A detailed experimental campaign into separated shear layers stemming from rectangular sections (having aspect ratios of 5 : 1, 3 : 1 and 1 : 1) was carried out at Reynolds numbers range between $1.34\times 10^{4}$ and $1.18\times 10^{5}$ based on the body thickness. Particle image velocimetry was used to locate the highest concentration of fluctuations in the velocity field and subsequent hot-wire measurements at those locations provided adequate spectral resolution to follow the evolution of various instabilities that are active within the separated shear layer. Similar to recent findings by this same group, the shear layer behaviour is observed to contain a combination of Reynolds invariant characteristics, including its time-averaged position, while other properties demonstrate clear Reynolds number dependency, including the spatial amplification of turbulent kinetic energy. Additional results here show that the ratio of side lengths of the body is a key parameter in revealing these effects. One reason for this is the level of coupling between modes of instability, which is evaluated using two-point correlation methods. These findings indicate that the separated shear layer on a bluff body is highly nonlinear. A specific set of scales responsible for these unique behaviours is identified and discussed, along with their relationship to other scales in the flow.

Experimental measurements of the force and torque on freely settling fibres are compared with predictions of the slender-body theory of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462). Although the flow is viscous dominated at the scale of the fibre diameter, fluid inertia is important on the scale of the fibre length, leading to inertial torques which tend to rotate symmetric fibres toward horizontal orientations. Experimentally, the torque on symmetric fibres is inferred from the measured rate of rotation of the fibres using a quasi-steady torque balance. It is shown theoretically that fibres with an asymmetric radius or mass density distribution undergo a supercritical pitch-fork bifurcation from vertical to oblique settling with increasing Archimedes number, increasing Reynolds number or decreasing asymmetry. This transition is observed in experiments with asymmetric mass density and we find good agreement with the predicted symmetry breaking transition. In these experiments, the steady orientation of the oblique settling fibres provides a means to measure the inertial torque in the absence of transient effects since it is balanced by the known gravitational torque.

While numerous theoretical studies exist on microstreaming around acoustically excited, trapped gas bubbles, experimental approaches have mainly been conducted for bubbles attached to a solid boundary. One of the main difficulties lies in the positional stability of the microbubble. In the present work we trigger surface modes by bubble coalescence, with the advantage of limiting translational instabilities and controlling the orientation of the axisymmetric deformation. Furthermore, streaming is visualised by fluorescent tracer particles. In this way, bubble dynamics and streaming patterns can be studied together. Different types of streaming patterns are observed and correlated to the respective mode number. Besides the mode number, the bubble size and the phase difference between modal components are identified as important parameters in the definition of the pattern type.

Stratification due to salt or heat gradients greatly affects the distribution of inert particles and living organisms in the ocean and the lower atmosphere. Laboratory studies considering the settling of a sphere in a linearly stratified fluid confirmed that stratification may dramatically enhance the drag on the body, but failed to identify the generic physical mechanism responsible for this increase. We present a rigorous splitting scheme of the various contributions to the drag on a settling body, which allows them to be properly disentangled whatever the relative magnitude of inertial, viscous, diffusive and buoyancy effects. We apply this splitting procedure to data obtained via direct numerical simulation of the flow past a settling sphere over a range of parameters covering a variety of situations of laboratory and geophysical interest. Contrary to widespread belief, we show that, in the parameter range covered by the simulations, the drag enhancement is generally not primarily due to the extra buoyancy force resulting from the dragging of light fluid by the body, but rather to the specific structure of the vorticity field set in by buoyancy effects. Simulations also reveal how the different buoyancy-induced contributions to the drag vary with the flow parameters. To unravel the origin of these variations, we analyse the different possible leading-order balances in the governing equations. Thanks to this procedure, we identify several distinct regimes which differ by the relative magnitude of length scales associated with stratification, viscosity and diffusivity. We derive the scaling laws of the buoyancy-induced drag contributions in each of these regimes. Considering tangible examples, we show how these scaling laws combined with numerical results may be used to obtain reliable predictions beyond the range of parameters covered by the simulations.

