We revisit the theoretical solutions of the problem of the reflected flow from a vertical obstacle encountered by a current generated by dambreak of a Cartesian long reservoir over a dry bottom and with tailwaters. Previous investigations derived accurate solutions by quite complex manipulation of the balances along the characteristics, supported by numerical solutions, of the shallow-water equations (a significant extension is the recent paper Hogg & Skevington, Q. J. Mech. Appl. Maths, vol. 74, no. 4, 2021, pp. 441–465). Here we developed a simplified model, based on the major assumption that the fluid between the obstacle (wall) and the reflected jump is stagnant. This allows the solution of the problem by a straightforward numerical integration of one initial-value ordinary differential equation. The model provides the position, height and speed of the jump as functions of time. For long times a simple analytical approximation is also available. The model points out clear-cut effects of the presence of the tailwaters. The model has been validated by comparisons with exact solutions of the shallow-water equations of the recent study of Hogg & Skevington (Q. J. Mech. Appl. Maths, vol. 74, no. 4, 2021, pp. 441–465) (obtained by significantly more complex hodograph-plane methods). In all the tested cases the agreement is good, for long periods of time. This model provides reliable insights and fast quantitative predictions recommended for use in research and engineering problems where a fair approximation is sufficient for the application.

We present results of numerical simulations of a stratified reservoir with a three-layer stratification, subject to an oscillating surface shear stress. We investigate the effect of sloped endwalls on mixing and internal wave adjustment to forcing within the basin, for three different periods of forcing. The simulations are carried out at a laboratory scale, using large-eddy simulation. We solve the three-dimensional Navier–Stokes equations under the Boussinesq approximation using a second-order-accurate finite-volume solver. The model was validated by reproducing experimental results for the response of a reservoir to surface shear stress and resonant frequencies of internal waves. We find interesting combinations of wave modes and mixing under variation of the forcing frequencies and of the inclination of the endwalls. When the frequency of the forcing is close to the fundamental mode-one wave frequency, a resonant internal seiche occurs and the response is characterized by the first vertical mode. For forcing periods twice and three times the fundamental period, the dominant response is in terms of the second vertical mode. Adjustment to forcing via the second vertical mode is accompanied by the cancellation of the fundamental wave and energy transfer to higher-frequency waves. The study shows that the slope of the endwalls dramatically affects the location of mixing, which has a feedback on the wave field by promoting the generation of higher vertical modes.

Nonlinear three-dimensional dynamo equilibrium solutions of viscous-resistive magneto-hydrodynamic equations are continued to formally infinite magnetic and hydrodynamic Reynolds numbers. The external driving mechanism of the dynamo is a uniform shear, which constitutes the base laminar flow and cannot support any kinematic dynamo. Nevertheless, an efficient subcritical nonlinear instability mechanism is found to be able to generate large-scale coherent structures known as streaks, for both velocity and magnetic fields. A finite amount of magnetic field generation is identified at the self-consistent asymptotic limit of the nonlinear solutions, thereby confirming the existence of an effective nonlinear dynamo action at astronomically large Reynolds numbers.

In this paper, we consider two-dimensional steady free-surface flows where gravity is ignored, but the effects of surface tension are included. It is found that the existence of an additional solid boundary can allow for previously unseen limiting configurations as the surface tension tends to infinity. The free surface of these new solutions is formed of straight lines, arcs of circles and a point where the flow turns at a wall. These limiting configurations form endpoints of solution branches of capillary free-surface flows. Other endpoints of these branches include the surface tension free (i.e. free streamline) solution, and a solution whose free surface is composed simply of a straight line. The model we explore is flow incoming along a channel of constant width. One of the walls terminates, where the fluid is forcibly separated from the wall and a free boundary is formed. The other wall meets a second straight boundary with interior angle $\beta$. Far downstream the solution approaches a uniform stream. Making use of Cauchy's integral formula, the unknowns are expressed in terms of values on the boundary. The integral equations are then solved numerically. The solution space relative to the parameter values of the model is discussed.

We consider the time-dependent speed of an infinitely long cylinder that minimizes the net work done on the surrounding fluid to travel a given distance perpendicular to its axis in a fixed amount of time. The flow that develops is two-dimensional. An analytical solution is possible using calculus of variations for the case that the distance travelled and the viscous boundary layer thickness that develops are much smaller than the circle radius. If $t$ represents the time since the commencement of motion and $T$ the final time, then the optimum speed profile is $Ct^{1/4}(T-t)^{1/4}$, where $C$ is determined by the distance travelled. The result also holds for rigid-body translations and rotation of cylinders formed by extrusion of smooth but otherwise arbitrary curves.

