We investigate the derivation of Euler's equation from Hamilton's variational principle for flows decomposed into their mean and fluctuating parts. Our particular concern is with the flow decomposition used in the derivation of the Navier–Stokes–α equation which expresses the fluctuating velocity in terms of the mean flow and a small fluctuating displacement. In the past the derivation has retained terms up to second order in the Lagrangian which is then averaged. The variation is effected by incrementing the mean velocity, while holding the moments of the products of the displacements fixed. The process leads to a mean Euler equation for the mean velocity. The Navier–Stokes–α equation is only obtained after making a further closure approximation, which is not the concern of this paper. Instead attention is restricted here to the exact analysis of Euler's equation. We show that a proper implementation of Hamilton's principle, which concerns the virtual variation of particle paths, can only be achieved when the fluctuating displacement and mean velocity are varied in concert. This leads to an exact form of Euler's equation. If, on the other hand, the displacement is held fixed under the variation, a term in Euler's equation is lost. Averaging that erroneous form provides the basis of the Navier–Stokes–α equation. We explore the implications of the correct mean equation, particularly with regard to Kelvin's circulation theorem, comparing it with the so called GLM and glm-equations.

The formation of a singularity in a compressible gas, as described by the Euler equation, is characterized by the steepening and eventual overturning of a wave. Using self-similar variables in two space dimensions and a power series expansion based on powers of $|t_{0}-t|^{1/2}$, $t_{0}$ being the singularity time, we show that the spatial structure of this process, which starts at a point, is equivalent to the formation of a caustic, i.e. to a cusp catastrophe. The lines along which the profile has infinite slope correspond to the caustic lines, from which we construct the position of the shock. By solving the similarity equation, we obtain a complete local description of wave steepening and of the spreading of the shock from a point. The shock spreads in the transversal direction as $|t_{0}-t|^{1/2}$ and in the direction of propagation as $|t_{0}-t|^{3/2}$, as also found in a one-dimensional model problem.

It is well known that water slowly issued vertically downward exhibits a hysteresis phenomenon. A jetting-to-dripping transition appearing upon a stepwise decrease in jet issue speed was used to identify the origin of the Plateau–Rayleigh unstable wave elements which disintegrate the jetting liquid. In the present laboratory experiment using a stainless steel nozzle of inner radius 1 mm and length 30 mm, the transition occurred at a dimensionless jet issue speed of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\sqrt{\rho U^{2}a_{0} /\sigma } =0.8,$ where $\rho $ and $\sigma $ respectively denote the density and surface tension coefficient of the liquid issued at the speed $U$ from the nozzle of radius $a_0$. The jet length gradually shortened with an oscillation of considerably large amplitude and period. High-speed camera images show that this oscillation is caused by tip contraction capillary wave (TCCW) elements which are elongated by the gravitationally accelerating jet flow and become Plateau–Rayleigh unstable wave elements. The jet length increases while the jet tip experiences end-pinching and radiates TCCW elements upstream. Only those TCCW elements destabilized at appropriate locations can grow sufficiently to shorten the jet. Since the unstable wave elements produced nearer the nozzle exit have much smaller amplitude at the jet tip, the end-pinching becomes effective. Thus, these processes are repeatable and constitute a self-destabilizing loop. The observed jetting-to-dripping transition has nothing to do with the random nozzle disturbances which were believed to be the origin of the Plateau–Rayleigh unstable wave in conventional instability theories. It is also different from the feature conjectured from current absolute/convective instability analysis. The underling physics of the self-destabilizing loop are explored in detail by numerical simulations based on a one-dimensional model.

Instabilities of a viscous fluid between two fast but differentially rotating concentric spheres, the so-called spherical Couette flow, with a fixed radius ratio of $\unicode[STIX]{x1D702}=r_{i}/r_{o}=1/3$ are studied, where $r_{i}$ is the inner and $r_{o}$ the outer radius of the spherical shell. Of particular interest is the difference between cases where the Rossby number $Ro=(\unicode[STIX]{x1D6FA}_{i}-\unicode[STIX]{x1D6FA}_{o})/\unicode[STIX]{x1D6FA}_{o}>0$ and cases with $Ro<0$, where $\unicode[STIX]{x1D6FA}_{i}$ and $\unicode[STIX]{x1D6FA}_{o}$ are the inner- and outer-sphere angular velocities. The basic state in both situations is an axisymmetric shear flow with a Stewartson layer situated on the tangent cylinder. The tangent cylinder is given by a cylinder that touches the equator of the inner sphere with an axis parallel to the axis of rotation. The experimental results presented fully confirm earlier numerical results obtained by Hollerbach (J. Fluid Mech., vol. 492, 2003, pp. 289–302) showing that for $Ro>0$ a progression to higher azimuthal wavenumbers $m$ can be seen as the rotation rate $\unicode[STIX]{x1D6FA}_{0}$ increases, but $Ro<0$ gives $m=1$ over a large range of rotation rates. It is further found that in the former case the modes have spiral structures radiating away from Stewartson layer towards the outer shell whereas for $Ro<0$ the modes are trapped in the vicinity of the Stewartson layer. Further, the mean flow excited by inertial mode self-interaction and its correlation with the mode’s amplitudes are investigated. The scaling of the critical $Ro$ with Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{o}\,d^{2})$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $d$ the gap width, is well within the bounds that have been established in a number of experimental studies using cylindrical geometries and numerical studies using spherical cavities. However, the present work is the first that experimentally examines Stewartson-layer instabilities as a function of the sign of $Ro$ for the true spherical-shell geometry.

