We develop a general framework to describe the cubically nonlinear interaction of a degenerate quartet of deep-water gravity waves in one or two spatial dimensions. Starting from the discretised Zakharov equation, and thus without restriction on spectral bandwidth, we derive a planar Hamiltonian system in terms of the dynamic phase and a modal amplitude. This is characterised by two free parameters: the wave action and the mode separation between the carrier and the sidebands. For unidirectional waves, the mode separation serves as a bifurcation parameter, which allows us to fully classify the dynamics. Centres of our system correspond to non-trivial, steady-state nearly resonant degenerate quartets. The existence of saddle-points is connected to the instability of uniform and bichromatic wave trains, generalising the classical picture of the Benjamin–Feir instability. Moreover, heteroclinic orbits are found to correspond to discrete, three-mode breather solutions, including an analogue of the famed Akhmediev breather solution of the nonlinear Schrödinger equation.

Recent data-driven efforts have utilized spectral decomposition techniques to uncover the geometric self-similarity of dominant motions in the logarithmic layer, and thereby validate the attached eddy model. In this paper, we evaluate the predictive capability of the stochastically forced linearized Navier–Stokes equations in capturing such structural features in turbulent channel flow at $Re_\tau =2003$. We use the linear coherence spectrum to quantify the wall-normal coherence within the velocity field generated by the linearized dynamics. In addition to the linearized Navier–Stokes equations around the turbulent mean velocity profile, we consider an enhanced variant in which molecular viscosity is augmented with turbulent eddy-viscosity. We use judiciously shaped white- and coloured-in-time stochastic forcing to generate a statistical response with energetic attributes that are consistent with the results of direct numerical simulation (DNS). Specifically, white-in-time forcing is scaled to ensure that the two-dimensional energy spectrum is reproduced and coloured-in-time forcing is shaped to match normal and shear stress profiles. We show that the addition of eddy-viscosity significantly strengthens the self-similar attributes of the resulting stochastic velocity field within the logarithmic layer and leads to an inner-scaled coherence spectrum. We use this coherence spectrum to extract the energetic signature of self-similar motions that actively contribute to momentum transfer and are responsible for producing Reynolds shear stress. Our findings support the use of coloured-in-time forcing in conjunction with the dynamic damping afforded by turbulent eddy-viscosity in improving predictions of the scaling trends associated with such active motions in accordance with DNS-based spectral decomposition.

The present paper investigates theoretically and experimentally the boundary layer generated by a stably stratified fluid flowing horizontally along a surface tilted in the transverse direction and deformed by sinusoidal undulations with crests perpendicular to the flow direction. In the absence of undulations, a weak transverse velocity proportional to the normal velocity is created such that the flow remains purely horizontal. In the presence of undulations of amplitude $h$, a stronger transverse flow is generated that exhibits a singular behaviour at the critical altitude where the frequency of the perturbation matches the buoyancy frequency of the fluid. This baroclinic critical layer was previously analysed by Passaggia et al. (J. Fluid Mech., vol. 751, 2014, pp. 663–684) for a boundary layer flow with a small sliding velocity on the surface. Here, the no-slip boundary condition of the experimental flow is applied. For this purpose, we solve the viscous sub-layer to obtain a complete theoretical model for the solution in the critical layer without any adjusting parameter. The theoretical predictions for the transverse velocity are compared with experimental measurements, and a good quantitative agreement is demonstrated. Compared with the sliding case, the no-slip boundary condition on the surface reduces the amplitude of the critical layer solution by a factor $Re^{-1/3}$, where the Reynolds number Re is defined using the velocity at infinity and the thickness of the boundary layer. As a consequence, the transverse velocity has a maximum in the critical layer of order $h$, but it still induces a shear rate of order $h\,Re^{1/3}$.

