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 $U_{L}=0$.

A theoretical model is developed to predict the maximum spreading of liquid metal drops when impacting onto dry surfaces under the influence of a vertical magnetic field. This model, which is constructed based on the energy conversion principle, agrees very well with the numerical results, covering a wide range of impact speeds, contact angles and magnetic strengths. When there is no magnetic field, we found that the maximum spreading factor can be predicted well by an interpolating scheme between the viscous and capillary effects, as proposed by Laan et al. (Phys. Rev. Appl., vol. 2 (4), 2014, 044018). However, when gradually increasing the magnetic field strength, the induced Lorentz forces are dominant over the viscous and capillary forces, taking the spreading behaviour into the ‘Joule regime’, where the Joule dissipation is significant. For most situations of practical interest, namely when the strength of the magnetic field is less than 3 T, all three energy conversion routes are important. Therefore, we determine the correct scaling behaviours for the magnetic influence by first equating the loss of kinetic energy to the Joule dissipation in the Joule regime, then by interpolating it with the viscous dissipation and the capillary effects, which allows for a universal rescaling. By plotting the numerical results against the theoretical model, all the results can be rescaled onto a single curve regardless of the materials of the liquid metals or the contact angles of the surfaces, proving that our theoretical model is correct in predicting the maximum spreading factor by constructing a balanced formula between kinetic energy, capillary energy, viscous dissipation energy and Joule dissipation energy.

We analyse velocity fluctuations inside coherent vortices generated as a result of the inverse cascade in the two-dimensional (2-D) turbulence in a finite box. As we demonstrated in Kolokolov & Lebedev (Phys. Rev. E, vol. 93, 2016, 033104), the universal velocity profile, established in Laurie et al. (Phys. Rev. Lett., vol. 113, 2014, 254503), corresponds to the passive regime of the flow fluctuations. This property enables one to calculate correlation functions of the velocity fluctuations in the universal region. We present the results of the calculations that demonstrate a non-trivial scaling of the structure function. In addition the calculations reveal strong anisotropy of the structure function.

Simultaneous measurements of the phase-averaged velocity distribution and the underlying discharge quantities of a dielectric barrier discharge plasma actuator (PA) are performed at $10~\text{k}\text{Hz}$ discharge frequency to investigate the interplay of the discharge and the surrounding flow. The underlying velocity information for the force estimation is obtained by means of phase-averaged particle image velocimetry; the discharge quantities are determined from a Lissajous-figure analysis. The results uncover a clear cause–effect relation between the phase-dependent effective discharge capacitance of the PA and the resulting spatiotemporal volume-force distributions. From this novel insight, it must be concluded that the instantaneous effective discharge intensity dominates the momentum-transfer rate rather than the formerly assumed operating voltage.

We consider the slow motion generated when a body is set into motion relative to an incompressible, inviscid, non-diffusive rotating stratified fluid, showing that there is generated in general a topographic Rossby wave which leads to non-decaying fluctuations in the lift on the obstacle and a fluctuating non-zero drag. The problem is relevant to the flow patterns and forces excited when slow oceanic flows cross bottom topography, and suggests a mechanism for slow fluctuations observed in laboratory experiments.

We investigate numerically the rotational behaviour of a circular porous particle suspended in a two-dimensional (2D) simple shear flow with fluid inertia at particle shear Reynolds number up to 108. We use the volume-averaged macroscopic momentum equation to formulate the flow field inside and outside the moving porous particle, which is solved by a modified single relaxation time lattice Boltzmann method. The effects of fluid inertia, confinement of the bounding walls, and permeability of the particle are studied. Our two-dimensional simulation results confirm that the permeability has little effect on the rotation of a porous particle in unbounded shear flow without fluid inertia (Masoud, Stone & Shelley, J. Fluid Mech., vol. 733, 2013, R6), but also suggest that the role of permeability cannot be neglected when the confinement effect is significant, or the fluid inertia is not negligible. The fluid inertia and the confined walls have similar effects on the rotation of a porous particle as that on a solid impermeable particle. The angular velocity decays with an increase in fluid inertia, and the confinement effect suppresses the angular velocity to a shear rate ratio below 0.5. A simple scaling argument based on the balance of torque exerted by fluid flows adjacent to the two bounding walls and that due to the flow recirculation can explain our results.

An asymptotic theory is developed for the linear stability curve of rapidly rotating Taylor–Couette flow. The analytic curve obtained by the theory excellently explains the limiting Navier–Stokes stability result for general disturbances. When the cylinders are corotating, the asymptotic theory describes the gap between the neutral curve and the Rayleigh stability criterion. For the case when the cylinders are counter-rotating, it is found that, along the stability boundary, the Reynolds number based on the inner cylinder speed is proportional to that based on the outer cylinder speed to the power of $3/5$.

