Statistical properties of the subgrid-scale stress tensor, the local energy flux and filtered velocity gradients are analysed in numerical simulations of forced three-dimensional homogeneous turbulence. High Reynolds numbers are achieved by using hyperviscous dissipation. It is found that in the inertial range the subgrid-scale stress tensor and the local energy flux allow simple parametrization based on a tensor eddy viscosity. This parametrization underlines the role that negative skewness of filtered velocity gradients plays in the local energy transfer. It is found that the local energy flux only weakly correlates with the locally averaged energy dissipation rate. This fact reflects basic difficulties of large-eddy simulations of turbulence, namely the possibility of predicting the locally averaged energy dissipation rate through inertial-range quantities such as the local energy flux is limited. Statistical properties of subgrid-scale velocity gradients are systematically studied in an attempt to reveal the mechanism of local energy transfer.

A new stability approach is developed for a wide class of strongly non-parallel axisymmetric flows of a viscous incompressible fluid. This approach encompasses all conical flows, and all steady and weakly unsteady disturbances, while prior studies were limited to specific flows and particular disturbances. A specially derived form of the Navier–Stokes equations allows the exact reduction of the linear stability problem to a system of ordinary differential equations. We found that disturbances originating at the boundaries of a similarity region cause a variety of steady bifurcations. Consideration of the still fluid allows disturbances to be classified into inner, outer and global modes, depending on the boundary conditions perturbed. Then we identify and study modes which cause bifurcation as the Reynolds number increases. The study provides improved understanding of (a) azimuthal symmetry breaking, (b) genesis of swirl, (c) onset of heat convection, (d) hydromagnetic dynamo, (e) hysteretic transitions, and (f) jump flow separation. We also discover and analyse two new bifurcations: (g) fold catastrophes and (h) appearance of radial oscillations in swirl-free jets. The stability analysis reveals that bifurcations (a), (c) and (f) are caused by inner perturbations, bifurcations (b), (d), (e) and (g) by outer perturbations, and bifurcation (h) by global perturbations. We deduce amplitude equations to describe the nonlinear spatiotemporal development of disturbances near the critical Reynolds numbers for (b) and (g). Disturbances switching between the basic and secondary steady states are found to grow monotonically with time.

A distinctive property of Lagrangian accelerations in geostrophic turbulence is that they are governed by the large and intermediate scales of the flow, both in time and space, so that the inertial part of the dynamics plays a much larger role than in three-dimensional turbulence where viscous effects are stronger. For the case of geostrophic turbulence on a β-plane, three terms contribute to the Lagrangian accelerations: the ageostrophic pressure gradient which often is the largest term, a meridional acceleration due to the β-effect, and an acceleration due to horizontally divergent ageostrophic motions. Both their spectral characteristics and patterns in physical space are studied in this paper. In particular the total accelerations field has an inertial spectrum slope which is identical to the geostrophic velocity field inertial slope.

The accelerations gradient tensor is shown to govern the topology of quasi-geostrophic stirring and transport properties. Its positive eigenvalues locate accurately the position of extrema of potential vorticity gradients. The three-dimensional distribution of tracer gradients is such that the vertical distribution is entirely constrained by the horizontal one, while the reverse is not true. We make explicit analytically their dependence on the three-dimensional accelerations gradient.

