The growth with fetch of young wind waves under steady wind forcing that is commonly attributed to shear flow instability results in a spatially inhomogeneous wave field with a spectrum evolving along the tank. The present laboratory study accounts for multiple co-existing statistically stationary random frequency harmonics. Single-point synchronous measurements of the instantaneous surface elevation and of its along-wind slope component are performed by optical methods at numerous locations. Assuming exponential spatial growth, the phase shift between the surface elevation and surface slope at each frequency is related to the spatial growth rate of each harmonic. The validity of the assumption that the wave energy varies exponentially with fetch is examined in a separate set of experiments; the instantaneous surface elevation at various wind-forcing conditions is measured at multiple locations along the tank. The spatial variation of the energy of individual frequency harmonics is determined. It is found that, below the local peak frequency, the energy of each harmonic grows exponentially, while the evolution of waves at frequencies approaching and exceeding the local peak is strongly affected by sheltering by the dominant wave, as well as by nonlinear bound waves. The outcomes of two independent methods of determination of spatial growth rates at a range of young wave frequencies are compared. The accumulated data also enable quantitative analysis of the sheltering phenomenon. The essential difference between the spatial and the temporal wind-wave evolution cases is discussed.

Numerical analysis of a steady monatomic gas flow about the point of the regular reflection of a strong oblique shock wave from the symmetry plane is conducted with the Navier–Stokes–Fourier (NSF) equations, the regularized Grad 13-moment (R13) equations and the direct simulation Monte Carlo (DSMC) method. In contrast to the inviscid solution to this problem completely defined by the Rankine–Hugoniot (RH) relations, all three models predict a complicated flow structure with strong thermal non-equilibrium and a long wake with flow parameters not predicted by the RH relations. The temperature $T_y$ related to thermal motion of molecules in the direction normal to the symmetry plane has a maximum inside the reflection zone while in a planar shock wave the maximum is observed for the $T_x$ temperature. The R13 equations predict these features much better than the NSF equations and are in good agreement with the benchmark DSMC results. An analysis of the flow with the conservation equations was conducted in order to evaluate the effects of various processes on a fluid element moving along the symmetry plane. In contrast to the shock wave where effects of viscosity and heat conduction are one-dimensional with zeroth net contribution to the fluid-element energy across the shock, the flow across the zone of the shock reflection is dominated by two-dimensional effects with positive net contribution of viscosity and negative contribution of heat conduction to the fluid-element energy. These effects are believed to be the main source of the wake with parameters deviating from the RH values.

This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ($u_{i}$,$p$) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$. Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$. If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

The study investigates the impact of a vertically layered bathymetry, consisting of submerged vertical plates, on a ship wake through theoretical analysis and experimental realization. For subwavelength distances between the plates, the analysis relies on a homogenized model that provides an effective, anisotropic, dispersion relation for the propagation of water waves. Our findings reveal that a highly asymmetric wake can be achieved, with the degree of asymmetry contingent upon the ship propagation direction in relation to the plate orientation. This anisotropy is characterized with respect to water depth and to ship length using the dimensionless depth and hull Froude numbers. Laboratory experiments align closely with theoretical predictions, confirming that the asymmetry of the wake can indeed be managed through manipulation of bathymetric conditions.

The nexus between turbulence, particle interaction and interfacial tension is virtually unexplored, despite being highly relevant to a wealth of industrial and environmental settings. Here we investigate it by conducting experiments on non-Brownian spherical particles at the interface of turbulent liquid layers. The latter are electromagnetically stirred in a quasi-two-dimensional apparatus, while the particles are individually tracked. By systematically varying interfacial conditions, turbulence intensity, particle size and concentration from dilute to dense, we map the system behaviour over a wide parameter space. We reveal how the dynamics is governed by the balance of drag, capillarity and lubrication. Based on their scaling, we propose a phase diagram comprising three distinct regimes, characterized by widely different levels of clustering and fluctuating energy of the particles. This is quantitatively confirmed by the experimental results.

