The total enthalpy of a uniform flow is redistributed behind a thermally insulated body such that a considerable cooling of fluid in terms of total temperature occurs near the wake centreline. This energy separation phenomenon is also observed after time averaging and can lead to the recovery temperature at the rear part of the body becoming lower than the static temperature of the incoming flow (the Eckert–Weise effect). By the numerical solution of the Navier–Stokes equations, the present work studies energy separation in the wake, the Eckert–Weise effect and their sensitivity on the flow interference behind side-by-side cylinders. The considered distances between the cylinders cover the key regimes of interference: single, bistable and coupled vortex streets. The first and second regimes amplify the efficiency of energy separation in the wake and the intensity of the Eckert–Weise effect, respectively. In the developed wake, a clear relation between the vortex structure and the total enthalpy redistribution is demonstrated as well as a limitation on the intensity of this redistribution imposed by the deficit of velocity. The connection of the Eckert–Weise effect with the vortex formation process is clarified. Among the mechanisms changing the total enthalpy in fluid particles, the non-stationarity of pressure defines the averaged distribution: the decrease of pressure in forming vortices is mainly responsible for the cooling of fluid behind the body. The balance with the heating of the rear surface by the involvement of fluid from the sides defines the intensity of the effect.

A framework for the computation of linear global modes, based on time stepping of a linearised Navier–Stokes solver with an Eulerian interface representation, is presented. The method is derived by linearising the nonlinear solver Basilisk, capable of computing immiscible two-phase flows, and offers several advantages over previous, matrix-based, multi-domain approaches to linear global stability analysis of interfacial flows. Using our linear solver, we revisit the study of Tammisola et al. (J. Fluid Mech., vol. 713, 2012, pp. 632–658), who found a counter-intuitive, destabilising effect of surface tension in planar wakes. Since their original study does not provide any validation, we further compute nonlinear results for the studied flows. We show that a surface-tension-induced destabilisation of plane wakes is observable which leads to periodic, quasiperiodic or chaotic oscillations depending on the Weber number of the flow. The predicted frequencies of the linear global modes, computed in the present study, are in good agreement with the nonlinear results, and the growth rates are comparable to the disturbance growth in the nonlinear flow before saturation. The bifurcation points of the nonlinear flow are captured accurately by the linear solver and the present results are as well in correspondence with the study of Tammisola et al. (J. Fluid Mech., vol. 713, 2012, pp. 632–658).

In this article, the objective is to characterize the influence of a continuous stratification on wake-induced vibrations of a circular cylinder. Experimental results are obtained by towing a horizontal cylinder in a horizontal direction perpendicular to its axes, at a constant speed in a linearly stratified fluid made of salty water. The cylinder can move vertically since it is fixed to free-to-rotate arms. The diameter of the cylinder, the length of the arms and the translating speed are varied. Two flow-induced vibrations modes are observed. The first one is analogous to a vortex-induced vibration (VIV) mode: it is associated with a resonance between the vortex shedding frequency and its frequency is proportional to the natural frequency of the mechanical system, as can be predicted by the VIV theory generalized to low mass ratios. Large amplitude oscillations of the cylinder are found to occur in a large range of velocities also in agreement with the low-mass-ratio VIV theory. The second mode is interpreted as a galloping mode. It has a low frequency and a large amplitude, and occurs for low Froude numbers and long arms. It corresponds to the regimes when the buoyancy forces are larger than the inertial and vertical drag forces. By computing the forces on the cylinder, it is shown that stratification is the source of a destabilizing lift when the cylinder departs from its horizontal motion. Only a weak effect of the Reynolds number on the stability characteristics has been observed in the considered range ($500 < Re < 15\,000$).