We describe the long-term dynamics of sustained stratified shear flows in the laboratory. The stratified inclined duct (SID) experiment sets up a two-layer exchange flow in an inclined duct connecting two reservoirs containing salt solutions of different densities. This flow is primarily characterised by two non-dimensional parameters: the tilt angle of the duct with respect to the horizontal, $\unicode[STIX]{x1D703}$ (a few degrees at most), and the Reynolds number $Re$, an input parameter based on the density difference driving the flow. The flow can be sustained with constant forcing over arbitrarily long times and exhibits a wealth of dynamical behaviours representative of geophysically relevant sustained stratified shear flows. Varying $\unicode[STIX]{x1D703}$ and $Re$ leads to four qualitatively different regimes: laminar flow; mostly laminar flow with finite-amplitude, travelling Holmboe waves; spatio-temporally intermittent turbulence with substantial interfacial mixing; and sustained, vigorous interfacial turbulence (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242–253). We seek to explain the scaling of the transitions between flow regimes in the two-dimensional plane of input parameters $(\unicode[STIX]{x1D703},Re)$. We improve upon previous studies of this problem by providing a firm physical basis and non-dimensional scaling laws that are mutually consistent and in good agreement with the empirical transition curves we inferred from 360 experiments spanning $\unicode[STIX]{x1D703}\in [-1^{\circ },6^{\circ }]$ and $Re\in [300,5000]$. To do so, we employ state-of-the-art simultaneous volumetric measurements of the density field and the three-component velocity field, and analyse these experimental data using time- and volume-averaged potential and kinetic energy budgets. We show that regime transitions are caused by an increase in the non-dimensional time- and volume-averaged kinetic energy dissipation within the duct, which scales with $\unicode[STIX]{x1D703}Re$ at high enough angles. As the power input scaling with $\unicode[STIX]{x1D703}Re$ is increased above zero, the two-dimensional, parallel-flow dissipation (power output) increases to close the budget through an increase in the magnitude of the exchange flow, incidentally triggering Holmboe waves above a certain threshold in interfacial shear. However, once the hydraulic limit of two-layer exchange flows is reached, two-dimensional dissipation plateaus and three-dimensional dissipation at small scales (turbulence) takes over, at first intermittently, and then steadily, in order to close the budget and follow the $\unicode[STIX]{x1D703}Re$ scaling. This general understanding of regime transitions and energetics in the SID experiment may serve as a basis for the study of more complex sustained stratified shear flows found in the natural environment.

The recirculation zone length behind a bluff body is influenced by the turbulence intensity at the base of the body in isothermal flows and also the heat release and its interaction with turbulence in reacting flows. This relationship is observed to be nonlinear and is controlled by the balance of forces acting on the recirculation zone, which arise from the pressure and turbulence fields. The pressure force is directly influenced by the volumetric expansion resulting from the heat release, whereas the change in the turbulent shear force depends on the nonlinear interaction between turbulence and combustion. This behaviour is elucidated through a control volume analysis. A scaling relation for the recirculation zone length is deduced to relate the turbulence intensity and the amount of heat release. This relation is verified using the large eddy simulation data from 20 computations of isothermal flows and premixed flames that are stabilised behind the bluff body. The application of this scaling to flames in an open environment and behind a backward facing step is also explored. The observations and results are explained on a physical basis.

A comprehensive study of the two-dimensional incompressible shear-driven flow in an open square cavity is carried out. Two successive bifurcations lead to two limit cycles with different frequencies and different numbers of structures which propagate along the top of the cavity and circulate in its interior. A branch of quasi-periodic states produced by secondary Hopf bifurcations transfers the stability from one limit cycle to the other. A full analysis of this scenario is obtained by means of nonlinear simulations, linear stability analysis and Floquet analysis. We characterize the temporal behaviour of the limit cycles and quasi-periodic state via Fourier transforms and their spatial behaviour via the Hilbert transform. We address the relevance of linearization about the mean flow. Although here the nonlinear frequencies are not very far from those obtained by linearization about the base flow, the difference is substantially reduced when eigenvalues are obtained instead from linearization about the mean and in addition, the corresponding growth rate is small, a combination of properties called RZIF (real zero imaginary frequency). Moreover growth rates obtained by linearization about the mean of one limit cycle are correlated with relative stability to the other limit cycle. Finally, we show that the frequencies of the successive modes are separated by a constant increment.

