We present experimental results for the wake structure of spheres moving in homogeneous and stratified fluid. In homogeneous fluid, the results of Kim & Durbin (1988) are confirmed and it is shown that the two characteristic frequencies of the wake correspond to two instability modes, the Kelvin–Helmholtz instability and a spiral instability. For the stratified wake four general regimes have been identified, depending principally on the Froude number F. For F > 4.5 the near wake is similar to the homogeneous case, and for F < 0.8 it corresponds to a triple-layer flow with two lee waves, of amplitude linear in F, surrounding a layer dominated by quasi-two-dimensional motion. Froude numbers close to one (F∈]0.8, 1.5[) give rise to a saturated lee wave of amplitude equal to half the sphere radius, which suppresses the separation region or splits it into two. Between F = 1.5 and 4.5 a more complex regime exists where the wake recovers progressively its behaviour in homogeneous fluid: the axisymmetry of the recirculating zone, the Kelvin–Helmholtz instability and, finally, the spiral instability.

The internal gravity wave field generated by a sphere towed in a stratified fluid was studied in the Froude number range 1.5 ≤ F ≤ 12.7, where F is defined with the radius of the sphere. The Reynolds number was sufficiently large for the wake to be turbulent (Re∈[380, 30000]). A fluorescent dye technique was used to differentiate waves generated by the sphere, called lee waves, from the internal waves, called random waves, emitted by the turbulent wake. We demonstrate that the lee waves are well predicted by linear theory and that the random waves due to the turbulence are related to the coherent structures of the wake. The Strouhal number of these structures depends on F when F [lsim ] 4.5. Locally, these waves behave like transient internal waves emitted by impulsively moving bodies.

The hydrodynamic problem of a circular cylinder submerged below a free surface and undergoing large-amplitude oscillation is investigated based on the velocity potential theory. The body-surface boundary condition is satisfied on its instantaneous position while the free-surface condition is linearized. The solution is obtained by writing the potential in terms of the multipole expansion. Various interesting results associated with the circular cylinder are obtained.

Explicit algebraic stress models that are valid for three-dimensional turbulent flows in non-inertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope (1975) who based his analysis on the Launder, Reece & Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models – as well as anisotropic eddy viscosity models – is theoretically established. A need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.

The motion of spherical bubbles in a box containing an incompressible and irrotational liquid with periodic boundary conditions is studied. Equations of motion are deduced using a variational principle. When the bubbles have approximately the same velocity their configuration is Lyapunov stable, provided they are arranged in such a way so as to minimize the effective conductivity of a composite material where the bubbles are treated as insulators. The minimizing configurations become asymptotically stable with the addition of gravity and liquid viscosity. This suggests that a randomly arranged configuration of bubbles, all with approximately the same velocity, cannot be stable. Explicit equations of motion for two bubbles are deduced using Rayleigh's method for solving Laplace's equation in a periodic domain. The results are extended for more than two bubbles by considering pairwise interactions. Numerical simulations with gravity and liquid viscosity ignored show the bubbles form clusters if initially they are randomly arranged with identical velocities. The clustering is inhibited, however, if the initial velocities are sufficiently different. The variance of the bubbles’ velocities is observed to act like a temperature; a lesser amount of clustering occurs with increasing temperature. Finally, in considering the effects of gravity and liquid viscosity, the long time behaviour is found to result in the formation of bubble clusters aggregated in a plane perpendicular to the direction of gravity.

A horizontal channel of infinite length and depth and of constant width contains inviscid, incompressible, two-layer fluid under gravity. The upper layer has constant finite depth and is occupied by a fluid of constant density ρ*. The lower layer has infinite depth and is occupied by a fluid of constant density ρ > ρ*. The parameter ε = (ρ/ρ*)–1 is assumed to be small. The lower fluid is bounded internally by an immersed horizontal cylinder which extends right across the channel and has its generators normal to the sidewalls. The free, time-harmonic oscillations of fluid, which have finite kinetic and potential energy (such oscillations are called trapped modes), are investigated. Trapped modes in homogeneous fluid above submerged cylinders and other obstacles are well known. In the present paper it is shown that there are two sets of frequencies of trapped modes for the two-layer fluid. The frequencies of the first finite set are close to the frequencies of trapped modes in the homogeneous fluid (when ρ* = ρ). They correspond to the trapped modes of waves on the free surface of the upper fluid. The frequencies of the second finite set are proportional to ε, and hence, are small. These latter frequencies correspond to the trapped modes of internal waves on the interface between two fluids. To obtain these results the perturbation method for a quadratic operator family was applied. The quadratic operator family with bounded, symmetric, linear, integral operators in the space L2(−∞, +∞) arises as a result of two reductions of the original problem. The first reduction allows to consider the potential in the lower fluid only. The second reduction is the same as used by Ursell (1987).

