The extended mild-slope equation and the modified mild-slope equation have been used successfully to study refraction–diffraction of linear water waves by steep bottom roughness. Their consistency has been questioned. A systematic derivation of these model equations exposes and illuminates their rationale. Their good performance stems from an accurate representation of (Class I) Bragg resonance. As a benchmark test case, we consider scattering by a sloping bottom with random roughness. The rates of scattering found for the mean field in both of the approximate models agree exactly with the full theory for scattering by small roughness. This greatly improves the limited agreement which was found for the mild-slope equation, and establishes the validity of the above model equations. The study involves operator calculus, a powerful method for simplifying problems with variable coefficients. The augmented mild-slope equation serves to consistently derive accurate model equations.

A liquid-metal flow driven by a rotating magnetic field in a finite-length cylinder is studied numerically as a function of the field frequency. In the high-frequency case, the magnetic field is expelled from the liquid-metal except in a skin-depth layer along the side and top walls of the cylinder. In the corner region, where the skin-depth layers intersect, the body force exhibits a large positive and negative azimuthal component as well as inward radial and axial components which are rotational. The flows for various frequencies are compared to the low-frequency flow.

Stratified flow in nocturnal boundary layers is studied using direct numerical simulation (DNS) of the Ekman layer, a model problem that is useful to understand atmospheric boundary-layer (ABL) turbulence. A stabilizing buoyancy flux is applied for a finite time to a neutral Ekman layer. Based on previous studies and the simulations conducted here, the choice of $L_{\mathit{cri}}^{+}=Lu_{\ast }/\unicode[STIX]{x1D708}\approx 700$ ($L$ is the Obukhov length scale and $u_{\ast }$ is the friction velocity) provides a cooling flux that is sufficiently strong to cause the initial collapse of turbulence. The turbulent kinetic energy decays over a time scale of $4.06L/u_{\ast }$ during the collapse. The simulations suggest that imposing $L_{\mathit{cri}}^{+}\approx 700$ on the neutral Ekman layer results in turbulence collapse during the initial transient, independent of Reynolds number, $Re_{\ast }$. However, the long-time state of the flow, i.e. turbulent with spatial intermittency or non-turbulent, is found to depend on the initial value of $Re_{\ast }$ since the cooling flux and resultant stratification increase with $Re_{\ast }$ for a given $L^{+}$. The lower-$Re_{\ast }$ cases have sustained turbulence with shear and stratification profiles that evolve in a manner such that the gradient Richardson number, $Ri_{g}$, in the near-surface layer, including the low-level jet, remains subcritical. The highest $Re_{\ast }$ case has supercritical $Ri_{g}$ in the low-level jet and turbulence does not recover. A theoretical discussion is performed to infer that the bulk Richardson number, $Ri_{b}$, is more suitable than $L^{+}$ to determine the fate of stratified Ekman layers at late time. DNS results support the implications of $Ri_{b}$ for the effect of initial $Re_{\ast }$ and $L^{+}$ on the flow.

Vortical-pattern visualization for the starting flow behind airfoils is extended from a chordwise or two-dimensional mode to a spanwise or three-dimensional mode. We demonstrate the capabilities and limits of the extended method by means of selected photographic frames and sequences.

The normal modes of a horizontally uniform, vertically sheared flow over a sloping bottom are considered in two active layers underneath a deep motionless third layer. The variations of the layer thickness are assumed to be small to analyse the sixth-order eigenvalue problem for finite-Froude-number typical for oceanic currents. The dispersion curves for the Rossby waves and the Poincaré modes of inertia–gravity waves (IGW) are investigated to identify the different types of instabilities that occur if there is a pair of wave components which have almost the same Doppler-shifted frequency related to crossover of the branches when the Froude number increases. Simple criteria for ageostrophic instabilities due to a resonance between the IGW and the Rossby wave because of the thickness gradient in either the lower or middle layer, are derived. They exactly correspond to violation of sufficient Ripa's conditions for the flow stability. In both cases the growth rate and the interval of unstable wavenumbers are shown to be proportional to the square root of the corresponding gradient of the layer thickness. These types of ageostrophic instability can coexist (and with Kelvin–Helmholtz instability). However, their role in generating unbalanced motions and mixing processes in geophysical fluids appears limited due to small growth rates and narrow intervals of the unstable wavenumbers in comparison to Kelvin–Helmholtz instability.

