the closed transport equation agree well with both measurements and numerical simulations, when $Pr$ is not too different from 1. However, the agreement deteriorates when $Pr$ differs significantly from 1. Nevertheless, the calculations capture the effect expected when $Pr$ varies. For example, as $Pr$ increases, a range of scales emerges where $\overline {( \delta \phi )^{2}}$ remains constant. This emerging scaling range should correspond to the $k^{-1}$ spectral range in the three-dimensional scalar spectrum commonly denoted the viscous–convective range. Also when $Pr$ decreases below 1, the calculations reproduce the emergence of an expected inertial–diffusive range for scales larger than the Kolmogorov length scale.

The collapse of cavitation bubbles in channel flows can give rise to structural damage along neighbouring walls. Although the collapse of a bubble near a single wall has been studied extensively, less is known about bubble collapse between two walls, e.g. as in a channel. We conduct highly resolved, direct simulations of the Navier–Stokes equations to investigate the bubble dynamics and pressures produced by the collapse of a bubble between two parallel rigid walls. We examine the dependence of the dynamics and pressures on the initial bubble location, confinement and driving pressure. For a fixed initial stand-off distance, as the channel width increases the bubble volume, migration distance and re-entrant jet speed approach their single-wall counterparts. We obtain an expression for the minimum channel width at which the confinement does not affect the bubble dynamics depending on the driving pressure difference and initial stand-off distance. For a fixed channel width, varying stand-off distance reduced the maximum wall pressures in the channel relative to the single wall; the trend was consistent for three different driving pressures. Two different jetting behaviours are seen when the bubble is centred in the channel, depending on the channel width. Under significant confinement, wall-parallel re-entrant jets impinge upon each other and further intensify the collapse of the vortex ring.

In this paper, we report that reversals of the large-scale circulation in two-dimensional Rayleigh–Bénard (RB) convection can be suppressed by imposing sinusoidally distributed heating to the bottom plate. We examine how the frequency of flow reversals depends on the dimensionless wavenumber $k$ of the spatial temperature modulation with various modulation amplitude $A$. For sufficiently large $k$, the flow reversal frequency is close to that in the standard RB convection under uniform heating. However, when $k$ decreases, the frequency of flow reversal gradually becomes lower and can even be largely reduced. Furthermore, we examine the growth of the corner roll and the global flow structure based on Fourier mode decomposition, and reveal that the size of the corner roll diminishes as the wavenumber decreases. The reason is that the regions occupied by the cold phase can absorb heat from the hot plumes and thus lower their temperature, which reduces the corner roll's kinetic energy input provided by the buoyancy force, and weakens the feeding process of the corner rolls. This results in the locking of the corner roll into a smaller region near the corner, making it harder for a reversal to occur. Using the concept of horizontal convection caused by non-uniform heating, we find a relevant parameter $k/A$ to describe briefly how the reversal frequency depends on wavenumber and modulation amplitude. The present work provides a new way to control the flow reversals in RB convection through modifying temperature boundary conditions.

We study experimentally the impact of substrate topology on shear-flow-induced motion of a single bead at low particle Reynolds numbers. The substrates are regular quadratic and triangular arrangements of fixed spherical particles. Their topology is varied by using different spacings between the spheres. Here, we show that it has a strong impact not only on the critical Shields number for incipient bead motion but also on its motion above threshold. We focus on Shields numbers where the bead velocity is smaller than the settling velocity. For the different substrates, the data on the average bead velocity collapse on a master curve, showing the impact of the critical Shields number on the bead motion. To describe the bead motion, we develop a model for creeping flows based on expressions by Goldman, Cox and Brenner for the flow-induced forces and torques on a moving bead near a plane. Our model considers rolling and sliding motion. The bead detaches from the substrate on the downhill side at larger substrate spacing or higher Shields numbers, and flies through the interstices of the substrate until hitting the neighbouring substrate spheres. While sliding has only a minor effect on the average bead velocity, detachment has a strong impact. At large substrate spacings, it leads to a bistability, usually associated with inertial flows, even for adhesionless particles under creeping-flow conditions. The model shows good agreement with the experimental results.

