A method is reported for predicting the shape of the phase boundary in two-phase isothermal constant-volume constant-mass rotating fluid systems. In contrast to previous methods that have employed the continuum concept of pressure, the proposed method uses the thermodynamic concept. The latter requires, in addition to the usual condition of a force balance existing at the boundary, that the equilibrium phase boundary shape be such that there is no net mass flux. The latter condition is imposed by requiring that the chemical potentials in the different phases be equal at the phase boundary. A non-dimensional parameter is defined that allows one to determine when the effects of a gravitational field acting at 90° to the axis of rotation may be neglected. Experiments have been performed under conditions where this restriction is satisfied. With known values of the experimentally controllable variables, the proposed method has been used to predict the length of the vapour phase. To within the experimental error, the predicted lengths are found to be in agreement with the measurements. If, however, a gravitational field of a sufficient magnitude is imposed the vapour phase has been found to become unstable and to break into two or more separate bubbles. Using the variable-gravity environment of an aircraft following a parabolic flight path, this instability has been investigated. By approximating the gravitational effects, the theoretical description has been extended and a method proposed to determine the conditions under which the phase boundary becomes unstable. If the angle of action of the net viscous shear force on the bubble were known, a prediction of the breakup could be made entirely in terms of experimentally controllable parameters. Using arguments for the value of this angle, bounds on the breakup condition are compared with experimental results.

We performed a theoretical investigation of the progression of a finger of air through a liquid-filled flexible-walled channel - an initial model of pulmonary airway reopening. Positive pressure, Pb* drives the bubble forward, and separates flexible walls that are modelled as membranes under tension, T, supported by linearly elastic springs with elasticity K. The gap width between the walls under stress-free conditions is 2H, and the liquid has constant surface tension, γ, and viscosity, μ. Three parameters define the state of the system: Ca = μU/γ is a dimensionless velocity that represents the ratio of viscous to capillary stresses; η = T/γ is the wall tension to surface tension ratio, and γ = KH2/γ is the wall elastance parameter. We examined steady-state solutions as a function of these parameters using lubrication analysis and the boundary element method.

These studies showed multiple-branch behaviour in the Pb-Ca relationship, where Pb = Pb*/(γ/H) is the dimensionless bubble pressure. Low Ca flows (Ca [Lt ] min (1, (Γ3/η)1/2)) are dominated by the coupling of surface tension and elastic stresses. In this regime, Pb decreases as Ca increases owing to a reduction in the downstream resistance to flow, caused by the shortening of the section connecting the open end of the channel to the fully collapsed region. High Ca behaviour (max (1, (γ3/η)1/2) [Lt ] Ca [Lt ] η) is dominated by the balance between fluid viscous and longitudinal wall tension forces, resulting in a monotonically increasing Pb–Ca relationship. Increasing η or decreasing Γ reduces the Ca associated with the transition from one branch to the other. Low Ca streamlines show closed vortices at the bubble tip, which disappear with increasing Ca.

Start-up yield pressures are predicted to range from 1 [les ] Pyield*/(γ/L*) [les ] 2, which is less than the minimum pressure for steady-state reopening, Pmin/(γ/L*), where L* is the upstream channel width. Since Pyield* < Pmin*, the theory implies that low Ca reopening may be unsteady, a behaviour that has been observed experimentally. Our results are consistent with experimental observations showing that Pb* in highly compliant channels scales with γ/L*. In contrast, we find that wall shear stress scales with γ/H. These results imply that wall shear and normal stresses during reopening are potentially very large and may be physiologically significant.

It is well known that in any conservative system that admits resonant triad interactions, a uniform (test) wavetrain that participates in a single triad is unstable if it has the highest frequency in the triad, and neutrally stable otherwise. We show that this result changes significantly in the presence of coupled triads: with coupling, the test wave can be unstable to a high-frequency perturbation. The coupling sends energy from the (weak) high-frequency source into particular low-frequency waves that grow even though they had zero amplitudes initially. This mechanism thereby selects these low-frequency waves from the spectrum of low-frequency waves available for triad interactions. Moreover, the instability persists in the presence of weak damping, provided the wave amplitudes exceed two thresholds. First, the initial amplitude of the test wavetrain must be large enough for the instability to dominate the damping. Secondly, the (small) initial amplitudes of the high-frequency perturbations must exceed a threshold in order for the low-frequency waves to grow to a prescribed amplitude.

