We generalise the classical Benjamin solution (Benjamin, J. Fluid Mech., vol. 31, 1968, pp. 209–248) modelling the flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the surface detaches from the ceiling of the duct as a free surface. It is shown that the Benjamin solution belongs to a one-parameter family of similar solutions, which are divided into two types: solutions that describe potential flows where the free surface of the fluid is deflected from the duct ceiling at a zero angle; and solutions that admit the formation of a vortex flow region in the vicinity of the point of fluid separation from the duct ceiling. It is shown that this one-parameter family of solutions is the limit of a two-parameter family of solutions in which part of the uniform flow energy is converted into energy of the small-scale fluid motion. Based on the local hydrostatic approximation, the applicability of the constructed solutions is discussed.

This work presents an experimental validation study of Isaacs’ incompressible unsteady-airfoil theory at Reynolds numbers above $10^{6}$, and explores the validity of the classical Kutta condition applied to surging flows. Harmonic variation of the free-stream velocity was produced by rotating choke vanes in an unsteady transonic wind tunnel, with time-resolved lift coefficients determined from surface pressure measurements on a NACA 0018 airfoil. Unsteady lift results demonstrate the same trends with reduced frequency and velocity amplitude ratio that are predicted by Isaacs’ theory. However, significant deviations of the lift magnitude and phase angle are observed. In order to understand the cause of these deviations, the background-oriented schlieren technique was used to visualize density gradients in the immediate vicinity of the airfoil trailing edge. The time-resolved background-oriented schlieren displacement field indicates oscillatory behaviour of the trailing-edge stagnation streakline, which violates the classical Kutta condition for this unsteady surging flow.

Double diffusion occurs when the fluid density depends on two components that diffuse at different rates (e.g. heat and salt in the ocean). Double diffusion can lead to an up-gradient buoyancy flux and drive motion at the expense of potential energy. Here, we follow the work of Lorenz (Tellus, vol. 7 (no. 2), 1955, pp. 157–167) and Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128) for a single-component fluid and define the background potential energy (BPE) as the energy associated with an adiabatically sorted density field and derive its budget for a double-diffusive fluid. We find that double diffusion can convert BPE into available potential energy (APE), unlike in a single-component fluid, where the transfer of APE to BPE is irreversible. We also derive an evolution equation for the sorted buoyancy in a double-diffusive fluid, extending the work of Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193) and Nakamura (J. Atmos. Sci., vol. 53 (no. 11), 1996, pp. 1524–1537). The criterion we develop for a release of BPE can be used to analyse the energetics of mixing and double diffusion in the ocean and other multiple-component fluids, and we illustrate its application using two-dimensional simulations of salt fingering.

We consider the time-dependent speed of an infinitely long cylinder that minimizes the net work done on the surrounding fluid to travel a given distance perpendicular to its axis in a fixed amount of time. The flow that develops is two-dimensional. An analytical solution is possible using calculus of variations for the case that the distance travelled and the viscous boundary layer thickness that develops are much smaller than the circle radius. If $t$ represents the time since the commencement of motion and $T$ the final time, then the optimum speed profile is $Ct^{1/4}(T-t)^{1/4}$, where $C$ is determined by the distance travelled. The result also holds for rigid-body translations and rotation of cylinders formed by extrusion of smooth but otherwise arbitrary curves.

The control of bluff-body wakes for reduced drag and enhanced stability has traditionally relied on the so-called direct-wake control approach. By the use of actuators or passive devices, one can manipulate the aerodynamic loads that act on the rear of the model. An alternative approach for the manipulation of the flow is to move the position of the actuator upstream, hence interacting with an easier-to-manipulate boundary layer. The present paper comprises a bluff-body flow study via large-eddy simulations to investigate the effectiveness of an upstream actuator (positioned at the leading edge) with regard to the manipulation of the wake dynamics and its aerodynamic loads. A rectangular cylinder with rounded leading edges, equipped with actuators positioned at the front curvatures, is simulated at $Re=40\,000$. A genetic algorithm (GA) optimization is performed to find an effective actuation that minimizes drag. It is shown that the GA selects superharmonic frequencies of the natural vortex shedding. Hence, the induced disturbances, penetrating downstream in the wake, significantly reduce drag and lateral instability. A comparison with a side-recirculation-suppression approach is also presented, the latter case being worse in terms of reduced drag (only 8 % drag reduction achieved), despite the total suppression of the side recirculation bubble. In contrast, the GA optimized case contributes to a 20 % drag reduction with respect to the unactuated case. In addition, the large drag reduction is associated with a reduced shedding motion and an improved lateral stability.