Numerical simulations for flow past a finite rectangular wing with a NACA 0012 section at $Re=1000$ for various semi-aspect ratios ($0.25\le sAR \le 7.5$) over a range of angles of attack ($0^{\circ }\le \alpha \le 14^{\circ }$) reveal streamwise vortices, which increase in strength and number to occupy an increasing spanwise extent with increase in $\alpha$. They result in non-monotonic spanwise variation of local force coefficients and increased strength of wing-tip vortex for $\alpha >8^{\circ }$. Viscous and pressure drag dominate for low and high sAR, respectively. The time-averaged drag coefficient first decreases and then increases with increase in $sAR$. Vortex shedding for $\alpha =14^{\circ }$ is single cell and parallel for $sAR<3$. Shedding is in two cells with an oblique angle that varies with time, leading to large spanwise variation in the root mean square of local force coefficients for higher $sAR$. Various types of dislocations, reported earlier in wakes of bluff bodies, are seen for different $\alpha$ and $sAR$. Dislocations for $\alpha =14^{\circ }$ appear at the same spanwise location for $sAR=3$ and at different spanwise locations for $sAR\ge 4$. Vortex shedding for $\alpha =12^{\circ }$ and $sAR=5$ exhibits one cell structure in the near wake and two cells in the far wake due to splitting and reconnection of vortices near the mid-span in the moderate wake. Linkages form between counter-rotating spanwise vortices for $sAR\ge 1$. Additional linkages between shed- and wing-tip vortices are observed for lower $sAR$. At each $\alpha$, the strength of the wing-tip vortex and radius of its core, estimated using Rankine and Lamb–Oseen models, increases up to a certain $sAR$ beyond which it is approximately constant.

We introduce maps of Cauchy–Green strain tensor eigenvalues to barycentric coordinates to quantify and visualize the full geometry of three-dimensional deformation in stationary and non-stationary fluid flows. As a natural extension of Lagrangian coherent structure diagnostics, which provide separate scalar fields and a one-dimensional quantification of fluid deformation, our barycentric mapping visualizes the role of all three Cauchy–Green eigenvalues (or rates of stretching) in a single plot through a novel stretching coordinate system. The coordinate system is based on the distance from three distinct limiting states of deformation that correspond with the dimension of the underlying invariant manifolds. One-dimensional axisymmetric deformation (sphere to rod deformation) corresponds to one-dimensional unstable manifolds, two-dimensional axisymmetric deformation (sphere to disk deformation) corresponds to two-dimensional unstable manifolds and the rare three-dimensional isometric case (sphere to sphere translation and rotation) corresponds to shear-free elliptic Lagrangian coherent structures (LCSs). We provide methods to visualize the degree to which fluid deformation approximates these limiting states, and tools to quantify differences between flows based on the compositional geometry of invariant manifolds in the flow. We also develop a simple analogue for bilinearly representing and plotting both rates of stretching and rotation as a single vector. As with other LCS techniques, these diagnostics define frame-indifferent material features in the flow. We provide multiple computed examples of LCS and momentum transport barriers, and show advantages over other coherent structure diagnostics.

A new scaling of the mean momentum equation is developed for the outer region of turbulent boundary layers (TBLs) under adverse pressure gradient (APG). The maximum Reynolds shear stress location, denoted as $y_{m}$, is employed to determine the proper scales for the outer region of an APG TBL. An outer length scale is proposed as $\delta _e - y_{m}$, where $\delta _e$ is the boundary layer thickness. An outer velocity scale for the mean streamwise velocity deficit is proposed as $U_e - U_{m}$, where $U_e$ and $U_m$ are the mean streamwise velocities at the boundary layer edge and $y_{m}$, respectively. An outer velocity scale for the mean wall-normal velocity deficit is proposed as $V_e - V_{m}$, where $V_e$ and $V_{m}$ are the wall-normal velocities at $\delta _e$ and $y_{m}$, respectively. The maximum Reynolds shear stress is found to scale as $(\delta _e - y_{m}) U_e \,{\rm d}U_e/{{\rm d}x}$. The new outer scaling collapses well the experimental and numerical data on APG TBLs over a wide range of Reynolds numbers and strengths of pressure gradient. Approximations of the new scaling are developed for TBLs under strong APG and at high Reynolds numbers. The relationships between the new scales and previously proposed scales are discussed.