Strongly nonlinear vortex-Tollmien-Schlichting-wave interaction equations are derived for the case where the undisturbed motion represents the developing flow in a circular pipe. The effect upon the equations of moving the wave input position further downstream is investigated and the development of the flow is found to be accelerated by increasing the size of the wave disturbance. Numerical solutions of the three-dimensional interaction equations are presented and indicate that the form of interaction considered here appears to promote the three-dimensionality as the flow develops downstream. It is shown that one of the interactions considered here can develop within an initially two-dimensional Blasius boundary layer.

There are a number of papers devoted to the construction of the exact solitary wave solution using a series. Power series in amplitude or Fourier series have usually been used. In the present paper we accomplish the exact summation of the Witting (1975) series and show that this series describes other flows, not solitary waves. One such flow is fluid suction under a curvilinear roof. The left half of it is similar to the left half of a maximal-amplitude solitary wave flow.

Helicity generation and vorticity dynamics in a helically symmetric flow are studied numerically by solving the Navier–Stokes equations in an unbounded domain. The helical symmetry reduces the problem to one in two dimensions, which makes it practicable to use an analytical expression for vortex surfaces. Furthermore, the field still retains three-dimensional aspects, such as helicity and vortex stretching. To every vortex surface there corresponds an inviscid invariant of helicity.

Our initial conditions are chosen as two cases of twisted elliptical tubes of high vorticity. The first case has elliptical vortex surfaces, which is a helically symmetric version of the initial condition employed by Aref & Zawadzki (1991), but the second case has axisymmetric vortex surfaces. The total helicity inside every vortex surface is zero in both the initial fields. It is found that vortex stretching plays an important role in the time evolution of the first case, but not of the second case.

We examine the relation of the vorticity dynamics to the helicity generation by using the representation of the vortex lines and the vortex surfaces rather than the equi-vorticity surfaces. This leads to new concepts for the mechanisms of formation of the spiral vortex structures observed for the two cases. The detailed investigation of the helicity generation is done by examining the distribution of the helicity on each vortex surface and the Fourier spectrum of helicity. The processes of helicity generation due to the vortex stretching are different for each initial condition. The viscosity dependence of ‘inviscid’ invariants shows that with smaller viscosity, only in the first case is more helicity generated. This is because in the first case where there is vortex stretching the contact zones of the adjacent vortex layers are elongated and the local vorticity is intensified to an extent limited by viscosity. Thus with smaller viscosity a more intense vortex is reconnected to generate more helicity. As expected, in both cases more of the energy is preserved with smaller viscosity.

A class of unsteady boundary layers that form on flat extensible surfaces of finite but increasing length in otherwise stagnant surroundings is considered. The surface length $\overline{R}$ is assumed to grow with time as tp where p > 0 and the velocity at any location $0 \le \overline{r} \le \overline{R}$ on the surface as tp-np-1r-n, where n [ges ] 0. The problem is cast into similarity variables and the governing parabolic differential equation shown to exhibit, for various combinations of n and p, regions of mixed mathematical diffusivity and reversals in the direction of convection of vorticity. Equations depicting such behaviour are usually termed singular parabolic and are here classified as follows: type-0, in which the mathematical diffusivity may be either positive or mixed but in which there are no reversals in the direction of convection of vorticity; type-1, in which the mathematical diffusivity may be either positive or mixed but in which there are reversals in the direction of convection of vorticity. Both types are shown to occur. Moreover while type-0 flows occur only when n = 1 and form with an unsteady separated stagnation point at the origin, type-1 flows occur only for 0 [les ] n < 1 and form with a steady stagnation point at the origin. Type-1 flows are further characterized by boundary layers with zero displacement thickness both at the origin and leading edge. Because singular parabolic equations require two initial conditions plus boundary conditions to ensure uniqueness, they are here treated numerically in a manner akin to elliptic boundary value problems. A successive-approximation implicit scheme was thus used and a wide range of cases solved in the parameter range n ∈ [0,1], p ∈ (0,2]. Amongst other things, it is shown that type-0 flows have lower drag than their type-1 counterparts. It is further shown that the drag on a flat rigid surface of finite length moving in its own plane at constant velocity and being continuously produced at the origin is higher than on a corresponding length of either a semi-infinite surface likewise produced or a semi-infinite plate in an aligned uniform stream; however if the surface is extensible and $n > \frac{1}{2}$ the converse is true.

