Branching tube flows are examined, for one mother to two, three or more daughter tubes. The case of many daughters (abrupt multi-branching) models blood flow through an arteriovenous malformation in the brain, while that of very few daughters (gradual branching) applies elsewhere in physiology and surgical grafting, as well as other applications including industrial ones. Theory and computation are presented for two- and three-dimensional motions, under the viscous and inviscid effects of small changes in mass flux between the daughter tubes, area expansion and turning of the flow. Specific configurations for which flow solutions are obtained are (a) with two large daughters, (b) with one small daughter/side branch, and (c) with multiple small daughters.

The numerous physical mechanisms acting concern overall upstream influence and through-flow, and flow separation and criteria for its avoidance, as well as criteria for the amount of turning and area expansion possible without energy loss and other factors associated with separation, and the role of the branching geometry versus that of the mass-flux distribution in the daughters. In particular, configuration (a) allows substantial separation-free turning and expansion only with certain shaping of the outer wall and an area expansion ratio typically less than 1.2, whereas more daughters involve a balance between geometry and mass flux. In (b), an abrupt pressure jump is induced at the mouth of the small daughter, near which mass-flux effects tend to dominate over geometrical shaping effects. In (c), as the number of daughters increases, the amount of separation-free turning and expansion is found to increase substantially, and the distributed mass-flux influence readily overrides the geometrical influence throughout the branching; there is also an integrated upstream effect of the multi-branching on the incident mother flow even though each daughter flow acts as if independent. Tentative designs based on wall shaping, flux distributions and divider placement are considered for flow improvement/surgery.

The primary and secondary instabilities of the sphere wake are investigated from the viewpoint of nonlinear dynamical systems theory. For the primary bifurcation, a theory of axisymmetry breaking by a regular bifurcation is given. The azimuthal spectral modes are shown to coincide with nonlinear modes of the instability, which provides a good reason for using the azimuthal expansion as an optimal spectral method. Thorough numerical testing of the implemented spectral–spectral-element discretization allows corroboration of existing data concerning the primary and secondary thresholds and gives their error estimates. The ideal axisymmetry of the numerical method makes it possible to confirm the theoretical conclusion concerning the arbitrariness of selection of the symmetry plane that arises. Investigation of computed azimuthal modes yields a simple explanation of the origin of the so-called bifid wake and shows at each Reynolds number the coexistence of a simple wake and a bifid wake zone of the steady non-axisymmetric regime. At the onset of the secondary instability, basic linear and nonlinear characteristics including the normalized Landau constant are given. The periodic regime is described as a limit cycle and the power of the time Fourier expansion is illustrated by reproducing experimental r.m.s. fluctuation charts of the streamwise velocity with only the fundamental and second harmonic modes. Each time–azimuthal mode is shown to behave like a propagating wave having a specific spatial signature. Their asymptotic, far-wake, phase velocities are the same but the waves keep a fingerprint of their passing through the near-wake region. The non-dimensionalized asymptotic phase velocity is close to that of an infinite cylinder wake. A reduced-accuracy discretization is shown to allow qualitatively satisfactory unsteady simulations at extremely low cost.

The problem of lateral heat/buoyancy transport in localized turbulent convection dominated by rotation in continuously stratified fluids of finite depth is considered. We investigate the specific mechanism of the vortex-dominated lateral spreading of anomalous buoyancy created in localized convective regions owing to outward propagation of intense heton-like vortices (pairs of vortices of equal potential vorticity (PV) strength with opposite signs located at different depths), each carrying a portion of buoyancy anomaly. Assuming that the quasi-geostrophic form of the PV evolution equation can be used to analyse the spreading phenomenon at fast rotation, we develop an analytical theory for the dynamics of a population of three-dimensional hetons. We analyse in detail the structure and dynamics of a single three-dimensional heton, and the mutual interaction between two hetons and show that the vortices can be in confinement, splitting or reconnection regimes of motion depending on the initial distance between them and the ratio of the mixing-layer depth to the depth of fluid (local to bulk Rossby radii). Numerical experiments are made for ring-like populations of randomly distributed three-dimensional hetons. We found two basic types of evolution of the populations which are homogenizing confinement (all vortices are predominantly inside the localized region having highly correlated wavelike dynamics) and vortex-dominated spreading (vortices propagate out of the region of generation as individual hetons or heton clusters). For the vortex-dominated spreading, the mean radius of heton populations and its variance grow linearly with time. The law of spreading is quantified in terms of both internal (specific for vortex dynamics) and external (specific for convection) parameters. The spreading rate is proportional to the mean speed of propagation of individual hetons or heton clusters and therefore depends essentially on the strength of hetons and the ratio of local to bulk Rossby radii. A theoretical explanation for the spreading law is given in terms of the asymptotic dynamics of a single heton and within the frames of the kinetic equation derived for the distribution function of hetons in collisionless approximation. This spreading law gives an upper ‘advective’ bound for the superdiffusion of heat/buoyancy. A linear law of spreading implies that diffusion parameterizations of lateral buoyancy flux in non-eddy-resolving models are questionable, at least when the spreading is dominated by heton dynamics. We suggest a scaling for the ‘advective’ parameterization of the buoyancy flux, and quantify the exchange coefficient in terms of the mean propagation speed of hetons. Finally, we discuss the perspectives of the heton theories in other problems of geophysical fluid dynamics.