We report on the modification of the spectrum of a passive scalar inside a turbulent flow by the injection of large bubbles. Although the spectral modification through bubbles is well known and well analysed for the velocity fluctuations, little is known on how bubbles change the fluctuations of an approximately passive scalar, in our case temperature. Here we uncover the thermal spectral scaling behaviour of a turbulent multiphase thermal mixing layer. The development of a $-3$ spectral scaling is triggered. By injecting large bubbles (${Re}_{{bub}} = {O}(10^2)$) with gas volume fractions $\alpha$ up to 5 %. For these bubbly flows, the $-5/3$ scaling is still observed at intermediate frequencies for low $\alpha$ but becomes less pronounced when $\alpha$ further increases and it is followed by a steeper $-3$ slope for larger frequencies. This $-3$ scaling range extends with increasing gas volume fraction. The $-3$ scaling exponent coincides with the typical energy spectral scaling for the velocity fluctuations in high-Reynolds-number bubbly flows. We identify the frequency scale of the transition from the $-5/3$ scaling to the $-3$ scaling and show how it depends on the gas volume fraction.

Direct numerical simulations (DNS) of a jet in cross-flow (JICF) with a triangular tab at two positions are performed at jet-to-cross-flow velocity ratios of $R = 2$ and $4$ with a jet Reynolds number of 2000 based on the jet's bulk velocity and exit diameter. The DNS and dynamic mode decomposition show the sensitivity of the tab's effect on the jet upstream shear layer (USL) structure and cross-section to $R$, echoing the experimental discoveries of Harris et al. (J. Fluid Mech., vol. 918, 2021). Furthermore, DNS reveals that the presence of a tab placed on the upstream side of the nozzle significantly modifies the USL through production of streamwise vortices that curl around the spanwise vortex tubes originating from the primary instability of the USL. This provides an explanation for the improvement in mixing that has been associated with an upstream tab. The streamwise vortex structure shows remarkable similarities to the ‘strain-oriented vortex tubes’ observed for disturbed plane shear layers by Lasheras & Choi (J. Fluid Mech., vol. 189, 1988, pp. 53–86). For both $R$ cases, the USL instability is delayed, the jet penetration is reduced, and the jet cross-section is flattened, although the tab has a less pronounced effect on the USL structure at higher velocity ratios, where the formation of the streamwise vortices is delayed. In contrast, a tab placed 45$^\circ$ from the upstream position produces significantly different effects compared with the upstream tab. At $R = 4$, the jet cross-section is significantly skewed away from the tab and a tertiary vortex is formed, as observed in past studies of round JICFs at relatively high $R$ and low Reynolds numbers. The ability of the tab to produce a controllable steady-state tertiary vortex has implications for a variety of applications. The 45$^\circ$ tab produces asymmetric effects in the wake of the jet at $R = 2$, but the effect on the jet cross-section is much smaller, highlighting the sensitivity of jets at high $R$ to asymmetric perturbations.

In this paper, we investigate the flow past a circular cylinder confined in a channel at a blockage ratio of $\beta =0.7$ (the ratio of the cylinder diameter and the channel height) for Reynolds numbers between ${\textit {Re}}=300$ and $3900$ using direct numerical simulation (DNS). We show for varying Reynolds numbers a wide range of wake dynamics occur as the spanwise domain length is changed. At a lower Reynolds number of ${\textit {Re}}=300$, a reverse von Kármán wake alongside either a top- or bottom-biased asymmetry was observed at different spanwise locations. The asymmetry was structurally similar the two-dimensional asymmetry studied by prior investigators, and was found to be a result of the confinement effect. Further, wake-jumping between the two intermittent states was present. For larger Reynolds numbers, ${\textit {Re}}=1000$ and $3900$, these asymmetric structures were found to become dominant. We also examine the dependence of the asymmetries on the spanwise domain. For small spanwise domains the asymmetries were uniformly orientated across the span. In contrast, for sufficiently large spanwise domains, the asymmetry flips its orientation at different spanwise locations. Comparisons of flow statistics demonstrate good agreement between the different spanwise domains, which suggests the same mechanism maintains the asymmetry in both cases. Further analysis at ${\textit {Re}}=1000$ found the number of times the wake flips is dependent on the initial conditions, with a wake that flips zero (purely asymmetric), two and four times being observed. These structures were also determined to remain stable over time scales of $1000D/U$.

