Inviscid models for vortex rings and dipoles are constructed using nested patches of vorticity. These models constitute more realistic approximations to experimental vortex rings and dipoles than the single-contour models of Norbury and Pierrehumbert, and nested contour dynamics algorithms allow their simulation with low computational cost. In two dimensions, nested-contour models for the analytical Lamb dipole are constructed. In the axisymmetric case, a family of models for vortex rings generated by a piston–cylinder apparatus at different stroke ratios is constructed from experimental data. The perturbation response of this family is considered by the introduction of a small region of vorticity at the rear of the vortex, which mimics the addition of circulation to a growing vortex ring by a feeding shear layer. Model vortex rings are found to either accept the additional circulation or shed vorticity into a tail, depending on the perturbation size. A change in the behaviour of the model vortex rings is identified at a stroke ratio of three, when it is found that the maximum relative perturbation size vortex rings can accept becomes approximately constant. We hypothesise that this change in response is related to pinch-off, and that pinch-off might be understood and predicted based on the perturbation responses of model vortex rings. In particular, we suggest that a perturbation response-based framework can be useful in understanding vortex formation in biological flows.

Wake formation and heat transfer from a rotationally oscillating circular cylinder in cross-flow at $\mathit{Re}= 750$ are studied. Two aspects, the effect of cylinder forcing on vortex shedding and the effect of the wake structures on convective heat transfer, are studied. Cylinder forcing conditions range between $0.09 \leq \theta _{PP} \leq 2.09$, where $\theta _{PP}$ is the peak-to-peak oscillation amplitude in radians and $0.70 \leq F_{R} \leq 3.16$, where $F_{R}$ is the ratio of forcing frequency to natural shedding frequency. Digital particle image velocimetry (DPIV) is used to obtain quantitative wake structure information. Wake modes, and regions of the parameter space in which they occur, are identified for both heated and unheated cylinders. For the heated cylinder, cylinder forcing is found to affect the convective heat-transfer rate. Certain wake modes, including newly discovered wake modes synchronized over multiple oscillation cycles, are found to correlate with significant heat-transfer enhancement. Cylinder tangential velocity is also found to affect the heat-transfer rate in certain regions of the parameter space.

The jet flow of a shear-thinning power-law fluid is examined theoretically as it emerges from a channel at moderate Reynolds number. Poiseuille flow conditions are assumed to prevail far upstream from the exit. The problem is solved using the method of matched asymptotic expansions. A similarity solution is obtained in the inner layer near the free surface, with the outer layer extending to the jet centreline. An inner thin viscous sublayer is introduced to smooth out the singularity in viscosity at the free surface, allowing the inner algebraically decaying solutions to be matched smoothly with the solution near the free surface. A Newtonian jet is found to contract more than a shear-thinning jet. While both the inner-layer thickness and the free-surface height are $O(\mathit{Re}^{-1/3})$, and grow with downstream distance, the sublayer thickness is smaller, $O(\mathit{Re}^{-(1+n)/3})$, growing with distance for $n < 0.5$, and decaying for $n > 0.5$. The relaxation downstream distance for the jet is found to grow logarithmically with $\mathit{Re}$.

Previous studies have shown that the behaviour of Rayleigh–Taylor (RT) unstable flames depends on the boundary conditions. If the boundary conditions at the domain walls are impermeable/adiabatic or reflecting then the flame assumes a stable parabolic shape. On the other hand, periodic boundary conditions can produce unstable pulsating solutions. In this paper, we explore why periodic boundary conditions allow unstable solutions by showing the results of two-dimensional direct numerical simulations of model flames. We show that RT unstable premixed model flames pulsate at low gravity because of a shear instability of the vorticity layers behind the flame front. The resulting vortex shedding is controlled by a region of absolute-like instability which moves closer to the flame front as gravity is increased, ultimately disturbing the flame and leading to pulsations. We demonstrate that this region is ‘absolutely unstable’ by showing that the wake is dominated by pure frequency oscillations. In addition, the shear instability can be described by the Landau equation and can be represented dynamically by a Hopf bifurcation. The applicability of the Landau equation allows the apparently complex spatio-temporal behaviour of the vortex shedding to be described by a simple temporal model with only a secondary spatial dependence. We show that the flame behaviour is analogous to the initial instability downstream of a circular cylinder, which leads to the von Kármán vortex street for large enough values of the Reynolds number.