Buoyancy-driven exchange flows arise in the natural and built environment wherever bodies of fluids at different densities are connected by a narrow constriction. In this paper we study these flows in the laboratory using the canonical stratified inclined duct experiment, which sustains an exchange flow in an inclined duct of rectangular cross-section over long time periods (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242–253). We study the behaviour of these sustained stratified shear flows by focusing on three dependent variables of particular interest: the qualitative flow regime (laminar, wavy, intermittently turbulent or fully turbulent), the mass flux (net transport of buoyancy between reservoirs) and the interfacial thickness (thickness of the layer of intermediate density between the two counter-flowing layers). Dimensional analysis reveals five non-dimensional independent input parameters: the duct aspect ratios in the longitudinal direction $A$ and spanwise direction $B$, the tilt angle $\unicode[STIX]{x1D703}$, the Reynolds number $Re$ (based on the initial buoyancy difference driving the flow) and the Prandtl number $Pr$ (we consider both salt and temperature stratifications). After reviewing the literature and open questions on the scaling of regimes, mass flux and interfacial thickness with $A,B,\unicode[STIX]{x1D703},Re,Pr$, we present the first extensive, unified set of experimental data where we varied systematically all five input parameters and measured all three output variables with the same methodology. Our results in the $(\unicode[STIX]{x1D703},Re)$ plane for five sets of $(A,B,Pr)$ reveal a variety of scaling laws, and a non-trivial dependence of all three variables on all five parameters, in addition to a sixth elusive parameter. We further develop three classes of candidate models to explain the observed scaling laws: (i) the recent volume-averaged energetics of Lefauve et al. (J. Fluid Mech., vol. 848, 2019, pp. 508–544); (ii) two-layer frictional hydraulics; (iii) turbulent mixing models. While these models provide significant qualitative and quantitative descriptions of the experimental results, they also highlight the need for further progress on shear-driven turbulent flows and their interfacial waves, layering, intermittency and mixing properties.

Transport of viscous fluid through porous media is a direct consequence of the pore structure. Here we investigate transport through a specific class of two-dimensional porous geometries, namely those formed by fluid-mechanical erosion. We investigate the tortuosity and dispersion by analyzing the first two statistical moments of tracer trajectories. For most initial configurations, tortuosity decreases in time as a result of erosion increasing the porosity. However, we find that tortuosity can also increase transiently in certain cases. The porosity-tortuosity relationships that result from our simulations are compared with models available in the literature. Asymptotic dispersion rates are also strongly affected by the erosion process, as well as by the number and distribution of the eroding bodies. Finally, we analyze the pore size distribution of an eroding geometry. The simulations are performed by combining a boundary integral equation solver for the fluid equations, a second-order stable time-stepping method to simulate erosion, and high-order numerical methods to stably and accurately resolve nearly touching eroded bodies and particle trajectories near the eroding bodies.

The problem of the transmission of wave energy through small gaps arises in a variety of physical contexts. Here we consider the problem of Rossby waves encountering a barrier with two small gaps. In contrast to waves encountering a barrier with one small gap, in which very little wave energy is predicted to transmit across the barrier, when there are two or more gaps linear theory predicts that the barrier may be surprisingly inefficient at blocking the transmission of Rossby wave energy, owing to the requirement that circulation be conserved around individual segments of the barrier. However, the theory neglects viscosity in the main basin interiors and nonlinear effects in the basins and the gaps. To examine these effects, here we present the results of a series of laboratory experiments in which Rossby basin modes interact with a barrier with zero, one or two gaps. We find that the large-scale waves are able to transmit across the barrier with two gaps as predicted by the theory. However, while the linear theory captures the large-scale flow structures, viscosity and nonlinearity significantly affect the flow along the boundaries and near the gaps in the barrier.