We consider the hydrodynamic stability of viscoelastic films flowing over inclined structured substrates with rectangular trenches. This topography allows investigating independently the effects of trench unit length, depth, width and inclination angle and complements the earlier work by Pettas et al. (Phys. Rev. Fluids, vol. 4, 2019, 33). We account for material rheology by employing the ePTT model. We perform a parametric study of the steady flow and its linear stability, assuming two-dimensional perturbations along the streamwise direction of arbitrary wavelength via the Floquet–Bloch theory. Our predictions for Newtonian liquids are in excellent agreement with previous results. We demonstrate that even for Newtonian liquids, the trench depth has a non-trivial effect on the flow stability. In viscoelastic solutions, the interaction of fluid elasticity with substrate morphology may have a significant impact on the flow dynamics, leading to either the enhancement or suppression of instabilities. Topography characteristics combined with enough material elasticity stabilize the flow. However, beyond a specific trench depth, this effect saturates by eddy formation inside the cavity. Moreover, the stability is also affected by the aspect ratio and shape of the trenches: flows over substrates with a pillar-like configuration are stabilized significantly. On the other hand, flows are destabilized by material shear-thinning. This study helps identifying the shape of the substrate that maximizes/minimizes the viscoelastic mechanisms. This is impossible when considering substrates with sinusoidal topography, but could be of utmost importance for several technological applications, by providing the potential for instability control through the development of appropriately tailored substrates.

Colloids which adsorb to and straddle a fluid interface form monolayers that are paradigms of particle dynamics on a two dimensional fluid landscape. The dynamics is typically inertialess (Stokes flows) and dominated by interfacial tension so the interface is undeformed by the flow, and pairwise drag coefficients can be calculated. Here the hydrodynamic interaction between identical spherical colloids on a planar gas/liquid interface is calculated as a function of separation distance and immersion depth. Drag coefficients (normalized by the coefficient for an isolated particle on the surface) are computed numerically for the four canonical interactions. The first two are motions along the line of centres, either with the particles mutually approaching each other or moving in the same direction (in tandem). The second two are motions perpendicular to the line of centres, either oppositely directed (shear) or in the same direction (tandem). For mutual approach and shear, the normalized coefficients increase with a decrease in separation due to lubrication forces, and become infinite on contact when the particle is more than half immersed. However, they remain bounded at contact when the particles are less than half immersed because they do not contact underneath the liquid. For in-tandem motion, the normalized coefficients decrease with a decrease in separation; they collapse, for all immersion depths, to the dependence of the drag coefficient on separation for two particles moving in tandem in an infinite medium. The coefficients are used to compute separation against time for colloids driven together by capillary attraction.

In a recent publication, Hornung et al. (J. Fluid Mech., vol. 871, 2019, pp. 1097–1116) showed that the shock wave stand-off distance and the drag coefficient of a cone in the inviscid hypersonic flow of a perfect gas can be expressed as the product of a function of the inverse normal-shock density ratio $\varepsilon$ and a function of the cone-angle parameter $\eta$, thus reducing the number of independent parameters from three (Mach number, specific heat ratio and angle) to two. Analytical forms of the functions were obtained by performing a large number of Euler computations. In this article, the same approach is applied to a symmetrical flow over a wedge. It is shown that the same simplification applies and corresponding analytical forms of the functions are obtained. The functions of $\varepsilon$ are compared with the newly determined corresponding functions for flow over a circular cylinder.

We analyse linear stability of interfacial waves in an idealised model of an aluminium reduction cell consisting of two stably stratified liquid layers which carry a vertical electric current in a collinear external magnetic field. If the product of electric current and magnetic field exceeds a certain critical threshold depending on the cell design, the electromagnetic coupling of gravity wave modes can give rise to a self-amplifying rotating interfacial wave which is known as the metal pad instability. Using the eigenvalue perturbation method, we show that, in the inviscid limit, rectangular cells of horizontal aspect ratios $\alpha =\sqrt {m/n}$, where $m$ and $n$ are any two odd numbers, can be destabilised by an infinitesimally weak electromagnetic interaction while cells of other aspect ratios have finite instability thresholds. This fractal distribution of critical aspect ratios, which form an absolutely discontinuous dense set of points interspersed with aspect ratios with non-zero stability thresholds, is confirmed by accurate numerical solution of the linear stability problem. Although the fractality vanishes when viscous friction is taken into account, the instability threshold is smoothed out gradually and its principal structure, which is dominated by the major critical aspect ratios corresponding to moderate values of $m$ and $n$, is well-preserved up to relatively large dimensionless viscous friction coefficients $\gamma \sim 0.1$. With a small viscous friction, the most stable are cells with $\alpha ^{2}\approx 2.13$ which have the highest stability threshold corresponding to the electromagnetic interaction parameter $\beta \approx 4.7$.