We investigate experimentally the spontaneous motion of drops and bubbles confined between two plates forming a narrow wedge. Such discoidal objects migrate under the gradient in interfacial energy induced by the non-homogeneous confinement. The resulting capillary driving force is balanced by viscous resistance. The viscous friction on a drop bridging parallel plates is estimated by measuring its sliding velocity under gravity. The viscous forces are the sum of two contributions, from the bulk of the liquid and from contact lines, the relative strength of which depends on the drop size and velocity and the physical properties of the liquid. The balance of capillarity and viscosity quantitatively explains the dynamics of spontaneous migration of a drop in a wedge. Close the tip of the wedge, bulk dissipation dominates and the migrating velocity of drops is constant and independent of drop volume. The distance between the drop and the tip of the wedge is thus linear with time $t$, $x(t) \sim t_0-t$, where $t_0$ is the time at which the drop reaches the tip of the wedge. Far away from the apex, contact lines dominate the friction, the motion is accelerated toward the tip of the wedge and velocities are higher for larger drops. In this regime, it is shown that $x(t) \sim (t_0-t)^{4/13}$. The position and time of the crossover between the two dissipation regimes are used to write a dimensionless equation of motion. Plotted in rescaled variables, all experimental trajectories collapse to the prediction of our model. In contrast to drops, gas bubbles in a liquid-filled wedge behave as non-wetting objects. They thus escape the confinement of the wedge to reduce their surface area. The physical mechanisms involved are similar for drops and bubbles, so that the forces acting have the same mathematical structures in both cases, except for the sign of the capillary driving force and a numerical factor. We thus predict and show experimentally that the trajectories of drops and bubbles obey the same equation of motion, except for a change in the sign of $t_0-t$.

The effects of plasma heating and thermal non-equilibrium on the statistical properties of a low-Reynolds-number ($Re_{\tau } = 49$) turbulent channel flow were experimentally quantified using particle image velocimetry, two-line planar laser-induced fluorescence, coherent anti-Stokes Raman spectroscopy and emission spectroscopy. Tests were conducted at two radiofrequency plasma settings. The nitrogen, in air, was vibrationally excited to $T_{vib} \sim 1240\ \mathrm{K}$ and 1550 K for 150 W and 300 W plasma settings, respectively, while the vibrational temperature of the oxygen and the rotational/translational temperatures of all species remained near room temperature. The peak axial turbulence intensities in the shear layers were reduced by 15 and 30 % in moving across the plasma for the 150 and 300 W cases, respectively. The plasma did not alter the transverse intensities. The Reynolds shear stresses were reduced by 30 and 50 % for the 150 and 300 W cases. The corresponding Reynolds shear stress correlation coefficient was also reduced, which indicates that the large-scale structures were diminished. Finally, the plasma enhanced the turbulence decay in the zero-shear regions, where the power law decay $t^{-1/n}$ exponential factor $n$ decreased from 1.0 to 0.8.