Floquet stability analysis and direct simulations of a circular cylinder undergoing vortex-induced vibration (VIV) are presented. Simulation predictions are examined for the reduced velocity range over which there is a strong and periodic resonant response: $U_r \in [4.0, 8.0]$, focusing on a mass ratio of $m^* = 2.546$ matching a number of previous investigations. Over most of this range, the dominant wake modes present are analogous to modes A, B and QP (quasi-periodic) observed in a stationary circular cylinder wake. However, at $U_r = 4.5$, the dominant modes are B, QP and a subharmonic mode (SH), whereas at $U_r = 4.0$, the two-dimensional base state switches to a $P+S$ wake. The critical Reynolds number for two- to three-dimensional transition is observed to decrease with an increase of $U_r$, in line with a decreasing response amplitude. Over this range, the minimum ${\textit {Re}}$ for which the wake remains two-dimensional is 202, which occurs at $U_r = 7.5$, but this increases to ${\textit {Re}}_{cr} = 300$ at $U_r = 4.5$, noting the critical Reynolds number for a stationary circular cylinder is ${\textit {Re}}_{cr}=189$. The corresponding critical spanwise wavelengths for $U_r = 4.5$ and 8 are $1.4D$ (mode B) and $4.0D$ (mode A), respectively. Simulations indicate that even at $Re=300$, flow three-dimensionality increases the amplitude of the lower branch considerably. The investigation establishes the role of oscillation amplitude and reduced velocity in three-dimensional transition for elastically mounted systems.

A new model of the steady boundary layer flow around a rotating sphere is developed that includes the widely observed collision and subsequent eruption of boundary layers at the equator. This is derived following the Segalini & Garrett (J. Fluid Mech., vol. 818, 2017, pp. 288–318) asymptotic approach for large Reynolds numbers but replacing the Smith & Duck (Q. J. Mech. Appl. Maths, vol. 30, issue 2, 1977, pp. 143–156) correction with a higher-order version of the Stewartson (Grenzschichtforschung/Boundary Layer Research, 1958, pp. 59–71. Springer) model of the equatorial flow. The Stewartson model is then numerically solved, for the first time, via a geometric multigrid method that solves the steady planar Navier–Stokes equations in streamfunction-vorticity form on large rectangular domains in a quick and efficient manner. The results are then compared with a direct numerical simulation of the full unsteady problem using the Semtex software package where it is found that there is broad qualitative agreement, namely the separation and reattachment of the boundary layer at the equator. However, the presence of unobserved behaviour such as a large area of reverse flow seen at lower Reynolds numbers than those observed in other studies, and that the absolute error increases with Reynolds number suggest the model needs improvement to better capture the physical dynamics.

Nonlinear machine learning for turbulent flows can exhibit robust performance even outside the range of training data. This is achieved when machine-learning models can accommodate scale-invariant characteristics of turbulent flow structures. This study presents a data-driven approach to reveal scale-invariant vortical structures across Reynolds numbers that provide insights for supporting nonlinear machine-learning-based studies of turbulent flows. To uncover conditions for which nonlinear models are likely to perform well, we use a Buckingham-Pi-based sparse nonlinear scaling to find the influence of the Pi groups on the turbulent flow data. We consider nonlinear scalings of the invariants of the velocity gradient tensor for an example of three-dimensional decaying isotropic turbulence. The present scaling not only enables the identification of vortical structures that are interpolatory and extrapolatory for the given flow field data but also captures non-equilibrium effects of the energy cascade. As a demonstration, the present findings are applied to machine-learning-based super-resolution analysis of three-dimensional isotropic turbulence. We show that machine-learning models reconstruct vortical structures well in the interpolatory space with reduced performance in the extrapolatory space revealed by the nonlinearly scaled invariants. The present approach enables us to depart from labelling turbulent flow data with a single parameter of Reynolds number and comprehensively examine the flow field to support training and testing of nonlinear machine-learning techniques.