We present a new adaptive control strategy to isolate and stabilize turbulent states in transitional, stably stratified plane Couette flow in which the gravitational acceleration (non-dimensionalized as the bulk Richardson number $Ri$) is adjusted in time to maintain the turbulent kinetic energy (TKE) of the flow. We demonstrate that applying this method at various stages of decaying stratified turbulence halts the decay process and allows a succession of intermediate turbulent states of decreasing energy to be isolated and stabilized. Once the energy of the initial flow becomes small enough, we identify a single minimal turbulent spot, and lower-energy states decay to laminar flow. Interestingly, the turbulent states which emerge from this process have very similar time-averaged $Ri$, but TKE levels different by an order of magnitude. The more energetic states consist of several turbulent spots, each qualitatively similar to the minimal turbulent spot. This suggests that the minimal turbulent spot may well be the lowest-energy turbulent state which forms a basic building block of stratified plane Couette flow. The fact that a minimal spot of turbulence can be stabilized, so that it neither decays nor grows, opens up exciting opportunities for further study of spatiotemporally intermittent stratified turbulence.

Linear stability of a laminar boundary-layer flow in a streamwise corner can only be treated with an ansatz that considers two-dimensional eigenfunctions with inhomogeneous boundary conditions in cross-flow directions. It is common practice to use Sommerfeld’s radiation condition with a certain wavenumber $\unicode[STIX]{x1D6FD}$ at the lateral domains of the integration domain which are at the same time the far-field domains for each wall. So far, this radiation condition has been exclusively used in a ‘symmetrical’ way, i.e. with the same $\unicode[STIX]{x1D6FD}$ on either far-field boundary plane. This has led to wave patterns that either enter or leave the corner region from the lateral sides for $\unicode[STIX]{x1D6FD}<0$ or $\unicode[STIX]{x1D6FD}>0$ respectively. Here, an ‘asymmetric’ use of Sommerfeld’s radiation condition is suggested, i.e. $\unicode[STIX]{x1D6FD}<0$ on one far side of the corner and $\unicode[STIX]{x1D6FD}>0$ on the other. With this modification, waves enter the corner area from one side and leave it through the other, i.e. they travel obliquely through the corner. In contrast to before, their amplification rate is always symmetric with respect to $\unicode[STIX]{x1D6FD}=0$ and there is no amplification-rate increase or decrease due to information that either continuously enters the corner from both sides or continuously leaves it through the far sides. The present analysis also shows that the inviscid corner modes are unaffected by the parameters of the far-field radiation boundary conditions. Nevertheless, superposition of two oppositely running single waves obtained by the modified application of the radiation condition leads to a similar wave pattern to that in the case with $\unicode[STIX]{x1D6FD}<0$ on both sides; however, with a slightly smaller amplification rate and a strictly streamwise propagation direction.

We present a new robust method for identifying three dynamically distinct regions in a stratified turbulent flow, which we characterise as quiescent flow, intermittent layers and turbulent patches. The method uses the cumulative filtered distribution function of the local density gradient to identify each region. We apply it to data from direct numerical simulations of homogeneous stratified turbulence, with unity Prandtl number, resolved on up to $8192\times 8192\times 4096$ grid points. In addition to classifying regions consistently with contour plots of potential enstrophy, our method identifies quiescent regions as regions where $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(1)$, layers as regions where $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(10)$ and patches as regions where $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}N^{2}\sim O(100)$. Here, $\unicode[STIX]{x1D716}$ is the dissipation rate of turbulence kinetic energy, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $N$ is the (overall) buoyancy frequency. By far the highest local dissipation and mixing rates, and the majority of dissipation and mixing, occur in patch regions, even when patch regions occupy only 5 % of the flow volume. We conjecture that treating stratified turbulence as an instantaneous assemblage of these different regions in varying proportions may explain some of the apparently highly scattered flow dynamics and statistics previously reported in the literature.

In order to better understand the mechanism governing segregation in dense granular flows, the force experienced by a large particle embedded in a granular flow made of small particles is studied using discrete numerical simulations. Accurate force measurements have been obtained in a large range of flow parameters by trapping the large particle in a harmonic potential well to mimic an optical tweezer. Results show that positive or negative segregation lift forces (perpendicular to the shear) exist depending on the stress inhomogeneity. An empirical expression of the segregation force is proposed as a sum of a term proportional to the gradient of pressure and a term proportional to the gradient of shear stress, which both depend on the local friction and particle size ratio.

The nonlinear stability of two-fluid Couette flows is studied using a novel evolution equation whose dynamics is validated by direct numerical simulation (DNS). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the experiments of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53). Related low-inertia models have been used in qualitative predictions rather than the direct comparisons carried out here, and ad hoc modifications appear to be necessary in order to predict asymmetry and bistability. Comparisons between model solutions and DNS show excellent agreement at Reynolds numbers of $O(10^{3})$ found in the experiments. Direct comparisons are also made with the available experimental results of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53) when the thin layer occupies $1/5$ of the channel height. Pointwise comparisons of the travelling wave shapes are carried out, and once again the agreement is very good.