We use numerical simulations of the reactive Euler equations to analyse the nonlinear stability of steady-state one-dimensional solutions for gaseous detonations in the presence of both momentum and heat losses. Our results point to a possible stabilization mechanism for the low-velocity detonations in such systems. The mechanism stems from the existence of a one-parameter family of solutions found in Semenko et al. (Shock Waves, vol. 26 (2), 2016, pp. 141–160).

High-speed synchronized stereo particle-imaging velocimetry and OH planar laser-induced fluorescence (PIV/OH-PLIF) measurements are performed on multiple $R{-}\unicode[STIX]{x1D703}$ planes downstream of a high-Reynolds-number swirling jet. Dynamic-mode decomposition (DMD) – a frequency-resolved data-reduction technique – is used to identify and characterize recurrent flow structures. Illustrative results are presented in a swirling flow field for two cases – the nominal flow dynamics and where self-excited combustion driven oscillations provide strong axisymmetric narrowband forcing of the flow. The robust constituent of the nominal reacting swirl flow corresponds to a helical shear-layer disturbance at a Strouhal number ($St$) of ${\sim}0.30$, $St=fD/U_{0}$, where $f$, $D$ and $U_{0}$ denote the precessing vortex core (PVC) frequency (${\sim}800~\text{Hz}$), the swirler exit diameter (19 mm) and the bulk velocity at the swirler exit ($50~\text{m}~\text{s}^{-1}$) respectively. Planar projections of the PVC reveal a pair of oscillating skew-symmetric regions of velocity, vorticity and OH-PLIF intensity that rotate in the same direction as the mean tangential flow. During combustion instabilities, the large-amplitude acoustics-induced axisymmetric forcing of the flow results in a fundamentally different flow response dominated by a nearly axisymmetric disturbance and almost complete suppression of the large-scale helical shear-layer disturbances dominating the nominal flow. In addition, reverse axial flows around the centreline are significantly reduced. Time traces of the robust constituent show reverse axial flows around the centreline and negative axial vorticity along the inner swirling shear layer when the planar velocity is in the same direction as the mean tangential flow. For both stable and unstable combustion, recurrent flow structures decay rapidly downstream of the air swirler, as revealed by the decreasing amplitude of the velocity, axial vorticity and OH-PLIF intensity.

The formation, evolution and co-existence of jets and vortices in turbulent planetary atmospheres is examined using a two-layer quasi-geostrophic $\unicode[STIX]{x1D6FD}$-channel shallow-water model. The study in particular focuses on the vertical structure of jets. Following Panetta & Held (J. Atmos. Sci., vol. 45 (22), 1988, pp. 3354–3365), a vertical shear arising from latitudinal heating variations is imposed on the flow and maintained by thermal damping. Idealised convection between the upper and lower layers is implemented by adding cyclonic/anti-cyclonic pairs, called hetons, to the flow, though the qualitative flow evolution is evidently not sensitive to this or other small-scale stochastic forcing. A very wide range of simulations have been conducted. A characteristic simulation which exhibits alternation between two different phases, quiescent and turbulent, is examined in detail. We study the energy transfers between different components and modes, and find the classical picture of barotropic/baroclinic energy transfers to be too simplistic. We also discuss the dependence on thermal damping and on the imposed vertical shear. Both have a strong influence on the flow evolution. Thermal damping is a major factor affecting the stability of the flow while vertical shear controls the number of jets in the domain, qualitatively through the Rhines scale $L_{Rh}=\sqrt{U/\unicode[STIX]{x1D6FD}}$.

The dynamics of the coupled Kelvin–Helmholtz (KH) and Rayleigh–Taylor (RT) instability (referred to as KHRT instability or KHRTI) is studied using statistically steady experiments performed in a multi-layer gas tunnel. Experiments are performed at four density ratios ranging in Atwood number $A_{t}$ from 0.035 to 0.159, with varying amounts of shear and $\unicode[STIX]{x0394}U/U$ ranging from 0 to 0.48, where $\unicode[STIX]{x0394}U$ is the speed difference between the two flow streams being investigated and $U$ is the mean velocity of these two streams. Three types of diagnostics – back-lit visualization, hot-wire anemometry and particle image velocimetry (PIV) – are employed to obtain the mixing widths, velocity field and density field. The flow is found to be governed by KH dynamics at early times and RT dynamics at late times. This transition from KH-instability-like to RT-instability-like behaviour is quantified using the Richardson number. Transitional Richardson number magnitudes obtained for the present KHRT flows are found to be in the range 0.17–0.56 similar to the critical Richardson numbers for stably stratified free shear flows. Comparing the evolution of density and velocity mixing widths, the density mixing layer is found to be approximately two times as thick as the velocity mixing layer. Scaling of velocity fluctuations is attempted using combinations of KH and RT scales. It is found that the proposed KHRT velocity scale, obtained using the combined mixing-layer growth equation, is appropriate for intermediate stages of the flow when both KH and RT dynamics are comparable. Probability density functions (p.d.f.s) for different fluctuating quantities are presented. Multiple peaks in p.d.f.s are qualitatively explained from the development of coherent KH roll-ups and their subsequent transition into turbulent pockets. The evolution of energy spectra indicates that density fluctuations start to show an inertial subrange from earlier times compared to velocity fluctuations. The spectra exhibit a slightly steeper slope than the Kolmogorov–Obukhov five-thirds law.