Shear significantly influences turbulence in the energy-containing range of shear-dominated flows, and the longitudinal structure functions do not have a universal form as they do in homogeneous isotropic turbulence. Despite this, the relative scaling of structure functions exhibits universal sub-Gaussian behaviour in shear-dominated flows, in particular for turbulent boundary layers, channels and Taylor–Couette flows. Our investigation of a turbulent vertical buoyancy layer at $Pr = 0.71$ using direct numerical simulation shows this universality even in moderate-Reynolds-number buoyancy-driven but shear-dominated boundary layers. It is demonstrated that the universality is related to the energy density of the eddies, which attains a hierarchical equilibrium in the energy-containing range of shear-dominated turbulence. We conjecture that the universal sub-Gaussian behaviour of the energy density of the energy-containing range, which was considered to be non-trivial in prior studies, is related to the universal anomalous scaling exponents of the inertial subrange turbulence. Based on this conjecture, we propose a hypothesis that relates large-scale eddies and the intermittent dissipation field in shear-dominated turbulence, highlighting a relationship between large and small scales. A phenomenological model is also developed to predict the scaling, which is verified using data from a turbulent boundary layer, half-channel and vertical buoyancy layer at friction Reynolds numbers spanning four orders of magnitude. Excellent agreement is observed.

Numerical simulations have been conducted to examine the structure of diffusive-convection staircases in the presence of vortical-mode-induced turbulent forcing. By modulating the input power $P$ and the background density ratio $R_\rho$, we have identified three distinct types of staircase structures in these simulations: namely staircases maintained in the system driven by double-diffusion, by turbulence or by a combination of both double-diffusion and turbulence. While we showed that staircases maintained in the double-diffusion-dominated system are accurately characterised by the existing model originally proposed by Linden & Shirtcliffe (J. Fluid Mech., vol. 87, no. 3, 1978, pp. 417–432), we introduced new physical models to describe the staircase structures maintained in the turbulence-dominated system and the system driven by both turbulence and double-diffusion. Our integrated model reveals that turbulence fundamentally governs the entire life cycle of the diffusive-convection staircases, encompassing their formation, maintenance and eventual disruption in the Arctic Ocean's thermohaline staircases. While our previous work of Ma & Peltier (J. Fluid Mech., vol. 931, 2022b) demonstrated that turbulence could initiate the formation of Arctic staircases, these staircases are sustained by both turbulence and double-diffusion acting together after formation has occurred. Strong turbulence may disrupt staircase structures; however, the presence of weak turbulence could lead to unstable stratification within mixed layers of the staircases, as well as enhancing vertical heat and salt fluxes. Turbulence can even sustain a stable staircase structure factor when $R_\rho$ is relatively large, following a similar mechanism to the density staircases observed in laboratory experiments. Consequently, previous parameterisations (e.g. Kelley, J. Geophys. Res.: Oceans, vol. 95, no. C3, 1990, pp. 3365–3371) on the vertical heat flux across the diffusive-convection staircases may provide a significant underestimation of the heat transport by ignoring the influences of turbulence.

The dynamics of turbulent flows past lateral cavities is relevant for multiple environmental applications. In rivers and coastal environments, these lateral recirculating regions constitute surface storage zones, where large-scale turbulent coherent structures control the transport and fate of contaminants. Mass transport in these flows is typically represented by one-dimensional first-order equations that predict the evolution of the spatially integrated concentration between the cavity and the main channel. These models, however, cannot represent the long-term evolution of the concentration or incorporate memory effects induced by turbulence. In this investigation, we carry out large-eddy simulations (LES) of the open-channel flow with a lateral square cavity of Mignot et al. (Phys. Fluids, vol. 28, issue 4, 2016, 045104). The model is coupled with an advection–diffusion equation and a Lagrangian particle model to investigate the transport mechanisms in the cavity and across the interface. From the simulations we provide quantitative comparisons of the physical processes from both perspectives, and investigate the effects of turbulent coherent structures on residence times and trajectories from finite-time Lyapunov exponents. From the Lagrangian results, we identify general spatial distributions of time scales in the cavity associated with the dynamics of coherent structures, providing new insights into the mechanisms that drive the global transport. We also show that an upscaled model informed by LES and based on a fractional derivative captures the evolution of concentration, and the exchange between the cavity and the main channel, providing accurate predictions of mass transport and reproducing the temporal dependence observed at larger scales.