The dynamics of vertically compensated two-layer vortices (hetons) with finite cores are examined within the context of the quasi-geostrophic approximation on the f-plane. The two-layer version of the contour dynamics method is used, and complemented by the so-called contour surgery technique. Special attention is paid to two-heton interactions when the initial locations of the vortex patches are symmetrical. A classification of the different regimes observed is made according to external parameters, namely geometrical parameters and stratification. In this parameter space, novel quasi-stationary states resulting from collisions between two hetons may be observed: (i) formation of a configuration consisting of two-layer vortices moving in opposite directions and, as a special case, a configuration analogous to the ‘klapstoss’ billiard shot, (ii) absorption of one heton by the other and subsequent movement of a new dipole, (iii) formation of two-layer dipoles, larger than the original hetons, associated with rotating peripheral satellite eddies in their wakes. Some of these results may have implications for the analysis of mesoscale vortices in the ocean.

We consider, using linear water-wave theory, three-dimensional problems concerning the interaction of waves with structures in a fluid which contains a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. In a single-layer fluid there are a number of reciprocity relations that exist connecting the various hydrodynamic quantities that arise, and these relations are systematically extended to the two-fluid case. The particular problems of wave radiation and scattering by a submerged sphere in either the upper or lower layer are then solved using multipole expansions.

A three-dimensional shear-driven turbulent boundary layer over a flat plate generated by moving a section of the wall in the transverse direction is studied using large-eddy simulations. The configuration is analogous to shear-driven boundary layer experiments on spinning cylinders, except for the absence of curvature effects. The data presented include the time-averaged mean flow, the Reynolds stresses and their budgets, and instantaneous flow visualizations. The near-wall behaviour of the flow, which was not accessible to previous experimental studies, is investigated in detail. The transverse mean velocity profile develops like a Stokes layer, only weakly coupled to the streamwise flow, and is self-similar when scaled with the transverse wall velocity, Ws. The axial skin friction and the turbulent kinetic energy, K, are significantly reduced after the imposition of the transverse shear, due to the disruption of the streaky structures and of the outer-layer vortical structures. The turbulent kinetic energy budget reveals that the decrease in production is responsible for the reduction of K. The flow then adjusts to the perturbation, reaching a quasi-equilibrium three-dimensional collateral state. Following the cessation of the transverse motion, similar phenomena take place again. The flow eventually relaxes back to a two-dimensional equilibrium boundary layer.

Developed interfacial dynamics of thin film flows with moderate Reynolds numbers exhibits a remarkable feature: the evolution is dominated by solitary-like pulses with a natural large wavelength between them. This phenomenon is robust and resembles the inverse energy cascade in two-dimensional turbulence. A new simple evolution equation is proposed to describe the film flow dynamics which captures such an inverse cascade. The equation combines the simplest kinematic nonlinearity with the exact linear term. The spectral kernel of the linear term is found from the numerical solution of the associated linear stability problem.

A boundary layer maintained as close as possible to separation over an extended distance was produced, in accordance with the concept of Stratford. The resulting layer was two-dimensional in the mean, had nearly a constant shape factor of 2.5 and approximately linear streamwise growth of its integral length scales. The flow exhibited a definite non-equilibrium character, indicated by the different scales required for collapse of the mean velocity and turbulence intensity profiles. It was also very sensitive to the thickness of the upstream boundary layer. External excitation was imposed for diagnostic purposes and as a tool for delaying separation. The oscillatory momentum level of cμ ≈ 0.1% was tested for its ability to increase the skin friction cf at the prescribed geometry. Various frequencies, corresponding to the Strouhal number 0.008 < fθ0/Uref < 0.064, were used for the free stream reference velocity of Uref = 15 m s−1 and for two different inflow conditions. Notable increase (close to 60%) in cf was observed at higher frequencies that did not undergo maximum amplification. The increase in cf was accompanied by a reduction in the boundary layer thickness and in the shape factor H. The latter decreased in one case from 2.5 to 2.1. The overall turbulence level in the boundary layer decreased due to the addition of plane external perturbations.

This paper deals with linear waves on the beta-plane over topography. The main assumption is that the topography consists of isolated radially symmetric irregularities (random or periodic), such that their characteristic radii are much smaller than the distances between them. This approximation allows one to obtain the dispersion relation for the frequency of wave modes; and in order to examine the properties of those, we consider a particular case where bottom irregularities are cylinders of various heights and radii. It is demonstrated that if all irregularities are of the same height, h, there exist two topographic and one Rossby modes. The frequency of one of the topographic modes is ‘locked’ inside the band (−fh/2H0, fh/2H0), where f is the Coriolis parameter and H0 is the mean depth of the ocean. The frequencies of the other topographic mode and the barotropic Rossby mode are ‘locked’ above and below the band, respectively. It is also demonstrated that if the heights of cylinders are distributed within a certain range, (−h0, h0), no harmonic modes exist with frequencies inside the interval (−fh0/2H0, fh0/2H0). The topographic and Rossby modes are ‘pushed’ out of the ‘prohibited’ band.

