The near-wall physics and turbulence structure of a planar shear-driven three-dimensional turbulent boundary layer (3DTBL) with varying strengths of crossflow are examined in a specialized facility. Spanwise shear modifies the near-wall turbulence structure. Flow visualization reveals a reduction of mean streak length by as much as 50%, while streak spanwise spacing remains constant. Power spectra of velocity confirm this shift towards higher temporal frequencies, corresponding to decreased streamwise length scales. Particle image velocimetry (PIV) measurements indicate that the spanwise shear increases the number and strength of flow structures that interact in the inner region of the boundary layer. This leads to increased momentum transfer between low- and high-speed fluids, resulting in a thickening of the inner region of the boundary layer. In addition, the organized two-dimensional boundary layer spanwise vorticity layer along the wall surface is disrupted and forms strong vortical structures that are lifted off the wall surface and diffuse into the boundary layer with streamwise distance. The transverse vorticity experiences a similar alteration. Streamwise velocity profiles exhibit an increasing velocity deficit with increased crossflow, consistent with the thickening of the inner region of the boundary layer. Increases in the normal Reynolds stresses are associated with interaction of the secondary flow structures and modifications to vortical structures with increasing three-dimensionality and higher relative wall speed. Significant increases are observed for the most highly sheared cases with translating wall velocities 2.0 and 2.75 times the free-stream velocity. This increase in the normal stresses leads to an increase in the turbulent kinetic energy over the belt surface, which diffuses into the inner region of the boundary layer with streamwise distance. The Reynolds shear stresses also exhibit a significant increase in magnitude over the belt surface owing to enhanced turbulence production over the translating wall section. In general, three-dimensional effects are confined to the area close to the wall and the crossflow increases the interaction of secondary flow structures leading to modifications of the vorticity in the inner region of the boundary layer. These flow distortions disrupt the near-wall streak structure and result in increased levels of turbulent kinetic energy and Reynolds stresses, particularly over the translating wall section.

This paper presents a comprehensive analysis of the transport processes that control the self-pressurization of a cryogenic storage tank in normal gravity. A lumped thermodynamic model of the vapour region is coupled with the Navier–Stokes and energy equations governing heat, mass and momentum transport in the liquid. These equations are discretized using a Galerkin finite-element method with implicit time integration. Three case studies are considered based on three different heating configurations imposed on the tank wall: liquid heating, vapour heating and uniform heating. For each case, the pressure and temperature rise in the vapour and the flow and temperature distributions in the liquid are determined. Results are compared to a lumped thermodynamic model of the entire tank. It is shown that the final rate of pressure rise is about the same in each case and close to that predicted by thermodynamics even though the actual pressures are different because of varying degrees of thermal stratification. Finally, a subcooled liquid jet is used to mix the liquid and limit the pressure rise. Even so, there is still some thermal stratification in the liquid, and as a result the final vapour pressure depends on the particular heat distribution.

Superhydrophobic surfaces generate very high contact angles as a result of their microstructure. The impact of a water drop on such a surface shows unusual features, such as total rebound at low impact speed. We report experimental and numerical investigations of the impact of approximately spherical water drops. The axisymmetric free surface problem, governed by the Navier–Stokes equations, is solved numerically with a front-tracking marker-chain method on a square grid. Experimental observations at moderate velocities and capillary wavelength much less than the initial drop radius show that the drop evolves to a staircase pyramid and eventually to a torus. Our numerical simulations reproduce this effect. The maximal radius obtained in numerical simulations precisely matches the experimental value. However, the large velocity limit has not been reached experimentally or numerically. We discuss several complications that arise at large velocity: swirling motions observed in the cross-section of the toroidal drop and the appearance of a thin film in the centre of the toroidal drop. The numerical results predict the dry-out of this film for sufficiently high Reynolds and Weber numbers. When the drop rebounds, it has a top-heavy shape. In this final stage, the kinetic energy is a small fraction of its initial value.

The electrophoretic motion of a sphere resulting from an applied electric field directed parallel to a nearby plane wall is analysed for the case of a small sphere–wall gap width. The thin-Debye-layer approximation is employed. Using matched asymptotic expansions, the fluid domain is separated into an ‘inner’ (gap-scaled) region, wherein the electric field and velocity gradients are large, and an ‘outer’ (sphere-scaled) region, wherein field variations are moderate. Asymptotic expressions for the force and torque acting on the sphere are obtained using a reciprocal theorem, thereby avoiding the need to explicitly solve the pertinent Stokes equations. These expressions, as well as the sphere's electrophoretic mobility, become unbounded for near-contact separations. The present asymptotic solution complements existing ‘exact’ bipolar-coordinate eigenfunction expansions, which are numerically suitable only for $O(1)$ gap thicknesses.

