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The article "Relative fluid stretching and rotation for sparse trajectory observations" in the Journal of Fluid Mechanics introduces a new method to quantify fluid stretching and rotation using relative Lagrangian velocities. The research focuses on identifying fluid structures in unsteady and sparsely sampled flows, like ocean buoy data and Lagrangian particle tracking, surpassing existing limits of sparse diagnostics. The study compares the proposed method with traditional metrics, demonstrating its advantages in various numerical and experimental flows. It also discusses the use of relative trajectory diagnostics to identify coherent flow features in sparse trajectory data, showcasing their effectiveness in identifying elliptic and hyperbolic Lagrangian coherent structures in different flow scenarios. [Extracted from the article]

The statistics of breaking wave fields are characterised within a novel multi-layer framework, which generalises the single-layer Saint-Venant system into a multi-layer and non-hydrostatic formulation of the Navier–Stokes equations. We simulate an ensemble of phase-resolved surface wave fields in physical space, where strong nonlinearities, including directional wave breaking and the subsequent highly rotational flow motion, are modelled, without surface overturning. We extract the kinematics of wave breaking by identifying breaking fronts and their speed, for freely evolving wave fields initialised with typical wind wave spectra. The $\varLambda (c)$ distribution, defined as the length of breaking fronts (per unit area) moving with speed $c$ to $c+{\rm d}c$ following Phillips (J. Fluid Mech. , vol. 156, 1985, pp. 505–531), is reported for a broad range of conditions. We recover the $\varLambda (c) \propto c^{-6}$ scaling without wind forcing for sufficiently steep wave fields. A scaling of $\varLambda (c)$ based solely on the root-mean-square slope and peak wave phase speed is shown to describe the modelled breaking distributions well. The modelled breaking distributions are in good agreement with field measurements and the proposed scaling can be applied successfully to the observational data sets. The present work paves the way for simulations of the turbulent upper ocean directly coupled to a realistic breaking wave dynamics, including Langmuir turbulence, and other sub-mesoscale processes. [ABSTRACT FROM AUTHOR]

The pressure ratios are circumferentially distributed in polar coordinate systems, where the origins are the intersection points between the cross-sections and the Graph $x$ axis and the angular coordinates (Graph $\varphi$) begin from Graph $0^{\circ }$ where the rays coincide with the positive Graph $z$ direction. Although at the end of convergence Graph $\textrm {IS} u$ and Graph $\textrm {IS} l$ are still terminated by DiMR (at Graph $T u$) and InMR (at Graph $T l$), respectively, at approximately Graph $x/L = 2.66$, it is worth mentioning that MS is evidently shrunk and leaned forward farther compared with that of Graph $\alpha = 2^{\circ }$. For comparison, figure 7( I a i ) gives the numerical pressure ratio variations for Graph $\textrm {IS} u$ and Graph $\textrm {IS} l$, together with those of the AIS at Graph $\theta w = 8^{\circ }$, Graph $10^{\circ }$ and Graph $12^{\circ }$. The new coefficients Graph $c u$ and Graph $c l$ have slight differences from Graph $c$ in the axisymmetric flow since Graph $c u$ is raised by the stronger windward leading shock, whereas Graph $c l$ is reduced by the weaker leeward leading shock. [Extracted from the article]

The force balance of rotating Rayleigh–Bénard convection regimes is investigated using direct numerical simulation on a laterally periodic domain, vertically bounded by no-slip walls. We provide a comprehensive view of the interplay between governing forces both in the bulk and near the walls. We observe, as in other prior studies, regimes of cells, convective Taylor columns, plumes, large-scale vortices (LSVs) and rotation-affected convection. Regimes of rapidly rotating convection are dominated by geostrophy, the balance between Coriolis and pressure-gradient forces. The higher-order interplay between inertial, viscous and buoyancy forces defines a subdominant balance that distinguishes the geostrophic states. It consists of viscous and buoyancy forces for cells and columns, inertial, viscous and buoyancy forces for plumes, and inertial forces for LSVs. In rotation-affected convection, inertial and pressure-gradient forces constitute the dominant balance; Coriolis, viscous and buoyancy forces form the subdominant balance. Near the walls, in geostrophic regimes, force magnitudes are larger than in the bulk; buoyancy contributes little to the subdominant balance of cells, columns and plumes. Increased force magnitudes denote increased ageostrophy near the walls. Nonetheless, the flow is geostrophic as the bulk. Inertia becomes increasingly more important compared with the bulk, and enters the subdominant balance of columns. As the bulk, the near-wall flow loses rotational constraint in rotation-affected convection. Consequently, kinetic boundary layers deviate from the expected behaviour from linear Ekman boundary layer theory. Our findings elucidate the dynamical balances of rotating thermal convection under realistic top/bottom boundary conditions, relevant to laboratory settings and large-scale natural flows. [ABSTRACT FROM AUTHOR]

