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.

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.

For the first time, a physical mechanism is identified to explain the phase lag term in Powell’s impinging feedback loop equation (Powell, J. Acoust. Soc. Am., vol. 83 (2), 1988, pp. 515–533). Ultra-high-speed schlieren reveals a previously unseen periodic transient shock in the wall jet region of underexpanded impinging flows. The motion of this shock appears to be responsible for the production of the acoustic waves corresponding to the impingement tone. It is suggested that the delay between the inception of the shock and the formation of the acoustic wave explains the phase lag in the aeroacoustic feedback process. This suggestion is quantitatively supported through an assessment of Powell’s feedback equation, using high-resolution particle image velocimetry and acoustic measurements.

The modified Townsend–Perry attached-eddy model of Vassilicos et al. (J. Fluid Mech., vol. 774, 2015, pp. 324–341) combines the outer peak/plateau behaviour of root-mean-square streamwise turbulence velocity profiles and the Townsend–Perry log decay of these profiles at higher distances from the wall. This model was validated by these authors for high-Reynolds-number turbulent pipe flow data and is shown here to describe equally well, and with approximately the same parameter values, turbulent boundary layer flow data from four different facilities and a wide range of Reynolds numbers. The model has predictive value as, when extrapolated to the extremely high Reynolds numbers of the SLTEST data obtained at the Great Salt Lake Desert atmospheric test facility, it matches these data quite well.

In this experimental study, we investigate the interaction of gravity–capillary solitary waves generated by two surface pressure sources moving side by side at constant speed. The nonlinear response of a water surface to a single source moving at a speed just below the minimum phase speed of linear gravity–capillary waves in deep water ( $c_{min}\approx 23~\text{cm}~\text{s}^{-1}$ ) consists of periodic generation of pairs of three-dimensional solitary waves (or lumps) in a V-shaped pattern downstream of the source. In the reference frame of the laboratory, these unsteady lumps propagate in a direction oblique to the motion of the source. In the present experiments, the strengths of the two sources are adjusted to produce nearly identical responses and the free-surface deformations are visualized using photography-based techniques. The first lumps generated by the two sources move in intersecting directions that make a half-angle of approximately $15^{\circ }$ and collide in the centreplane between the sources. A steep depression is formed during the collision, but this depression quickly decreases in amplitude while radiating small-amplitude radial waves. After the collision, a quasi-stable pattern is formed with several rows of localized depressions that are qualitatively similar to lumps but exhibit periodic amplitude oscillations, similar to a breather. The shape of the wave pattern and the period of oscillations depend strongly on the distance between the sources.

It is well known that when a flexible or flexibly mounted structure is placed perpendicular to the flow of a Newtonian fluid, it can oscillate due to the shedding of separated vortices. Here, we show for the first time that fluid–structure interactions can also be observed when the fluid is viscoelastic. For viscoelastic fluids, a flexible structure can become unstable in the absence of fluid inertia, at infinitesimal Reynolds numbers, due to the onset of a purely elastic flow instability. Nonlinear periodic oscillations of the flexible structure are observed and found to be coupled to the time-dependent growth and decay of viscoelastic stresses in the wake of the structure.

The effect of angular misalignment on the low-frequency dynamics of the near wake of a blunt-based axisymmetric body is investigated at a Reynolds number of $Re=67\,000$ . While for axisymmetric boundary conditions all azimuthal orientations of the wake are explored with equal probability, resulting in a statistical axisymmetry, an angular offset as small as $0.2^{\circ }$ is found to suppress the low-frequency large-scale behaviour that is associated with the erratic meandering of the region of reversed flow. As a result of the misalignment, the centroid of this backflow region is displaced from the model axis and remains confined around an average off-centre position. Spectral and modal analysis provides evidence that the erratic backflow behaviour occurs within a narrow angular range of deviations from axisymmetric conditions.

We numerically investigate the direct initiation of detonations driven by the propagation of a blast wave into a unconfined gaseous combustible mixture to study the role played by multidimensional instabilities in direct initiation of stable and unstable detonations. To this end, we first model the dynamics of unsteady propagation of detonation using the one-dimensional compressible Euler equations with a one-step chemical reaction model and cylindrical geometrical source terms. Subsequently, we use two-dimensional compressible Euler equations with just the chemical reaction source term to directly model cylindrical detonations. The one-dimensional results suggest that there are three regimes in the direct initiation for stable detonations, that the critical energy for mildly unstable detonations is not unique, and that highly unstable detonations are not self-sustainable. These phenomena agree well with one-dimensional theories and computations available in the literature. However, our two-dimensional results indicate that one-dimensional approaches are valid only for stable detonations. In mildly and highly unstable detonations, one-dimensional approaches break down because they cannot take the effects and interactions of multidimensional instabilities into account. In fact, instabilities generated in multidimensional settings yield the formation of strong transverse waves that, on one hand, increase the risk of failure of the detonation and, on the other hand, lead to the initiation of local over-driven detonations that enhance the overall self-sustainability of the global process. The competition between these two possible outcomes plays an important role in the direct initiation of detonations.