This paper presents a mathematical derivation using the classical theory of fluid dynamics for the force on an arbitrarily shaped body in a linear shear flow. To make the analysis tractable, the problem is linearized by assuming that the strain rate is weak and neglecting terms of the order of the strain rate squared. The argument generalizes previous established analytical results due to Darwin regarding the drift-volume and Lighthill for the asymptotic form of the rotational velocity field induced by the body. The final expression for the force is determined by generalizing an analytical argument due to Auton for the sphere. The results identify for the first time a rotational lift force component that occurs only when the body shape is truly asymmetric.

An explicit solution is provided for the scattering of an obliquely incident flexural-gravity wave by a narrow straight-line crack separating two semi-infinite thin elastic plates floating on water of finite depth. By first separating the solution into the sum of symmetric and antisymmetric parts it is shown that a simple form for each part can be derived in terms of a rapidly convergent infinite series multiplied by a fundamental constant of the problem. This constant is simply determined by applying an appropriate edge condition. Curves of reflection and transmission coefficients are presented, showing how they vary with plate properties and angle of incidence. It is also shown that in the absence of incident waves and for certain relations between their wavelength and frequency, symmetric edge waves exist which travel along the crack and decay in a direction normal to the crack.

The capability of rapid distortion theory to predict the long-time evolution of shearless turbulence close to an impermeable surface has been seriously questioned in recent years. However, experiments and large-eddy simulations performed at high Reynolds number show that second-order turbulence statistics follow closely the predictions of the theory elaborated by Hunt & Graham (1978). To clarify this issue, a theoretical analysis is carried out in order to determine the relative magnitude of the vortical corrections which were not taken into account in the original theory. By evaluating the various terms of the enstrophy balance in the near-surface region, it is shown that this relative magnitude is a decreasing function of the turbulent Reynolds number, an argument reconciling most existing results. Hence the Hunt & Graham theory appears to be a leading-order approximation capable of describing short- and long-time evolutions of shear-free boundary layers in the limit of large Reynolds number. The expression for the pressure fluctuation corresponding to this approximation is then derived and approximate Reynolds stress budgets are obtained. These budgets are used to predict and discuss the characteristics of the intercomponent energy transfer near a flat surface in both time-decaying and spatially decaying turbulence. In agreement with available results, predictions reveal that tangential velocity components transfer energy towards the normal component in the former case, while they generally receive energy from this component in the latter case.

Direct numerical simulation is used to study the turbulent flow over a smooth wavy wall undergoing transverse motion in the form of a streamwise travelling wave. The Reynolds number based on the mean velocity $U$ of the external flow and wall motion wavelength $\lambda$ is 10 170; the wave steepness is $2\pi a/\lambda=0.25$ where $a$ is the travelling wave amplitude. A key parameter for this problem is the ratio of the wall motion phase speed $c$ to $U$, and results are obtained for $c/U$ in the range of $-1.0$ to $2.0$ at $0.2$ intervals. For negative $c/U$, we find that flow separation is enhanced and a large drag force is produced. For positive $c/U$, the results show that as $c/U$ increases from zero, the separation bubble moves further upstream and away from the wall, and is reduced in strength. Above a threshold value of $c/U\approx 1$, separation is eliminated; and, relative to small- $c/U$ cases, turbulence intensity and turbulent shear stress are reduced significantly. The drag force decreases monotonically as $c/U$ increases while the power required for the transverse motion generally increases for large $c/U$; the net power input is found to reach a minimum at $c/U\approx 1.2$ (for fixed $U$). The results obtained in this study provide physical insight into the study of fish-like swimming mechanisms in terms of drag reduction and optimal propulsive efficiency.

This work studies the momentum and energy transport mechanisms in the corner between a free surface and a solid wall. We perform large-eddy simulations of the incompressible fully developed turbulent flow in a square duct bounded above by a free-slip wall, for Reynolds numbers based on the mean friction velocity and the duct width equal to 360, 600 and 1000. The flow in the corner is strongly affected by the advection due to two counter-rotating secondary-flow regions present immediately below the free surface. Because of the convection of the inner eddy, as the free surface is approached, the friction velocity on the sidewall first decreases, then increases again. A similar behaviour is observed for the surface-parallel Reynolds-stress components, which first decrease and then increase again very close to the surface. The budgets of the Reynolds stresses show a strong reduction of all terms of the dissipation tensor in both the inner and outer near-corner regions. They exhibit a reduction in both production and dissipation towards the free surface. Very close to the solid boundary, within 15–20 viscous lengths of the sidewall, the turbulent kinetic energy production and the surface-parallel fluctuations rebound in the thin layer adjacent to the free surface. The Reynolds-stress anisotropy appears to be the main factor in the generation of the mean secondary flow. The multi-layer structure of the boundary layer near the free surface is also discussed.

