The three-dimensional Green's function of water waves in variable-bathymetry regions, associated with the problem of propagation of water waves emitted from a monochromatic point source, is derived and studied. The solution is of interest in its own right but also provides useful information for the formulation and treatment of complex wave–body–seabed interaction problems in variable-bathymetry regions, especially as regards the hydrodynamic characteristics of large structures installed in the nearshore and coastal environment. Assuming a parallel-contour bathymetry, with a continuous depth function of the form $h(x,y) \,{=}\, h(x)$, attaining constant, but possibly different, values at the semi-infinite regions $x \,{<}\, a$ and $x \,{>}\, b$, the problem is reduced to a two-dimensional one, by using Fourier transform. The transformed problem is treated by applying domain decomposition and reformulating it as a transmission problem in the finite domain containing the bottom irregularity. An appropriate decomposition of the wave potential is introduced, permitting the singular part to be solved analytically, and the problem for the regular part to be reformulated as a variational problem. An enhanced local-mode series representation is used for the regular wave potential in the variable-bathymetry region, including the propagating mode, the sloping-bottom mode (see Athanassoulis & Belibassakis 1999), and a number of evanescent modes. Using this representation, in conjunction with the variational principle, a forced system of horizontal coupled-mode equations is derived for the determination of the complex modal-amplitude functions of the regular wave potential. This system is numerically solved by using a second-order central finite-difference scheme. The source-generated water-wave potential is, finally, obtained by an efficient numerical Fourier inversion based on FFT. Numerical results are presented and discussed for various bottom topographies, including smooth but steep underwater trenches and ridges, putting emphasis on the identification of the important features of the near- and far-field patterns on the horizontal plane and on the vertical plane containing the point source. Characteristic patterns of trapped (well-localized) wave propagation over ridges are predicted and discussed.

In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.

The three-dimensional transition of the flow behind a circular cylinder with a near-wake wire disturbance has been investigated experimentally. The asymmetric placement of a wire in the near-wake region of the cylinder causes an unnatural mode of shedding to occur, namely mode C. We performed flow visualization and particle image velocimetry (PIV) experiments to investigate the influence of the wire on various properties of the flow, such as the dynamics of the streamwise secondary vortices. Experiments were performed at the Reynolds number range of Re = 165–300. From these experiments, it can be concluded that mode C structures are formed as secondary streamwise vortices around the primary von Kármán vortices. The spanwise wavelength of those mode C structures is determined to be approximately two cylinder diameters. The presence of the wire also triggered the occurrence of period doubling in the wake. Each new set of mode C structures is out of phase with the previous set, i.e. doubling the shedding period. This period-doubling phenomenon is due to a feedback mechanism between the consecutively shed upper vortices.

A transformation technique is used to solve the problem of steady nonlinear surface waves where the restoring force is either gravity or surface tension. An exact nonlinear integro-differential equation is found which yields known approximate solutions. Extensions to the method to account for more complicated geometries are also illustrated. The equation is solved numerically and results in agreement with previous solutions are obtained. In the case of capillary waves, the existence of two types of wave of greatest height is clearly indicated.

A three-layer intrusion flow is considered, in which all three layers are in motion, with different speeds, relative to the observer. Shear is present in the middle layer, and the lowest fluid may even move oppositely to the upper two (so giving an exchange flow). Two thin interfaces are present, above and below the moving middle layer. A linearized analysis is presented for small wave amplitudes. Nonlinear periodic solutions are then obtained using a Fourier technique, and reveal a range of nonlinear phenomena, including limiting waves, multiple solutions and resonances.