A complete semi-analytical solution is given for second-order diffraction of monochromatic waves by a truncated vertical circular cylinder in water of uniform finite depth. The methodology presented in detail elsewhere (Eatock Taylor & Huang 1996) is adopted to find a particular solution which exactly satisfies the governing equation, the inhomogeneous free-surface condition and the seabed condition. In order to satisfy the boundary condition on the cylinder bottom, the fluid domain around the cylinder is divided into two regions. First- and second-order velocity potentials are described separately in the two regions and matched on the interface by the pressure and normal-velocity continuity conditions. Based on the formulation, the second-order wave field in the vicinity of the cylinder and the corresponding wave forces and overturning moments on the cylinder are studied in detail. Numerical results for the double frequency forces obtained by using the present semi-analytical approach are compared with those computed with a higher-order boundary element method (Eatock Taylor & Chau 1992). As well as the exact solution, an approximate solution is also given for the second-order potential and the corresponding forces. Numerical results show that the approximate solution possesses excellent accuracy for the total second-order heave force over a wide range of conditions. When kb > 1.2 (where k, b are the incident wavenumber and the draught of the cylinder respectively), the accuracy for total second-order surge force and pitch moment is also satisfactory. These results could lead to the development of very efficient solutions and corresponding algorithms for the analysis of second-order wave diffraction by more complicated structures such as tension leg platforms. Numerical results based on the present solution show that in many cases, both the first- and the second-order-free surface elevation in the vicinity of a truncated cylinder is very close to that of a bottom-seated cylinder. For waves with larger amplitudes, the maximum free-surface elevation around a vertical cylinder predicted with the second-order theory can significantly exceed that given by linear theory. There is also a considerable difference in the location of the maximum elevation predicted by the linear and nonlinear theories.

The far-field sound of an unstable wave packet undergoing transition in a low-Mach-number, flat-plate boundary layer is investigated in the framework of Lighthill's acoustic analogy. Detailed accounts of the wave packet evolution are obtained by solving the full incompressible Navier–Stokes equations at Reδ = 1000. The numerically simulated flow structures show qualitative agreement with experimental observations of the fundamental breakdown type. The acoustic calculations are focused on the quadrupole source functions arising from Reynolds stress fluctuations. The wave packet is shown to produce negligible sound throughout the primary and secondary instability stages. Dramatic amplification of the Reynolds stress quadrupoles occurs as a result of the disintegration of the detached high-shear layer and the associated vortex shedding near the boundary layer edge. The dominant frequency of source oscillations coincides with that of vortex shedding.

We consider the slow axial motion of a symmetric particle or drop in a bounded rotating fluid for small Rossby and Ekman numbers, Ro and E. Previous investigations pointed out that the available linear-theory results, based on the assumption of a dominant geostrophic core and infinitesimally thin viscous layers, yield a drag force larger than the available relevant experimental results, and are unable to explain some of the observed flow-field properties, for both solid and deformable particles.

Here we attempt to improve the drag calculation model and the interpretation of the flow field by incorporating shear effects in the core (outside the E1/2 Ekman and E1/3 Stewartson layers), first in the linear (Ro = 0) formulation, then with keeping some influential nonlinear inertial terms for small but finite Ro. The major equation for the angular velocity in the core, ω(r), was usually solved by a finite-differences method, because in the practical parameter range the available analytical results are sufficiently accurate only for a disk particle or for a bubble. Results for various $\epsilon = (\frac{1}{2}H)^{1/2}$, no-slip parameter of particle surface κ, and half container height H are presented for both spherical and disk particles. The drag is below the geostrophic value, typically, by 25% for ε ≈ 0.1 and by 50% for ε ≈ 0.5. The inclusion of the inertial terms causes the lateral (‘vertical’) shear regions to contract and expand on the upstream and downstream sides, respectively, and an inertial sublayer appears in the latter when Ro ≈ O(E3/4), but the net contribution to the drag is smaller than expected. Compared with more accurate solutions and experiments the present results underestimate the drag (the reasons are discussed) but are qualitatively consistent in many respects, which indicates that many of the observed flow-field features that have been traditionally attributed to inertial effects (not sufficiently small Ro) are, rather, by-products of the lateral shear (finite value of ε).