Long-time evolution of a weakly perturbed wavetrain near the modulational instability (MI) threshold is examined within the framework of the compact Zakharov equation for unidirectional deep-water waves (Dyachenko and Zakharov, JETP Lett., vol. 93, 2011, pp. 701–705). Multiple-scale solutions reveal that a perturbation to a slightly unstable uniform wavetrain of steepness $\mu $ slowly evolves according to a nonlinear Schrodinger equation (NLS). In particular, for small carrier wave steepness $\mu <\mu _{1}\approx 0.27$ the perturbation dynamics is of focusing type and the long-time behaviour is characterized by the Fermi–Pasta–Ulam recurrence, the signature of breather interactions. However, the amplitude of breathers and their likelihood of occurrence tend to diminish as $\mu $ increases while the Benjamin–Feir index (BFI) decreases and becomes nil at $\mu _{1}$. This indicates that homoclinic orbits persist only for small values of wave steepness $\mu \ll \mu _{1}$, in agreement with recent experimental and numerical observations of breathers. When the compact Zakharov equation is beyond its nominal range of validity, i.e. for $\mu >\mu _{1}$, predictions seem to foreshadow a dynamical trend to wave breaking. In particular, the perturbation dynamics becomes of defocusing type, and nonlinearities tend to stabilize a linearly unstable wavetrain as the Fermi–Pasta–Ulam recurrence is suppressed. At $\mu =\mu _{c}\approx 0.577$, subharmonic perturbations restabilize and superharmonic instability appears, possibly indicating that wave dynamical behaviour changes at large steepness, in qualitative agreement with the numerical simulations of Longuet-Higgins and Cokelet (Proc. R. Soc. Lond. A, vol. 364, 1978, pp. 1–28) for steep waves. Indeed, for $\mu >\mu _{c}$ a multiple-scale perturbation analysis reveals that a weak narrowband perturbation to a uniform wavetrain evolves in accord with a modified Korteweg–de Vries/Camassa–Holm type equation, again implying a possible mechanism conducive to wave breaking.

An oblate spheroid, the respective hemispheroids of which are kept at different uniform temperatures, placed in a rarefied gas at rest is considered. The explicit formula for the force acting on the spheroid (radiometric force) is obtained for small Knudsen numbers. This is a model of a vane of the Crookes radiometer. The analysis is performed for a general axisymmetric distribution of the surface temperature of the spheroid, allowing abrupt changes. Although the generalized slip flow theory, established by Sone (Rarefied Gas Dynamics, vol. 1, 1969, pp. 243–253), is available for general rarefied gas flows at small Knudsen numbers, it cannot be applied to the present problem because of the abrupt temperature changes. However, if it is combined with the symmetry relations for the linearized Boltzmann equation developed recently by Takata (J. Stat. Phys., vol. 136, 2009, pp. 751–784), one can bypass the difficulty. To be more specific, the force acting on the spheroid in the present problem can be generated from the solution of the adjoint problem to which the generalized slip flow theory can be applied, i.e. the problem in which the same spheroid with a uniform surface temperature is placed in a uniform flow of a rarefied gas. The analysis of the present paper follows this strategy.

The hydrodynamics of a swimming jellyfish depends on the morphology of the species. For example, oblate jellyfish appear to generate wide vortex structures near the bell margin. The vortex structures affect both the propulsion system and the feeding structure because the swimming and prey capturing activities are interrelated processes in these taxa. A three-dimensional computational model was established for an oblate jellyfish to analyse how the vortex structures present in the wake affect the swimming mechanism and the propulsion efficiency, which is defined as the ratio of power output (thrust multiplied by centre velocity) to power input (energy rate required for bell contraction). An improved penalty immersed boundary method was adopted to consider the interactions between the swimming jellyfish and the ambient fluid. The vortex structures in the wake of the swimming jellyfish were investigated in detail. The vortices generated by the contraction and expansion of the jellyfish bell interact with the vortex structures generated by the forward-moving behaviour of the jellyfish. The resulting vortex structures not only transfer momentum to the swimming jellyfish via the fluid, thereby providing the main source of thrust, but also have an implication for feeding. The effects of the elastic properties of the jellyfish on the propulsion were examined. The propulsion efficiency reaches its optimum value at particular elastic properties. We also investigated the effect of the swimming pattern of jellyfish on the propulsion efficiency. The efficiency increases with the flapping frequency and force duration.

Torque measurements in Taylor–Couette flow, with large radius ratio and large aspect ratio, over a range of velocities up to a Reynolds number of 24 000 are presented. Following a specific procedure, nine states with distinct numbers of vortices along the axis were found and the aspect ratios of the vortices were measured. The relationship between the speed and the torque for a given number of vortices is reported. In the turbulent Taylor vortex flow regime, at relatively high Reynolds number, a change in behaviour is observed corresponding to intersections of the torque–speed curves for different states. Before each intersection, the torque for a state with a larger number of vortices is higher. After each intersection, the torque for a state with a larger number of vortices is lower. The exponent, from the scaling laws of the torque, always depends on the aspect ratio of the vortices. When the Reynolds number is rescaled using the mean aspect ratio of the vortices, only a partial collapse of the exponent data is found.