This paper is concerned with the classical, well-studied problem of convective instabilities in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres. Sanchez et al. (J. Fluid Mech., vol. 791, 2016, R1) recently found, unexpectedly via careful numerical simulation, that non-magnetic convection in the form of axially symmetric, equatorially antisymmetric torsional oscillation is physically preferred in a special range of small Prandtl number for rapidly rotating fluid spheres with the stress-free boundary condition. We derive an asymptotic solution describing convection-driven torsional oscillation – whose flow velocity and pressure are fully analytical and in closed form – that confirms the result of the numerical analysis and is in quantitative agreement with the numerical solution. We also demonstrate, through the derivation of a different asymptotic solution, that convection-driven torsional oscillation cannot occur in rapidly rotating fluid spheres with the no-slip boundary condition.

Incompressible three-dimensional Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work, we propose an exact solution of the Euler equations for the asymptotic pancake evolution. This solution combines a shear flow aligned with an asymmetric straining flow, and is characterized by a single asymmetry parameter and an arbitrary transversal vorticity profile. The analysis is based on detailed comparison with numerical simulations performed using a pseudospectral method in anisotropic grids of up to $972\times 2048\times 4096$ .

Swimming micro-organisms often have to propel themselves in complex non-Newtonian fluids. We carry out experiments with self-propelling helical swimmers driven by an externally rotating magnetic field in shear-thinning inelastic fluids. Similarly to swimming in a Newtonian fluid, we obtain for each fluid a locomotion speed that scales linearly with the rotation frequency of the swimmer, but with a prefactor that depends on the power index of the fluid. The fluid is seen to always increase the swimming speed of the helix, up to 50 % faster, and thus the strongest of such type reported to date. The maximum relative increase is for a fluid power index of approximately 0.6. Using simple scalings, we argue that the speed increase is not due directly to the local decrease of the flow viscosity around the helical filament, but hypothesise instead that it originates from confinement-like effect due to viscosity stratification around the swimmer.

In liquid rope coiling, a slender jet of viscous fluid falling onto a rigid surface builds a rotating corkscrew-like structure. Here, I use a numerical continuation method to construct a complete regime diagram for liquid rope coiling. I first consider the onset of coiling, and show that a suitable onset criterion is that the radius $a_{1}$ of the rope itself be just equal to the radius $R$ of the coil. Numerical calculation of the critical surface $a_{1}=R$ in the space of the dimensionless fall height $\unicode[STIX]{x1D6F1}_{H}$ , flow rate $\unicode[STIX]{x1D6F1}_{Q}$ and nozzle diameter $\unicode[STIX]{x1D6F1}_{d}$ shows that the surface has four distinct asymptotic limits corresponding to a viscous (V) mode, a gravitational (G) mode and two inertial modes (I1, I2) which are distinguished by how much the tail of the jet is stretched by gravity. Exact expressions for the onset frequencies in each of these four modes are determined. Finally, the regime diagram is constructed in the form of contour plots of the dimensionless coiling frequency as a function of $\unicode[STIX]{x1D6F1}_{H}$ and $\unicode[STIX]{x1D6F1}_{Q}$ for several values of $\unicode[STIX]{x1D6F1}_{d}$ . The diagram exhibits a total of six modes: V, G, I1, I2, a multivalued inertio-gravitational (IG) mode and a third inertial mode I3 with viscosity-dominated stretching of the tail. The regime diagram permits prediction of the coiling frequency for given values of the fall height, flow rate, viscosity and nozzle diameter, and should therefore be useful in practical applications ranging from non-woven textile production to 3D printing.

We present a combined numerical (particle vortex method) and experimental (soap film tunnel) study of a symmetric foil undergoing prescribed oscillations in a two-dimensional free stream. We explore pure pitching and pure heaving, and contrast these two generic types of kinematics. We compare measurements and simulations when the foil is forced with pitching oscillations, and we find a close correspondence between flow visualisations using thickness variations in the soap film and the numerically determined vortex structures. Numerically, we determine wake maps spanned by oscillation frequency and amplitude, and we find qualitatively similar maps for pitching and heaving. We determine the drag–thrust transition for both pitching and heaving numerically, and we discuss it in relation to changes in wake structure. For heaving with low oscillation frequency and high amplitude, we find that the drag–thrust transition occurs in a parameter region with wakes in which two vortex pairs are formed per oscillation period, in contrast to the common transition scenario in regions with inverted von Kármán wakes.