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$, compares to the basic-state amplitude parameter, $\epsilon \ll 1$. For $\mu ={O}(1)$, the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$, this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

The interaction of steady free-surface flows of viscoplastic material with a surface-piercing obstruction of square cross-section on an inclined plane is investigated theoretically. The flow thickness increases upstream of the obstruction and decreases in its lee. The flow depends on two dimensionless parameters: an aspect ratio that relates the flow thickness, the obstruction width and the plane inclination; and a Bingham number that quantifies the magnitude of the yield stress relative to the gravitationally induced stresses. Flows with a non-vanishing yield stress always form a static ‘dead’ zone in a neighbourhood of the upstream and downstream stagnation points. For relatively wide obstructions, a deep ‘ponded’ region develops upstream with a small dead zone, while the deflected flow reconnects over relatively long distances downstream. The depth of the upstream pond increases with both the dimensionless yield stress and width of the obstruction, while the unyielded dead zone varies primarily with the yield stress. Both are predicted asymptotically by balancing the volume flux of fluid into and out of the ponded region. When the obstruction is narrow, the perturbation to the depth of the oncoming flow is reduced. It exhibits fore–aft antisymmetry, while the dead zone is symmetric to leading order. Increasing the yield stress leads to larger dead zones that eventually encompass all of the upstream- and downstream-facing boundaries of the obstruction and fully divert the flow. Results for obstructions with circular and rhomboidal cross-sections are also presented and illustrate the effects of boundary shape on the properties of the steady flow.

In Part 1 (Wienkers, Thomas & Taylor, J. Fluid Mech., vol. 926, 2021, A6), we described the theory for linear growth and weakly nonlinear saturation of symmetric instability (SI) in the Eady model representing a broad frontal zone. There, we found that both the fraction of the balanced thermal wind mixed down by SI and the primary source of energy are strongly dependent on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Strong fronts with steep isopycnals develop a flavour of SI we call ‘slantwise inertial instability’ by extracting kinetic energy from the background flow and rapidly mixing down the thermal wind profile. In contrast, weak fronts extract more potential energy from the background density profile, which results in ‘slantwise convection.’ Here, we extend the theory from Part 1 using nonlinear numerical simulations to focus on the adjustment of the front following saturation of SI. We find that the details of adjustment and amplitude of the induced inertial oscillations depend on the front strength. While weak fronts develop narrow frontlets and excite small-amplitude vertically sheared inertial oscillations, stronger fronts generate large inertial oscillations and produce bore-like gravity currents that propagate along the top and bottom boundaries. The turbulent dissipation rate in these strong fronts is large, highly intermittent and intensifies during periods of weak stratification. We describe each of these mechanisms and energy pathways as the front evolves towards the final adjusted state, and in particular focus on the effect of varying the dimensionless front strength.

Clustering of externally and evenly heated particles is enhanced by the increased viscosity of heated fluid in the vicinity of these clusters – a phenomenon known as viscous capturing (VC). Herein we study, via direct numerical simulations of decaying turbulence, the effect of temperature-driven viscosity on clustering with different particle loading densities. We employ a two-way momentum and energy coupling, and gas viscosity is modelled by a power law to understand the role of the increased drag and particle back-reaction force on the clustering intensity. For the continuum and dispersed phases, Eulerian and Lagrangian point particle schemes have been used, neglecting inter-particle collisions. We found that the enhanced viscosity-driven clustering is a strong function of particle loading density, as the increase in particle number density enables the formation of large uneven clusters before heating, which is the main condition for VC to take effect. Higher number density should result in greater turbulence modulation and negate local temperature-based viscous effects leading to VC. However, due to higher local particle number density in the clusters and interphase heat transfer, increased drag force prevails in such cases and delivers excessive clustering. By sampling conditionally the particle velocity and temperature inside the clusters, it is found that the thermodynamic and kinematic properties of the particles in the clusters are highly correlated, and this correlation increases with the particle loading density. Therefore, based on the particle number density, temperature-based viscosity can enhance considerably the clustering of heated particles and alter the effect of particles on the underlying turbulence.

We consider a theoretical model for the settling of rod-shaped particles of a dilute, initially homogeneous, suspension in rapid rotation. The particle Reynolds number and the particle Taylor number of the detailed flow around the particles are assumed small, representing a relevant limit for an industrial centrifugal separation process. By applying a statistical approach using the Fokker–Planck equation, and neglecting particle–particle interactions, we obtain an explicit, analytical solution for the time dependent, spatially uniform particle orientation distribution function. Not only does the volume fraction in the bulk of the suspension decrease with time due to the divergent centrifugal field, as similarly described in the literature for suspensions of spherical particles, the orientation of the rod particles also changes with time from an initially uniform distribution to one where the particles tend to align with a plane perpendicular to the axis of rotation. The corresponding particle trajectories, as also influenced by first-order effects from the Coriolis acceleration and gyroscopic effects, are obtained numerically for different initial particle orientation angles.

The dynamics of charge-free dielectric droplets under electric fields in weakly conducting fluids is relevant to a large range of technological and environmental applications. In Sorgentone & Vlahovska (J. Fluid Mech., vol. 951, 2022, R2), the authors investigate by asymptotic theory and boundary integral simulations the motility states of a pair of dissimilar droplets at an arbitrary orientation to the electric field and find a state of steady motility that is carefully balanced between repulsion and contact.