The runup of long strongly nonlinear waves impinging on a vertical wall can exceed six times the far-field amplitude of the incoming waves. This outcome stems from a precursory evolution process in which the wave height undergoes strong amplification due to the combined action of nonlinear steepening and dispersion, resulting in the formation of nonlinearly dispersive wave trains, i.e. undular bores. This part of the problem is first analysed separately, with emphasis on the wave amplitude growth rate during the development of undular bores within an evolving large-scale background. The growth of the largest wave in the group is seen to reflect the asymptotic time scaling provided by nonlinear modulation theory rather closely, even in the case of fully nonlinear evolution and moderately slow modulations. In order to address the effect of such a dynamics on the subsequent wall runup, numerical simulations of evolving long-wave groups are then carried out in a computational wave tank delimited by vertical walls. Conditions for optimal runup efficiency are sought with respect to the main physical parameters characterizing the incident waves, namely the wavelength, the length of the propagation path and the initial amplitude. Extreme runup is found to be strongly correlated to the ratio between the available propagation time and the shallow-water nonlinear time scale. The problem is studied in the twofold mathematical framework of the fully nonlinear free-surface Euler equations and the strongly nonlinear Serre–Green–Naghdi model. The performance of the reduced model in providing accurate long-time predictions can therefore be assessed.

Consistent shallow-water equations are derived on the rotating sphere with topography retaining the Coriolis force due to the horizontal component of the planetary angular velocity. Unlike the traditional approximation, this ‘non-traditional’ approximation captures the increase with height of the solid-body velocity due to planetary rotation. The conservation of energy, angular momentum and potential vorticity are ensured in the system. The caveats in extending the standard shallow-water wisdom to the case of the rotating sphere are exposed. Different derivations of the model are possible, being based, respectively, on (i) Hamilton’s principle for primitive equations with a complete Coriolis force, under the hypothesis of columnar motion, (ii) straightforward vertical averaging of the ‘non-traditional’ primitive equations, and (iii) a time-dependent change of independent variables in the primitive equations written in the curl (‘vector-invariant’) form, with subsequent application of the columnar motion hypothesis. An intrinsic, coordinate-independent form of the non-traditional equations on the sphere is then given, and used to derive hyperbolicity criteria and Rankine–Hugoniot conditions for weak solutions. The relevance of the model for the Earth’s atmosphere and oceans, as well as other planets, is discussed.

The structure of a screeching axisymmetric jet in the helical C mode at a nozzle pressure ratio of 3.4 issuing from a convergent nozzle is studied using high-resolution particle image velocimetry. Proper orthogonal decomposition (POD) is used to extract the dominant coherent structures within the jet. The first two modes produced by the POD are used to reconstruct a phase-averaged data sequence. A triple decomposition into mean, coherent and random velocity components is performed. The embedded shock structures within the jet are shown to strongly modulate the coherent axial stresses within the shear layer and to weakly modulate the random axial stresses. Analysis of the third and fourth moments of the velocity probability density function is used as an indicator of possible regions of shock–vortex interaction and thus screech tone generation. Peaks of kurtosis (flatness) occur at the second, third and fourth shock–boundary intersection points, with the radial position shifting towards the centreline with increasing downstream distance. Analysis of the coherent component of vorticity shows that the largest fluctuations in coherent vorticity occur at the high-speed side of the shear layer in an area extending from the second to the fourth shock cell. With reference to prior literature, the argument is made that it is this increased magnitude of coherent vorticity fluctuation that is the primary factor in the determination of which shock cells act as dominant screech sources.