A zero-pressure-gradient turbulent boundary layer flowing over a transitionally rough surface (24-grit sandpaper) with $k^{+}\approx 11$ and a momentum-thickness Reynolds number of approximately 2400 is studied using direct numerical simulation (DNS). Heat transfer between the isothermal rough surface and the turbulent flow with molecular Prandtl number $Pr=0.71$ is simulated. The dynamic multiscale approach developed by Araya et al. (J. Fluid Mech., vol. 670, 2011, pp. 581–605) is employed to prescribe realistic time-dependent thermal inflow boundary conditions. In general, the rough surface reduces mean and fluctuating temperature profiles with respect to the smooth surface flow when normalized by Wang & Castillo (J. Turbul., vol. 4, 2003, 006) inner/outer scaling. It is shown that the Reynolds analogy does not hold for $y^{+}<9$. In this region the value of the turbulent Prandtl number departs substantially from unity. Above this region the Reynolds analogy is only approximately valid, with the turbulent Prandtl number decreasing from 1 to 0.7 across the boundary layer for rough and smooth walls. In comparison with the smooth-wall case, the turbulent transport of heat per unit mass, $\overline{v^{\prime }v^{\prime }{\it\theta}^{\prime }}$, towards the wall is enhanced in the buffer layer, but the transport of $\overline{v^{\prime }v^{\prime }{\it\theta}^{\prime }}$ away from the wall is reduced in the outer layer for the rough case; similar behaviour is found for the vertical transport of turbulent momentum per unit mass, $\overline{v^{\prime }u^{\prime }v^{\prime }}$. Above the roughness sublayer (3$k$–5$k$) it is found that most of the temperature field statistics, including higher-order moments and conditional averages, are highly similar for the smooth and rough surface flow, showing that the Townsend’s Reynolds number similarity hypothesis applies for the thermal field as well as the velocity field for the Reynolds number and $k^{+}$ considered in this study.

In this paper flow-visualization techniques are used to study the flow instability which occurs at the orifice in the free convection open thermosyphon. The influence of the shape of the orifice on the measured heat transfer of the system is also studied under both laminar and turbulent conditions. The dominant mode of penetration of the hot stream by the cold fluid entering at the base of the reservoir is found to instigate a mixing region at the orifice. In turbulent flow this spreads into and eventually fills the tube as the Rayleigh number is increased. In such circumstances a sharp-edged orifice appears to give better overall heat transfer than a rounded orifice.

Two temporally developing isothermal round jets at a Mach number of 0.9 and Reynolds numbers of 3125 and 12 500 are simulated in order to investigate noise generation in high-subsonic jet flows. Snapshots and statistical properties of the flow and sound fields, including mean, root-mean-square and skewness values, spectra and auto- and cross-correlations of velocity and pressure, are presented. The jet at a Reynolds number of 12 500 develops more rapidly, exhibits more fine turbulent scales and generates more high-frequency acoustic waves than the other. In both cases, however, when the jet potential core closes, mixing-layer turbulent structures intermittently intrude on the jet axis and strong low-frequency acoustic waves are emitted in the downstream direction. These waves are dominated by the axisymmetric mode and are significantly correlated with centreline flow fluctuations. These results are similar to those obtained at the end of the potential core of spatially developing jets. They suggest that the mechanism responsible for the downstream noise component of these jets also occurs in temporal jets, regardless of the Reynolds number. This mechanism is revealed by averaging the flow and pressure fields of the present jets using a sample synchronization with the minimum values of centreline velocity at potential-core closing. A spot characterized by a lower velocity and a higher level of vorticity relative to the background flow field is found to develop in the interfacial region between the mixing layer and the potential core, to strengthen rapidly and reach a peak intensity when arriving on the jet axis, and then to break down. This is accompanied by the growth and decay of a hydrodynamic pressure wave, propagating at a velocity which, initially, is close to 65 per cent of the jet velocity and slightly increases, but quickly decreases after the collapse of the high-vorticity spot in the flow. During that process, sound waves are radiated in the downstream direction.

The experimental work of Franklin, Price & Williams (1973) shows that for moderately large driving amplitudes there are features of spatial resonance that are not predicted by the model representation of Mahony & Smith (1972). We here derive an alternative model, which remains valid for moderately large driving amplitudes, and we are able to obtain a theoretical description of both hysteresis and nonlinear detuning of the low frequency wave response. An experiment in which surface waves were generated by a sinusoidal pressure field at the free surface (and which corresponds almost exactly to the theoretical problem) was conducted in order to test these predictions.