Growth of a fluid-infused patch on a thin porous layer, e.g. on a piece of paper or cloth, is related to the transmission of virus particles through exhaled droplets and aerosols. We present a theoretical model to describe how a wet patch develops gradually through imbibition, once a sessile droplet attaches at a permeable surface and drains gradually into a thin porous layer. Two limiting cases are considered based on different assumptions on the motion of the contact line during the coupled process of drop drainage and patch growth: (i) the apparent contact angle remains unchanged, so the radius of a sessile droplet decreases with time; and (ii) the location of the contact line remains pinned, so the contact angle decreases as time progresses. The model leads to evolution pathways for both the droplet and the fluid film within the porous layer, without introducing arbitrary fitting parameters. Potential implications of the model and its solutions are also discussed briefly in the context of the outspread of COVID-19, employing physical parameters for exhaled droplets, paper and cloth.

Maas (J. Fluid Mech., vol. 684, 2011, pp. 5–24) showed that, for an oscillating two-dimensional barotropic tide flowing over sub-critical topography of compact support, some topographic forms existed that produced non-radiating baroclinic disturbances. The problem is related to ‘stealth’ and ‘cloaking’ problems. Here Maas's result is derived using a simpler approach, not involving complicated mappings, but formally restricted to perturbation topography. Wider results come from the discussion of nearly compact support topographic disturbances provided by Schwartz functions with weak high-wavenumber radiation and by exploiting both a known functional equation formulation and Fourier methods. The problem is extended to disturbances on uniform slopes. A variety of non-radiating topographies can be found, although they are mathematically delicate and unlikely to be found in nature. Topography with weak radiation at high wavenumber is a much wider class of structures. Application of these solutions would lie with the ability to estimate dissipation over and near the topography from motions observed at a distance.

We consider the axisymmetric displacement of an ambient fluid by a second input fluid of lower density and lower viscosity in a horizontal porous layer. If the two fluids have been segregated vertically by buoyancy, then the flow becomes self-similar with the input fluid preferentially flowing near the upper boundary. We show that this axisymmetric self-similar flow is stable to angular-dependent perturbations for any viscosity ratio. The Saffman–Taylor instability is suppressed due to the buoyancy segregation of the fluids. The radial extent of the segregated flow is inversely proportional to the viscosity ratio. This horizontal extension of the intrusion eliminates the discontinuity in the pressure gradient between the fluids associated with the viscosity contrast. Hence at late times, viscous fingering is shut down even for arbitrarily small density differences. The stability is confirmed through numerical integration of a coupled problem for the interface shape and the pressure gradient, and through complementary asymptotic analysis, which predicts the decay rate for each mode. The results are extended to anisotropic and vertically heterogeneous layers. The interface may have relatively steep shock-like regions, but the flow is always stable when the fluids have been segregated by buoyancy, as in a uniform layer.

We develop a new methodology to assess the streamwise inclination angles (SIAs) of the wall-attached eddies populating the logarithmic region with a given wall-normal height. To remove the influences originating from other scales on the SIA estimated via two-point correlation, the footprints of the targeted eddies in the vicinity of the wall and the corresponding streamwise velocity fluctuations carried by them are isolated simultaneously, by coupling the spectral stochastic estimation with the attached-eddy hypothesis. Datasets produced with direct numerical simulations spanning $Re_{\tau } \sim O(10^2)\unicode{x2013}O(10^3)$ are dissected to study the Reynolds number effect. The present results show, for the first time, that the SIAs of attached eddies are Reynolds-number-dependent in low and medium Reynolds numbers, and tend to saturate at $45^{\circ }$ as the Reynolds number increases. The mean SIA reported by vast previous experimental studies are demonstrated to be the outcomes of the additive effect contributed by multi-scale attached eddies. These findings clarify the long-term debate and perfect the picture of the attached-eddy model.