Experimental observations of an azimuthal electrolyte flow driven by Lorentz force in a thin annular fluid layer placed on top of a magnet show that it develops a robust vortical system near the outer cylindrical wall. It appears to be a result of instabilities developing on a background of steady axisymmetric flow. Therefore, the goal of this paper is to establish a scene for a future comprehensive stability analysis of such a flow. We discuss popular depth-averaged and quasi-two-dimensional approximate solutions that take advantage of the thin-layer assumption first, and argue that they cannot lead to the observed flow patterns. Thus, three-dimensional toroidal flows are considered. Their similarities to various other well-studied rotating flow configurations are outlined, but no close match is found. Multiple axisymmetric solutions are detected numerically for the same governing parameters, indicating the possibility of subcritical bifurcations, namely type 1, consisting of a single torus, and type 2, developing a second counter-rotating toroidal flow near the outer cylinder. It is suggested that the transition between these two axisymmetric solutions is likely to be caused by the centrifugal instability, while the shear-type instability of the type 2 solution may be responsible for the observed vortex structures. However, a dedicated stability analysis which is currently underway and will be reported in a separate publication is required to confirm these hypotheses.

Convection in the metallic cores of terrestrial planets is likely to be subjected to lateral variations in heat flux through the outer boundary imposed by creeping flow in the overlying silicate mantles. Boundary anomalies can significantly influence global diagnostics of core convection when the Rayleigh number, $Ra$, is weakly supercritical; however, little is known about the strongly supercritical regime appropriate for planets. We perform numerical simulations of rapidly rotating convection in a spherical shell geometry and impose two patterns of boundary heat flow heterogeneity: a hemispherical $Y_{1}^{1}$ spherical harmonic pattern; and one derived from seismic tomography of the Earth’s lower mantle. We consider Ekman numbers $10^{-4}\leqslant E\leqslant 10^{-6}$, flux-based Rayleigh numbers up to ${\sim}800$ times critical, and a Prandtl number of unity. The amplitude of the lateral variation in heat flux is characterised by $q_{L}^{\ast }=0$, 2.3, 5.0, the peak-to-peak amplitude of the outer boundary heat flux divided by its mean. We find that the Nusselt number, $Nu$, can be increased by up to ${\sim}25\,\%$ relative to the equivalent homogeneous case due to boundary-induced correlations between the radial velocity and temperature anomalies near the top of the shell. The $Nu$ enhancement tends to become greater as the amplitude and length scale of the boundary heterogeneity are increased and as the system becomes more supercritical. This $Ra$ dependence can steepen the $Nu\propto Ra^{\unicode[STIX]{x1D6FE}}$ scaling in the rotationally dominated regime, with $\unicode[STIX]{x1D6FE}$ for our most extreme case approximately 20 % greater than the equivalent homogeneous scaling. Therefore, it may be important to consider boundary heterogeneity when extrapolating numerical results to planetary conditions.

Weakly nonlinear theory is used to explore the dynamics of a single-mode internal tide in variable stratification with rotation. Nonlinear self-interaction in variable stratification generates a perturbation which is forced with double the original frequency and wavenumber. The dynamics of the perturbation is analogous to a forced harmonic oscillator, with the steady-state solution corresponding to a bound harmonic matching the forcing frequency and wavenumber. When the forcing frequency is near a natural frequency of the system, even a small-amplitude (nearly linear) internal tide may induce a significant harmonic response. Idealized stratification profiles are utilized to explore the relevance of this effect for oceanic $M_{2}$ baroclinic internal tides, and the results indicate that a rapidly growing harmonic may occur in some environments near the Equator, but is unlikely at higher latitudes. The results are relevant to recent observations of $M_{4}$ (harmonic) internal tides in the South China Sea and elsewhere. More generally, nonlinear self-interaction may contribute to the transfer of energy to smaller scales and the dissipation of baroclinic internal tides, especially in equatorial waters.