The structure of the turbulent shear flow over a propagating wave of fixed frequency is examined. The vertical and horizontal velocities were measured in a wind-wave facility at Stanford University. The structure of the wave-perturbed turbulence was found to depend significantly on the ratio of the local mean velocity to the wave speed. The results support the idea of ‘cat's-eye’ type flow about the mean critical height.

A simplified model problem has recently been suggested by Schaeffer (1980) in order to explain the results obtained by Benjamin (1978) in his investigation of Taylor vortices in short cylinders. In particular Schaeffer reproduces the results obtained by Benjamin for cylinders so short that only two-cell or four-cell flows are possible. The model given by Schaeffer has artificial conditions imposed on the fluid velocity field at the end walls. These conditions depend on a parameter α and reduce to the no-slip condition when α = 1. If α = 0 the conditions require that the normal component of the velocity and the normal derivative of the tangential velocity vanish at the ends. In this case the onset of Taylor vortex-like motion occurs as a bifurcation from purely circumferential flow. If α is now taken to be small and positive, there is no bifurcation and the circulatory flow develops smoothly. We shall use perturbations method for the case of small α. The imperfect bifurcation problem which we obtain predicts some results consistent with those of Benjamin.

The evolution of artificially generated localized disturbances in the shape of hairpin vortices, in laminar axisymmetric rotating shear flows, is investigated experimentally. The results are compared with the predictions of a theoretical model (Levinski & Cohen 1995) with respect to the growth of such disturbances. Hairpin vortices were generated at the surface of the inner cylinder of an axisymmetric Couette apparatus, employing an injection–suction technique. The flow field was analysed from flow visualization using top and side views and by measurements of the mean and instantaneous velocity fields, carried out using laser Doppler anemometry and particle image velocimetry. An instability domain, within the range of base flow parameters where the flow is known to be linearly stable, was found. The marginal ratio between the angular velocities of the inner and outer cylinders beyond which the flow is stable to finite-amplitude localized disturbances agrees with the theoretical prediction based on the measured mean flow in the region of the disturbance. The dependence of the hairpin's inclination angle on the ratio between the two angular velocities is fairly well predicted by the theoretical model.

Lagrangian coherent structures (LCS) of a turbulent boundary layer at ${\mathit{Re}}_{\theta } $ of 9800 are identified in a plane parallel to the wall at ${y}^{+ } = 50$. Three-component high-speed stereo particle image velocimetry measurements on a two-dimensional rectangular plane are used for the analysis. The velocity field is extended in the streamwise direction, using Taylor’s frozen field hypothesis. A computational approach utilizing the variational theory of hyperbolic Lagrangian coherent structures is applied to the domain and trajectories are computed using the extended field. The method identified both attracting and repelling Lagrangian coherent structures. There are no apparent differences in distribution of size, orientation and location of attracting and repelling structures. Hyperbolic behaviour appeared in the fluid at and around points of intersection between the attracting and repelling Lagrangian coherent structures. The network of curves identifying distinct regions of coherent flow patterns is displayed in observed relationship between the arrangement of Lagrangian coherent structures and various Eulerian fields.

This paper considers the structure and linear stability of two-dimensional hollow vortex equilibria. Equilibrium solutions for a single hollow vortex in linear and nonlinear straining flows are derived in analytical form using free streamline theory. The linear stability properties of this solution class are then determined numerically and a new type of resonance-induced displacement instability is identified. It is found to be a consequence of the fact that one of the shape distortion modes of a circular hollow vortex has the same frequency as one of the modes corresponding to displacement of the vortex centroid. The instability is observed in the case of an isolated hollow vortex situated in straining flow of order three. We also revisit the hollow vortex row solution due to Baker, Saffman & Sheffield (J. Fluid Mech., vol. 74, 1976, p. 1469), and since it is currently lacking in the literature, we present a full linear stability analysis of this solution using Floquet analysis.