The physics and spectral content of cloud cavitation about a sphere are investigated in a variable-pressure water tunnel using dynamic surface pressure measurement and high-speed imaging. Experiments are conducted using a polyvinyl chloride sphere at a Reynolds number of $1.5\times 10^{6}$ with cavitation numbers, $\unicode[STIX]{x1D70E}$ , ranging from inception to supercavitation. Three distinct shedding regimes are identified: a uni-modal regime for $\unicode[STIX]{x1D70E}>0.9$ and two bi-modal regimes for $0.9>\unicode[STIX]{x1D70E}>0.675$ and $0.675>\unicode[STIX]{x1D70E}>0.3$ . For small cavity lengths ( $\unicode[STIX]{x1D70E}>0.9$ ), Kelvin–Helmholtz instability and transition to turbulence in the overlying separated boundary layer form the basis for cavity breakup and coherent vortex formation. At greater lengths ( $\unicode[STIX]{x1D70E}<0.9$ ), larger-scale shedding ensues, driven by coupled re-entrant jet formation and shockwave propagation. Strong adverse pressure gradients about the sphere lead to accumulation and radial growth of re-entrant flow, initiating breakup, from which, in every case, a condensation shockwave propagates upstream causing cavity collapse. When the shedding is most energetic, shockwave propagation upstream may cause large-scale leading edge extinction. The bi-modal response is due to cavity shedding being either axisymmetric or asymmetric. The two bi-modal regimes correspond to $\unicode[STIX]{x1D70E}$ ranges where the cavity and re-entrant jet either remain attached or become detached from the sphere. There is a distinct frequency offset at transition between regimes in both shedding modes. Despite the greater cavity lengths at lower $\unicode[STIX]{x1D70E}$ values, the second bi-modal regime initially exhibits shorter shedding periods due to increased cavity growth rates. The second regime persists until supercavitation develops for $\unicode[STIX]{x1D70E}<0.3$ .

The cylindrical lock-release laboratory experiments of Sutherland & Nault (J. Fluid Mech., vol. 586, 2007, pp. 109–118) showed that a radially advancing symmetric intrusive gravity current spreads not as an expanding annulus (as is the case for bottom-propagating gravity currents), but rather predominantly along azimuthally periodic radial ‘spokes’. Here, we investigate whether the spokes are associated with azimuthal perturbations that undergo ‘optimal’ growth. We use a nonlinear axisymmetric numerical simulation initialised with the experimental parameters to compute the time-evolving axisymmetric base state of the collapsing lock fluid. Using fields from this rapidly evolving base state together with the linearised perturbation equations and their adjoint, the ‘direct–adjoint looping’ method is employed to identify, as a function of the azimuthal wavenumber  $m$ , the vertical–radial structure of the set of initial perturbations that exhibit the largest total perturbation energy gain over a target time  $T$ . Of this set of perturbations, the one that extracts energy fastest, and so is expected to be observed to emerge first from the base flow, has azimuthal wavenumber comparable to the number of spokes observed in the experiment.

Previous studies have predicted $We^{-2/5}$ dependence of the streamwise location at which primary breakup of turbulent liquid jets begins and $We^{-3/5}$ dependence of the Sauter mean diameter (SMD) of droplets released at that location, where $We$ is the jet Weber number. Measured deviations from these predictions were attributed to measurement uncertainties and to the simplicity of the analysis, which invoked turbulence inertial-range phenomenology. Here, it is proposed that breakup onset is instead controlled by the residual presence of the boundary-layer structure of the nozzle flow in the near field of the jet. Assuming that the size of the breakup-inducing eddy is within the scale range of the log-law region, $We^{-1}$ dependence of both the onset location and the SMD at onset is predicted. These dependences agree with the available measurements more closely than those previously predicted. To predict the dependences on the Reynolds number $Re$ , either the friction velocity in conjunction with the Blasius friction law or the bulk velocity can be used, where the former yields $Re^{3/8}$ and $Re^{1/4}$ dependence of the onset location and the SMD at onset respectively, while the latter implies no $Re$ dependence of either. The latter result is consistent with the available measurements, but the boundary-layer analysis indicates that the velocity scaling should be based on the friction velocity rather than the bulk velocity, so the origin of the measured lack of $Re$ dependence merits further investigation. A plausible hypothesis is that pressure effects associated with the transition from wall-bounded nozzle flow to jet free-slip boundary conditions induce a transient large-scale flow modification that counteracts the $Re$ dependence of the nozzle flow while preserving the logarithmic flow structure near the jet surface. Notwithstanding the absence of direct evidence supporting this hypothesis, the new analysis and comparisons of its predictions with measurements suggest that transient effects such as the residual influence of the nozzle-flow structure are the likely explanations of the observed parameter dependences.