Numerical simulations of rapid granular flows in large periodic domains have demonstrated the existence of an ‘inelastic microstructure’ characterized by agglomerations of particles into clusters. In the present work the phenomenon of particle clustering is considered to be a manifestation of hydrodynamic instability of the equations governing the dynamics of rapid granular flows. A linear stability analysis of the simple shear flow of smooth, slightly, inelastic disks and spheres is carried out utilizing the governing equations formulated by the kinetic theory. The results indicate that disturbances of large wavelengths initially grow at an exponential rate for any value of the restitution coefficient. The results also demonstrate that this phenomenon will not be exposed in numerical simulations within small periodic domains, because the range of possible disturbance wavelengths is limited by the size of the periodic domain.

The partially cavitating two-dimensional hydrofoil problem is treated using nonlinear theory by employing a low-order potential-based boundary-element method. The cavity shape is determined in the framework of two independent boundary-value problems; in the first, the cavity length is specified and the cavitation number is unknown, and in the second the cavitation number is known and the cavity length is to be determined. In each case, the position of the cavity surface is determined in an iterative manner until both a prescribed pressure condition and a zero normal velocity condition are satisfied on the cavity. An initial approximation to the nonlinear cavity shape, which is determined by satisfying the boundary conditions on the hydrofoil surface rather than on the exact cavity surface, is found to differ only slightly from the converged nonlinear result.

The boundary element method is then extended to treat the partially cavitating three-dimensional hydrofoil problem. The three-dimensional kinematic and dynamic boundary conditions are applied on the hydrofoil surface underneath the cavity. The cavity planform at a given cavitation number is determined via an iterative process until the thickness at the end of the cavity at all spanwise locations becomes equal to a prescribed value (in our case, zero). Cavity shapes predicted by the present method for some three-dimensional hydrofoil geometries are shown to satisfy the dynamic boundary condition to within acceptable accuracy. The method is also shown to predict the expected effect of foil thickness on the cavity size. Finally, cavity planforms predicted from the present method are shown to be in good agreement to those measured in a cavitating three-dimensional hydrofoil experiment, performed in MIT's cavitation tunnel.

This paper presents a general theory and physical interpretation of the interaction between a solid body and a Newtonian fluid flow in terms of the vorticity ω and the compression/expansion variable Π instead of primitive variables, i.e. velocity and pressure. Previous results are included as special simplified cases of the theory. The first part of this paper shows that the action of a solid wall on a fluid can be exclusively attributed to the creation of a vorticity-compressing ω–Π field directly from the wall, a process represented by respective boundary fluxes. The general formulae for these fluxes, applicable to any Newtonian flow over an arbitrarily curved surface, are derived from the force balance on the wall. This part of the study reconfirms and extends Lighthil's (1963) assertion on vorticity-creation physics, clarifies some currently controversial issues, and provides a sound basis for the formulation of initial boundary conditions for the ω-Π variables.

The second part of this paper shows that the reaction of a Newtonian flow to a solid body can also be exclusively attributed to that of the ω-Π field created. In particular, the integrated force and moment formulae can be expressed solely in terms of the boundary vorticity flux. This implies an inherent unity of the action and reaction between a solid body and a ω-Π field.

In both action and reaction phases the ω-Π coupling on the wall plays an essential role. Thus, once a solid wall is introduced into a flow, any theory that treats ω and Π separately will be physically incomplete.

Significant simplifications and minor corrections are made to a previous second-order solution of Graham & Kullar for the lift on a flat-plate airfoil encountering a sinusoidal gust in compressible flow. The related cases of a skewed gust in incompressible flow, a parallel gust in compressible flow and the generalized case of a skewed gust in compressible flow are considered. In addition to the simplifications, the solutions are combined into a composite solution that is more accurate than the solutions from which it is composed, making it useful for numerical calculations.