The effects of leading-edge tubercles, based on the flippers of humpback whales, on the flow field around an airfoil undergoing large-amplitude dynamic changes in the angle of attack have been studied experimentally. Airfoils were pitched from an initial angle of attack of $0^{\circ }$ to $50^{\circ }$ at constant pitch rates with a chord Reynolds number of 12 000. Velocity and vorticity fields around a standard NACA 0012 airfoil and NACA 0012 modified with leading-edge tubercles were quantified using molecular tagging velocimetry. Vortex dynamics were characterized by tracking the location, core radius and circulation. The resulting velocity fields showed that the dynamics of the formation and separation of the leading-edge vortex were fundamentally different between the straight leading-edge airfoil and the tubercled airfoil. The tubercled airfoil also showed spanwise variation in dynamics of the dynamic stall vortex (DSV) formation. The characteristics of the DSV, specifically the circulation and proximity of the DSV to the airfoil suction surface, are known to have an impact on lift during dynamic pitching. The results showed that the DSV was stronger and remained closer to the airfoil longer for the modified airfoil. The baseline DSV convected away from the airfoil faster than the DSV on the tubercled airfoil once it began to separate from the airfoil. Shear-layer vortices, which form during dynamic stall near the mid-cord region, appeared to affect the convective behaviour of the DSV. The results suggest that the leading-edge tubercles observed on Humpback whale flippers act as passive flow-control mechanisms to control or delay dynamic stall.

Wall-resolved and wall-modelled large-eddy simulations are performed to study the localizing effect of Langmuir cells (LCs) on small-scale background turbulence in shallow water. The total velocity fluctuations are decomposed into an LC content extracted by streamwise averaging and a background turbulence part. Based on the large-scale motions of LCs, the spanwise domain is divided into three regions dominated by the upwelling, spanwise and downwelling flows of LCs, respectively. The localized Reynolds stresses $\langle u_{i}^{T}u_{j}^{T}\rangle _{xt}$ in different spanwise regions are compared to show the localizing effects of the LCs on the background turbulence quantitatively, where $u_{1}^{T}$ (or $u^{T}$), $u_{2}^{T}$ (or $v^{T}$) and $u_{3}^{T}$ (or $w^{T}$) represent the streamwise, vertical and spanwise components of the background turbulence velocity, respectively, and $\langle \cdot \rangle _{xt}$ denotes time and streamwise averaging. It is shown that the magnitudes of the localized Reynolds stresses in different spanwise regions vary significantly. The transport equations of the localized Reynolds stresses are then analysed to investigate the mechanisms underlying the localizing effects. It is discovered that the difference in the energy production correlated to the shear of the LC content among different regions is the key factor that leads to the localization of background turbulence. In addition, the energy production correlated to the shear of the mean flow, the energy redistribution due to the pressure–strain correlation, and the interaction between the localized Reynolds stresses and the shear of the Stokes drift also play important roles. Based on the results obtained from the analysis of the transport equations, predictive models are proposed for the localizing effects, which assess the spatial dependence of the Boussinesq model for background turbulence in coastal Langmuir turbulence. These models show good scaling performance of $\langle u^{T}u^{T}\rangle _{xt}$ near the water bottom and of $\langle -u^{T}v^{T}\rangle _{xt}$, $\langle -u^{T}w^{T}\rangle _{xt}$ and $\langle -v^{T}w^{T}\rangle _{xt}$ in the central region of the water column under various flow conditions with different values of the Reynolds number, turbulent Langmuir number and wavenumber.