Under Stokesian conditions, a neutrally buoyant non-Brownian spheroid in simple shear flow rotates indefinitely in any of a one-parameter family of closed (Jeffery) orbits characterized by an orbit constant $C$ . The limiting values, $C=0$ and $C=\infty$ , correspond to spinning and tumbling modes respectively. Hydrodynamics alone does not determine the distribution of spheroid orientations across Jeffery orbits in the absence of interactions, and the rheology of a dilute suspension of spheroids remains indeterminate. A combination of inertia and stochastic orientation fluctuations eliminates the indeterminacy. The steady-state Jeffery-orbit distribution arising from a balance of inertia and thermal fluctuations is shown to be of the Boltzmann equilibrium form, with a potential that depends on $C$ , the particle aspect ratio ( $\unicode[STIX]{x1D705}$ ), and a dimensionless shear rate ( $Re\,Pe_{r}$ , $Re$ and $Pe_{r}$ being the Reynolds and rotary Péclet numbers), and therefore lends itself to a novel thermodynamic interpretation in $C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. In particular, the transition of the potential from a single to a double-well structure, below a critical $\unicode[STIX]{x1D705}$ , has similarities to a thermodynamic phase transition, and the small- $C$ and large- $C$ minima are therefore identified with spinning and tumbling phases. The hysteretic dynamics within the two-phase tumbling–spinning envelope renders the rheology sensitively dependent on the precise shear rate history, the signature in simple shear flow being a multivalued viscosity at a given shear rate. The tumbling–spinning transition identified here is analogous to the coil–stretch transition in the polymer physics literature. It should persist under more general circumstances, and has implications for the suspension stress response in inhomogeneous shearing flows.

We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as $\unicode[STIX]{x1D716}^{-1}$ , where $\unicode[STIX]{x1D716}$ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.

Two transfer functions for the unsteady lift response of an airfoil under attached flow conditions are experimentally investigated: the Theodorsen function for an airfoil oscillating in a constant free stream and the Sears function for a steady airfoil encountering a sinusoidal vertical gust. A two-dimensional airfoil with a Clark Y profile is submitted to two different unsteady excitations of distinct frequencies: a pitching oscillation around the leading edge and a sinusoidal vertical gust. The reduced frequency of the perturbation is in the range of $0.025<k<0.3$ and the Reynolds number of the undisturbed flow is in the range of $120\,000<\mathit{Re}<300\,000$ . While the Theodorsen function is found to be a good estimator for the unsteady lift at moderate mean angles of attack, the Sears function does not capture the experimental transfer functions in frequency dependence or in limiting values. A second-order model provided by Atassi (J. Fluid Mech., vol. 141, 1984, pp. 109–122) agrees well with the experimental transfer function.

The Lagrangian paths, horizontal Lagrangian drift velocity, $U_{L}$ , and the Lagrangian excess period, $T_{L}-T_{0}$ , where $T_{L}$ is the Lagrangian period and $T_{0}$ the Eulerian linear period, are obtained by particle tracking velocimetry (PTV) in non-breaking periodic laboratory waves at a finite water depth of $h=0.2~\text{m}$ , wave height of $H=0.49h$ and wavenumber of $k=0.785/h$ . Both $U_{L}$ and $T_{L}-T_{0}$ are functions of the average vertical position of the paths, $\bar{Y}$ , where $-1<\bar{Y}/h<0$ . The functional relationships $U_{L}(\bar{Y})$ and $T_{L}-T_{0}=f(\bar{Y})$ are very similar. Comparisons to calculations by the inviscid strongly nonlinear Fenton method and the second-order theory show that the streaming velocities in the boundary layers below the wave surface and above the fluid bottom contribute to a strongly enhanced forward drift velocity and excess period. The experimental drift velocity shear becomes more than twice that obtained by the Fenton method, which again is approximately twice that of the second-order theory close to the surface. There is no mass flux of the periodic experimental waves and no pressure gradient. The results from a total number of 80 000 experimental particle paths in the different phases and vertical positions of the waves show a strong collapse. The particle paths are closed at the two vertical positions where  $U_{L}=0$ .