Eigenmodes of averaged small-amplitude perturbations to a turbulent channel flow – which is one of the most fundamental canonical flows – are identified for the first time via an extensive set of high-fidelity graphics processing unit accelerated direct numerical simulations. While the system governing averaged small-amplitude perturbations to turbulent channel flow remains unknown, the fact such eigenmodes can be identified constitutes direct evidence that it is linear. Moreover, while the eigenvalue associated with the slowest-decaying anti-symmetric eigenmode mode is found to be real, the eigenvalue associated with the slowest-decaying symmetric eigenmode mode is found to be complex. This indicates that the unknown linear system governing the evolution of averaged small-amplitude perturbations cannot be self-adjoint, even for the case of a uni-directional flow. In addition to elucidating aspects of the flow physics, the findings provide guidance for development of new unsteady Reynolds-averaged Navier–Stokes turbulence models, and constitute a new and accessible benchmark problem for assessing the performance of existing models, which are used widely throughout industry.

Acoustic microstreaming due to an oscillating microbubble, either coated or free, is analytically investigated. The detailed flow field is obtained and the closed streamlines of the ring vortex generated by microstreaming are plotted in both Eulerian and Lagrangian descriptions. Analytical expressions are found for the ring vortex showing that its length depends only on the separation of the microbubble from the wall and the dependence is linear. The circulation as a scalar measure of the vortex is computed quantitatively identifying its spatial location. The functional dependence of circulation on bubble separation and coating parameters is shown to be similar to that of the shear stress.

We propose a Boussinesq-type model to study the surface/interfacial wave manifestation of an underlying, slowly varying, long-wavelength baroclinic flow in a two-layer, density-stratified system. The results of our model show numerically that, under strong nonlinearity, surface waves, with their typical wavenumber being the resonant $k_{res}$, can be generated locally at the leading edge of the underlying, slowly varying, long-wavelength baroclinic flow. Here, the resonant $k_{res}$ satisfies the class 3 triad resonance condition among two short-mode waves and one long-mode wave in which all waves propagate in the same direction. Moreover, when the slope of the baroclinic flow is sufficiently small, only one spatially localized large-amplitude surface wave packet can be generated at the leading edge. This localized surface wave packet becomes high in amplitude and large in group velocity after the interaction with its surrounding waves. These results are qualitatively consistent with various experimental observations including resonant surface waves at the leading edge of an internal wave. Subsequently, we propose a mechanism, referred to as the modulation-resonance mechanism, underlying these surface phenomena, based on our numerical simulations. The proposed modulation-resonance mechanism combines the linear modulation, ray-based, theory for the spatiotemporal asymmetric behaviour of surface waves and the nonlinear class 3 triad resonance theory for the energy focusing of surface waves around the resonant wavenumber $k_{res}$ in Fourier space.

We herein report an experimental study on the morphological evolution of a vortex ring formed inside a liquid pool after it is impacted and penetrated by a coalescing drop of the same liquid. The dynamics of the penetrating vortex ring along with the deformation of the pool surface has been captured using simultaneous high-speed laser induced fluorescence and shadowgraph techniques. It is identified that the motion of such a vortex ring can be divided into three stages, during which inertial, capillary and viscous effects alternatingly play dominant roles to modulate the penetration process, resulting in linear, non-monotonic and decelerating motion in these three stages respectively. Furthermore, we also evaluate the relevant time and length scales of these three stages and subsequently propose a unified description of the downward motion of the penetrating vortex ring. Finally, we use the experimental data for a range of drop diameters and impact speeds to validate the proposed scaling.