We study the fragmentation of a liquid drop that is hit by a laser pulse. The drop expands into a thin sheet that breaks by the radial expulsion of ligaments from its rim and the nucleation and growth of holes on the sheet. By combining experimental data from two liquid systems with vastly different time and length scales, we show how the early-time laser–matter interaction affects the late-time fragmentation. We identify two Rayleigh–Taylor instabilities of different origins as the prime cause of the fragmentation and derive scaling laws for the characteristic breakup time and wavenumber. The final web of ligaments results from a subtle interplay between these instabilities and deterministic modulations of the local sheet thickness, which originate from the drop deformation dynamics and spatial variations in the laser-beam profile.

Two-dimensional direct numerical simulations of a cylinder undergoing forced streamwise oscillations in steady approaching flow are conducted over ranges of oscillation amplitude, oscillation frequency and gap distance between the cylinder and the wall at a Reynolds number of 175. The flow characteristics are found to be strongly affected by the gap distance, compared to those observed around an isolated cylinder (Tang et al., J. Fluid Mech., vol. 832, 2017, pp. 146–169). The synchronisation modes are mapped out in the parameter ranges. The existence of the plane wall leads to an increased chance of occurrence of high-order modes with the denominator being an odd number. Two new flow phenomena, namely the period doubling and transition to quasi-periodic states through cascade of period doubling within the primary synchronisation region, are observed. The interaction of the plane-wall boundary layer with vortices shed from the cylinder and the asymmetry of the flow through the gap and around the top side of the cylinder are identified as the primary physical mechanisms responsible for the observed behaviours. The influence of velocity gradient in the plane-wall boundary layer on the two new phenomena is quantified through a numerical test involving linear shear flow around an isolated cylinder. The period-doubling phenomenon occurs only when the velocity gradient is larger than a critical value. The results obtained through three-dimensional simulations suggest that the synchronisation modes identified through two-dimensional simulations are not significantly affected by the three-dimensionality of the flow over the parameter ranges covered in the present study.

The generation mechanism of screech harmonics in supersonic exhausts is revealed using shadowgraph imaging and acoustic far-field measurements for a rectangular, high aspect-ratio nozzle. The coherent information associated with screech and its harmonics, i.e. flow structures and acoustic radiation pattern, is extracted from the time-resolved shadowgraph images. We show that, for large lateral distortions of the jet plume, the passage of screech associated flow structures triggers the screech-cyclic formation of shocks, which travel downstream and merge with the original shocks. The interaction of the shock waves with the flow structures associated with screech alters the appearance of the perturbations in the mixing layer, which constitute the higher harmonics of screech. Visualisations of the acoustic radiation pattern expose that the third and higher screech tone harmonics originate from these interaction locations. Further, the occurrence of mode resonance between the screech and its harmonics is demonstrated, where the mode resonance location coincides with the screech tone origin.

This study investigates the effects of a Bethe–Zel’dovich–Thompson (BZT) dense gas (FC-70) on the development of a turbulent compressible mixing layer at a convective Mach number $M_{c}=1.1$. Three-dimensional direct numerical simulations are performed with both FC-70 and air. The initial thermodynamic state for FC-70 lies inside the inversion region where the fundamental derivative of gas dynamics ($\unicode[STIX]{x1D6E4}$) becomes negative. The complex Martin–Hou thermodynamic equation of state is used to reproduce thermodynamic peculiarities of the BZT dense gas (DG). The unstable growth phase in the mixing layer development shows an increase of $xy$-turbulent stress tensors in DG compared to perfect gas (PG). The following self-similar period has been carefully defined from the time evolution of the integrated streamwise production and transport terms. During the self-similar stage, DG and PG mixing layers at $M_{c}=1.1$ display close values of the momentum thickness growth rate, which seems similarly affected by the well-known compressibility-related reduction for PG. The same mechanisms are at stake, related to the reduction of pressure–strain terms. Turbulent kinetic energy (TKE) spectra show a slower decrease of TKE at small scales for DG compared with PG. The filtered kinetic energy equation balance developed by Aluie (Physica D, vol. 247 (1), 2013, pp. 54–65) is applied for the first time to a compressible mixing layer. The equation is reshaped to better account for TKE transport across the mixing layer. This new formulation brings out the role played by $\unicode[STIX]{x1D6F4}_{l}$, the pressure strengths power. A detailed comparison of the contributions to the filtered TKE equation is provided for both PG and DG mixing layers.