The wetting kinetics of a drop on a solid surface is measured by observing the movement of the contact line, which is often seen to be unstable, showing a scalloped profile. Many factors have been cited, which, although they can cause instability, can also be eliminated from the experiments, but still the instabilities appear. The basic shape of a spreading drop has a large curvature localized in the vicinity of the contact line as determined by microscopy. It is shown here using linear stability analysis that this curvature can destabilize the contact line region. When the drop profile is disturbed from a basic thickness of h to h + h′, there are two contributions from h′ in the form of added Laplace pressure. One of these is commonly accounted for in the stability analyses. The other is not, and occurs only if the basic shape has a curvature, and the drop has a large curvature near the apparent dynamic contact line, but only for a wetting liquid. This is why instability is not reported in the case of spreading of drops of non-wetting liquids. It also explains why instability gives rise to the changed spreading kinetics of drops that are sometimes reported in the literature, and suggests that as larger curvatures are expected in forced spreading those cases are probably accompanied quite frequently by unstable contact lines.

The convergence rate and the structures of the proper orthogonal decomposition (POD) reconstruction are re-examined. The relatively slow convergence rate of the wall-normal velocity and the over-prediction of the Reynolds shear stress from POD-based representations of near-wall turbulence suggest that coherent structures are too well correlated. This is a consequence of the directional preference of the eigenfunctions toward the most energetic data (the streamwise motion at $x^+_2 \,{=}\, 13$) which may be relaxed by extracting the POD eigenfunctions from the streak modes and the streamwise vortex modes separately. The rate of convergence and representation of structures may be improved by performing the POD in Squire's coordinate system (PODS). The statistical reconstruction of the localized ($x^+_2 \,{\leq}\, 84$) PODS eigen-functions from the large-eddy simulation (LES) database suggests that the convergence rate of component energy, Reynolds shear stress and the correlation coefficient of streamwise and wall-normal velocity improve significantly. The PODS eigenvalue spectra and the spatial structures of the eigenfunctions indicate the presence of observed streaks, streamwise vortices, and ejection/sweep events.

Dynamical models of the full channel turbulence are constructed by projecting the Navier–Stokes equations onto the PODS eigenfunctions. The unresolved modes are taken into account by closure models adapted from a constant eddy-viscosity model, dynamic Smagorinsky, and dynamic one-coefficient mixed subgrid-scale (SGS) model. Statistical moments obtained from PODS-based simulations are much more accurate than their counterparts from POD-based simulations. This is particularly true for the mean velocity, streamwise turbulent intensity, Reynolds shear stress and the correlation coefficient of streamwise and wall-normal velocity.

Convection plays a major role in a variety of natural hydrodynamic systems. Those in which convection drives exchange flows through a lateral contraction and/or over a sill form a special class with typical examples being the Red and Mediterranean Seas, the Persian Gulf, and the fjords that indent many coastlines. The present work focuses on the spatial distribution and scaling of the density difference between the inflowing and outflowing fluid layers. Using a long water-filled channel, fitted with buoyancy sources at its upper surface, experiments were conducted to investigate the influence of the geometry of the strait and the channel as well as the magnitude of the buoyancy flux. Two different scaling laws, one by Phillips (1966), and one by Maxworthy (1994, 1997) were compared with the experimental results. It has been shown that a scaling law for which g′ = kB02/3x/h4/3 best describes the distribution of the observed density difference along the channel, where B0 is the buoyancy flux, x the distance from the closed end of the channel, h its height at the open end (sill) and k a constant that depends on the details of the channel geometry and flow conditions. This result holds for the experimental results and appears to be valid for a number of natural systems as well.

The motion of a light particle in an eccentrically rotating cylinder provides a method for verifying stationary history lift force effects at low but non-zero particle Reynolds numbers. We examine the flow in detail using a Lagrangian equation of motion for constant, non-zero-vorticity flows, and we predict a measurable and stationary contribution of history lift effects that can be verified experimentally with current experimental techniques. Because the history lift contribution is relevant only under certain conditions (which are determined in this work), the present flow configuration also allows one to isolate history drag effects under normal gravitation conditions without resorting to the tethered-particle arrangement used in previous works. We formulate and solve the trajectory problem for light particles that attain stable orbital motion, and we propose an experimental concept that makes possible the study of individual contributions of Lagrangian forces to the motion of small particles in viscous flows.

The control of streamwise vortices in high Reynolds number boundary layer flows often aims at reducing the vortex energy as a means of mitigating the growth of secondary instabilities, which eventually delays the transition from laminar to turbulent flow. In this paper, we aim at utilizing such an energy reduction strategy using optimal control theory to limit the growth of Görtler vortices developing in an incompressible laminar boundary layer flow over a concave wall, and excited by a row of roughness elements with spanwise separation of the same order of magnitude as the boundary layer thickness. Commensurate with control theory formalism, we transform a constrained optimization problem into an unconstrained one by applying the method of Lagrange multipliers. A high Reynolds number asymptotic framework is utilized, wherein the Navier–Stokes equations are reduced to the boundary region equations, in which wall deformations enter the problem through an appropriate Prandtl transformation. In the optimal control strategy, the wall displacement or the wall transpiration velocity serves as the control variable, while the cost functional is defined in terms of the wall shear stress. Our numerical results indicate, among other things, that the optimal control algorithm is very effective in reducing the amplitude of the Görtler vortices, especially for the control based on wall displacement.