A new system of equations for calculating time-dependent motions of deep-water gravity waves (Balk 1996) is here developed analytically and set in a form suitable for practical applications. The method is fully nonlinear, and has the advantage of essential simplicity. Both the potential and the kinetic energy involve polynomial expressions of low degree in the Fourier coefficients Yn(t). This leads to equations of motion of correspondingly low degree. Moreover the constants in the equations are very simple. In this paper the equations of motion are specialized to standing waves, where the coefficients Yn are all real. Truncation of the series at low values of [mid ]n[mid ], say n < N, leads to ‘partial waves’ with solutions apparently periodic in the time t. For physical applications N must however be large. The method will be applied to the breaking of standing waves by the forming of sharp corners at the crests, and the generation of vertical jets rising from the wave troughs.

Exact and numerical similarity solutions for compressible perturbations to an incompressible, two-dimensional, axisymmetric vortex reference flow are presented. The reference flow consists of a set of two-dimensional, self-similar, incompressible vortices. Similarity variables, which give explicit expressions for the decay rates of the velocities and thermodynamic variables in the vortex flows, are used to reduce the governing partial differential equations to a set of ordinary differential equations. The ODEs are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are employed to study the dependences of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Additionally, several integral relations, which allow one to trace the energy transfer in a slightly compressible vortex, are derived.

The effect of inertia on the dynamics of a solid particle (a circular cylinder, an elliptical cylinder, and an ellipsoid) suspended in shear flow is studied by solving the discrete Boltzmann equation. At small Reynolds number, when inertia is negligible, the behaviour of the particle is in good agreement with the creeping flow solution showing periodic orbits. For an elliptical cylinder or an ellipsoid, the results show that by increasing the Reynolds number, the period of rotation increases, and eventually becomes infinitely large at a critical Reynolds number, Rec. At Reynolds numbers above Rec, the particle becomes stationary in a steady-state flow. It is found that the transition from a time-periodic to a steady state is through a saddle-node bifurcation, and, consequently, the period of oscillation near this transition is proportional to [mid ]p−pc[mid ]−1/2, where p is any parameter in the flow, such as the Reynolds number or the density ratio, which leads to this transition at p = pc. This universal scaling law is presented along with the physics of the transition and the effect of the inertia and the solid-to-fluid density ratio on the dynamics. It is conjectured that this transition and the scaling law are independent of the particle shape (excluding body of revolution) or the shear profile.

The separation of the laminar boundary layer from a convex corner on a rigid body contour in transonic flow is studied based on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number. It is shown that the flow in a small vicinity of the separation point is governed, as usual, by strong interaction between the boundary layer and the inviscid part of the flow. Outside the interaction region the Kármán–Guderley equation describing transonic inviscid flow admits a self-similar solution with the pressure on the body surface being proportional to the cubic root of the distance from the separation point. Analysis of the boundary layer driven by this pressure shows that as the interaction region is approached the boundary layer splits into two parts: the near-wall viscous sublayer and the main body of the boundary layer where the flow is locally inviscid. It is interesting that contrary to what happens in subsonic and supersonic flows, the displacement effect of the boundary layer is primarily due to the inviscid part. The contribution of the viscous sublayer proves to be negligible to the leading order. Consequently, the flow in the interaction region is governed by the inviscid–inviscid interaction. To describe this flow one needs to solve the Kármán–Guderley equation for the potential flow region outside the boundary layer; the solution in the main part of the boundary layer was found in an analytical form, thanks to which the interaction between the boundary layer and external flow can be expressed via the corresponding boundary condition for the Kármán–Guderley equation. Formulation of the interaction problem involves one similarity parameter which in essence is the Kármán–Guderley parameter suitably modified for the flow at hand. The solution of the interaction problem has been constructed numerically.

In a recent study by Yeung & Parkinson (1997), a wake width was proposed which allowed the bluff-body potential-flow model by Parkinson & Jandali (1970) to be extended to include the flow around an oblique flat plate. By incorporating this wake width in the momentum equation originally derived by Eppler (1954) for separated flow, the drag of the plate is related to its inclination and base pressure through a simple analytical condition. It allows the base pressure, which is usually treated as an empirical input, to be determined theoretically and thus the model becomes self-contained. Predictions of the base pressure, drag and width of wake are found to be in reasonable agreement with the experimental data. When applied to the symmetrical flow around a wedge of arbitrary vertex angle, similar agreement with experimental measurements is obtained as well. It is also demonstrated that this condition is compatible with the free-streamline models by Wu (1962) and Wu & Wang (1964) such that the corresponding predictions are in good agreement with experiment.