The cylinder with a slanted base is a simplified, canonical bluff body geometry that shares similarities to aircraft fuselages, which are known to produce a strong vortex pair due to their upswept afterbody. This work will examine in detail the surface flow and near-field characteristics of the flow over the slanted cylinder for slant angles of $20^{\circ }$, $32^{\circ }$ and $45^{\circ }$ with spatially dense measurements. Principal flow features of the mean flow field are identified, showing the connection between the main counter-rotating vortex pair observed in the wake and the separation bubble observed at the leading edge of the slant. A full reconstruction of the three-dimensional mean flow field using stacked stereoscopic particle image velocimetry reveals intricate details of this flow and clearly shows the direct connection between the two features. To our knowledge, this is the first direct measurement of the full three-dimensional flow topology for this geometry. The separation bubble length is found to be directly proportional to the slant angle and inversely proportional to the Reynolds number. Furthermore, the circulation within the primary vortex pair increases with increasing slant angle. This strengthening of the vortices is correlated to the form drag of this body in the vortex-dominated regime. A bi-stable steady-state wake is also observed in this flow at a low Reynolds number for the slant angle of $45^{\circ }$, where the formation of either a separated wake flow state or a vortex-dominated flow state is dependent upon initial conditions, i.e. the presence of overshoot of the free-stream velocity during wind tunnel start-up.

Previous measurements and numerical simulations of buoyant turbulent plumes that develop from area sources provide convincing evidence that entrainment varies locally in response to an imbalance from the asymptotic state of equilibrium, a state referred to as a pure plume. Across the wide spectrum of possible source conditions, that span forced and lazy plume releases, this variation of entrainment has been successfully captured by a single, or universal, description in which the entrainment function $\unicode[STIX]{x1D6FC}$ varies linearly with the local Richardson number. Herein, an analytical solution for the virtual origin of forced, pure and lazy turbulent plumes from circular sources in unstratified environments is derived based on this universal description of entrainment. Prior to this, the analytical solutions reported were limited to those based on the simplifying assumption of invariant entrainment, so-called constant-$\unicode[STIX]{x1D6FC}$ solutions of the plume conservation equations. Analytical solutions for the fluxes of volume and specific momentum are first developed. These solutions highlight the deficit in near-field entrainment in forced plumes and enable the general imbalance from the equilibrium state to be predicted via the streamwise variation of the local Richardson number. Focus then turns to the virtual origin due to the practical benefits that a knowledge of this location offers experimentalists (e.g. in comparing measurement with theory) and theoretical modellers (e.g. in incorporating a turbulent plume within a broader modelling framework).

In the present study, a compressible low-Reynolds-number flow over a circular cylinder was investigated using a low-density wind tunnel with time-resolved schlieren visualizations and pressure and force measurements. The Reynolds number ($Re$) based on freestream quantities and the diameter of a circular cylinder was set to be between 1000 and 5000, and the freestream Mach number ($M$) between 0.1 and 0.5. As a result, we have clarified the effect of $M$ on the aerodynamic characteristics of flow over a circular cylinder at $Re=O(10^{3})$. The results of the schlieren visualization showed that the trend of $M$ effect on the flow field, that are the release location of the Kármán vortices, the Strouhal number of vortex shedding and the maximum width of the recirculation, is changed at approximately $Re=3000$. In addition, the spanwise phase difference of the surface pressure fluctuation was captured by the measurement using pressure-sensitive paint at approximately $Re=3000$ of higher-$M$ cases. The observed spanwise phase difference is considered to relate to the spanwise phase difference of the vortex shedding due to the oblique instability wave on the separated shear layer caused by the compressibility effects. The Strouhal number of the vortex shedding is influenced by $M$ and $Re$, and those effects are nonlinear. However, the effects of $M$ and $Re$ can approximately be characterized by the maximum width of the recirculation. In addition, the $M$ effect on the drag coefficient can be characterized by the maximum width of the recirculation region and the Prandtl–Glauert transformation.