A general constitutive model is constructed and validated for highly concentrated monodisperse emulsions of deformable drops with insoluble surfactant through long-time, large-scale and high-resolution multidrop simulations. There is the same amount of surfactant on each drop, and the linear model is assumed for the surface tension versus the surfactant concentration. The surfactant surface transport is coupled to multidrop hydrodynamics through the convective–diffusive equation and the interfacial stress balance. Only the limit of small surfactant diffusivities is addressed, when this parameter does not affect the rheology. An Oldroyd constitutive equation is postulated, with five variable coefficients depending on one instantaneous flow invariant (chosen as the drop-phase contribution to the dissipation rate). These coefficients are found by fitting the model to five precise rheological functions from two steady base flows at arbitrary deformation rates. One base flow is planar extension (PE) ($\dot{\unicode[STIX]{x1D6E4}}x_{1},-\dot{\unicode[STIX]{x1D6E4}}x_{2},0$), the other one is planar mixed flow (PM) ($\dot{\unicode[STIX]{x1D6FE}}x_{2}$, $\dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D712}x_{1}$, 0) with $\unicode[STIX]{x1D712}=0.16$. A small but finite $\unicode[STIX]{x1D712}$ (a precise choice in the range $\unicode[STIX]{x1D712}\sim 0.1$ is unimportant) provides a necessarily perturbation to exclude severe ergodic difficulties and abnormal, kinked behaviour inherent in simple shear for high drop volume fractions $c$, especially at small capillary numbers $Ca$ and small drop-to-medium viscosity ratios $\unicode[STIX]{x1D706}$. The database rheological functions are obtained for $c=0.45{-}0.6$, $\unicode[STIX]{x1D706}=0.25{-}3$ and surfactant elasticities $\unicode[STIX]{x1D6FD}=0.05{-}0.2$ (based on the equilibrium surfactant concentration) from long-time simulations by a multipole-accelerated boundary-integral code with $N=100{-}200$ drops in a periodic cell and 2000–4000 boundary elements per drop. The code is an extension from Zinchenko & Davis (J. Fluid Mech., vol. 779, 2015, pp. 197–244) to account for surfactant transport and Marangoni stresses. Massive drop cusping or (sometimes) drop break-up limit the range of $Ca$ from above in the base flows, but there is no substantial lower limitation owing to the absence of phase transition difficulties. At small $\unicode[STIX]{x1D706}$, even minimal surface contamination may have a strong effect on the rheology. The simulations remain accurate for quite strong drop interactions, when the PE emulsion viscosity is nine times that for the carrier fluid. The model validation against a steady PM flow with a different $\unicode[STIX]{x1D712}=0.5$ shows a very good agreement for various $Ca$, $c$ and $\unicode[STIX]{x1D706}$. In the three PE and PM time-dependent flow tests, the quasi-steady approximation is found to predict stresses poorly. In contrast, the combination of the steady-state results for PE and PM used in the present method to generate the Oldroyd parameters gives a model with much better predictions for these time-dependent flows.

We analyse numerically a pinch-type instability in a semi-infinite planar layer of inviscid conducting liquid bounded by solid walls and carrying a uniform electric current. Our model is as simple as possible but still captures the salient features of the instability which otherwise may be obscured by the technical details of more comprehensive numerical models and laboratory experiments. Firstly, we show the instability in liquid metals, which are relatively poor conductors, differs significantly from the astrophysically relevant Tayler instability. In liquid metals, the instability develops on the magnetic response time scale, which depends on the conductivity and is much longer than the Alfvén time scale, on which the Tayler instability develops in well conducting fluids. Secondly, we show that this instability is an edge effect caused by the curvature of the magnetic field, and its growth rate is determined by the linear current density and independent of the system size. Our results suggest that this instability may affect future liquid-metal batteries when their size reaches a few metres.