The characteristics of the initial formation of streamwise vortices and their relation to the spanwise structures are experimentally investigated at several phases of the initial roll-up and pairing in a two-stream mixing layer at high Reynolds numbers (up to Reθ = 1.25 × 103). The spanwise structures were stabilized by acoustically exciting the mixing layer at the natural instability frequency and its first subharmonic. No artificial spanwise forcing was applied. Quantitative information was obtained through conditional sampling with the forcing signal as the phase reference. Time traces of the three velocity components were recorded by hot-wire anemometry at a grid of locations on planes normal to the flow at four successive streamwise locations which include the roll-up and first pairing of the spanwise structures. From these measurements, three-dimensional distributions of ensemble-average vorticity components were computed and analysed.

The results show that the spanwise structures remain mostly two-dimensional from roll-up to the end of the first pairing, except for small oscillations in curvature of their axes. Concentrated streamwise vortices evolve at locations corresponding to small ‘kinks’ in the spanwise rollers and first appear at the beginning of the pairing process. Their subsequent development bears many similarities with the results of small-perturbation simulations. They expand laterally, forming counter-rotating rib vortices that grow almost exponentially in the braid region. In the core region, a complex but organized streamwise vorticity pattern emerges. The pattern consists of layered streamwise vortices whose orientation suggests that the oscillation of the spanwise roller core is π radians out of phase with that of the streamwise vortex lines that join the rib vortices. The peak streamwise vorticity and circulation in the core region increase until the completion of the spanwise structure pairing when the layered pattern collapses. Measurements at large spanwise locations conducted downstream show that the ‘kink’ is not the only location of streamwise vorticity generation: streamwise vortices develop at nearly periodic spanwise locations other than that of the initial ‘kink’. The initial spacing between streamwise vortices agrees with previous stability analysis. At the end of the spanwise structure pairing, isolated pairings between streamwise vortices take place, but a global doubling in spanwise wavelength does not occur.

The stability with respect to two- and three-dimensional perturbations of natural-convection flow of air in a square enclosure with differentially heated vertical walls and periodic boundary conditions in the lateral direction has been investigated. The horizontal walls are either conducting or adiabatic. The solution is numerically approximated by Chebyshev–Fourier expansions. In contrast to the assumption made in earlier studies, three-dimensional perturbations turn out to be less stable than two-dimensional perturbations, giving a lower critical Rayleigh number in the three-dimensional case for the onset of transition to turbulence. Both the line-symmetric and line-skew-symmetric three-dimensional perturbations are found to be unstable. The most unstable wavelengths in the lateral direction typically are of the same size as the enclosure. In the nonlinear solution new symmetry breaking occurs, giving either a steady or an oscillating final state. The three-dimensional structures in the nonlinear saturated solution consist of counter-rotating longitudinal convection rolls along the horizontal walls. The energy balance shows that the three-dimensional instabilities have a combined thermal and hydrodynamic nature. Besides the stability calculations, two- and three-dimensional direct numerical simulations of the weakly turbulent flow were performed for the square conducting enclosure at the Rayleigh number 108. In the two-dimensional case, the time-dependent temperature shows different dominant frequencies in the horizontal boundary layers, vertical boundary layers and core region, respectively. In the three-dimensional case almost the same frequencies are found, except for the horizontal boundary layers. The strong three-dimensional mixing leaves no, or only very weak, three-dimensional structures in the time-averaged nonlinear solution. Three-dimensional effects increase the maximum of the time- and depth-averaged wall-heat transfer by 15%.

Steady, inviscid, incompressible two-dimensional flow in a quarter-circular cavity containing two vortex patches is investigated. A two-parameter family of solutions, characterized by any two out of the positions of the separation and reattachment points of the main eddy, the tangential velocity at separation and the ratio of the core vorticities, is identified and computed numerically. It is found that solutions can only be obtained for a rather narrow band of combinations of these parameters; the reasons for this constraint are discussed. Finally, we consider whether any of the coupled Batchelor flow solutions actually does represent the limit of high Reynolds number flow by comparing the inviscid results with those of earlier Navier–Stokes computations (Vynnycky & Kimura 1994). Agreement for the position of the dividing streamline and the location of the centre of the main core proves to be very encouraging, and suggestions are made as to the possible future development of such a two-eddy model.