Small viscous effects in high-Reynolds-number rotational flows always accumulate over time to have a leading-order effect. Therefore, the high-Reynolds-number limit for the Navier–Stokes equations is singular. It is important to investigate whether a solution of the Euler equations can approximate a real flow at large Reynolds number. These facts are often overlooked and, as a result, the Euler equations are used to simulate laminar rotational flows at large Reynolds number. Based on the Fredholm alternative, an asymptotic perturbation theory is described to establish secularity conditions determined by viscosity for an inviscid solution to approximate a real viscous fluid. Four important classical inviscid solutions are investigated using the theory with the following conclusions. The Stuart cats' eyes and Mallier–Maslowe vortices are inconsistent with any real fluid at high Reynolds number; whereas Hill's spherical vortex is confirmed to be consistent with a steady state in the spherical core region and the Lamb–Chaplygin dipole is found to be consistent with a quasi-steady state in the circular core region. These solutions have been widely used for analysing the stability of vortex flows and wakes, and their interactions with shock waves or bubbles. Serendipitously, we have revealed an original exact solution of the Navier–Stokes equations which is time dependent, has non-zero nonlinear convective terms and is restricted to a finite domain with the decay rate depending on dipole radius. [ABSTRACT FROM AUTHOR]

Electrohydrodynamics of drops is a classic fluid mechanical problem where deformations and microscale flows are generated by application of an external electric field. In weak fields, electric stresses acting on the drop surface drive quadrupolar flows inside and outside and cause the drop to adopt a steady axisymmetric shape. This phenomenon is best explained by the leaky-dielectric model under the premise that a net surface charge is present at the interface while the bulk fluids are electroneutral. In the case of dielectric drops, increasing the electric field beyond a critical value can cause the drop to start rotating spontaneously and assume a steady tilted shape. This symmetry-breaking phenomenon, called Quincke rotation, arises due to the action of the interfacial electric torque countering the viscous torque on the drop, giving rise to steady rotation in sufficiently strong fields. Here, we present a small-deformation theory for the electrohydrodynamics of dielectric drops for the complete Melcher–Taylor leaky-dielectric model in three dimensions. Our theory is valid in the limits of strong capillary forces and highly viscous drops and is able to capture the transition to Quincke rotation. A coupled set of nonlinear ordinary differential equations for the induced dipole moments and shape functions are derived whose solution matches well with experimental results in the appropriate small-deformation regime. Retention of both the straining and rotational components of the flow in the governing equation for charge transport enables us to perform a linear stability analysis and derive a criterion for the applied electric field strength that must be overcome for the onset of Quincke rotation of a viscous drop. [ABSTRACT FROM AUTHOR]

In a theoretical analysis, we generalise well-known asymptotic results to obtain expressions for the rate of transfer of material from the surface of an arbitrary, rigid particle suspended in an open pathline flow at large Péclet number, ${Pe}$. The flow may be steady or periodic in time. We apply this result to numerically evaluate expressions for the surface flux to a freely suspended, axisymmetric ellipsoid (spheroid) in Stokes flow driven by a steady linear shear. We complement these analytical predictions with numerical simulations conducted over a range of ${Pe} = 10^1\text {--}10^4$ and confirm good agreement at large Péclet number. Our results allow us to examine the influence of particle shape upon the surface flux for a broad class of flows. When the background flow is irrotational, the surface flux is steady and is prescribed by three parameters only: the Péclet number, the particle aspect ratio and the strain topology. We observe that slender prolate spheroids tend to experience a higher surface flux compared to oblate spheroids with equivalent surface area. For rotational flows, particles may begin to spin or tumble, which may suppress or augment the convective transfer due to a realignment of the particle with respect to the strain field. [ABSTRACT FROM AUTHOR]