Bubble interaction with turbulence has a number of applications in engineering processes and nature. The complex interplay between the vortical structures present in a turbulent flow and the bubbles drives their deformation dynamics, which may lead to bubble rupture under the appropriate conditions. Such a process includes nonlinear interaction among the turbulent eddies and between the eddies and the bubbles. Thus, the coupled evolution of a single vortex ring with a bubble represents an idealized scenario that can provide a framework to shed light on understanding such a common and complex mechanism. Jha & Govardhan (J. Fluid Mech., vol. 773, 2015, pp. 460–497) have performed elegant experiments generating controlled vortex rings and injecting bubbles of known volume. They have reported interesting results on the elongation process of the bubble and its impact on vortex dynamics.

The low value of the speed of sound in dilute granular media permits the study of the properties of supersonic flows for a wide range of Mach numbers. In this paper, we report the experimental observation of a subsonic–supersonic transition in a vibrated granular gas. The shock fronts studied are obtained by simply pushing a rectangular obstacle into the granular gas for different obstacle velocities. The supersonic regime is characterized by the formation of normal shock waves whose width increases when the Mach number decreases to values close to 1. The bimodal model proposed by Mott-Smith in the 1950s provides a good description for the velocity distributions as well as the macroscopic quantities for shock waves in molecular gases but remains inadequate for dissipative media like granular gases and plasmas. Here by examining the shock front structure for a wide range of Mach numbers, we adapt the Mott-Smith bimodal description to a dissipative medium. By using balance equations from granular kinetic theory and taking into account different dissipation sources, the proposed model allows us to understand how this dissipation modifies temperature, mean velocity and volume fraction profiles through the shock front.

Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe (Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6E5}$, for $\unicode[STIX]{x1D716}$ the linear prediction of the maximum slope at focusing and $\unicode[STIX]{x1D6E5}$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.

We consider the problem of weakly nonlinear buoyant convection in horizontal mushy layers with permeable mush–liquid interface during the solidification of binary alloys. We analyse the effects of several parameters of the problem on the stationary modes of convection in the form of either a hexagonal pattern or a non-hexagonal pattern such as rolls, rectangles and squares. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous theoretical investigations of the problem are relaxed here in order to study the problem based on a more realistic model. Using both analytical and numerical methods, we determine the steady solutions for the nonlinear problem in a range of the Rayleigh number R near its critical value. Both the nonlinear basic state and variable permeability of the present problem favour hexagon-pattern convection. The results of the analyses and computations indicate that depending on the range of values of the parameters, bifurcation to hexagonal or non-hexagonal convection can be either supercritical or subcritical. However, among all the computed solutions in the particular range of values of the parameters that are most relevant to those of the experiments, only convection in the form of down-hexagons with downflow at the cell centres and upflow at the cell boundaries, was found to be realizable, in the sense that its amplitude increases with R.

For two-dimensional periodic water waves or sound waves, the kinetic energy per wavelength is ½mdc2, and the momentum per wavelength is ±mdc, where c is the wave velocity, and md is the drift mass per wavelength. These results also hold for three-dimensional periodic waves, for which the kinetic energy, momentum, and drift mass are all for one wave cell, the area of which is the product of the wavelengths in two perpendicular directions.

The results obtained are rigorous, and not restricted to linear waves or even to nonlinear symmetric waves. For linear water waves, in particular, the kinetic energy can be shown to be equal to the sum of the potential energy and the surface energy (due to surface tension), so that the total energy E is twice the kinetic energy, and

formula here

McIntyre's (1981) contention that wave momentum is a myth is discussed at length for both water waves and sound waves.

Davis has shown by means of a lengthy calculation that, for two-dimensional oscillations in a canal of width 2a, the mth eigenvalue has the form \[ N_m = {\textstyle\frac{1}{2}}m\pi - \frac{\lambda_1+\lambda_2}{4m\pi}+o\bigg(\frac{1}{m}\bigg), \] where λ1/a and λ2/a are the curvatures of the bounding cross-sectional curve C at its vertical intersections with the free surface. Here the same result is obtained more simply.

From Whitham's ray-shock theory and the Brinkley-Kirkwood theory of shock propagation, a general theory for the propagation of asymmetrical blast waves of arbitrary shapes and strengths is developed in this paper. The general theory requires the simultaneous numerical solution of a set of partial differential equations and a pair of ordinary differential equations. If the shock shape is assumed to be known and remains invariant with time then the geometrical and the dynamical relationships in the theory can be decoupled. In this case the solution simply requires the integration of the ordinary differential equations governing the dynamics of the blast motion since the geometry is already known. As a specific example the asymmetrical blast waves generated by the rupture of a pressurized ellipsoid are studied. The peak pressure is calculated by assuming that the shock surface remains ellipsoidal for all times and that the peak overpressure decay rate of the blast depends on the local curvature. For weak shocks, it is found that the degree of directionality is more pronounced than for stronger shocks. For weak blasts the present theory agrees with the solution based on acoustic theory. Experimental results on the shock trajectories for asymmetrical blast waves generated by exploding wires are found to agree with the present theory.