Direct numerical simulation (DNS) of a turbulent boundary layer over the Gaussian (Boeing) bump is performed. This boundary layer exhibits a series of adverse and favourable pressure gradients and convex and concave curvature effects before separating. These effects on turbulent boundary layers are characterised and compared with a lower-Reynolds-number flow over the same geometry. The momentum budgets are analysed in the streamline-aligned coordinate system upstream of the separation region. These momentum budgets allow the simplification of equations to facilitate an integral analysis. Integral-analysis-based approximations for Reynolds stresses in the inner and outer regions of the boundary layer are also formulated. The shear and wall-normal Reynolds stress profiles normalised by these approximations exhibit a better collapse compared with friction velocity and Zagarola–Smits normalisations in the strong favourable pressure gradient region and in the mild adverse pressure region that precedes it in this flow. Simplification of these Reynolds stress approximations along with results from the DNS are used to obtain semi-empirical approximations that are able to provide stress closure in terms of wall solution fields for the turbulent boundary layer under consideration.

Evaporation of multicomponent droplets is important, albeit complex, in a wide range of applications, and requires a careful investigation. We investigate experimentally and numerically the evaporation characteristics of spherical, ethanol–water droplets with different initial concentration ratios in the acoustic levitation field. Imaging techniques and infrared thermometry are used for acquiring volume and surface temperature variations of droplets, reflecting their mass and heat transfer characteristics. Numerical simulations are conducted using modified parameters based on a theoretical model to consider the effect of the acoustic field. The calculation results show good agreement with the experimental data. The concentration and temperature distribution within the droplet is further investigated based on the numerical results.

The presence of a dispersed phase can significantly modulate the drag in turbulent systems. We derived a conserved quantity that characterizes the radial transport of azimuthal momentum in the fluid–fluid two-phase Taylor–Couette turbulence. This quantity consists of contributions from advection, diffusion and two-phase interface, which are closely related to density, viscosity and interfacial tension, respectively. We found from interface-resolved direct numerical simulations that the presence of the two-phase interface consistently produces a positive contribution to the momentum transport and leads to drag enhancement, while decreasing the density and viscosity ratios of the dispersed phase to the continuous phase reduces the contribution of local advection and diffusion terms to the momentum transport, respectively, resulting in drag reduction. Therefore, we concluded that the decreased density ratio and the decreased viscosity ratio work together to compete with the presence of a two-phase interface for achieving drag modulation in fluid–fluid two-phase turbulence.

A neural-network-based large eddy simulation is performed for flow over a circular cylinder. To predict the subgrid-scale (SGS) stresses, we train two fully connected neural network (FCNN) architectures with and without fusing information from two separate single-frame networks (FU and nFU, respectively), where the input variable is either the strain rate (SR) or the velocity gradient (VG). As the input variables, only the grid-filtered variables are considered for the SGS models of G-SR and G-VG, and both the grid- and test-filtered variables are considered for the SGS models of T-SR and T-VG. The training data are the filtered direct numerical simulation (fDNS) data at $Re_d=3900$ based on the free-stream velocity and cylinder diameter. Using the same grid resolution as that of the training data, the performances of G-SR and G-VG (grid-filtered inputs) and T-SR-FU and T-VG-FU (grid- and test-filtered inputs with fusion) are better than those of the dynamic Smagorinsky model and T-SR-nFU and T-VG-nFU (grid- and test-filtered inputs without fusion). These FCNN-based SGS models are applied to untrained flows having different grid resolutions from that of training data. Although the performances of G-SR and G-VG are degraded, T-SR-FU and T-VG-FU still provide good performances. Finally, T-SR-FU and T-VG-FU trained at $Re_d = 3900$ are applied to higher-Reynolds-number flows ($Re_d = 5000$ and 10 000) and their results are also in good agreements with those of fDNS and previous experiment, indicating that adding the test-filtered variables and fusion increases the prediction capability even for untrained Reynolds number flows.