The Brownian motion of a nanoparticle in fluid depends on the molecular forces acting on it. Because of the small size and the high frequency, it is difficult to make experimental measurements of these forces. In the present work, Brownian forces acting on a nanoparticle are numerically investigated with the molecular dynamics method. Some new phenomena are disclosed. (i) The probability distribution shows that the Brownian forces conform to the Gaussian distribution and self-similarity of the probability distribution is also found for different $1/Kn$ numbers which are characterized with the particle radius and the mean path $\unicode[STIX]{x1D706}$ of the gas molecule $(1/Kn=R/\unicode[STIX]{x1D706})$. (ii) The frequency spectrum distribution of the Brownian force is not a white noise spectrum, which is different from the assumption commonly used in Langevin model. The preferential frequency of the Brownian force is found. (iii) The size effect relating to the Brownian forces is not monotonically varying with $1/Kn=R/\unicode[STIX]{x1D706}$ and is also found. It first increases and then decreases after it reaches the maximum value at $1/Kn\approx 250$. The variation process for $1/Kn<250$ observed in the present work has not been reported in previous research to date.

Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (Dysthe Proc. R. Soc. Lond. A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.

Fluid dynamics induced by periodically forced flow around a cylinder is analysed computationally for the case when the forcing frequency is much lower than the von Kármán vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman mode decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric – i.e. not an additive – forcing effect. We find that the dynamics of the flow in this regime is characterized by alternating instances of quiescent and strong oscillatory behaviour and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon – that we name quasi-periodic intermittency – using the new normal form model and show that the dynamics is caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomenon is also characterized by positive finite-time Lyapunov exponents that, over a long period of time, asymptotically approach zero.

In this work, an exact Eulerian model is used to describe the steady-state motion of a bidirectional vortex in a conical chamber. This particular model is applicable to idealized representations of cyclone separators and liquid rocket engines with slowly expanding chamber cross-sections. The corresponding bulk motion is assumed to be non-reactive, rotational, inviscid and incompressible. Then, following Bloor & Ingham (J. Fluid Mech., vol. 178, 1987, pp. 507–519), the spherical Bragg–Hawthorne equation is used to construct a mathematical model that connects the solution to the swirl number and the cone divergence angle. Consequently, a self-similar formulation is obtained independently of the cone’s finite body length. This enables us to characterize the problem using closed-form approximations of the principal flow variables. Among the cyclonic parameters of interest, the mantle divergence angle and the maximum cross-flow velocity are obtained explicitly. The mantle consists of a spinning cone that separates the circumferential inflow region from the central outflow. This interfacial layer bisects the fluid domain at approximately 60 per cent of the cone’s divergence half-angle. Its accurate determination is proven asymptotically using two different criteria, one being preferred by experimentalists. Finally, recognizing that the flow in question is of the Beltramian type, results are systematically described over a range of cone angles and spatial locations in both spherical and cylindrical coordinates; they are also compared to available experimental and numerical data.

Previous work on solute transport with sorption in Poiseuille flow has reached contradictory conclusions. Some have concluded that sorption increases mean solute transport velocity and decreases dispersion relative to a tracer, while others have concluded the opposite. Here we resolve this contradiction by deriving a series solution for the transient evolution that recovers previous results in the appropriate limits. This solution shows a transition in solute transport behaviour from early to late time that is captured by the first- and zeroth-order terms. Mean solute transport velocity is increased at early times and reduced at late times, while solute dispersion is initially reduced, but shows a complex dependence on the partition coefficient $k$ at late times. In the equilibrium sorption model, the time scale of the early regime and the duration of the transition to the late regime both increase with $\ln k$ for large $k$. The early regime is pronounced in strongly sorbing systems ($k\gg 1$). The kinetic sorption model shows a similar transition from the early to the late transport regime and recovers the equilibrium results when adsorption and desorption rates are large. As the reaction rates slow down, the duration of the early regime increases, but the changes in transport velocity and dispersion relative to a tracer diminish. In general, if the partition coefficient $k$ is large, the early regime is well developed and the behaviour is well characterized by the analysis of the limiting case without desorption.

