The method of energy is used to discuss the stability of time-dependent diffusive temperature profiles in fluid layers subject to impulsive changes in surface temperature.

Bounds for the ratio of disturbance energy production to dissipation are found to be parametric functions of time because the basic temperature develops through diffusion. This time dependence leads to the demarcation of regions of stability in a Rayleigh number-time plane and the interpretation of these regions is given. Numerical results are presented for the cases of impulsive heating and cooling of initialty isothermal fluid layers. New global stability results which give the Rayleigh number below which the diffusive solution to the Boussinesq equations is unique are reported for these cases.

An efficient algorithm for hydrodynamical interaction of many deformable drops subject to shear flow at small Reynolds numbers with triply periodic boundaries is developed. The algorithm, at each time step, is a hybrid of boundary-integral and economical multipole techniques, and scales practically linearly with the number of drops N in the range N < 1000, for NΔ ∼ 103 boundary elements per drop. A new near-singularity subtraction in the double layer overcomes the divergence of velocity iterations at high drop volume fractions c and substantial viscosity ratio γ. Extensive long-time simulations for N = 100–200 and NΔ = 1000–2000 are performed up to c = 0.55 and drop-to-medium viscosity ratios up to λ = 5, to calculate the non-dimensional emulsion viscosity μ* = Σ12/(μeγ˙), and the first N1 = (Σ11−Σ22)/(μe[mid ]γ˙[mid ]) and second N2 = (Σ22−Σ33)/(μe[mid ]γ˙[mid ]) normal stress differences, where γ˙ is the shear rate, μe is the matrix viscosity, and Σij is the average stress tensor. For c = 0.45 and 0.5, μ* is a strong function of the capillary number Ca = μe[mid ]γ˙[mid ]a/σ (where a is the non-deformed drop radius, and σ is the interfacial tension) for Ca [Lt] 1, so that most of the shear thinning occurs for nearly non-deformed drops. For c = 0.55 and λ = 1, however, the results suggest phase transition to a partially ordered state at Ca [les ] 0.05, and μ* becomes a weaker function of c and Ca; using λ = 3 delays phase transition to smaller Ca. A positive first normal stress difference, N1, is a strong function of Ca; the second normal stress difference, N2, is always negative and is a relatively weak function of Ca. It is found at c = 0.5 that small systems (N ∼ 10) fail to predict the correct behaviour of the viscosity and can give particularly large errors for N1, while larger systems N [ges ] O(102)show very good convergence. For N ∼ 102 and NΔ ∼ 103, the present algorithm is two orders of magnitude faster than a standard boundary-integral code, which has made the calculations feasible.

In this paper a two-dimensional free boundary arising from the steady thermo-capillary flow in a viscous incompressible fluid is studied numerically. The problem is considered in the context of the open-boat crystal-growth technique. The motion of the fluid is governed by the Navier-Stokes equations coupled with the heat equation. The problem is solved numerically by a finite-element-method discretization. Three iterative methods are introduced for the computation of the free boundary. The non-dimensional form of the problem gives rise to the following characteristic parameters: Reynolds, Grashof, Prandtl, Marangoni, Bond, Ohnesorge, Biot numbers. The influence of these parameters on the flow field, the temperature distribution and the shape of the free boundary is studied.

The generation of indirect combustion noise by compositional inhomogeneities is examined theoretically. For this, the compact-nozzle theory of Marble & Candel (J. Sound Vib., vol. 55 (2), 1977, pp. 225–243) is extended to a multi-component gas mixture, and the chemical potential function is introduced as an additional acoustic source mechanism. Transfer functions for subcritical and supercritical nozzle flows are derived, and the contribution of compositional noise is compared to entropy noise and direct noise by considering an idealized nozzle downstream of the combustor exit. It is shown that compositional noise is dependent on the local mixture composition and can exceed entropy noise for fuel-lean conditions and supercritical nozzle flows. This suggests that the compositional indirect noise requires potential consideration with the implementation of low-emission combustors.

The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant and equal to unity is rederived using ‘boundary layer’ techniques together with the von Mises variables x and ψ. A method of successive approximations then gives a closer approximation to this solution for ε (λ − 1)/(λ + 1) small and the free-strea Mach number infinite. Formulae for the streamlines, shock shape and pressure distribution are determined to this approximation. These formulae are valid for any plane or axially symmetric shape, giving the ‘stand-off’ distance of the shock wave from the body as ½εlog(4|3ε) and ε times the nose radius of curvature for plane and axially-symmetric flows respectively. Particular results are computed for a number of special shapes. For certain shapes, the theory has a singular point where the first approximation to the pressure vanishes (θ = 60° for a sphere). Actually, the theory is not applicable where the pressure becomes too small. The corresponding theory for gases of general thermodynamic properties is deduced, the approximation being valid provided the total energy of the gas is large compared with the energy contained in the translational modes of the gas molecules.