The unsteady two-dimensional heat-transfer problem of melting around a horizontal heated cylinder is studied numerically. The cylinder is heated isothermally. A physical model is introduced which accounts for the effects of density change upon melting and subcooling effect, as well as natural convection. Most previous work has assumed that the density number (ratio of solid density to liquid density) is unity. In practice, all solid materials exhibit some density change upon melting. If the density number is greater than one, this induces a blowing effect at the phase-change boundary. If the density number is less than one, a suction effect is produced. This study indicates that the density-change effect on heat transfer during melting is minor. Subcooling results when the solid is at a temperature below the melting temperature. When the melting process begins, some of the available thermal energy must be used as sensible heat, to raise the solid's temperature to the melting point. As a result, less thermal energy is available for melting. Subcooling effects are found to have a substantial effect on the heat-transfer process. The effects of natural convection have been clearly documented, and indicate that natural convection must be included in any realistic model of the melting process. Detailed predictions of the effects of density change and subcooling on the melting process are given. Information on the temperature and flow fields for representative values of Stefan, Rayleigh, Prandtl, subcooling and density number is given. Further results from the numerical solutions include information on local and average heat-transfer rates and sensibleheat gain as well as melt volume as a function of time. Comparisons are made with earlier numerical and analytical results.

We study the breakup of confined fluid threads at low flow rates to understand instability mechanisms. To determine the critical conditions between the earlier quasi-stable necking stage and the later unstable collapse stage, simulations and experiments are designed to operate at an extremely low flow rate. The critical mean radii at the neck centres are identified by the stop-flow method for elementary microfluidic configurations. Two distinct origins of capillary instabilities are revealed for different confinement situations. One is the gradient of capillary pressure induced by the confinements of geometry and external flow, whereas the other is the competition between the capillary pressure and internal pressure determined by the confinements.

The flow past circular cylinders of finite length, supported at one end and lying with their axes perpendicular to a uniform stream, has been investigated in a supersonic stream at Mach number 1.96 and also in a low-speed stream. In both stream it was found that the flow past the cylinders could be divided into three regions: (a) a central region, (b) that near the free end of the cylinder, and (c) that near the supported end. The locations of the second and third regions were found to be almost independent of the cylinder length-to-diameter ratio, provided that this exceeded 4, while the flow within and the extent of the first region were dependent on this ratio. Form-drag coefficients determined in the central region in the supersonic flow were in close agreement with the values determined at the same Mach number by other workers. In the low-speed flow the local form-drag coefficients were dependent on length-to-diameter ratio and were always less than that of an infinite-length cylinder at the same Reynolds number.

Continuum modelling of dense gas–solid flows strongly depends on the constitutive relations used, including the interphase drag force, particle phase stress and the boundary condition for particle–wall interactions. The lack of scale separation is usually claimed to cause the breakdown of the Navier–Stokes (NS) order continuum theory. In this study, computational fluid dynamics–discrete element method (CFD-DEM) simulations of bubbling, turbulent and fast fluidization of smooth, inelastic spheres were conducted to systematically analyse the valid range of NS theory. An entropy-based criterion $I_{s}$ and Knudsen numbers defined using different characteristic length scales ($Kn_{frac}$, $Kn_{gran}$ and $Kn_{vel}$) were quantified. It was found that (i) except at the centre of bubbles where the solid concentration is quite low, NS theory for discrete particles was valid for bubbling fluidization irrespective of the breakdown criterion. Even at the boundary of the bubbles, values of $I_{s}$ and $Kn$ were small; and (ii) the conclusion depended on the criterion used in turbulent and fast fluidization. If $Kn_{frac}$ was used, NS theory would be generally valid. If $I_{s}$ was chosen, NS theory would be still valid but with lower confidence. However, if $Kn_{vel}$ or $Kn_{gran}$ was selected, NS theory broke down. Because $I_{s}$ includes the non-equilibrium effects caused by the gradient of hydrodynamic fields and particle inelasticity, we may conclude that NS theory was valid for all tested cases. This means that the continuum description of discrete particles is not the main source of the breakdown of NS theory.