We study the transition of a Mach 6 laminar boundary layer due to an isolated cylindrical roughness element using large-scale direct numerical simulations (DNS). Three flow conditions, corresponding to experiments conducted at the Purdue Mach 6 quiet wind tunnel are simulated. Solutions are obtained using a high-order, low-dissipation scheme for the convection terms in the Navier–Stokes equations. The lowest Reynolds number ($Re$) case is steady, whereas the two higher $Re$ cases break down to a quasi-turbulent state. Statistics from the highest $Re$ case show the presence of a wedge of fully developed turbulent flow towards the end of the domain. The simulations do not employ forcing of any kind, apart from the roughness element itself, and the results suggest a self-sustaining mechanism that causes the flow to transition at a sufficiently large Reynolds number. Statistics, including spectra, are compared with available experimental data. Visualizations of the flow explore the dominant and dynamically significant flow structures: the upstream shock system, the horseshoe vortices formed in the upstream separated boundary layer and the shear layer that separates from the top and sides of the cylindrical roughness element. Streamwise and spanwise planes of data were used to perform a dynamic mode decomposition (DMD) (Rowley et al., J. Fluid Mech., vol. 641, 2009, pp. 115–127; Schmid, J. Fluid Mech., vol. 656, 2010, pp. 5–28).

High-resolution numerical simulations of statistically steady convection in a three-dimensional porous medium are presented for Rayleigh numbers $Ra \leqslant 2 \times 10^4$. Measurements of the Nusselt number $Nu$ in the range $1750 \leqslant Ra \leqslant 2 \times 10^4$ are well fitted by a relationship of the form $Nu = \alpha _3 Ra + \beta _3$, for $\alpha _3 = 9.6 \times 10^{-3}$ and $\beta _3 = 4.6$. This fit indicates that the classical linear scaling $Nu \sim Ra$ is attained, and that $Nu$ is asymptotically approximately $40\, \%$ larger than in two dimensions. The dynamical flow structure in the range $1750 \leqslant Ra \leqslant 2\times 10^4$ is analysed, and the interior of the flow is found to be increasingly well described as $Ra \to \infty $ by a heat-exchanger model, which describes steady interleaving columnar flow with horizontal wavenumber $k$ and a linear background temperature field. Measurements of the interior wavenumber are approximately fitted by $k\sim Ra^{0.52 \pm 0.05}$, which is distinguishably stronger than the two-dimensional scaling of $k\sim Ra^{0.4}$.

The current study focuses on a transition scenario in which the linear transient growth mechanism is initiated by four decaying normal modes. It is shown that the four modes, the initial structure of which corresponds to counter-rotating vortex pairs, are sufficient to capture the transient growth mechanism. More importantly, it is demonstrated that the kinetic energy growth of the initial disturbance is not the key parameter in this transition mechanism. Rather, it is the ability of the transient growth process to generate an inflection point in the wall-normal direction and consequently to make the flow susceptible to a three-dimensional disturbance leading to transition to turbulence. Because of the minimal number of modes participating in the transition process, it is possible to follow its earlier key stages analytically and to compare them with the results of direct numerical simulation. This procedure reveals the role of various flow parameters during the transition, such as the difference between symmetric and antisymmetric transient growth scenarios. Moreover, it is shown that the resulting modified base flow of the linear process is not sufficient to produce a significant localized maximum of the base-flow vorticity (i.e. a ‘strong’ inflection point), and it is only due to nonlinear effects that the base flow becomes unstable with respect to an infinitesimal three-dimensional disturbance. Finally, the physical mechanism during key stages of transition is well captured by the analytical expressions. Furthermore, the vortex dynamics during these stages is very similar to the model proposed by Cohen, Karp & Mehta (J. Fluid Mech., vol. 747, 2014, pp. 30–43) according to which streamwise variation of the initial counter-rotating vortex pair is required to generate concentrated spanwise vorticity, which together with the lift-up by the induced velocity and shear of the base flow generates packets of hairpins.

