The dynamics of quasi-geostrophic flow with uniform potential vorticity reduces to the evolution of buoyancy, or potential temperature, on horizontal boundaries. There is a formal resemblance to two-dimensional flow, with surface temperature playing the role of vorticity, but a different relationship between the flow and the advected scalar creates several distinctive features. A series of examples are described which highlight some of these features: the evolution of an elliptical vortex; the start-up vortex shed by flow over a mountain; the instability of temperature filaments; the ‘edge wave’ critical layer; and mixing in an overturning edge wave. Characteristics of the direct cascade of the tracer variance to small scales in homogeneous turbulence, as well as the inverse energy cascade, are also described. In addition to its geophysical relevance, the ubiquitous generation of secondary instabilities and the possibility of finite-time collapse make this system a potentially important, numerically tractable, testbed for turbulence theories.

The planar flow arising in an initially quiescent viscous fluid under the action of a localized dipolar-type forcing has been studied analytically and experimentally. The force dipole, with non-dimensional forcing amplitude Re, brings net zero momentum into the fluid and gives rise to the formation of a quadrupolar vortex: a system of two dipolar vortices moving apart. Experimentally, the action of a force dipole was modelled by a vertical cylinder oscillating horizontally in the shallow upper layer of a two-layer fluid. Two cases were studied: single quadrupoles and an array of quadrupoles. It is found that single quadrupoles develop in a self-similar manner: the length L and the translation velocity Ū of the quadrupolar vortex change with time as L ∼ t1/2 and Ū ∼ t-1/2. These quantities are characterized by non-dimensional functions α(Re) and β(Re), respectively, which have been determined theoretically for small Re-values and experimentally for Re-values in the range 160–2200.

To produce an array of quadrupoles an array of oscillating vertical rods was used. Two stages in the flow evolution were studied experimentally: the initial stage, when the interactions between the quadrupoles are weak, and the intermediate stage when the interactions play an essential role and the flow is (two-dimensionally) turbulent. It is found that at both stages the width H of the region with intense vortical motions increases with time as H ∼ t1/2. A theoretical explanation of the experimental results is given.

An analysis is given for the electro-kinetic transport properties in a system consisting of a line of identical spheres placed equidistantly with their centres on the axis of a cylindrical tube containing a viscous fluid. Both the spheres and the wall of the tube are charged and a two-species symmetrical electrolyte with valence Z is present in the system. As a result of the charges on the surface of the spheres and on the surface of the tube electrical double layers will develop. When an electrical field is applied to the system an electrokinetic motion is induced. We will use the thin double layer theory (Dukhin & Derjaguin 1974; O'Brien 1983), valid for sufficiently high electrolyte concentration and where the polarization of the electrical double layer is included. Using a multipole expansion an infinite set of linear equations for the multipoles will be derived from which the electro-kinetic transport coefficients may be determined. These coefficients depend on the system parameters, such as the radius of the tube R, the radius of the sphere a, the separation between the spheres d, the Debije radius κ-1, the zeta-potentials of the spheres ζp and of the wall of the tube ζw and the valency Z of the electrolyte. From these coefficients a relation is found between the pressure drop Δp per unit length and the drag force D on the spheres on one side and with the velocity U of the spheres, the total discharge Q and the applied electrical field E0 on the other side. For some values for the system parameters we have numerically solved the infinite set of linear equations by truncation and calculated the transport coefficients. We have also calculated the streamlines for some situations. The plots of these streamlines show that depending on the conditions on the system vortices may appear.

Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann–Enskog equation. The Chapman–Enskog method for gas kinetic theory is modified to derive the Euler-like hydrodynamic equations for a system of moving spheres, possessing constant roughness and inelasticity. The solution is obtained by employing a general isotropic expression for the singlet distribution function, dependent upon the spatial gradients of averaged hydrodynamic properties. This solution form is shown to be appropriate for description of rapid shearless motions of granular materials, in particular vibrofluidized regimes induced by external vibrations.

The existence of the hydrodynamic state of evolution of a granular medium, where the Euler-like equations are valid, is delineated in terms of the particle roughness, β, and restitution, e, coefficients. For perfectly elastic spheres this state is shown to exist for all values of particle roughness, i.e. − 1≤β≤1. However, for inelastically colliding granules the hydrodynamic state exists only when the particle restitution coefficient exceeds a certain value em(β)< 1.