Tomographic background oriented Schlieren (Tomo-BOS) imaging measures density or temperature fields in three dimensions using multiple camera BOS projections, and is particularly useful for instantaneous flow visualizations of complex fluid dynamics problems. We propose a new method based on physics-informed neural networks (PINNs) to infer the full continuous three-dimensional (3-D) velocity and pressure fields from snapshots of 3-D temperature fields obtained by Tomo-BOS imaging. The PINNs seamlessly integrate the underlying physics of the observed fluid flow and the visualization data, hence enabling the inference of latent quantities using limited experimental data. In this hidden fluid mechanics paradigm, we train the neural network by minimizing a loss function composed of a data mismatch term and residual terms associated with the coupled Navier–Stokes and heat transfer equations. We first quantify the accuracy of the proposed method based on a two-dimensional synthetic data set for buoyancy-driven flow, and subsequently apply it to the Tomo-BOS data set, where we are able to infer the instantaneous velocity and pressure fields of the flow over an espresso cup based only on the temperature field provided by the Tomo-BOS imaging. Moreover, we conduct an independent PIV experiment to validate the PINN inference for the unsteady velocity field at a centre plane. To explain the observed flow physics, we also perform systematic PINN simulations at different Reynolds and Richardson numbers and quantify the variations in velocity and pressure fields. The results in this paper indicate that the proposed deep learning technique can become a promising direction in experimental fluid mechanics.

The lift and drag forces acting on a small spherical particle moving with a finite slip in single-wall-bounded flows are investigated via direct numerical simulations. This study is an extension of our previous work that considered the lift and drag forces acting on a sphere moving near a wall in the presence shear, but in the absence of slip (Ekanayake et al., J. Fluid Mech., vol. 904, 2020, A6). The effect of slip velocity on the particle force is analysed as a function of separation distance for low slip and shear Reynolds numbers ($10^{-3} \leq Re_{{slip}} \leq 10^{-1}$ and $10^{-3} \leq Re_{\gamma } \leq 10^{-1}$) in both quiescent and linear shear flows. A generalised lift model valid for arbitrary particle–wall separation distances and $Re_{\gamma }, Re_{{slip}} \leq 10^{-1}$ is developed based on the results of the simulations. The proposed model can now predict the lift forces in linear shear flows in the presence or absence of slip, and in quiescent flows when slip is present. Existing drag models are also compared with numerical results for both quiescent and linear shear flows to determine which models capture near-wall slip velocities most accurately for low particle Reynolds numbers. Finally, we compare the results of the proposed lift model to previous experimental results of negatively buoyant particles and to numerical results of neutrally buoyant (force-free) particles moving near a wall in quiescent and linear shear flows. The generalised lift model presented can be used to predict the behaviour of particle suspensions in biological and industrial flows where the particle Reynolds numbers based on slip and shear are ${O}(10^{-1})$ and below.

Data from an energetic tidal flow are used to investigate the appropriateness of the classical scaling of wall turbulence for very high Reynolds numbers and fully rough conditions. The boundary layer occupied the entire channel depth and the Reynolds numbers were several orders of magnitude larger than typically attained in the laboratory. Profiles of the mean velocity, turbulence quantities and spectra exhibit consistencies with the wall-similarity hypothesis and the attached-eddy model, as well as the universality of the near-wall turbulence structure for geophysical flows. The wall-normal velocity spectra and intensities near the free surface are also strongly attenuated at low wavenumbers, consistent with the constraints imposed by any boundary which may be approximated as a plane slip surface. The results can be used to inform the development of turbulence inflow models for assessing the dynamic loading on tidal turbines and the layout of tidal energy farms.