High-fidelity, direct numerical simulations (DNSs) are conducted to examine the vortex structure and aerodynamic loading of unidirectionally revolving wings in quiescent fluid. Wings with aspect ratios $({\mathit{AR}}) = 1$, 2 and 4 are considered at a fixed root-based Reynolds number of 1000. Each wing is shown to generate a coherent leading-edge vortex (LEV) that remains in close proximity to the surface and provides persistent suction throughout the motion. Towards the tip, the LEV lifts off as an arch-like structure and reorients itself along the chord through its connection with the tip vortex. The substantial and sustained aerodynamic loads achieved during the motion saturate with aspect ratio resulting from the chordwise growth of the LEV along the span eventually becoming geometrically constrained by the trailing edge. Further, for ${\mathit{AR}}=4$, substructures develop in the feeding sheet of the LEV, which appear to directly correlate with the local, span-based Reynolds number achieved during rotation. The lower-aspect-ratio wings do not have sufficient spans for these transitional elements to manifest. In contrast, vortex breakdown, which occurs around midspan for each aspect ratio, shows a strong dependence on the spanwise pressure gradient established between the root and tip of the wing and not local Reynolds number. This independent development of shear-layer substructures and vortex breakdown parallels very closely with what has been observed in delta wing flow. Next, the centrifugal, Coriolis and pressure gradient forces are also analysed at several spanwise locations across each wing, and the centrifugal and pressure gradient forces are shown to be responsible for the spanwise flow above the wing. The Coriolis force is directed away from the surface at the base of the LEV, indicating that it is not a contributor to LEV attachment, which is contrary to previous hypotheses. Finally, as a means of emphasizing the importance of the centrifugal force on LEV attachment, the ${\mathit{AR}}=2$ wing is simulated with the addition of a source term in the governing equations to oppose and eliminate the centrifugal force near the surface. The initial formation and development of the LEV is unhindered by the absence of this force; however, later in the motion, the outboard lift-off of the LEV moves inboard. Without the opposing outboard-directed centrifugal force to keep the separation past midspan, the entire vortex eventually separates and moves away from the surface.

The characteristics of a turbulent boundary layer overlying a complex roughness topography were explored with stereo particle-image velocimetry measurements in the wall-normal–spanwise $(y\mbox{--}z)$ plane. The roughness under consideration was replicated from a turbine blade damaged by deposition of foreign materials containing a broad range of topographical scales arranged in a highly irregular manner. The single-point turbulence statistics displayed strong spanwise heterogeneity, in particular spanwise-alternating low- and high-momentum flow pathways in the mean flow marked by enhanced Reynolds stresses and turbulent kinetic energy. The spanwise regions between high- and low-momentum flow pathways were occupied by swirling motions, suggesting the generation and sustainment of turbulent secondary flows due to the spanwise heterogeneity of the complex roughness under consideration. Similar observations were recently reported for more ordered spanwise roughness transitions by Nugroho, Hutchins & Monty (Intl J. Heat Fluid Flow vol. 41, 2013, pp. 90–102) and Willingham et al. (Phys. Fluids vol. 26, 2014, 025111).

On the basis of a two-point similarity analysis, the well-known power-law variations for the mean kinetic energy dissipation rate $\overline{\epsilon }$ and the longitudinal velocity variance $\overline{u^2}$ on the axis of a round jet are derived. In particular, the prefactor for $\overline{\epsilon } \propto (x-x_0)^{-4}$, where $x_0$ is a virtual origin, follows immediately from the variation of the mean velocity, the constancy of the local turbulent intensity and the ratio between the axial and transverse velocity variance. Second, the limit at small separations of the two-point budget equation yields an exact relation illustrating the equilibrium between the skewness of the longitudinal velocity derivative $S$ and the destruction coefficient $G$ of enstrophy. By comparing the latter relation with that for homogeneous isotropic decaying turbulence, it is shown that the approach towards the asymptotic state at infinite Reynolds number of $S+2G/R_{\lambda }$ in the jet differs from that in purely decaying turbulence, although $S+2G/R_{\lambda } \propto R_{\lambda }^{-1}$ in each case. This suggests that, at finite Reynolds numbers, the transport equation for $\overline{\epsilon }$ imposes a fundamental constraint on the balance between $S$ and $G$ that depends on the type of large-scale forcing and may thus differ from flow to flow. This questions the conjecture that $S$ and $G$ follow a universal evolution with $R_{\lambda }$; instead, $S$ and $G$ must be tested separately in each flow. The implication for the constant $C_{\epsilon 2}$ in the $k-\overline{\epsilon }$ model is also discussed.