We present high-resolution implicit large eddy simulation (iLES) of the turbulent air-entraining flow in the wake of three-dimensional rectangular dry transom sterns with varying speeds and half-beam-to-draft ratios $B/D$. We employ two-phase (air/water), time-dependent simulations utilizing conservative volume-of-fluid (cVOF) and boundary data immersion (BDIM) methods to obtain the flow structure and large-scale air entrainment in the wake. We confirm that the convergent-corner-wave region that forms immediately aft of the stern wake is ballistic, thus predictable only by the speed and (rectangular) geometry of the ship. We show that the flow structure in the air–water mixed region contains a shear layer with a streamwise jet and secondary vortex structures due to the presence of the quasi-steady, three-dimensional breaking waves. We apply a Lagrangian cavity identification technique to quantify the air entrainment in the wake and show that the strongest entrainment is where wave breaking occurs. We identify an inverse dependence of the maximum average void fraction and total volume entrained with $B/D$. We determine that the average surface entrainment rate initially peaks at a location that scales with draft Froude number and that the normalized average air cavity density spectrum has a consistent value providing there is active air entrainment. A small parametric study of the rectangular geometry and stern speed establishes and confirms the scaling of the interface characteristics with draft Froude number and geometry. In Part 2 (Hendrikson & Yue, J. Fluid Mech., vol. 875, 2019, pp. 884–913) we examine the incompressible highly variable density turbulence characteristics and turbulence closure modelling.

We analyse the turbulence characteristics and consider the closure modelling of the air entraining flow in the wake of three-dimensional, rectangular dry transom sterns obtained using high-resolution implicit large eddy simulations (iLES) (Hendrickson et al., J. Fluid Mech., vol. 875, 2019, pp. 854–883). Our focus is the incompressible highly variable density turbulence (IHVDT) in the near surface mixed-phase region ${\mathcal{R}}$ behind the stern. We characterize the turbulence statistics in ${\mathcal{R}}$ and determine it to be highly anisotropic due to quasi-steady wave breaking. Using unconditioned Reynolds decomposition for our analysis, we show that the turbulent mass flux (TMF) is important in IHVDT for the production of turbulent kinetic energy and is as relevant to the mean momentum equations as the Reynolds stresses. We develop a simple, regional explicit algebraic closure model for the TMF based on a functional relationship between the fluxes and tensor flow quantities. A priori tests of the model show mean density gradients and buoyancy effects are the main driving parameters for predicting the turbulent mass flux and the model is capable of capturing the highly localized nature of the TMF in ${\mathcal{R}}$.

The spectrum of turbulence kinetic energy for homogeneous turbulence is generally computed using the Fourier transform of the velocity field from physical three-dimensional space to wavenumber $k$. This analysis works well for single-phase homogeneous turbulent flows. In the case of multiphase turbulent flows, instead, the velocity field is non-smooth at the interface between the carrier fluid and the dispersed phase; thus, the energy spectra computed via Fourier transform exhibit spurious oscillations at high wavenumbers. An alternative definition of the spectrum uses the wavelet transform, which can handle discontinuities locally without affecting the entire spectrum while additionally preserving spatial information about the field. In this work, we propose using the wavelet energy spectrum to study multiphase turbulent flows. Also, we propose a new decomposition of the wavelet energy spectrum into three contributions corresponding to the carrier phase, droplets and interaction between the two. Lastly, we apply the new wavelet-decomposition tools in analysing the direct numerical simulation data of droplet-laden decaying isotropic turbulence (in absence of gravity) of Dodd & Ferrante (J. Fluid Mech., vol. 806, 2016, pp. 356–412). Our results show that, in comparison to the spectrum of the single-phase case, the droplets (i) do not affect the carrier-phase energy spectrum at high wavenumbers ($k_{m}/k_{min}\geqslant 128$), (ii) increase the energy spectrum at high wavenumbers ($k_{m}/k_{min}\geqslant 256$) by increasing the interaction energy spectrum at these wavenumbers and (iii) decrease the energy at low wavenumbers ($k_{m}/k_{min}\leqslant 16$) by increasing the dissipation rate at these wavenumbers.

This paper presents a combined experimental and large-eddy simulation study to characterise the effect of aspect ratio on the near-wake structure of a square finite wall-mounted cylinder (FWMC). The cylinder aspect ratios (span $L$ to width $W$) investigated in the experiments were $1.4\leqslant L/W\leqslant 21.4$ and the oncoming boundary-layer thicknesses were $1.3W$ and $0.9W$ at a Reynolds number based on cylinder width of $1.4\times 10^{4}$ and $1.1\times 10^{4}$, respectively. In complementary simulations, the cylinder aspect ratios investigated were 1.4, 4.3, 10 and 18.6. The cylinder wake structure was visualised in three-dimensional space using a vortex core detection method and decomposed to its oscillation modes using the spectral proper orthogonal decomposition (SPOD) technique. A parametric diagram is proposed to predict whether the time-averaged wake structure is a dipole or a quadrupole pattern, based on oncoming boundary-layer height and aspect ratio. Cellular shedding occurs when the aspect ratio is high with up to three shedding cells occurring across the span for aspect ratios $L/W>18$. Each of these cells sheds at a distinct frequency, as evidenced by the spectral content of the surface pressure measured on the side face and the near-wake velocity. Amplitude modulation is also observed in the vortex shedding, which explains the amplitude modulation of the acoustic pressure emitted by square FWMCs. SPOD is shown to be a viable method to identify the occurrence of cellular shedding in the wake.