The Lamb–Chaplygin dipole (Lamb, Hydrodynamics, 2nd edn, 1895, Cambridge University Press; Lamb, Hydrodynamics, 3rd edn, 1906, Cambridge University Press; Chaplygin, Trudy Otd. Fiz. Nauk Imper. Mosk. Obshch. Lyub. Estest., vol. 11, 1903, pp. 11–14) is one of the few closed-form relative equilibrium solutions of the two-dimensional (2-D) Euler equation characterized by a continuous vorticity distribution. We consider the problem of its linear stability with respect to 2-D circulation-preserving perturbations. It is demonstrated that this flow is linearly unstable, although the nature of this instability is subtle and cannot be fully understood without accounting for infinite-dimensional aspects of the problem. To elucidate this, we first derive a convenient form of the linearized Euler equation defined within the vortex core which accounts for the potential flow outside the core while making it possible to track deformations of the vortical region. The linear stability of the flow is then determined by the spectrum of the corresponding operator. Asymptotic analysis of the associated eigenvalue problem shows the existence of approximate eigenfunctions in the form of short-wavelength oscillations localized near the boundary of the vortex and these findings are confirmed by the numerical solution of the eigenvalue problem. However, the time integration of the 2-D Euler system reveals the existence of only one linearly unstable eigenmode and since the corresponding eigenvalue is embedded in the essential spectrum of the operator, this unstable eigenmode is also shown to be a distribution characterized by short-wavelength oscillations rather than a smooth function. These findings are consistent with the general results known about the stability of equilibria in 2-D Euler flows and have been verified by performing computations with different numerical resolutions and arithmetic precisions.

Solid particles trapped in an acoustic standing wave have been observed to undergo propulsion. This phenomenon has been attributed to the generation of a steady streaming flow, with a reversal in the propulsion direction at a distinct frequency. We explain the mechanism underlying this reversal by considering the canonical problem of a sphere executing oscillatory rotation in an unbounded fluid that undergoes rectilinear oscillation; these two oscillations occur at identical frequency but with an arbitrary phase difference. Two distinct bifurcations in the flow field occur: (1) a stagnation point first forms with increasing frequency, which (2) splits into a saddle node and a vortex centre. Reversal in the propulsion direction is driven by reversal in the flow far from the sphere, which coincides with the second bifurcation. This flow is identified with that of a Stokeslet whose strength is the net force exerted on the particle, which has implications for studying the flow field around particles of non-spherical geometries and for modelling suspensions of particles in acoustic fields.

Direct numerical simulations (DNSs) of three-dimensional cylindrical release gravity currents in a linearly stratified ambient are presented. The simulations cover a range of stratification strengths $0< S\leq 0.8$ (where $S=(\rho _b^*-\rho _0^*)/(\rho _c^*-\rho _0^*), \rho _b^*, \rho _0^*$ and $\rho _c^*$ are the dimensional density at the bottom of the domain, top of the domain and the dense fluid, respectively) at two different Reynolds numbers. A comparison between the stratified and unstratified cases illustrates the influence of stratification strength on the dynamics of cylindrical gravity currents. Specifically, the front velocity in the slumping phase decreases with increasing stratification strength whereas the duration of the slumping phase increases with increments of $S$. The Froude number calculated in this phase shows a good agreement with models proposed by Ungarish & Huppert (J. Fluid Mech., vol. 458, 2002, pp. 283–301) and Ungarish (J. Fluid Mech., vol. 548, 2006, pp. 49–68), originally developed for planar gravity currents in a stratified ambient. In the inertial phase, the front velocity across cases with different stratification strengths adheres to a power-law scaling with an exponent of $-$1/2. Higher Reynolds numbers led to more frequent lobe splitting and merging, with lobe size diminishing as stratification strength increased. Strong interactions among inner vortex rings occurred during the slumping phase, leading to the early formation of hairpin vortices in weakly stratified cases, while strongly stratified cases exhibited delayed vortex formation and less turbulence.

Boundary-layer disturbances are analysed on a $5^{\circ }$ half-angle blunted cone in Mach 5, high-enthalpy flow ($h_0 = 9\ {\rm MJ}\ {\rm kg}^{-1}$) with a low wall-to-edge temperature ratio, $T_w/T_e = 0.18$. Schlieren and focused laser differential interferometry (FLDI) are utilized to assess the structures and frequency content associated with disturbances. Wave packets are identified from bursts of modal content on time-resolved spectrograms. Bandpass filtering, proper orthogonal decomposition (POD) and space–time POD are then applied to the schlieren data. Bandpass filtering suggests the presence of wave packet dispersion and elongation indicative of slow-acoustic-wave synchronization. Modal reconstruction techniques indicate the radiation of content outside the boundary layer and distinct orientation changes within disturbances, potentially the first experimental evidence of the supersonic-mode instability in such a flow field. Cross-bicoherence computations are carried out for discrete time segments of data from both schlieren and FLDI data. They demonstrate that the most dominant nonlinear interactions are the fundamental–first-harmonic and the fundamental–low-frequency interactions.