The finite-amplitude instability of mixed convection of air in a vertical concentric annulus with each cylinder maintained at a different temperature is studied by use of weakly nonlinear instability theory and by direct numerical simulation. A strictly shear instability and two thermally induced instabilities exist in the parameter space of Reynolds and Grashof numbers. The first thermal instability occurs at low Reynolds numbers as the rate of heating increases, and is called a thermal-shear instability because it is a shear-driven instability induced by thermal effects. The second thermal instability occurs at larger Reynolds number as heating increases, and is also a thermally induced shear instability called the interactive instability. The weakly nonlinear results demonstrate that the thermal-shear instability is supercritical at all wavenumbers. With the shear and interactive instabilities, however, both subcritical and supercritical branches appear on the neutral curves. The validity of the weakly nonlinear calculations are verified by comparison with a direct simulation. The results for subcritical instabilities show that the weakly nonlinear calculations are accurate when the magnitude of the amplification rate is small, but the accuracy deteriorates for large amplification rates. However, the trends predicted by the weakly nonlinear theory agree with those predicted by the direct simulations for a large portion of the parameter space. Analyses of the energy sources for the disturbance show that subcritical instability of the shear and interactive modes occurs at larger wavenumbers because of increased gradient production of disturbance kinetic energy. This is because, at shorter wavelengths, the growth of the wave causes the shape of the fundamental disturbance to change from that predicted by linear instability theory to a shape more favourable for shear-energy production. The results also show that many possibly unstable modes may be present simultaneously. Consequently, all of these modes, as well as all of the possible wave interactions among the modes, must be considered to obtain a complete picture of mixed-convection instability.

We consider a floating or submerged body in deep water translating parallel to the undisturbed free surface with a steady velocity U while undergoing small oscillations at frequency ω. It is known that for a single source, the solution becomes singular at the resonant frequency given by τ ≡ Uω/g=¼, where g is the gravitational acceleration. In this paper, we show that for a general body, a finite solution exists as τ → ¼ if and only if a certain geometric condition (which depends only on the frequency ω but not on U) is satisfied. For a submerged body, a necessary and sufficient condition is that the body must have non-zero volume. For a surface-piercing body, a sufficient condition is derived which has a geometric interpretation similar to that of John (1950). As an illustration, we provide an analytic (closed-form) solution for the case of a submerged circular cylinder oscillating near τ = ¼. Finally, we identify the underlying difficulties of existing approximate theories and numerical computations near τ = ¼, and offer a simple remedy for the latter.

The propagation of acoustic energy from a sound source to the far field is a fundamental problem of acoustics. In this paper the use of computational fluid dynamics (CFD) to directly calculate the acoustic field is investigated. The two-dimensional, compressible, inviscid flow about an accelerating circular cylinder is used as a model problem. The time evolution of the energy transfer from the cylinder surface to the fluid, as the cylinder is moved from rest to some non-negligible velocity, is shown. Energy is the quantity of interest in the calculations since various components of energy have physical meaning. By examining the temporal and spatial characteristics of the numerical solution, a distinction can be made between the propagating acoustic energy, the convecting energy associated with the entropy change in the fluid, and the energy following the body. In the calculations, entropy generation is due to a combination of physical mechanisms and numerical error. In the case of propagating acoustic waves, entropy generation seems to be a measure of numerical damping associated with the discrete flow solver.

A substantial amount of drilling fluid can invade a permeable bed during the drilling of an oil well. The presence of this fluid, often referred to as filtrate, can greatly influence the performance of instruments lowered into the wellbore for the purpose of locating these permeable beds. The invaded filtrate can also substantially alter the physical properties of the porous rock. For these reasons, it is of great interest to known where the filtrate goes upon entering the bed. The objective of this study is to quantify the influence of the difference in density between the filtrate and the naturally occurring formation fluid on the shape of the filtrate front as the filtrate invades the formation. This type of phenomenon is often referred to as buoyancy or gravity segregation. In this study, Part 1, we determine the behaviour of the filtrate as it accumulates (and spreads out) at a horizontal impermeable barrier within the formation. This is a combined theoretical and experimental study in which an X-ray CT scanner is extensively used to determine the appropriateness and limitations of the simplifying assumptions used in the theory. In Part 2, the flow of the invading filtrate within the entire bed will be presented. The problem addressed in Part 1 may be viewed from the broader, more fundamental, perspective, as a well-defined model fluid mechanics problem for flow in porous media. One fundamental issue infrequently addressed concerns the consequence on the dynamics of the fluids of heterogeneities, always present to some degree, in consolidated porous solids. The X-ray CT scanner enables the assessment of the appropriateness of modelling such porous solids as spatially homogeneous, a very popular assumption. This study also addresses the limitation of the small-slope approximation when a fluid–fluid interface occurs in a porous solid, an approximation which has enjoyed great success in free-surface fluid mechanics problems when no porous media is present.