According to the celebrated Bolgiano–Obukhov (Bolgiano, J. Geophys. Res., vol. 64 (12), 1959, pp. 2226–2229; Obukhov, Dokl. Akad. Nauk SSSR, vol. 125, 1959, p. 1246) phenomenology for moderately stably stratified turbulence, the energy spectrum in the inertial range shows a dual scaling: the kinetic energy follows (i) ${\sim}k^{-11/5}$ for $k<k_{B}$, and (ii) ${\sim}k^{-5/3}$ for $k>k_{B}$, where $k_{B}$ is the Bolgiano wavenumber. The $k^{-5/3}$ scaling, akin to passive scalar turbulence, is a direct consequence of the assumption that buoyancy is insignificant for $k>k_{B}$. We revisit this assumption, and using the constancy of kinetic and potential energy fluxes and simple theoretical analysis, we find that the $k^{-5/3}$ spectrum is absent. This is because the velocity field at small scales is too weak to establish a constant kinetic energy flux as in passive scalar turbulence. A quantitative condition for the existence of the second regime is also derived in the paper.

Bulk-to-shear viscosity ratios of three orders of magnitude are often reported in carbon dioxide but are always neglected when predicting aerothermal loads in external (Mars exploration) or internal (turbomachinery, heat exchanger) turbulent flows. The recent (and first) numerical investigations of that matter suggest that the solenoidal turbulence kinetic energy is in fact well predicted despite this seemingly arbitrary simplification. The present work argues that such a conclusion may reflect limitations from the choice of configuration rather than provide a definite statement on the robustness of kinetic-energy transfers to the use of Stokes’ hypothesis. Two distinct asymptotic regimes (Euler–Landau and Stokes–Newton) in the eigenmodes of the Navier–Stokes equations are identified. In the Euler–Landau regime, the one captured by earlier studies, acoustic and entropy waves are damped by transport coefficients and the dilatational kinetic energy is dissipated, even more rapidly for high bulk-viscosity fluids and/or forcing frequencies. If the kinetic energy is initially or constantly injected through solenoidal motions, effects on the turbulence kinetic energy remain minor. However, in the Stokes–Newton regime, diffused bulk compressions and advected isothermal compressions are found to prevail and promote small-scale enstrophy via vorticity–dilatation correlations. In the absence of bulk viscosity, the transition to the Stokes–Newton regime occurs within the dissipative scales and is not observed in practice. In contrast, at high bulk viscosities, the Stokes–Newton regime can be made to overlap with the inertial range and disrupt the enstrophy at small scales, which is then dissipated by friction. Thus, flows with substantial inertial ranges and large bulk-to-shear viscosity ratios should experience enhanced transfers to small-scale solenoidal kinetic energy, and therefore faster dissipation rates leading to modifications of the heat-transfer properties. Observing numerically such transfers is still prohibitively expensive, and the present simulations are restricted to two-dimensional turbulence. However, the theory laid here offers useful guidelines to design experimental studies to track the Stokes–Newton regime and associated modifications of the turbulence kinetic energy, which are expected to persist in three-dimensional turbulence.