We present some preliminary results from using large-eddy simulation to compute the late wake of a sphere towed at constant speed through a non-stratified and a uniformly stratified fluid. The wake is computed in each case for two values of the Reynolds number: Re = 104, which is comparable to that used in laboratory experiments, and Re = 105. An important aspect of the simulation is the use of an iterative procedure to relax the initial turbulence field so that the normal and shear turbulent stresses are properly correlated and the turbulent production and dissipation are in equilibrium. For the lower Reynolds number our results compare well with existing laboratory experimental results. For the higher Reynolds number we find that even though the turbulence is more developed and the wake contains finer structure, most of the similarity properties of the wake are unchanged compared with those observed at the lower Reynolds number.

Cavitation inception in a turbulent shear layer was studied at Reynolds numbers up to 2 × 106. Flash photography, high-speed motion pictures and holography were used to study the relation of cavitation inception to the shear-layer turbulent structure. Both spanwise and streamwise vortices were clearly visualized by the cavitation. Cavitation inception consistently occurred in the streamwise vortices and more fully developed cavitation was visible in both structures, with the streamwise cavities typically confined to the braid regions between adjacent spanwise vortices. The strength of the streamwise vortices was estimated using a Rankine vortex model, which showed that their strength was always less than 10% of that of the spanwise vortices. Measurements of fluctuating pressures were made by holographically monitoring the size of air bubbles injected into the non-cavitating shear flow. The measured pressure fluctuations had positive and negative peaks as high as 3 times the free-stream dynamic pressure, sufficient to explain cavitation inception at high values of the inception index. The occurrence of inception in the streamwise vortices of the shear layer, combined with previous reports of velocity dependence of the streamwise vortex strength, may explain the commonly observed Reynolds-number scaling of the cavitation inception index in shear flows.

Recent experiments by Pihler-Puzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular Hele-Shaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Non-axisymmetric instabilities are only observed at much larger flow rates, and the large-amplitude fingers that develop are fundamentally different from the highly branched fingers in rigid-walled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depth-averaged lubrication equation for the fluid flow to the Föppl–von Kármán equations, which describe the deformation of the elastic membrane. We show that fluid–structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air–liquid interface to propagate into a converging fluid-filled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigid-walled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.

The upper surface of a large gas bubble rising steadily through liquid under gravity is a statically unstable interface, and if the liquid were stationary small sinusoidal disturbances to the interface with wavelength exceeding the critical value Λc determined by surface tension would grow exponentially. The existence of the deforming motion of the liquid adjoining the interface of a steadily rising bubble changes the nature of the problem of stability. It is shown that a small sinusoidal disturbance of the part of the interface that is approximately plane and horizontal remains sinusoidal, although with exponentially increasing wavelength. The amplitude of such a disturbance increases, from the instant at which Λ = Λc until Λ becomes comparable with the radius of curvature of the interface (R), and the largest amplification occurs for a disturbance whose initial wavelength is approximately equal to Λc. With a plausible guess at the disturbance amplitude and wavelength at which bubble break-up due to nonlinear effects is inevitable, it is possible to obtain an approximate numerical relation between the initial magnitude of the disturbance and the maximum value of R for which a bubble remains intact. This relation applies both to a spherical-cap bubble in a large tank and a bubble rising in a vertical tube in which the liquid far ahead of the bubble is stationary. The few published observations of the maximum size of spherical-cap bubbles are not incompatible with the theory, but lack of information about the magnitude of the ambient disturbances in the liquid precludes any close comparison.

This paper is concerned with the spherically symmetric collapse of an empty cavity in water. The effects of viscosity and surface tension are neglected, but the compressibility of the water is allowed for and a suitable equation of state for the water assumed. The object of this is to clarify the effect of compressibility on the flow by considering it in isolation and thus to describe the formation of a pressure pulse by the collapse and its subsequent propagation.

The exact flow equations are integrated numerically and it is found that very high flow speeds develop in the neighbourhood of the collapse point. The radius of the cavity is found to be proportional to (-t)n for small t, where t = 0 is the instant of collapse, and n is some positive number less than unity.

The flow in the neighbourhood of the collapse point can be described by means of a similarity solution, and the value of n is determined by regularity properties of the similarity solution (the value of n depends on the equation of state assumed for the water). The similarity theory, which is valid only at very high pressures and velocities, can be continued beyond the instant of collapse to describe the formation of a shock wave after the collapse is completed, and the initial propagation of this shock.