We numerically investigate the flow structure of periodic steady water waves of fixed relative mass flux propagating on rotational flows with piece-wise constant vorticity. We show that, for wave solutions along the global bifurcation diagram, the stagnation point can first occur internally or at the bottom, and then again occurs at the crest with further increase in wave height. We observe that the bifurcation diagram has a new branch which is not connected to the trivial solution. Furthermore, we present an in-depth discussion of the results on pressure distributions and particle trajectories beneath large-amplitude steady waves near stagnation. We also expand on previous results concerning the amplitude and mass flux of steady water waves travelling on rotational flows with discontinuous vorticity. [ABSTRACT FROM AUTHOR]

Direct numerical simulations are used to study the three-dimensional, incompressible and viscous flow dynamics of a base solid-body rotation flow with a uniform axial velocity entering a rotating, finite-length, straight circular pipe. Steady in time profiles of the axial, radial and circumferential velocities are prescribed along the pipe inlet. The convective boundary conditions for each velocity flux component is set at the pipe outlet. The simulation results describe the neutral stability line in response to either axisymmetric or three-dimensional perturbations in a diagram of Reynolds number (Re, based on inlet axial velocity and pipe radius) versus the incoming flow swirl ratio (ω). This line is in good agreement with the neutral stability line recently predicted by the linear stability theory of Wang et al. (J. Fluid Mech., vol. 797, 2016, pp. 284-321). The computed time history of the velocity components at a certain point in the flow is used to describe three-dimensional phase portraits of the flow global dynamics and its long-term behaviour. They show three types of flow evolution scenarios. First, the Wang & Rusak (Phys. Fluids, vol. 8 (4), 1996, pp. 1007-1016) axisymmetric instability mechanism and evolution to a stable axisymmetric breakdown state is recovered at certain operational conditions in terms of Re and ω. However, at other operational conditions with same ω but with a higher Re, a second scenario is found. The axisymmetric breakdown state continues to evolve and a spiral instability mode appears on it and grows to a rotating spiral breakdown state. Moreover, at higher levels of ω a third scenario is found where there exists a dominant three-dimensional spiral type of instability mode that agrees with the linear stability theory of Wang et al. (J. Fluid Mech., vol. 797, 2016, pp. 284-321). The growth of this mode leads directly to a spiral type of flow roll-up and nonlinearly saturates on a rotating spiral type of vortex breakdown. The Reynolds-Orr equation is used to reveal the mechanism that drives all the instabilities as well as the nonlinear global flow evolution. At high swirl ratios, the confined kinetic energy in the swirling flow can be triggered to be released through various physical agents, such as the asymmetric inlet-outlet conditions, that eliminate axial homogeneity along the pipe and induce flow instabilities and evolution to breakdown states. It is also shown that local instability analysis or its extension using the assumption of a weakly non-parallel flow to conduct convective instability-absolute instability analyses is definitely not related to any of the instability modes found in the present study. Moreover, a stability study based on the strongly non-parallel flow character, including axial inhomogeneity due to a finite-domain boundary conditions, must be conducted to reveal instabilities in such flows. [ABSTRACT FROM AUTHOR]