The behaviour of a miniature calorimetric sensor, which is under consideration for catheter-based coronary-artery-flow assessment, is investigated in both steady and pulsatile tube flows. The sensor is composed of a heating element operated at constant power and two thermopiles that measure flow-induced temperature differences over the sensor surface. An analytical sensor model is developed, which includes axial heat conduction in the fluid and a simple representation of the solid wall, assuming a quasi-steady sensor response to the pulsatile flow. To reduce the mathematical problem, described by a two-dimensional advection–diffusion equation, a spectral method is applied. A Fourier transform is then used to solve the resulting set of ordinary differential equations and an analytical expression for the fluid temperature is found. To validate the analytical model, experiments with the sensor mounted in a tube have been performed in steady and pulsatile water flows with various amplitudes and Strouhal numbers. Experimental results are generally in good agreement with theory and show a quasi-steady sensor response in the coronary-flow regime. The model can therefore be used to optimize the sensor design for coronary-flow assessment.

This paper discusses a potential-vorticity-conserving approach to modelling nonlinear internal gravity waves in a rotating Boussinesq fluid. The focus of the work is on the pseudo-plane motion (motion in the x, z-plane), for which we present a broad range of numerical results. In this case there are two material coordinates, the density and the y-component of the velocity in the inertial frame of reference, which are related to the x and z displacements of fluid particles relative to a reference configuration. The amount of potential vorticity within a fluid region bounded by isosurfaces of these material coordinates is proportional to the area within this region, and is therefore conserved as well. Two new potentials, defined in terms of the displacements and combining the vorticity and density fields, are introduced as new dependent variables. These potentials entirely govern the dynamics of internal gravity waves for the linearized system when the basic state has uniform potential vorticity. The final system of equations consists of three prognostic equations (for the potential vorticity and the Laplacians of the two potentials) and one diagnostic equation, of Monge–Ampère type, for a third potential. This diagnostic equation arises from the nonlinear definition of potential vorticity. The ellipticity of the Monge–Ampère equation implies both inertial and static stability. In three dimensions, the three potentials form a vector, whose (three-dimensional) Laplacian is equal to the vorticity plus the gradient of the perturbation density.

Numerical simulations are carried out using a novel algorithm which directly evolves the potential vorticity, in a Lagrangian manner (following fluid particles), without diffusion. We present results which emphasize the way in which potential vorticity anomalies modify the characteristics of internal gravity waves, e.g. the propagation of internal wave packets, including reflection, refraction, and amplification. We also show how potential vorticity anomalies may generate internal gravity waves, along with the subsequent ‘geostrophic adjustment’ of the flow to a ‘balanced’ wave-less state. These examples, and the straightforward extension of the theoretical and numerical approach to three dimensions, point to a direct and accurate means to elucidate the role of potential vorticity in internal gravity wave interactions. As such, this approach may help a better understanding of the observed characteristics of internal gravity waves in the oceans.

Flow visualization has shown that the interaction of line vortices is a combination of tearing, elongation and rotation, the extent of each depending upon the flow conditions. A discrete-vortex model is used to study the interaction of two and three growing line vortices of different strengths and to assess the suitability of the method for such simulation.

Many of the features observed in experimental studies of shear layers are reproduced. The controlled study shows the importance and rapidity of the tearing process under certain conditions.

Measurements of concentration fluctuations in a passive scalar plume released within a turbulent boundary layer are utilised to ascertain the scaling of concentration spectra. It is observed that the concentration spectra in a narrow meandering plume has a self-similar behaviour in both transverse ($y$) and vertical ($z$, i.e. wall-normal) directions. Experimental data reveal self-similarity when the magnitude of concentration spectra is scaled by the local concentration variance whereas frequency is suitably scaled utilising the integral length scale of the streamwise velocity or the boundary layer thickness and the source velocity as length and velocity scales, respectively. Furthermore, our data show that at each frequency, the concentration energy is distributed across the $y$ and $z$ directions that is proportional to concentration variance at that location. These results are consistent with our non-dimensional analysis. Based on these observations, if the mean plume statistics are known, a model is proposed with which concentration spectrum at any position within the plume can be calculated using the spectrum at any another location as the input. The model is tested extensively for point-source plumes released at various heights and streamwise distances in a turbulent boundary layer, and is found to predict spectra at different $y$ and $z$ locations in close agreement with measurements.