The nonlinear development of the Batchelor–Nitsche instability (of a periodically stratified fluid) is considered, utilizing the disparity between vertical and horizontal scales of motion. The resulting evolution equation is used to show that the preferred pattern of convection takes the form of rolls, and that the motion evolves to larger and larger horizontal scales as time increases.

We consider the instability of the steady, axisymmetric base flow past a sphere, and a circular disk (oriented broadside-on to the incoming flow). Finite-element methods are used to compute the steady axisymmetric base flows, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for the sphere and the circular disk, the first instability of the base flow is through a regular bifurcation, and the critical Reynolds number (based on the body radius) is 105 for the sphere, and 58.25 for the circular disk. In both cases, the unstable mode is non-axisymmetric with azimuthal wavenumber m = 1. These computational results are consistent with previous experimental observations (Magarvey & Bishop 1961 a, b; Nakamura 1976; Willmarth, Hawk & Harvey 1964).

The results of an investigation on the effect of a weak heterogeneity of a porous medium on natural convection are presented. A medium heterogeneity is represented by spatial variations of the permeability and of the effective thermal conductivity. As a general rule the existence of horizontal thermal gradients in heterogeneous porous media provides a sufficient condition for the occurrence of natural convection. The implications of this condition are investigated for horizontal layers or rectangular domains subject to isothermal top and bottom boundary conditions. Results lead to a restriction on the classes of thermal conductivity functions which allow a motionless solution. Analytical solutions for rectangular weak heterogeneous porous domains heated from below, consistent with a basic motionless solution, are obtained by applying the weak nonlinear theory. The amplitude of the convection is obtained from an ordinary non-homogeneous differential equation, with a forcing term representative of the medium heterogeneity with respect to the effective thermal conductivity. A smooth transition through the critical Rayleigh number is obtained, thus removing a bifurcation which usually appears in homogeneous domains with perfect boundaries, at the critical value of the Rayleigh number. Within a certain range of slightly supercritical Rayleigh numbers, a symmetric thermal conductivity function is shown to reinforce a symmetrical flow while antisymmetric functions favour an antisymmetric flow. Except for the higher-order solutions, the weak heterogeneity with respect to permeability plays a relatively passive role and does not affect the solutions at the leading order. In contrast, the weak heterogeneity with respect to the effective thermal conductivity does have a significant effect on the resulting flow pattern.

Flow visualization studies of plane laminar bubble plumes have been conducted to yield quantitative data on transition height, wavelength and wave velocity of the most unstable disturbance leading to transition. These are believed to be the first results of this kind. Most earlier studies are restricted to turbulent bubble plumes. In the present study, the bubble plumes were generated by electrolysis of water and hence very fine control over bubble size distribution and gas flow rate was possible to enable studies with laminar bubble plumes. Present observations show that (a) the dominant mode of instability in plane bubble plumes is the sinuous mode, (b) transition height and wavelength are related linearly with the proportionality constant being about 4, (c) wave velocity is about 40% of the mean plume velocity, and (d) normalized transition height data correlate very well with a source Grashof number. Some agreement and some differences in transition characteristics of bubble plumes have been observed compared to those for similar single-phase flows.

The stationary instabilities of flow patterns associated with Rayleigh–Bénard convection in a 3 × 1 × 9 rectangular container are extensively investigated by numerical simulation. Two types of spatial instabilities of the base convection rolls are predicted in the transition from steady two-dimensional flow to the unsteady oscillatory regime; these instabilities depend on the Prandtl number. For Pr = 0.71 the soft-roll instability is found at moderate Rayleigh number Ra. The results obtained confirm the importance of this flow pattern as a continuous mechanism for steady transition from one wavenumber to another. For Pr = 15, cross-roll instability is obtained, which at larger Ra leads to bimodal convection. For this value of Pr the soft-roll flow pattern is found at intermediate Ra. At higher Ra a new flow structure in which cross-rolls are superimposed on the soft roll is obtained. The effects of the various flow structures on the heat transfer are given. A quantitative comparison with previous experimental and theoretical findings is also presented and discussed.

The way in which energy propagates away from a two-dimensional oscillatory disturbance in a thermocline is considered theoretically and experimentally. It is shown how the St. Andrew's-cross-wave is modified by reflections and how the cross-wave can develop into thermocline waves. A linear shear flow is then superimposed on the thermocline. Ray theory is used to evaluate the wave shapes and these are compared to finite-difference solutions of the full Navier–Stokes equations.