In this paper we propose a strategy, entirely relying on available experimental data, to estimate the effect of a small control rod on the frequency of vortex shedding in the wake past a thick perforated plate. The considered values of the flow Reynolds number range between $Re\simeq 6.6\times 10^{3}$ and $Re=5.3\times 10^{4}$. By means of particle image velocimetry, an experimental database consisting of instantaneous flow fields is collected for different values of suction through the body surface. The strategy proposed here is based on classical stability and sensitivity analysis applied to mean flow fields and on the formulation of an original ad hoc model for the mean flow. The mean flow model is obtained by calibrating the closure of the Reynolds averaged Navier–Stokes equations on the basis of the available experimental data through an optimisation algorithm. As a result, it is shown that the predicted control map agrees reasonably well with the equivalent one measured experimentally. Moreover, it is shown that even when turbulence effects are neglected, the stability analysis applied to the mean flow fields provides a reasonable estimation of the vortex shedding frequency, confirming what is known in the literature and extending it up to $Re=5.3\times 10^{4}$. It is also shown that, when turbulence is taken into account in the stability analysis using the same closure that is calibrated for the corresponding mean flow model, the prediction of the vortex shedding frequency is systematically improved.

We derive a time-averaged ‘hydrostatic wave equation’ from the hydrostatic Boussinesq equations that describes the propagation of inertia–gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes the wave field has a well-defined non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, assumes that the wave field and quasi-geostrophic flow have comparable spatial scales and neglects nonlinear wave–wave dynamics. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large-scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation model and illustrates how the model fails when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial. The hydrostatic wave equation provides a first step toward a coupled model for energy transfer between oceanic internal tides and quasi-geostrophic eddies and currents.

The stability of low-speed jets in cross-flow (JICF) is studied using tri-global linear stability analysis (GLSA). Simulations are performed at a Reynolds number of 2000, based on the jet exit diameter and the average velocity. A time stepper method is used in conjunction with the implicitly restarted Arnoldi iteration method. GLSA results are shown to capture the complex upstream shear-layer instabilities. The Strouhal numbers from GLSA match upstream shear-layer vertical velocity spectra and dynamic mode decomposition from simulation (Iyer & Mahesh, J. Fluid Mech., vol. 790, 2016, pp. 275–307) and experiment (Megerian et al., J. Fluid Mech., vol. 593, 2007, pp. 93–129). Additionally, the GLSA results are shown to be consistent with the transition from absolute to convective instability that the upstream shear layer of JICFs undergoes between $R=2$ to $R=4$ observed by Megerian et al. (J. Fluid Mech., vol. 593, 2007, pp. 93–129), where $R=\overline{v}_{jet}/u_{\infty }$ is the jet to cross-flow velocity ratio. The upstream shear-layer instability is shown to dominate when $R=2$, whereas downstream shear-layer instabilities are shown to dominate when $R=4$.

We present a combined experimental and theoretical study of the primary and secondary instabilities in a Kolmogorov-like flow. The experiment uses electromagnetic forcing with an approximately sinusoidal spatial profile to drive a quasi-two-dimensional (Q2D) shear flow in a thin layer of electrolyte suspended on a thin lubricating layer of a dielectric fluid. Theoretical analysis is based on a two-dimensional (2D) model (Suri et al., Phys. Fluids, vol. 26 (5), 2014, 053601), derived from first principles by depth-averaging the full three-dimensional Navier–Stokes equations. As the strength of the forcing is increased, the Q2D flow in the experiment undergoes a series of bifurcations, which is compared with results from direct numerical simulations of the 2D model. The effects of confinement and the forcing profile are studied by performing simulations that assume spatial periodicity and strictly sinusoidal forcing, as well as simulations with realistic no-slip boundary conditions and an experimentally validated forcing profile. We find that only the simulation subject to physical no-slip boundary conditions and a realistic forcing profile provides close, quantitative agreement with the experiment. Our analysis offers additional validation of the 2D model as well as a demonstration of the importance of properly modelling the forcing and boundary conditions.