We present an experimental study of bubble coalescence at an air–water interface and characterize the evolution of both the underwater neck and the surface bridge. We explore a wide range of Bond number, $Bo$, which compares gravity and capillary forces and is a dimensionless measure of the free surface's effect on bubble geometry. The nearly spherical $Bo\ll 1$ bubbles exhibit the same inertial–capillary growth of the classic underwater dynamics, with limited upper surface displacement. For $Bo>1$, the bubbles are non-spherical – residing predominantly above the free surface – and, while an inertial–capillary scaling for the underwater neck growth is still observed, the controlling length scale is defined by the curvature of the bubbles near their contact region. With it, an inertial–capillary scaling collapses the neck contours across all Bond numbers to a universal shape. Finally, we characterize the upper surface with a simple oscillatory model which balances capillary forces and the inertia of liquid trapped at the centre of the liquid-film surface.

We examine Lyman's (Appl. Mech. Rev., vol. 43, issue 8, 1990, pp. 157–158) proposed definition of the boundary vorticity flux, as an alternative to the traditional definition provided by Lighthill (Introduction: boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), chap. 2, 1963, pp. 46–109. Oxford University Press). While either definition may be used to describe the generation and diffusion of vorticity, Lyman's definition offers several conceptual benefits. First, Lyman's definition can be interpreted as the transfer of circulation across a boundary, due to the acceleration of that boundary, and is therefore closely tied to the dynamics of linear momentum. Second, Lyman's definition allows the vorticity creation process on a solid boundary to be considered essentially inviscid, effectively extending Morton's (Geophys. Astrophys. Fluid Dyn., vol. 28, 1984, pp. 277–308) two-dimensional description to three-dimensional flows. Third, Lyman's definition describes the fluxes of circulation acting in any two-dimensional reference surface, enabling a control-surface analysis of three-dimensional vortical flows. Finally, Lyman's definition more clearly illustrates how the kinematic condition that vortex lines do not end in the fluid is maintained, providing an elegant description of viscous processes such as vortex reconnection. The flow over a sphere, in either translational or rotational motion, is examined using Lyman's definition of the vorticity flux, demonstrating the benefits of the proposed framework in understanding the dynamics of vortical flows.

The low-frequency unsteady motions behind a backward-facing step (BFS) in a turbulent flow at $Ma=1.7$ and $Re_\infty =1.3718\times 10^7 \ {\rm m}^{-1}$ are investigated using a well-resolved large-eddy simulation. The instantaneous flow field illustrates the unsteady phenomena of the shock wave/boundary layer interaction (SWBLI) system, including vortex shedding in the shear layer, the flapping motions of the shock and breathing of the separation bubble, streamwise streaks near the wall and arc-shaped vortices in the turbulent boundary layer downstream of the separation bubble. A spectral analysis reveals that the low-frequency behaviour of the system is related to the interaction between shock wave and separated shear layer, while the medium-frequency motions are associated with the shedding of shear layer vortices. Using a three-dimensional dynamic mode decomposition (DMD), we analyse the individual contributions of selected modes to the unsteadiness of the shock and streamwise-elongated vortices around the reattachment region. Görtler-like vortices, which are induced by the centrifugal forces originating from the strong curvature of the streamlines in the reattachment region, are strongly correlated with the low-frequency unsteadiness in the current BFS case. Our DMD analysis and the comparison with an identical but laminar case provide evidence that these unsteady Görtler-like vortices are affected by fluctuations in the incoming boundary layer. Compared with SWBLI in flat plate and ramp configurations, we observe a slightly higher non-dimensional frequency (based on the separation length) of the low-frequency mode.

The propagation of wave disturbances from a complex multi-fault submarine earthquake of slender rectangular segments in a sea of constant depth is discussed, accounting for both water compressibility and gravity effects. It is found that including gravity effects the modal envelopes of the modified two-dimensional acoustic waves and the tsunami are governed by the Schrödinger equation. An explicit solution is derived using a multi-fault approach that allows capturing the main peak of the tsunami. Moreover, a linear superposition of the solution allows solving complicated multi-fault ruptures, in particular in the absence of dissipation due to large variations in depth. Consequently, the modulations of acoustic waves due to gravity, and of tsunami due to compressibility, are governed simultaneously and accurately, which is essential for practical applications such as tsunami early warning systems. The results are validated numerically against the mild-slope equation for weakly compressible fluids.