A simple model of rotating 2-lobed droplets is proposed by setting the outline shape of the droplet to the Cassinian oval, a mathematical curve that closely resembles in shape. By deriving the governing equation of the proposed model and obtaining its stationary solutions, the relationship between the angular velocity of rotation and the maximum deformation length is explicitly and precisely calculated. The linear stability analysis is performed for the stationary solutions, and it is demonstrated that the stability of the solutions depends only on the ratio of the deformation length to the radius of the central cross-section of the droplet, which is independent of the physical properties of the droplet. Via comparison with an experimental study, it is observed that the calculated result is consistent with the deformation behaviour of actual 2-lobed droplets in the range where the stationary solution of the proposed model is linearly stable. Therefore, the proposed model is a suitable model for reproducing the steady deformation behaviour of 2-lobed droplets in a wide range of viscosities, surface tensions, densities and initial radii of the droplet, and especially if the viscosity of the droplet is low, the entire process of deformation of the 2-lobed droplet, including the unsteady breakup process, can be very well reproduced by the proposed model. [ABSTRACT FROM AUTHOR]

Direct numerical simulations are carried out to investigate the role of the turbulent region in a self-sustaining system with a spiral vortex structure in the threedimensional boundary layer over a rotating disk by solving the full Navier-Stokes equations. Two computational domains with two different azimuthal sizes, 2π/68 and 2π/32, are used to deal with different initially dominant wavenumbers. An artificial disturbance is introduced by short-duration strong suction and blowing on the disk surface. After the flow field reaches a steady state, a turbulent region forms downstream of Re = 640. The turbulent region is then removed using two methods: a sponge region, and application of a slip condition at the wall. In both cases, the turbulent region disappears, leaving the spiral vortex structure upstream unaffected. The results suggest that the downstream turbulent region is not related to the velocity fluctuations that grow by the global instability. In addition, when the area where the slip condition is applied is changed from Re > 630 to Re > 610, the velocity fluctuations decay. The results indicate that the vibration source of the velocity fluctuations which grow by the global instability is located between Re = 611 and Re=630. [ABSTRACT FROM AUTHOR]

A model of a laminar viscous conducting flow, near a dielectric disc in a uniform magnetic field and in the presence of external rotation, is considered, where there is a uniform suction and an axial temperature gradient between the flow and the disc's surface. It is assumed that the parameters of the suction or the magnetohydrodynamic (MHD) interaction are such that the nonlinear inertial terms, related to the circulation flow, are negligible in the differential equations of the MHD boundary layer on a rotating disc. Analysis of the motion and energy equations, taking the dependence of density on temperature into account, is carried out using the Dorodnitsyn transformation. The exact analytical solution for the boundary layer and heat transfer equations is obtained and analysed, neglecting the viscous and Joule dissipation. The dependence of the flow characteristics in the boundary layer on the rate of suction and the magnetic field induction is studied. It is shown that the direction of the radial flow in the boundary layer on a disc can be changed, not only by variation of the ratio between the angular velocities in the external flow and the boundary layer, but also by changing the ratio of the temperatures in these two flows, as well as by varying the hydrodynamic Prandtl number. The approximate calculation of a three-dimensional flow in a rotating cylinder with a braking disc (or lid) is carried out, demonstrating that a magnetic field slows the circulation velocity in a rotating cylinder. [ABSTRACT FROM AUTHOR]

Under the action of solid body rotation, homogeneous neutrally stratified turbulence undergoes anisotropization and onset of the inverse energy cascade. These processes are investigated using a quasi-normal scale elimination (QNSE) theory in which successive coarsening of a flow domain yields scale-dependent eddy viscosity and diffusivity. The effect of rotation increases with increasing scale and manifests in anisotropization of the eddy viscosities, eddy diffusivities and kinetic energy spectra. Not only the vertical (in the direction of the vector of rotation Ω) and horizontal eddy viscosities and eddy diffusivities become different but, reflecting both directional and componental anisotropization, there emerge four different eddy viscosities. Three of them decrease relative to the eddy viscosity in non-rotating flows while one increases; the horizontal 'isotropic' viscosity decreases at the fastest rate. This behaviour is indicative of the increasing redirection of the energy flux to larger scales, the phenomenon that can be associated with the energy backscatter or inverse energy cascade. On scales comparable to the Woods's scale which is the rotational analogue of the Ozmidov length scale in stably stratified flows, the horizontal viscosity rapidly decreases, and in order to keep it positive, a weak rotation limit is invoked. Within that limit, an analytical theory of the transition from the Kolmogorov to a rotation-dominated turbulence regime is developed. It is shown that the dispersion relation of linear inertial waves is unaffected by turbulence while all one-dimensional energy spectra undergo steepening from the Kolmogorov -5=3 to the -3 slope. [ABSTRACT FROM AUTHOR]

We investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, $\mathit{Ro}$ and $\mathit{Fr}$, we show that the PEs can be obtained from the BQs by taking the limit ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} \rightarrow 0$. When ${\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} $ is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from ${ k}_{h}^{- 3} $ to ${ k}_{h}^{- 5/ 3} $ at the small-scale end. The vertical spectra show a similar transition as the horizontal spectra and we find that Charney isotropy is approximately valid also at larger wavenumbers than the transition wavenumber. The high resolutions employed allow us to capture both ranges within the same simulation. At the transition scale, kinetic energy in the rotational and in the horizontally divergent modes attain comparable values. The divergent energy is several orders of magnitude larger than the quasi-geostrophic divergent energy given by the $\Omega $-equation. The amount of energy cascading downscale is mainly controlled by the rotation rate, with a weaker dependence on the stratification. A larger degree of stratification favours a downscale energy cascade. For intermediate degrees of rotation and stratification, a constant energy flux and a constant enstrophy flux coexist within the same range of scales. In this range, the enstrophy flux is a result of triad interactions involving three geostrophic modes, while the energy flux is a result of triad interactions involving at least one ageostrophic mode, with a dominant contribution from interactions involving two ageostrophic and one geostrophic mode. Dividing the ageostrophic motions into two classes depending on the sign of the linear wave frequency, we show that the energy transfer is for the largest part supported by interactions within the same class, ruling out the wave–wave–vortex resonant triad interaction as a mean of the downscale energy transfer. The role of inertia-gravity waves is studied through analyses of time-frequency spectra of single Fourier modes. At large scales, distinct peaks at frequencies predicted for linear waves are observed, whereas at small scales no clear wave activity is observed. Triad interactions show a behaviour which is consistent with turbulent dynamics, with a large exchange of energy in triads with one small and two large comparable wavenumbers. The exchange of energy is mainly between the modes with two comparable wavenumbers. [ABSTRACT FROM PUBLISHER]

The linear stability of a uniform flow in an infinitely wide erodible channel is investigated with respect to disturbances of the bed that are periodic in both the transverse and the longitudinal directions. A rotational flow and sediment transport model, originally developed to study the formation of two-dimensional dunes and antidunes, is straightforwardly extended to cover variations in the lateral direction. Sediment is assumed to be transported as bed load, disregarding the role of suspension. Following a standard linearization procedure, a dispersion relationship is obtained that expresses the growth rate and the celerity of the sand wave as a function of the streamwise and spanwise wavenumbers and of the relevant flow and sediment parameters. Regions of instabilities in the space of the parameters are found, which can be associated with bed forms of different kinds, spanning from dunes and antidunes to alternate bars. Therefore, the present theory allows for a unified view of the formation of two- and three-dimensional bed forms in rivers in terms of the relevant flow and sediment parameters. [ABSTRACT FROM PUBLISHER]

The rotation of a neutrally buoyant spheroidal particle in a Couette flow is studied by a multi-relaxation-time (MRT) lattice Boltzmann method. We find several new periodic and steady rotation modes for a prolate spheroid for Reynolds numbers ($\mathit{Re}$) exceeding 305. The simulations cover the regime up to $\mathit{Re}\leq 700$. The rotational behaviour of the spheroid appears to be not only sensitive to the Reynolds number but also to its initial orientation. We discuss the effects of initial orientation in detail. For $305\lt \mathit{Re}\lt 345$ we find that the prolate spheroid reaches a periodic mode characterized by precession and nutation around an inclined axis which is located close to the middle plane where the velocity is zero. For $345\lt \mathit{Re}\lt 385$, the prolate spheroid precesses around the vorticity direction with a nutation. For $\mathit{Re}$ close to the critical ${\mathit{Re}}_{c} \approx 345$, a period-doubling phenomenon is observed. We also identify a motionless mode at higher Reynolds numbers ($\mathit{Re}\gt 445$) for the prolate spheroid. For the oblate spheroid the dynamic equilibrium modes found are log rolling, inclined rolling and different steady states for $\mathit{Re}$ increasing from 0 to 520. The initial-orientation effects are studied by simulations of 57 evenly distributed initial orientations for each $\mathit{Re}$ investigated. Only one mode is found for the prolate spheroid for $\mathit{Re}\lt 120$ and $385\lt \mathit{Re}\lt 445$. In other $\mathit{Re}$ regimes, more than one mode is possible and the final mode is sensitive to the initial orientation. However, the oblate spheroid dynamics are insensitive to its initial orientation. [ABSTRACT FROM PUBLISHER]