We discuss the linear stability of a cross-doubly-diffusive fluid layer with surface tension variation along the free surface. Two limiting cases of the mass flux basic state are considered in the presence of non-zero Soret and Dufour diffusivities. The first case, which has remained largely unexplored, is one where a temperature difference, ΔT¯, and a concentration difference, ΔC¯, are both imposed across the layer. The second case, which has greater significance to thermosolutal systems, is that where the imposed ΔT¯ across the layer induces a ΔC¯. We rescale the problem in the absence of buoyancy, which leads to a more concise representation of neutral stability results in and near the limit of zero gravity. We obtain exact solutions for stationary stability in both cases. One important consequence of the exact solutions is the validation of recently published numerical results in the limit of zero gravity. Moreover, the precise location of asymptotes in relevant parameter (Smc, Mac) space are computed from exact solutions. Both numerical and exact solutions are used to further examine stability behaviour. We also derive algebraic expressions for stationary stability, oscillatory stability, frequency, and codimension two point from a one-term Galerkin approximation. The one-term solutions qualitatively reflect the stability behaviour of the system over the parameter ranges in our investigation. A practical consequence is that the nature of the stability (oscillatory or stationary) for a given set of parameter values can be determined approximately, without solving the numerical eigenvalue problem.

A linear stability analysis has been implemented for hydromagnetic dissipative Couette flow, a viscous electrically conducting fluid between rotating concentric cylinders in the presence of a uniform axial magnetic field. The small-gap equations with respect to non-axisymmetric disturbances are derived and solved by a direct numerical procedure. Both types of boundary conditions, conducting and non-conducting walls, are considered. A parametric study covering wide ranges of μ, the ratio of angular velocity of the outer cylinder to that of inner cylinder, and Q, the Hartmann number which represents the strength of axial magnetic field, is conducted. Results show that the stability characteristics depend on the conductivity of the cylinders. For the case of non-conducting walls, it is found that the critical disturbance is a non-axisymmetric mode as the value of μ is sufficiently negative and the domain of Q where non-axisymmetric instability modes prevail is limited. Similar results are obtained for conducting walls at low Hartmann number. In addition, the transition of the onset of instability from non-axisymmetric modes to axisymmetric modes for the case μ=−1 with increasing strength of magnetic field are discussed in detail. For high values of the Hartmann number, the critical disturbance is always the axisymmetric stationary mode for non-conducting walls but not for conducting walls. For −1[les ]μ<1, it is demonstrated that non-axisymmetric instability modes prevail in a wide range of Q for conducting walls and axisymmetric oscillatory modes may, in fact, become more critical than both of the non-axisymmetric and axisymmetric stationary modes at higher values of the Hartmann number.

A small-scale experiment was conducted (in a 3 m long flume) to study interfacial long-waves in a two-immiscible-fluid system (water and petrol were used). Experiments and nonlinear theories are compared in terms of wave profiles, phase velocity and mainly frequency–amplitude relationships. As expected, the KdV solitary waves match the experiments for small-amplitude waves for all layer thickness ratios. The characteristics of ‘large’-amplitude waves (that is when the crest is close to the critical level – approximately located at mid-depth) asymptotically tend to be predicted by a ‘KdV-mKdV’ equation containing both quadratic and cubic nonlinear terms. In addition a numerical solution of the complete Euler equations, based on Fourier series expansions, is devised to describe solitary waves of intermediate amplitude. In all cases, solitary interfacial waves in this numerical theory tally with the experimental data. When the layer thicknesses are almost equal (ratio of lower layer to total depth equal to 0.4 or 0.63) both the KdV-mKdV and the numerical solutions match the experimental points.

For a convex equation of state, a general theorem on shock waves is proved: a sequence of two shocks has a lower entropy than a single shock to the same final pressure. We call this the triple-shock entropy theorem. This theorem has important consequences for shock interactions. In one dimension the interaction of two shock waves of the opposite family always results in two outgoing shock waves. In two dimensions the intersection of three shocks, such as a Mach configuration, must have a contact. Moreover, the state behind the Mach stem has a higher entropy than the state behind the reflected shock. For the transition between a regular and Mach reflection, this suggests that the von Neumann (mechanical equilibrium) criterion would be preferred based on thermodynamic stability, i.e. maximum entropy subject to the system constraint that the total specific enthalpy is fixed. However, to explain the observed hysteresis of the transition we propose an analogy with phase transitions in which locally stable wave patterns (regular or Mach reflection) play the role of meta-stable thermodynamic states. The hysteresis effect would occur only when the transition threshold exceeds the background fluctuations. The transition threshold is affected by flow gradients in the neighbourhood of the shock intersection point and the background fluctuations are due to acoustic noise. Consequently, the occurrence of hysteresis is sensitive to the experimental design, and only under special circumstances is hysteresis observed.