Wave diffraction, wave forces and wave drift damping due to a floating body performing a slow rotation about the vertical axis (yaw) is considered. The rotation angle of the body may be arbitrary. The angular velocity is assumed small compared to the wave frequency, however. The problem is formulated in the frame of reference following the slow rotation of the body, accounting for non-Newtonian forces. By applying the method of multiple timescales, the fluid flow is determined consistently to leading order in the slow angular velocity and to second order in the wave amplitude. Mathematical solution of the problem is obtained by means of integral equations that are applicable to geometries of arbitrary shape. The wave loads are found by applying conservation of linear and angular momentum. The wave drift damping is expressed by the far-field amplitudes of the wave field and the dipole moments of the time-averaged second-order potential. Numerical results are presented for a ship and a vertical cylinder describing a circular path in the horizontal plane. The results show that the wave drift damping due to a slow yaw motion of a floating body is one order of magnitude larger than the time-averaged forces and moment when there is no rotation. Wave drift damping due to slow rotation and slow translation are found to be of equal importance.

The dynamic model for large-eddy simulation of turbulence samples information from the resolved velocity field in order to optimize subgrid-scale model coefficients. When the method is used in conjunction with the Smagorinsky eddy-viscosity model, and the sampling process is formulated in a spatially local fashion, the resulting coefficient field is highly variable and contains a significant fraction of negative values. Negative eddy viscosity leads to computational instability and as a result the model is always augmented with a stabilization mechanism. In most applications the model is stabilized by averaging the relevant equations over directions of statistical homogeneity. While this approach is effective, and is consistent with the statistical basis underlying the eddy-viscosity model, it is not applicable to complex-geometry inhomogeneous flows. Existing local formulations, intended for inhomogeneous flows, are most commonly stabilized by artificially constraining the coefficient to be positive. In this paper we introduce a new dynamic model formulation, that combines advantages of the statistical and local approaches. We propose to accumulate the required averages over flow pathlines rather than over directions of statistical homogeneity. This procedure allows the application of the dynamic model with averaging to in-homogeneous flows in complex geometries. We analyse direct numerical simulation data to document the effects of such averaging on the Smagorinsky coefficient. The characteristic Lagrangian time scale over which the averaging is performed is chosen based on measurements of the relevant Lagrangian autocorrelation functions, and on the requirement that the model be purely dissipative, guaranteeing numerical stability when coupled with the Smagorinsky model. The formulation is tested in forced and decaying isotropic turbulence and in fully developed and transitional channel flow. In homogeneous flows, the results are similar to those of the volume-averaged dynamic model, while in channel flow, the predictions are slightly superior to those of the spatially (planar) averaged dynamic model. The relationship between the model and vortical structures in isotropic turbulence, as well as ejection events in channel flow, is investigated. Computational overhead is kept small (about 10% above the CPU requirements of the spatially averaged dynamic model) by using an approximate scheme to advance the Lagrangian tracking through first-order Euler time integration and linear interpolation in space.

A series of experiments has been carried out on low-viscosity fluid in a right-circular cylinder that rotates rapidly at a constant speed about its axis of symmetry. This axis in turn is made to undergo less rapid precession about a second axis passing through the centroid of the cylinder. The linear inviscid response of the fluid to such forcing can be expressed as a spectrum of inertial wave modes. However, there are several interesting features of the problem that are associated with nonlinear and viscous effects. One such phenomenon is the appearance of an azimuthal flow under conditions that are related to the underlying linear inertial wave behaviour. Results are presented concerning the manner in which this flow depends on the various experimental parameters. Dynamical properties of the circulation following the onset of forcing have also been investigated. The flow at forcing frequencies close to the fundamental inertial wave resonance was found to have a vortex-like structure, and this led to data that suggest that hydrodynamic instabilities may play a part in the observed breakdown to turbulent motion in regimes of strong forcing.