We investigate the slat narrowband peak noise generating mechanism. Unsteady flow data were generated by a lattice-Boltzmann-based commercial code for four configurations, accounting for variations of the airfoil angle of attack and slat overlap. Comparison with experimental results indicates that the aspects of the flow field relevant for the generation of the narrowband peaks were accurately captured. Frequency-domain proper orthogonal decomposition (POD) is applied to identify dominant large-scale structures in the frequency range dominated by the peaks. The combined use of the two POD metrics, namely, the turbulent kinetic energy in the turbulent flow region and the acoustic pressure in the far field, demonstrated that the structures most correlated with the noise resemble spanwise coherent Kelvin–Helmholtz vortices which dominate the slat cove only at the frequency of the narrowband peaks. Time evolution of the structures educed using the acoustic pressure correlation provides detailed evidence of the hydrodynamic and acoustic steps of a Rossiter-like feedback mechanism between the slat cusp and trailing edge. The combined analysis of results for the different slat configurations provides an explanation for the effect of the slat configuration on the amplitude of the narrowband peaks observed in previous studies, particularly the influence of the main-element suction peak.

In this study, the flow dynamics and heat transfer in partially filled pin-based microchannel heat sinks (MCHS) are examined. The lattice Boltzmann method is used to analyse the physics of these systems and examine the effects of the flow, pin configuration, size and porous medium height. The results of the study reveal that, unlike the fully filled pin-based MCHS, there is no unique behaviour for the pin configuration effects and the performance of partially filled pin-based MCHS depends on the porous medium size and structure as well as the inertial forces in the flow. In particular, it is found that there are hydrodynamic and thermal-based critical porous medium heights at which the best performance in terms of heat removal switches from the inline to the staggered configuration. The dependence of these critical heights on the Reynolds number and the porous medium properties are analysed and the effects of the flow dynamics are further unravelled through a particle tracing technique. Furthermore, a simple flow model is developed, and is shown to capture well the main trends obtained from the simulations and to bring to light more of the system physics that help explain the interplay between the different parameters.

Shallow granular avalanches on slopes close to repose exhibit hysteretic behaviour. For instance, when a steady-uniform granular flow is brought to rest it leaves a deposit of thickness $h_{stop}(\unicode[STIX]{x1D701})$ on a rough slope inclined at an angle $\unicode[STIX]{x1D701}$ to the horizontal. However, this layer will not spontaneously start to flow again until it is inclined to a higher angle $\unicode[STIX]{x1D701}_{start}$, or the thickness is increased to $h_{start}(\unicode[STIX]{x1D701})>h_{stop}(\unicode[STIX]{x1D701})$. This simple phenomenology leads to a rich variety of flows with co-existing regions of solid-like and fluid-like granular behaviour that evolve in space and time. In particular, frictional hysteresis is directly responsible for the spontaneous formation of self-channelized flows with static levees, retrogressive failures as well as erosion–deposition waves that travel through the material. This paper is motivated by the experimental observation that a travelling-wave develops, when a steady uniform flow of carborundum particles on a bed of larger glass beads, runs out to leave a deposit that is approximately equal to $h_{stop}$. Numerical simulations using the friction law originally proposed by Edwards et al. (J. Fluid Mech., vol. 823, 2017, pp. 278–315) and modified here, demonstrate that there are in fact two travelling waves. One that marks the trailing edge of the steady-uniform flow and another that rapidly deposits the particles, directly connecting the point of minimum dynamic friction (at thickness $h_{\ast }$) with the deposited layer. The first wave moves slightly faster than the second wave, and so there is a slowly expanding region between them in which the flow thins and the particles slow down. An exact inviscid solution for the second travelling wave is derived and it is shown that for a steady-uniform flow of thickness $h_{\ast }$ it produces a deposit close to $h_{stop}$ for all inclination angles. Numerical simulations show that the two-wave structure deposits layers that are approximately equal to $h_{stop}$ for all initial thicknesses. This insensitivity to the initial conditions implies that $h_{stop}$ is a universal quantity, at least for carborundum particles on a bed of larger glass beads. Numerical simulations are therefore able to capture the complete experimental staircase procedure, which is commonly used to determine the $h_{stop}$ and $h_{start}$ curves by progressively increasing the inclination of the chute. In general, however, the deposit thickness may depend on the depth of the flowing layer that generated it, so the most robust way to determine $h_{stop}$ is to measure the deposit thickness from a flow that was moving at the minimum steady-uniform velocity. Finally, some of the pathologies in earlier non-monotonic friction laws are discussed and it is explicitly shown that with these models either steadily travelling deposition waves do not form or they do not leave the correct deposit depth $h_{stop}$.