The instantaneous structure of a turbulent boundary layer that has developed beneath freestream turbulence (FST) was investigated with planar particle image velocimetry. Measurements were performed in the streamwise–wall-normal plane at a position $Re_x = U_\infty x / \nu = 2.3 \times 10^6$ downstream of the boundary layer origin. FST was generated with an active grid placed upstream of a boundary layer plate. Three cases were investigated with FST intensity increasing from approximately zero to 12.8 %. It is shown that internal interfaces, separating areas of approximately uniform momentum, are present for all cases and show a dependence on the FST. In particular, the number and length of the uniform-momentum zones (UMZs) decrease with increasing FST. The modal velocities associated with the UMZs are distributed in approximately the same way regardless of the condition of the outer flow, suggesting a degree of resilience of the instantaneous structure to the freestream. Tracking the top edge of the upper-most UMZ revealed that this contour approaches the wall for increasing FST. Finally, it is demonstrated that the observed first-order phenomena in a turbulent boundary layer subjected to FST can be modelled by superimposing an isotropic turbulence field on a turbulent boundary layer field, which supports the hypothesis that the underlying structure of the boundary layer appears to remain largely intact if the naturally occurring fluctuations in the turbulent boundary layer exceed the energy of the FST.

The possible ill conditioning of the Reynolds average Navier–Stokes (RANS) equations when an explicit data-driven Reynolds stress tensor closure is employed is a discussion of paramount importance. This matter has far-reaching consequences on the emerging field of data-driven turbulence modelling, as well as in Reynolds stress models and in epistemic uncertainty quantification. In the present work, we explore fundamental aspects of this problem with the aid of direct numerical simulation (DNS) databases of the turbulent flows in a plane channel, in a square duct and in periodic hill geometries. We show that the RANS equations are ill conditioned in the whole range of cases analysed, even when the linear term of the Reynolds stress tensor is treated implicitly, when no information with respect to the DNS mean velocity field is provided. That is, in more than ten different simulations varying Reynolds number and geometry, using the DNS Reynolds stress tensor solely, which carries a small error, the error propagation to the mean velocity field is significantly amplified. Accurate solutions can be obtained with implicit or explicit procedures that include information from the DNS mean velocity field. We propose a new strategy along these lines for solving the RANS equations that combines ideas put forward in the literature and show that this new procedure outperforms the ones previously adopted to mitigate the error propagation to the mean velocity field. We have shown advantages of adopting the Reynolds force vector, the divergence of the Reynolds stress tensor, as a quantity with its own identity, which can impact on the choice of the target quantity to be modelled in RANS equations.

In the field of gas-turbine engineering, entropy waves and fluctuations in fuel–air mixing are of significant importance. The impact of either mechanism on thermoacoustic stability of the engine and combustion noise considerably depends on how they are convected in the combustion chamber. In this work, a novel method is employed to analyse their convection. Both effects are modelled using a transport equation of a passive scalar linearized around the mean field. The linearized transport equation is discretized using finite elements. It is shown that turbulent passive scalar transport can be described by an eddy diffusivity in the linear framework. The method is furthermore validated against direct numerical simulation (DNS) of passive scalar transport in a turbulent channel flow. Taking the mean flow from the DNS as input, the method reproduces transport of periodic passive scalar fluctuations with high accuracy at negligible numerical expense. Previous studies investigated destructive interference of the passive scalar due to a non-uniform mean flow profile, a process termed mean flow shear dispersion. The method introduced in this study, however, allows us to additionally quantify the impact of molecular and turbulent diffusion. For the channel flow under investigation, mean flow shear dispersion is the dominant mechanism at low frequencies while, at higher frequencies, turbulent diffusion needs to be accounted for to reproduce the DNS results. Molecular diffusion, however, only has a minor effect on the overall convection in the turbulent channel flow.