The evolution of a perturbed vortex in a pipe to axisymmetric vortex breakdown is studied through numerical computations. These unique simulations are guided by a recent rigorous theory on this subject presented by Wang & Rusak (1997a). Using the unsteady and axisymmetric Euler equations, the nonlinear dynamics of both small- and large-amplitude disturbances in a swirling flow are described and the transition to axisymmetric breakdown is demonstrated. The simulations clarify the relation between our linear stability analyses of swirling flows (Wang & Rusak 1996a, b) and the time-asymptotic behaviour of the flow as described by steady-state solutions of the problem presented in Wang & Rusak (1997a). The numerical calculations support the theoretical predictions and shed light on the mechanism leading to the breakdown process in swirling flows. It has also been demonstrated that the fundamental characteristics which lead to vortex instability and breakdown in high-Reynolds-number flows may be calculated from considerations of a single, reduced-order, nonlinear ordinary differential equation, representing a columnar flow problem. Necessary and sufficient criteria for the onset of vortex breakdown in a Burgers vortex are presented.

Results of laboratory experiments are presented in which a fixed volume of homogeneous fluid is suddenly released into another fluid of slightly lower density, over a horizontal thin metallic grid placed a given distance above the solid bottom of a rectangular-cross-section channel. Dense liquid develops as a gravity current over the grid at the same time as it partially flows downwards. The results show that the gravity current loses mass at an exponential rate through the porous substrate with a time constant τ; the front velocity and the head of the current also decrease exponentially. The loss of mass dominates the flow and, in contrast to gravity currents running over solid bottoms, no self-similar inertial regime seems to be developed. A simple model is introduced to explain the scaling law of the loss of mass and the evolution of the front position. The flow evolution depends on the characteristic time of the initial (slumping) phase and the time constant τ, related to the initial conditions and the permeability of the porous substrate, respectively. Qualitative comparisons with other gravity currents with loss of mass, such as particle-driven gravity currents, are provided.

A lubrication analysis is presented for the near-contact axisymmetric motion of spherical drops covered with an insoluble non-diffusing surfactant. Detailed results are presented for the surfactant distribution, the interfacial velocity, and the gap width between the drop surfaces. The effect of surfactant is characterized by a dimensionless force parameter: the external force normalized by Marangoni stresses. Critical values of the force parameter have been established for drop coalescence and separation. Surfactant-covered drops are stable to rapid coalescence for external forces less than 4πkTac0, where c0 is the surfactant concentration at the edge of the near-contact region and a is the reduced drop radius.

For subcritical forces, the behaviour of surfactant-covered drops is described by two time scales: a fast time scale characteristic of near-contact motion between drops with clean interfaces and a slow time scale associated with rigid particles. The surfactant distribution evolves on the short time scale until Marangoni stresses approximately balance the external force. Supercritical values of the external force cannot be balanced; coalescence and separation occur on the fast time scale. The coalescence time normalized by the result for drops with clean interfaces is independent of the viscosity ratio and initial gap width.

Under subcritical force conditions, a universal long-time behaviour is attained on the slow time scale. At long times, the surfactant distribution scales with the near-contact region and the surface velocity is directed inward which impedes the drop approach and accelerates their separation compared to rigid particles. For drops pressed together with a sufficiently large subcritical force, a shrinking surfactant-free clean spot forms.

Surfactant-covered drops exhibit an elastic response to unsteady external forces because of energy stored in the surfactant distribution.