We consider the problem of predicting the rate of mass transfer to a fluid flowing parallel to the axes of randomly placed aligned tubes, a model of hollow-fibre contactors. The analysis is carried out for the limiting cases of short contactors, for which the concentration boundary layers remain thin compared with the radius of the tubes, and for the fully developed case corresponding to very long tubes. Numerical simulations for random arrays are carried out for $N$ randomly placed tubes within a unit cell of a periodic array. It is shown that the mass transfer coefficient for the fully developed case is vanishingly small in the limit $N\rightarrow \infty$. This suggests that the mass transfer coefficient for a random array of tubes of radius $a$ enclosed in a shell of radius $S$ will vanish logarithmically as the ratio $S/a$ is increased. This behaviour arises due to the logarithmically divergent nature of concentration disturbances caused by each tube in the plane normal to its axis. A theory is developed for determining conditionally averaged velocity and concentration fields and its predictions are shown to compare very well with the results of rigorous numerical computations. The predictions of the theory are also shown to compare well with the measurements of the mass transfer coefficients in hollow-fibre contactors reported in the literature.

Trapped modes in the linearized water-wave problem are free oscillations of the fluid which have finite energy. They are known to exist at isolated frequencies in the presence of certain special structures. The existence of a trapped mode implies the non-uniqueness, or non-existence, of the solution to physically relevant radiation and diffraction problems for such a structure.

Previous work on the three-dimensional problem has established the existence of vertically axisymmetric structures that support trapped modes with either a single interior free surface, or two concentric interior free surfaces. In the present work the existence of several new types of trapping structures is established. These include non-axisymmetric structures with a single interior free surface and various structures with multiple interior free surfaces. The method used is an indirect one in which flow fields without wave radiation are specified, and corresponding structures are found by constructing suitable stream surfaces. Computations of the added-mass coefficients for these structures provide independent support for the existence of a trapping mode and illustrate their hydrodynamic characteristics at other wavenumbers.

Under certain conditions, the free surface of a conducting liquid subject to an electric field elongates into a cone whose apex emits a thin stationary jet that carries an electric current. The structure of the flow in the cone-to-jet transition region is investigated here, assuming that the size of this region is small compared with any other length of the system where the conical meniscus is formed. The local problem depends then on three non-dimensional parameters, two of which are properties of the liquid while the third measures the flow rate injected through the meniscus. Numerical solutions are computed and the electric current is determined as a function of these parameters. A qualitative asymptotic analysis of the physically important limit of large non-dimensional flow rates gives an electric current increasing as the square root of the flow rate and independent of the dielectric constant of the liquid. When the inertia of the liquid is taken into account, the flow in this asymptotic limit is effectively inviscid in the bulk of the transition region, where the electric current is dominated by conduction in the liquid and the surface is close to an equipotential of the electric field in the gas. The effects of the viscosity of the liquid, the current transported by convection of the surface charge, and the electric shear at the surface come into play in a slender region of the jet. The limit of small non-dimensional flow rates is briefly discussed.

We examine the reaction fronts that develop when an undersaturated ‘injection fluid’ displaces a saturated ‘formation fluid’ in a chemically reactive porous medium. We allow the injection and formation fluids to differ both in temperature and in chemical composition. The undersaturation of the incoming fluid drives a dissolution reaction and leads to the formation of a ‘depletion’ front. Under certain circumstances, which we describe, the temperature difference drives a separate thermal reaction front. We develop long-time asymptotic solutions of the governing equations which illustrate the interaction between thermal reaction fronts and depletion fronts. Two distinct regimes arise. If the compositional difference between the injection and formation fluids exceeds a critical value, the depletion front travels faster than the thermal front, leaving the porous medium depleted of reactant from the source to a point downstream of the thermal front and no thermal reaction front develops. Conversely, if the compositional difference is smaller than the critical value, a thermal reaction front advances ahead of the depletion front and so there is a double reaction front structure. We illustrate the evolution of the thermal and compositional fields towards these asymptotic solutions with numerical simulations. We discuss the implications of this work for secondary mineralization in subsurface reservoirs.