The long-time evolution of a flow structure subject to background rotation and a coaxial uniform magnetic field is investigated in this paper. The conditions of magnetic Reynolds number $Rm\ll 1$ and Rossby number $Ro\ll 1$ apply, while the condition of magnetic interaction parameter $N\gg 1$ ensures that nonlinear inertial forces are small in the system. Cylindrical polar coordinates $(s,\unicode[STIX]{x1D719},z)$ are used, where the velocity and the induced magnetic field are axisymmetric. Two regimes are analysed in the inviscid limit, that of strong rotation, where the inertial wave frequency is much higher than the Alfvén wave frequency, and that of weak rotation, where the Alfvén wave frequency is dominant. In either regime, the evolution consists of a damped wave-dominated phase followed by a diffusion-dominated phase. For strong rotation, the laws of energy decay in the damped wave phase are obtained by considering the decay of the fast and slow magneto-Coriolis (MC) waves individually. The diffusion-dominated phase obeys the decay laws in the well-known quasistatic approximation. The wave–diffusion transition time scale indicates that the wave phase of decay is very long, so that small-scale turbulence is characterized by damped wave motions. The ratio of kinetic to magnetic energies of the slow MC wave in the early stages of evolution is $O(Le^{2})$, where $Le$ is the initial ratio of the inertial wave to Alfvén wave time scales. The induced magnetic field is hence far more efficient than the velocity in supporting slow MC waves for $Le\ll 1$. In the regime of weak rotation, the fast and slow MC wave solutions merge and tend to the classical damped Alfvén wave solution. Here, the decay laws in non-rotating magnetohydrodynamic turbulence (Moffatt, J. Fluid Mech., vol. 28, 1967, pp. 571–592) are recovered. Computations of the general solution for the long-time decay of an isolated vortex confirm the theoretical energy scalings as well as the wave–diffusion transition time scale of the kinetic energy. It is shown that a magnetically damped system that initially generates Alfvén waves because of relatively weak rotation can subsequently give rise to MC waves. Small-scale motions of $Rm\sim 1$ in the Earth’s core probably generate slow magnetostrophic waves only for $Le>0.1$, which suggests that a mean intensity of ${\sim}10~\text{mT}$ or higher is plausible for the toroidal magnetic field within the core.

In low-density axisymmetric jets, the onset of global instability is known to depend on three control parameters, namely the jet-to-ambient density ratio $S$, the initial momentum thickness $\unicode[STIX]{x1D703}_{0}$ and the Reynolds number $Re$. For sufficiently low values of $S$ and $\unicode[STIX]{x1D703}_{0}$, these jets bifurcate from a steady state (a fixed point) to a self-excited oscillatory state (a limit cycle) when $Re$ increases above a critical value corresponding to the Hopf point, $Re_{H}$. In the literature, this Hopf bifurcation is often regarded as supercritical. In this experimental study, however, we find that under some conditions, there exists a hysteretic bistable region at $Re_{SN}<Re<Re_{H}$, where $Re_{SN}$ denotes a saddle-node point. This shows that, contrary to expectations, the Hopf bifurcation can also be subcritical, which we explore by evaluating the coefficients of a truncated Landau model. The existence of subcritical bifurcations implies the potential for triggering and the need for weakly nonlinear analyses to be performed to at least fifth order if one is to be able to predict saturation and bistability. We conclude by proposing a universal scaling for $Re_{H}$ in terms of $S$ and $\unicode[STIX]{x1D703}_{0}$. This scaling, which is insensitive to the super/subcritical nature of the bifurcations, can be used to predict the onset of self-excited oscillations, providing further evidence to support Hallberg & Strykowski’s concept (J. Fluid Mech., vol. 569, 2006, pp. 493–507) of universal global modes in low-density jets.

A one-dimensional, unsteady nozzle flow is modelled to identify the sources of indirect noise in multicomponent gases. First, from non-equilibrium thermodynamics relations, it is shown that a compositional inhomogeneity advected in an accelerating flow is a source of sound induced by inhomogeneities in the mixture (i) chemical potentials and (ii) specific heat capacities. Second, it is shown that the acoustic, entropy and compositional linear perturbations evolve independently from each other and they become coupled through mean-flow gradients and/or at the boundaries. Third, the equations are cast in invariant formulation and a mathematical solution is found by asymptotic expansion of path-ordered integrals with an infinite radius of convergence. Finally, the transfer functions are calculated for a supersonic nozzle with finite spatial extent perturbed by a methane–air compositional inhomogeneity. The proposed framework will help identify and quantify the sources of sound in nozzles with relevance, for example, to aeronautical gas turbines.