The flow in a split cylinder with each half in exact counter rotation is studied numerically. The exact counter rotation, quantified by a Reynolds number $\mathit{Re}$ based on the rotation rate and radius, imparts the system with an $O(2)$ symmetry (invariance to azimuthal rotations as well as to an involution consisting of a reflection about the mid-plane composed with a reflection about any meridional plane). The $O(2)$ symmetric basic state is dominated by a shear layer at the mid-plane separating the two counter-rotating bodies of fluid, created by the opposite-signed vortex lines emanating from the two endwalls being bent to meet at the split in the sidewall. With the exact counter rotation, the additional involution symmetry allows for steady non-axisymmetric states, that exist as a group orbit. Different members of the group simply correspond to different azimuthal orientations of the same flow structure. Steady states with azimuthal wavenumber $m$ (the value of $m$ depending on the cylinder aspect ratio $\unicode[STIX]{x1D6E4}$) are the primary modes of instability as $\mathit{Re}$ and $\unicode[STIX]{x1D6E4}$ are varied. Mode competition between different steady states ensues, and further bifurcations lead to a variety of different time-dependent states, including rotating waves, direction-reversing waves, as well as a number of slow–fast pulse waves with a variety of spatio-temporal symmetries. Further from the primary instabilities, the competition between the vortex lines from each half-cylinder settles on either a $m=2$ steady state or a limit cycle state with a half-period-flip spatio-temporal symmetry. By computing in symmetric subspaces as well as in the full space, we are able to unravel many details of the dynamics involved.

Experimental and numerical studies have shown that, with sufficient nonlinearity, the theoretical capillary-wave power-law spectrum derived from the kinetic equation (KE) of weak turbulence theory can be realized. This is despite the fact that the KE is derived assuming an infinite domain with continuous wavenumber, while experiments and numerical simulations are conducted in realistic finite domains with discrete wavenumbers for which the KE theoretically allows no energy transfer. To understand this, we first analyse results from direct simulations of the primitive Euler equations to elucidate the role of nonlinear resonance broadening (NRB) in discrete turbulence. We define a quantitative measure of the NRB, explaining its dependence on the nonlinearity level and its effect on the properties of the obtained stationary power-law spectra. This inspires us to develop a new quasi-resonant kinetic equation (QKE) for discrete turbulence, which incorporates the mechanism of NRB, governed by a single parameter $\unicode[STIX]{x1D705}$ expressing the ratio of NRB and wavenumber discreteness. At $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D705}_{0}\approx 0.02$, the QKE recovers simultaneously the spectral slope $\unicode[STIX]{x1D6FC}_{0}=-17/4$ and the Kolmogorov constant $C_{0}=6.97$ (corrected from the original derivation) of the theoretical continuous spectrum, which physically represents the upper bound of energy cascade capacity for the discrete turbulence. For $\unicode[STIX]{x1D705}<\unicode[STIX]{x1D705}_{0}$, the obtained spectra represent those corresponding to a finite domain with insufficient nonlinearity, resulting in a steeper spectral slope $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{0}$ and reduced capacity of energy cascade $C>C_{0}$. The physical insights from the QKE are corroborated by direct simulation results of the Euler equations.

The fate of small particles in turbulent flows depends strongly on the velocity gradient properties of the surrounding fluid, such as rotation and strain rates. For non-inertial (fluid) particles, the restricted Euler model provides a simple low-dimensional dynamical system representation of Lagrangian evolution of velocity gradients in fluid turbulence, at least for short times. Here, we derive a new restricted Euler dynamical system for the velocity gradient evolution of inertial particles, such as solid particles in a gas, or droplets and bubbles in turbulent liquid flows. The model is derived in the limit of small (sub-Kolmogorov-scale) particles and low Stokes number. The system exhibits interesting fixed points, stability and invariant properties. Comparisons with data from direct numerical simulations show that the model predicts realistic trends such as the tendency of increased straining over rotation along heavy particle trajectories and, for light particles such as bubbles, the tendency of reduced self-stretching of the strain rate.

Because of the associated far-field logarithmic divergence, the transport problem governing two-dimensional phoretic self-propulsion lacks a steady solution when the Péclet number $\mathit{Pe}$ vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of $\mathit{Pe}$, however small. We consider that problem employing a first-order kinetic model of solute absorption, where the ratio of the characteristic magnitudes of reaction and diffusion is quantified by the Damköhler number $\mathit{Da}$. As $\mathit{Pe}\rightarrow 0$ the dominance of diffusion breaks down at distances that scale inversely with $\mathit{Pe}$; at these distances, the leading-order transport represents a two-dimensional point source in a uniform stream. Asymptotic matching between the latter region and the diffusion-dominated near-particle region provides the leading-order particle velocity as an implicit function of $\log \mathit{Pe}$. Another scenario involving weak advection takes place under strong reactions, where $\mathit{Pe}$ and $\mathit{Da}$ are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as $\mathit{Da}^{2}/\mathit{Pe}$.