Active and passive controls for drag reduction in flow around a cylinder are obtained by computing the sensitivity of drag with respect to surface velocity perturbations and roughness, respectively. Both controls are concentrated around the separation line and localized in the spanwise direction, producing suction effects to the separating boundary layers. In the wake, the control induces localized vertical displacements and streamwise stretches of the upper and lower vorticity sheets, and subsequently delay the vortex shedding and push the local pressure minimum away from the cylinder. Instead of suppressing separation and recirculation as commonly observed in two-dimensional controls, the present three-dimensional control extends the recirculation zone to produce a virtual surface converting the bluff body flow to a streamlined body flow. Through this mechanism, the control reduces drag by 20 % at maximum control velocity 2 % of the free-stream velocity (or momentum coefficient $10^{-4}$) at Reynolds number $Re=190$. The control is much more efficient than the previously tested spanwise uniform suction or periodic suction/blowing, both requiring maximum control velocity above 8 % (or momentum coefficient above $10^{-3}$) to achieve similar drag reduction effects. The power savings ratio, defined as the ratio of the control-reduced drag power and the maximum input power, is above 20, up to $Re=1000$, the highest Reynolds number considered in this work. This ratio reduces slightly to 17.8 when the control is simplified to spanwise localized suction around the separation lines in order to facilitate practical implementations.

Experiments were performed using an active flow control approach that has shown the ability to significantly reduce the viscous drag in turbulent boundary layers. The purpose of this work was to document the changes in the turbulence characteristics of the boundary layer with the drag reduction. The flow control involved generating a steady spanwise velocity component of the order of $u_{\tau }$, within the sublayer using an array of pulsed-DC plasma actuators. The intent was to reduce the wall-normal vorticity component, $\omega _y$, that is associated with the mean flow distortion caused by quasi-steady streamwise vorticity associated with the wall streak structure first observed by Kline et al. (J. Fluid Mech., vol. 30, 1967, pp. 741–773). The significance of the $\omega _y$ comes from Schoppa & Hussain (J. Fluid Mech., vol. 453, 2002, pp. 57–108), who proposed an autonomous mechanism for self-sustained wall turbulence generation of which the sublayer wall-normal vorticity component is a critical parameter. The results document the characteristics of a turbulent boundary layer in which the viscous drag was reduced by 68 %. This involved measurements of the $u$ and $v$ velocity components in a three-dimensional region within the boundary layer using a pair of dual (X) hot-wire probes. Under the reduced drag, these documented a decrease in $u$ and $v$ turbulence intensity levels through most of the boundary layer. When scaled by $u_{\tau }$, the impact on the $v$ fluctuations was larger than that on the $u$ fluctuations. Analysis based on [$uv$] quadrant splitting documented a decrease in duration, and an increase in the time between ‘ejections’ (Q2) and ‘sweep’ (Q4) events that substantially lowered the near-wall turbulence production in the drag-reduced boundary layers. Conditional averages used to reconstruct the two- and three-dimensional coherent motions including $\lambda _2$ vortical structures, indicate a suppression of coherent features in the wall layer. These results are consistent with an underlying mechanism for drag reduction that comes from a suppression of the turbulence producing events in the wall layer associated with the wall streak structure.

Precise prediction of unsteady flapping aerodynamics in insect flight is of potential importance in the analysis of maneuverability and flight control. While the quasi-steady model is a cheap while reasonable tool, accurate evaluation of unsteady dynamic effects in complex flight behaviours remains a challenge. Here we develop a computational fluid dynamics (CFD) data-driven aerodynamic model (CDAM), which is informed by high-fidelity CFD simulations using overset meshes to enable the precise and fast prediction of both cycle-averaged and transient aerodynamic force, torque and power with various flying motions and wing kinematics. The CDAM comprises a quasi-steady model for flapping wings and an aerodynamic model for a moving body. The least square method and a surrogate method are employed to achieve aerodynamic coefficient fitting through training using a CFD database. With comparison to CFD test data, the CDAM is validated to be capable of accurately evaluating the aerodynamic force, torque and power of a wing-body bumblebee model in various flight velocities. A genetic optimization algorithm embedded with CDAM is proposed to determine trimmed states for forward flight through adjusting wing kinematics, indicating that bumblebees likely fly in a minimized mass-specific aerodynamic power consumption. The CDAM is further applied to proportional-derivative-based longitudinal flight control of bumblebee hovering, with the control parameters optimized by Laplace transformation and the root locus method, which is implemented consistently in both CDAM and CFD environments. Our results demonstrate that CDAM provides a versatile tool to achieve fast and precise aerodynamical prediction for flying insects in various flight behaviours.