In contrast with the previous results obtained by approximate moment methods, the partition of the random-motion kinetic energy of inelastic rough particles between rotational and translational modes is shown to be strongly affected by the particle restitution coefficient. The effect of increasing inelasticity of particle collisions is to redistribute the kinetic energy of their random motion in favour of the rotational mode. This is shown to significantly affect the energy partition law, with respect to the one prevailing in a gas composed of perfectly elastic spheres of arbitrary roughness. In particular, the translational specific heat of a gas composed of inelastically colliding (e = 0.6) granules differs from its value for elastic particles by as much as 55 %.

It is shown that the hydrodynamic Euler-like equation, describing the transport and evolution of the kinetic energy of particle random motion, contains energy sink terms of two types (both, however, stemming from the non-conservative nature of particle collisions) : (i) the term describing energy losses in incompressibly flowing gas; (ii) the terms accounting for kinetic energy loss (or gain) associated with the work of pressure forces, leading to gas compression (or expansion). The approximate moment methods are shown to yield the Euler-like energy equation with an incorrect energy sink term of type (ii), associated with the ‘dense gas effect’. Another sink term of the same type, but associated with the energy relaxation process occurring within compressed granular gases, was overlooked in all previous studies.

The speed of sound waves propagating in a granular gas is analysed in the limits of low and high granular gas densities. It is shown that the particle collisional properties strongly affect the speed of sound in dense granular media. This dependence is manifested via the kinetic energy sink terms arising from gas compression. Omission of the latter terms in the evaluation of the speed of sound results in an error, which in the dense granular gas limit is shown to amount to a several-fold factor.

Taylor—Couette flow is ideal for studying drag-reducing polymer additives because, unlike turbulent boundary layers, the instabilities are better understood. Video records of laser-induced fluorescence experiments with and without polymers will be presented. Polyethylene-oxide (PEO) ‘oceans’ were used in concentrations of 20 and 100 p.p.m. In the Taylor number range, 3 × 104 ≤ Ta ≤ 108, Newtonian flow consisted of Taylor vortices which span the gap between cylinders and much smaller Görtler vortices at the inner cylinder wall. Measurements of core-to-core separation between counter-rotating vortices were made to estimate the Görtler instability wavenumber. These measurements show that PEO addition increases the Görtler wavenumber for a given Taylor number. At the lower Taylor numbers, Görtler vortex formation was suppressed by PEO. This implies that polymers directly affect the evolution of centrifugal instabilities.

An inverse potential methodology is introduced for the solution of the fully 3-D target pressure problem. The method is based on a potential function/stream function formulation, where the physical space is mapped onto a computational one via a body-fitted coordinate transformation. A potential function and two stream vectors are used as the independent natural coordinates, whilst the velocity magnitude, the aspect ratio and the skew angle of the elementary streamtube cross-section are assumed to be the dependent ones. A novel procedure based on differential geometry and generalized tensor analysis arguments is employed to formulate the method. The governing differential equations are derived by requiring the curvature tensor of the flat 3-D physical Eucledian space, expressed in terms of the curvilinear natural coordinates, to be zero. The resulting equations are discussed and investigated with particular emphasis on the existence and uniqueness of their solution. The general 3-D inverse potential problem, with ‘target pressure’ boundary conditions only, seems to be illposed accepting multiple solutions. This multiplicity is alleviated by considering elementary streamtubes with orthogonal cross-sections. The assumption of orthogonal stream surfaces reduces the number of dependent variables by one, simplifying the governing equations to an elliptic p.d.e. for the velocity magnitude and to a second-order o.d.e. for the streamtube aspect ratio. The solution of these two equations provides the flow field. Geometry is determined independently by integrating Frenet equations along the natural coordinate lines, after the flow field has been calculated. The numerical implementation as well as validation test cases for the proposed inverse methodology are presented in the companion paper (Paper 2).

A potential function/stream function formulation is introduced for the solution of the fully 3-D inverse potential ‘target pressure’ problem. In the companion paper (Part 1) it is seen that the general 3-D inverse problem is ill-posed but accepts as a particular solution elementary streamtubes with orthogonal cross-section. Under this simplification, a novel set of flow equations was derived and discussed. The purpose of the present paper is to present the computational techniques used for the numerical integration of the flow and geometry equations proposed in Part 1. The governing flow equations are discretized with centred finite difference schemes on a staggered grid and solved in their linearized form using the preconditioned GMRES algorithm. The geometry equations which form a set of first-order o.d.e.s are integrated numerically using a second-order-accurate space marching scheme. The resulting computational algorithm is applied to a double turning duct and a 3-D converging-diverging nozzle ‘reproduction’ test case.