Flexural-gravity wave interactions with multiple cracks in an ice sheet of infinite extent are considered, based on the linearized velocity potential theory for fluid flow and thin elastic plate model for an ice sheet. Both the shape and location of the cracks can be arbitrary, while an individual crack can be either open or closed. Free edge conditions are imposed at the crack. For open cracks, zero corner force conditions are further applied at the crack tips. The solution procedure starts from series expansion in the vertical direction based on separation of variables, which decomposes the three-dimensional problem into an infinite number of coupled two-dimensional problems in the horizontal plane. For each two-dimensional problem, an integral equation is derived along the cracks, with the jumps of displacement and slope of the ice sheet as unknowns in the integrand. By extending the crack in the vertical direction into the fluid domain, an artificial vertical surface is formed, on which an orthogonal inner product is adopted for the vertical modes. Through this, the edge conditions at the cracks are satisfied, together with continuous conditions of pressure and velocity on the vertical surface. The integral differential equations are solved numerically through the boundary element method together with the finite difference scheme for the derivatives along the crack. Extensive results are provided and analysed for cracks with various shapes and locations, including the jumps of displacement and slope, diffraction wave coefficient, and the scattered cross-section.

Large field-of-view (FOV) particle image velocimetry experiments are conducted in the vicinity of a shock wave boundary layer interaction (SWBLI) at Mach 2. The current FOV covers up to 30 boundary layer thicknesses ($\widehat{\unicode[STIX]{x1D6FF}}_{I}$), comprising of upstream and downstream regions relative to the SWBLI, thereby allowing the turbulent boundary layer and shock to be simultaneously captured. The relationship between the boundary layer features and the instantaneous shock location is directly quantified, with the aim of better understanding the mechanisms responsible for oscillation of the reflected shock. The results show that the reflected shock location is clearly influenced by the instantaneous state of the incoming boundary layer. It is found that passage of low-/high-momentum very-large-scale turbulent features through the SWBLI region causes the reflected shock to move upstream/downstream of the mean location. Moreover, interaction with the shock is found to introduce additional velocity fluctuations across a range of spanwise length scales within the boundary layer. The spanwise scales smaller than one $\widehat{\unicode[STIX]{x1D6FF}}_{I}$ recover within one $\widehat{\unicode[STIX]{x1D6FF}}_{I}$ downstream of the SWBLI region. However, at larger spanwise wavelengths, two persistent modes at approximately one and six $\widehat{\unicode[STIX]{x1D6FF}}_{I}$ are observed, where they remain correlated for a longer streamwise extent ($\gg \widehat{\unicode[STIX]{x1D6FF}}_{I}$) than the other spanwise modes. The wall pressure measurements indicate that the low-frequency fluctuations arising from the oscillating shock foot are due to dampening of high-frequency contents beyond the critical frequency associated with the unstable global mode. Thus, the results suggest that low-frequency pressure oscillations are not necessarily an independent phenomenon from the turbulent features entering the SWBLI region and interacting with the shock. Instead, a large scale separation between their dominant time scales is due to the critical frequency of the unstable global mode occurring at frequencies that are orders of magnitudes slower than the dominant frequency of the very-large-scale features.

Fines are smaller droplets produced from an auxiliary mechanism besides the formation of the standard drops in a fragmentation process. We report their formation in a controlled experiment which isolates an individual fragmentation protocol: the collision of two rims bordering growing adjacent holes on a liquid sheet. The standard drops come from the capillary breakup of the fused rims. Occasionally, the rims collision is strong enough to trigger a new, splash-like mechanism, producing an expanding lamellae perpendicular to the main sheet, which destabilizes into finer drops. We quantify the threshold condition for the onset of this mechanism first discovered by Lhuissier & Villermaux (J. Fluid Mech., vol. 714, 2013, pp. 361–392), we document the resulting lamellae dynamics and explain why it affects the mean drop size in the spray, broadening substantially the overall drop size distribution, which we determine. Possible applications of these findings are mentioned.