Fibre-level computer simulation is carried out to study the rotational diffusion and structural evolution of semidilute suspensions of non-Brownian, rigid-rod-like fibres under shear flow in a Newtonian fluid. The analyses use a hybrid approach where the lattice-Boltzmann method is coupled with the external boundary force method. The probability distribution of the orbit constant, $p({C}_{b} )$, in the semidilute regime is predicted with this method. The paper emphasizes assessment of the characteristics of a rotary diffusion model – anisotropic in nature (Koch, Phys. Fluids, vol. 7, 1995, pp. 2086–2088) – when used in suspensions with fibres of different aspect ratios (ranging from ${r}_{p} = 16$ to $72$) and with different volume concentrations (ranging from ${c}_{v} = 7. 58\ensuremath{\times} 1{0}^{\ensuremath{-} 3} $ to $6. 14\ensuremath{\times} 1{0}^{\ensuremath{-} 2} $). A measure of the scalar Folgar–Tucker constant, ${C}_{I} $, is extracted from the anisotropic diffusivity tensor, $ \mathbisf{C} $. The scalar ${C}_{I} $ is mostly $O(1{0}^{\ensuremath{-} 4} )$ in the semidilute regime and compares very well with the experimental observations of Stover (PhD thesis, School of Chemical Engineering, Cornell University, 1991) and Stover, Koch & Cohen (J. Fluid Mech., vol. 238, 1992, pp. 277–296). The ${C}_{I} $ values provide substantial numerical evidence that the range of ${C}_{I} $ (0.0038–0.0165) obtained by Folgar & Tucker (J. Rein. Plast. Compos., vol. 3, 1984, pp. 98–119) in the semidilute regime is actually overly diffusive. The paper also branches out to incorporate anisotropic diffusion (through the use of the Koch model) in the second-order evolution equation for $ \mathbisf{A} $ (a second-order orientation tensor). The solution of the evolution equation with the Koch model demonstrates unphysical behaviour at low concentrations. The most plausible explanation for this behaviour is error in the closure approximation; and the use of the Koch model in a spherical harmonics-based method (Montgomery-Smith, Jack & Smith, Compos. A: Appl. Sci. Manuf., vol. 41, 2010, pp. 827–835) to solve for the orientation moments corroborates this claim. [ABSTRACT FROM PUBLISHER]

Turbulent channel flow with spanwise rotation is studied by direct numerical simulation (DNS) and the so-called helical wave decomposition (HWD). For a wall-bounded channel domain, HWD decomposes the flow fields into helical modes with different scales and opposite polarities, which allows us to investigate the energy distribution and nonlinear transfer among various scales. Our numerical results reveal that for slow rotation, the fluctuating energy concentrates into large-scale modes. The flow visualizations show that the fine vortices at the unstable side of the channel form long columns, which are basically along the streamwise direction and may be related to the roll cells reported in previous studies. As the rotation rate increases, the concentration of the fluctuating energy shifts towards smaller scales. For strong rotation, an inverse energy cascade occurs due to the nonlinear interaction of the fluctuating modes. A possible mechanism for this inverse cascade is then proposed and attributed to the Coriolis effect. That is, under strong rotation the fluctuating Coriolis force tends to be parallel to the fluctuating vorticity in the region where the streamwise mean velocity has linear profile. Thus the force can induce strong axial stretching/shrinking of the vortices and change the scales of the vortical structures significantly. [ABSTRACT FROM PUBLISHER]