Direct numerical simulation of turbulent homogeneous shear flow is performed in order to clarify compressibility effects on the turbulence growth in the flow. The two Mach numbers relevant to homogeneous shear flow are the turbulent Mach number Mt and the gradient Mach number Mg. Two series of simulations are performed where the initial values of Mg and Mt are increased separately. The growth rate of turbulent kinetic energy is observed to decrease in both series of simulations. This ‘stabilizing’ effect of compressibility on the turbulent energy growth rate is observed to be substantially larger in the DNS series where the initial value of Mg is changed. A systematic comparison of the different DNS cases shows that the compressibility effect of reduced turbulent energy growth rate is primarily due to the reduced level of turbulence production and not due to explicit dilatational effects. The reduced turbulence production is not a mean density effect since the mean density remains constant in compressible homogeneous shear flow. The stabilizing effect of compressibility on the turbulence growth is observed to increase with the gradient Mach number Mg in the homogeneous shear flow DNS. Estimates of Mg for the mixing layer and the boundary layer are obtained. These estimates show that the parameter Mg becomes much larger in the high-speed mixing layer relative to the high-speed boundary layer even though the mean flow Mach numbers are the same in the two flows. Therefore, the inhibition of turbulent energy production and consequent ‘stabilizing’ effect of compressibility on the turbulence (over and above that due to any mean density variation) is expected to be larger in the mixing layer relative to the boundary layer, in agreement with experimental observations.

Linear stability of the incompressible flow along a streamwise corner is studied by solving the two-dimensional eigenvalue problem governed by partial differential equations. It is found that this fully three-dimensional flow is subject to inviscid instability due to the inflectional nature of the streamwise velocity profile. The higher growth rates for the inviscid instability mode, which is symmetric about the corner bisector, as compared to the viscous Tollmien-Schlichting instability operative away from the corner, is consistent with the experimental findings that the corner flow transitions to turbulence earlier than the two-dimensional Blasius flow away from the corner.

A new mechanism that could be responsible for excitation of long-period oscillations in partially enclosed harbours is discussed. This mechanism is based on the interaction between a shear flow and the harbour-basin natural mode and does not suppose any external exciting forces caused by wind waves, tsunami, etc. The growth rate of harbour oscillations is found in terms of a plane-wave reflection coefficient integrated on the wavenumber spectrum of the oscillating outflow field near the harbour entrance. Analytical considerations for simple shear flows (vortex sheet and jet) show that the growth rate changes its sign depending on the ratio of oscillation frequency to flow speed.

An analytical model for solving the flow field associated with regular reflections of straight shock waves over porous layers has been developed. The governing equations of the gas inside the porous material were obtained by simplifying the general macroscopic balance equations which were obtained by an averaging process over a representative elementary volume of the microscopic balance equations as originally done by Bear & Bachmat (1990). The analytical predictions of the proposed model were compared to experimental results of Skews (1992) and Kobayashi, Adachi & Suzuki (1993). Very good to excellent agreement was evident.

Perfect core-annular flows are two-phase flows, for example of oil and water, with the oil in a perfectly round core of constant radius and the water outside. Eccentric core flows can be perfect, but the centre of the core is displaced off the centre of the pipe. The flow is driven by a constant pressure gradient, and is unidirectional. This kind of flow configuration is a steady solution of the governing fluid dynamics equations in the cases when gravity is absent or the densities of the two fluids are matched. The position of the core is indeterminate so that there is a family of these eccentric core flow steady solutions. We study the linear stability of this family of flows using the finite element method to solve a group of PDEs. The large asymmetric eigenvalue problem generated by the finite element method is solved by an iterative Arnoldi's method. We find that there is no linear selection mechanism; eccentric flow is stable when concentric flow is stable. The interface shape of the most unstable mode changes from varicose to sinuous as the eccentricity increases from zero.

A theoretical and experimental investigation of drop motion in rotating fluids is presented. The theory describing the vertical on-axis translation of an axisymmetric rigid body through a rapidly rotating low-viscosity fluid is extended to the case of a buoyant deformable fluid drop of arbitrary viscosity. In the case that inertial and viscous effects are negligible within the bulk external flow, motions are constrained to be two-dimensional in compliance with the Taylor–Proudman theorem, and the rising drop is circumscribed by a Taylor column. Calculations for the drop shape and rise speed decouple, so that theoretical predictions for both are obtained analytically. Drop shapes are set by a balance between centrifugal and interfacial tension forces, and correspond to the family of prolate ellipsoids which would arise in the absence of drop translation. In the case of a drop rising through an unbounded fluid, the Taylor column is dissipated at a distance determined by the outer fluid viscosity, and the rise speed corresponds to that of an identically shaped rigid body. In the case of a drop rising through a sufficiently shallow plane layer of fluid, the Taylor column extends to the boundaries. In such bounded systems, the rise speed depends further on the fluid and drop viscosities, which together prescribe the efficiency of the Ekman transport over the drop and container surfaces.