Wall-bounded turbulent flows exhibit a zonal arrangement, in which streamwise velocity organizes into uniform momentum zones (UMZs), separated by thin layers of elevated interfacial shear. While significant research efforts have focused on these structural features in neutrally stratified flows, the effects of unstable thermal stratification on UMZs and on analogous uniform temperature zones (UTZs) have not been considered previously. In this article, statistical properties of UMZs and UTZs are investigated using a suite of large eddy simulations of unstably stratified turbulent channel flow spanning weakly to highly convective conditions. When normalized by the friction velocity and stability-dependent mixing length, the mean velocity gradient based on UMZ interfacial velocity jumps and the vorticity thickness exhibits good collapse for all stabilities, establishing a link between UMZ properties and scaling predictions from Monin–Obukhov similarity theory. A similar relationship is found between UTZ properties and surface-layer scaling of the mean temperature gradient. In the mixed layer, mean UMZ depth is quasi-constant with wall-normal distance, while the deepest UTZs are found in the centre of the boundary layer. These instantaneous structures are found to be linked to the well-mixed velocity and temperature profiles in the convective mixed layer. Conditional averaging indicates that both UMZ and UTZ interfaces are associated with ejections of momentum and warm updrafts below the interface and sweeps of momentum and cool downdrafts above the interface. These results demonstrate a tangible connection between instantaneous structural features, mean properties and scaling laws in unstably stratified flows.

The so-called coffee stain effect has been intensively studied over the past decades, but most of the studies have focused on sessile droplets. In this paper, we analyse the origin of the difference between the deposition of suspended particles in a sessile drop and in an axisymmetric drop deposited on a fibre. First, we model the shape of a drop on a fibre and its evaporative flux with some approximations to derive analytical calculations. Then, for pinned contact lines, we solve the hydrodynamics equations in the liquid phase under the lubrication approximation to determine the flow velocity toward the contact lines. We comment on these results by comparison to a sessile drop of similar evaporating conditions, and we show that the substrate curvature plays a role on the contact line depinning, the local evaporative flux and the liquid flow field. The competition between the advection and the Brownian motion indicates that the transport of the particles toward the contact line occurs in a volume localised in the close vicinity of the contact lines for a drop on a fibre. Thus, the fibre geometry induces a weaker accumulation of particles at the contact line compared to a sessile drop, leading to the more homogeneous deposit observed experimentally.

Tonal noise emitted from the trailing edge of an airfoil is considered using modal analysis techniques to investigate secondary quadrupole tones. We examine the origin of quadrupole sound generated from two-dimensional unsteady laminar flow over a NACA0012 airfoil. In this paper, we consider two flow configurations at Mach numbers of $M_\infty = 0.1$ and 0.05 that lead to different acoustic characteristics: the former has a significant high-frequency quadrupole noise source, whereas the latter does not. We use vortex sound theory, dynamic mode decomposition (DMD), and resolvent analysis to analyze the sound source. First, we employ DMD modes to reveal that the quadrupole sound is only observed in the higher Mach number case. Next, the vortex dynamics in the vicinity of the trailing edge are studied to identify the origin of quadrupole sound. It is found that the quadrupole sound is caused by vortex shedding in the vicinity of the trailing edge. The complex vortex interaction between both sides of the airfoil strengthens the quadrupole source in the higher Mach number case, while it is negligible in the lower one. Furthermore, we perform resolvent analysis to examine the vortex generation over the airfoil. The resolvent mode indicates that the interaction between the vortices on both sides of the airfoil causes a multi-scale vortex structure on the suction-side wall.