We study the respective effects of shear rate and of external field intensity and direction on the contribution to the bulk stress of Brownian dipolar axisymmetric particles suspended in a steady macroscopically homogeneous shear flow of an incompressible Newtonian fluid. Towards this end we obtain the steady orientational distribution and make use of existing general dynamic theories of dilute suspensions. The calculation focuses on the limit of weak rotary diffusion. Thus, unlike previous analyses, the present contribution is not restricted to weak shear effects.

Explicit results are presented for the bulk stress when the external field acts in the plane of the simple shear flow. In cases when the deterministic rotary motion possesses a single sufficiently stable node a simple unified description of the respective effects of both the intensity and azimuthal direction of the external field is provided by the boundary-layer approximation. This approximation enables a qualitative explanation of existing numerical results as well as furnishing quantitatively accurate analytical results at relatively moderate values of the rotary Peclét number and the Langevin parameter (∼10). Furthermore, at still larger values of these parameters use of the present asymptotic approximation is clearly preferable since the numerical schemes rapidly deteriorate when steep orientational gradients appear.

Singularities of the bulk stress are rationalized in terms of the corresponding deterministic rotary motion. This is particularly interesting because some of these singular phenomena (e.g. those associated with an ‘intermediate regime’ of the field intensity and direction, for which more than one stable attractor exists in the deterministic problem) have no counterparts in suspensions of dipolar spheres or torque-free axisymmetric particles.

Finally, the present results obtained for the orientational distribution are also applicable to the study of other aspects of the macroscale description of suspensions of dipolar axisymmetric particles. In this context we mention the extension of continuum modelling of suspensions of swimming micro-organisms so as to enable the analysis of fully developed bioconvection.

Secondary and tertiary states of fluid flow in a layer between two plates in relative motion and rotating about a normal axis of rotation are studied numerically for a wide range of parameters. Plane Couette flow without rotation and the single Ekman layer at a rigid plate above a quiescent fluid half-space are obtained as limiting cases. A Galerkin method is used for the investigation of the bifurcation structures of the problem. A Chebyshev collocation scheme is used for following the evolution of time-dependent states of the flow. Comparisons are made with experimental observations as well as with previous studies of particular parameter limits.

We study the behaviour of an upward vertical water jet of density, ρ, and surface tension, σ, injected through a tube of diameter, D, with a momentum-averaged velocity, V. These fountains are shown to exhibit large-amplitude oscillations in the range 0.1[les ]D/a[les ]1.6, and 20[les ]V2/(gD)[les ]400, where g is the acceleration due to gravity and a is the capillary length, a≡(2σ/(ρg))1/2. The characteristic frequency of the oscillations, f, and their limits of existence are studied experimentally. A model is developed, leading to the expression for the frequency:

formula here

This expression is shown to be in good agreement with existing data and with new measurements, conducted over a wide range of Bond (Bo≡D/a) and Froude (Fr≡V2/gD) numbers. The stability of the model is considered and the limits of the oscillatory regime are related to the hydrodynamic properties of the flow.

The differential equation describing the one-point joint probability density function for the wind velocity given by Lundgren (1967) in neutral turbulent flows is extended by a term which also takes into consideration the pressure–mean strain interaction. For the new equation a solution is given describing the one-point probability density function for the wind velocity fluctuations if the profile of the mean wind velocity is logarithmic. The properties of this solution are discussed to identify the differences to a Gaussian having the same first and second moments.

The problem of the inception of cavitation is formulated in terms of a comparison of the breaking strength or cavitation threshold at each point in a liquid sample with the principal stresses there. A criterion of maximum tension is proposed which unifies the theory of cavitation, the theory of maximum tensile strength of liquid filaments and the theory of fracture of amorphous solids. Liquids at atmospheric pressure which cannot withstand tension will cavitate when and where tensile stresses due to motion exceed one atmosphere. A cavity will open in the direction of the maximum tensile stress which is 45° from the plane of shearing in pure shear of a Newtonian fluid. Experiments which support these ideas are discussed and some new experiments are proposed.