A set of complementary experiments is also presented, which illustrate the effects of drop viscosity on steady drop motion in bounded rotating systems. The experimental results provide qualitative agreement with the theoretical predictions; in particular, the poloidal circulation observed inside low-viscosity drops is consistent with the presence of a double Ekman layer at the interface, and is opposite to that expected to arise in non-rotating systems. The steady rise speeds observed are larger than those predicted theoretically owing to the persistence of finite inertial effects.

The formation and evolution of the three-dimensional structure of straight and mildly curved ($b/\bar{R} < 2\%$) flat plate wakes at relatively high Reynolds numbers (Reb = 28 000) have been studied through detailed measurements of the mean and fluctuating velocities. In both cases, the role of initial conditions was examined by generating wakes from untripped (laminar) and tripped (turbulent) initial boundary layers. The curved wake was affected by the angular momentum instability such that the inside half of the wake was unstable, whereas the outside half was stable. In both the straight and curved untripped wakes, large spanwise variations, in the form of ‘pinches’ and ‘crests’, were observed in the contours of mean velocity and Reynolds stresses. Well-organized, ‘spatially stationary’ streamwise vorticity was generated in the near-field region in the form of quadrupoles, to which the spanwise variations in the velocity contours were attributed. The presence of mean streamwise vorticity had a significant effect on the wake growth and defect decay rates, mainly by providing additional entrainment. In the straight wake, the mean streamwise vorticity decayed on both sides of the wake such that it had decayed completely by the far-field region. However, in the curved case, the mean streamwise vorticity on the unstable side decayed at a rate significantly lower than that on the stable side. Despite the decay of mean streamwise vorticity, the spanwise variations persisted into the far wake in both cases. The effects of curvature were also apparent in the Reynolds stress results which showed that the levels on the unstable side were increased significantly compared to those on the stable side, with the effect much stronger in the initially laminar wake. With the initial boundary layers tripped, spatially stationary streamwise vortex structures were not observed in either the straight or curved wakes and the velocity contours appeared nominally two-dimensional. This result further confirms the strong dependency of the three-dimensional structure of plane wakes on initial conditions.

A modulational perturbation analysis is presented which shows that when a strained vortex layer becomes unstable, vorticity concentrates into steady tubular structures with finite amplitude, in quantitative agreement with the numerical simulations of Lin & Corcos (1984). Elaborated three-dimensional visualizations suggest that this process, due to a combination of compression and self-induced rotation of the layer, is at the origin of intense and long-lived vortex tubes observed in direct numerical simulations of homogeneous turbulence.

We study the nonlinear evolution of a pair of oblique instability waves in a supersonic boundary layer over a flat plate in the nonlinear non-equilibrium viscous critical layer regime. The instability wave amplitude is governed by the same integro-differential equation as that derived by Goldstein & Choi (1989) in the inviscid limit and by Wu, Lee & Cowley (1993) with viscous effects included, but the coefficient appearing in this equation depends on the mean flow and linear neutral stability solution of the supersonic boundary layer. This coefficient is evaluated numerically for the Mach number range over which the (inviscid) first mode is the dominant instability. Numerical solutions to the amplitude equation using these values of the coefficient are obtained. It is found that, for insulated and cooled wall conditions and angles corresponding to the most rapidly growing waves, the amplitude ends in a singularity at a finite downstream position over the entire Mach number range regardless of the size of the viscous parameter. The explosive growth of the instability waves provides a mechanism by which the boundary layer can break down. A new feature of the compressible problem is the nonlinear generation of a spanwise-dependent mean distortion of the temperature along with that of the velocity found in the incompressible case.

Care is needed with algorithms for computer simulations of the Brownian motion of complex systems, such as colloidal and macromolecular systems which have internal degrees of freedom describing changes in configuration. Problems can arise when the diffusivity or the inertia changes with the configuration of the system. There are some problems in replacing very stiff bonds by rigid constraints. These problems and their resolution are illustrated by some artificial models; firstly in one dimension, then in the neighbourhood of an ellipse in two dimensions and finally for the trimer polymer molecule.