We report an investigation of temperature profiles in turbulent Rayleigh–Bénard convection of water based on direct numerical simulations (DNS) for a cylindrical cell with unit aspect ratio for the same Prandtl number Pr and similar Rayleigh numbers Ra as used in recent high-precision measurements by Funfschilling et al. (J. Fluid Mech., vol. 536, 2005, p. 145). The Nusselt numbers Nu computed for Pr = 4.38 and Ra = 108, 3 × 108, 5 × 108, 8 × 108 and 109 are found to be in excellent agreement with the experimental data corrected for finite thermal conductivity of the walls. Based on this successful validation of the numerical approach, the DNS data are used to extract vertical profiles of the mean temperature. We find that near the heating and cooling plates the non-dimensional temperature profiles Θ(y) (where y is the non-dimensional vertical coordinate), obey neither a logarithmic nor a power law. Moreover, we demonstrate that the Prandtl–Blasius boundary layer theory cannot predict the shape of the temperature profile with an error less than 7.9% within the thermal boundary layers (TBLs). We further show that the profiles can be approximated by a universal stretched exponential of the form Θ(y) ≈ 1 − exp(−y − 0.5y2) with an absolute error less than 1.1% within the TBLs and 5.5% in the whole Rayleigh cell. Finally, we provide more accurate analytical approximations of the profiles involving higher order polynomials in the approximation.

In connection with the recent investigations of the instability of unbounded elliptical flows, some methods are discussed for the study of the instability of bounded flows. The stability of a ‘basic flow’ which is two-dimensional and rotating, with elliptical streamlines similar to the elliptical section of an experimentally studied cavity, is investigated in the framework of linear theory (for circular rotation, the flow discussed is stable). The regions of instability for three-dimensional disturbances are found in the plane of the parameters defining the geometry of the system (the height of the ellipsoidal cavity and the degree of ellipticity). It is shown that two types of instability exist, characterized by either monotone or oscillatory growth of the amplitudes of small disturbances.

The influence of the Coriolis force field on this instability mechanism is also studied. Rotation of the system as a whole changes the regions of instability in parameter space characterizing the geometry of the cavity and the wavenumbers of unstable disturbances. As a result, the Coriolis force may stabilize or destabilize the basic flow for a given geometry.

The instability of rotating density-stratified flow with elliptical streamlines is also considered.

The previously reported non-equilibrium dissipation law is investigated in turbulent flows generated by various regular and fractal square grids. The flows are documented in terms of various turbulent profiles which reveal their differences. In spite of significant inhomogeneity and anisotropy differences, the new non-equilibrium dissipation law is observed in all of these flows. Various transverse and longitudinal integral scales are measured and used to define the dissipation coefficient $C_{\varepsilon }$ . It is found that the new non-equilibrium dissipation law is not an artefact of a particular choice of the integral scale and that the usual equilibrium dissipation law can actually coexist with the non-equilibrium law in different regions of the same flow.

In constant area back pressured ducts, shock trains exhibit inherent unsteadiness where the shock system fluctuates about its time-averaged position despite constant bulk inflow and outflow conditions. In this work, the underlying causes of inherent unsteadiness are identified and the flow dynamics of the system is studied for a shock train in a Mach 2.0 ducted flow that is mechanically back pressured. High-speed schlieren movies and pressure measurements are collected to quantify the shock system fluctuations. Cross-spectral analysis of this data is used to identify specific perturbations, i.e. the fluid phenomena that impact the shock train motion. Key information about each perturbation is also obtained, including where it originates, what direction it travels and how it impacts each shock. Oil flow visualization and particle image velocimetry are then used to gain insight into the physical structure of perturbations and the flow phenomena that generate them. The results identify a complex, frequency-dependent dynamical system that is influenced by (i) upstream propagating acoustic waves that emanate from separation bubbles, (ii) vortices that shed from separation bubbles and convect downstream and (iii) upstream propagating acoustic waves generated in the diffuser. With this information, a scaling argument for the shock train inherent unsteadiness is presented.

Davey, Di Prima & Stuart's (1968) double amplitude expansion for disturbances in flow between concentric cylinders is formulated in matrix notation. The stability of the secondary equilibrium (Taylor-vortex) flow is calculated using fifth-order terms in amplitude, and using the full equations rather than the small-gap approximation. Qualitative confirmation is found of instabilities to the Taylor-vortex flow to non-a.xisymmetric disturbances at about 10 % above the first critical Taylor number.

A model is developed for turbulent natural convection in boundary layers formed next to isothermal vertical surfaces. A scaling analysis shows that the flow can be described by plume equations for an outer turbulent region coupled to a resolved near-wall laminar flow. On the laboratory scale, the inner layer is dominated by its own buoyancy and the Nusselt number scales as the one-third power of the Rayleigh number (Nu ∝ ). This gives a constant heat flux, consistent with previous experimental and theoretical studies. On larger geophysical scales the buoyancy is strongest in the outer layer and the laminar layer is driven by the shear imposed on it. The predicted heat transfer correlation then has the Nusselt number proportional to the one-half power of Rayleigh number (Nu ∝ ) so that a larger heat flux is predicted than might be expected from an extrapolation of laboratory-scale results. The criteria for transitions between flow regimes are consistent with a hierarchy of instabilities of the near-wall laminar flow, with a buoyancy-driven instability operating on the laboratory scale and a shear-driven instability operating on geophysical scales.

The properties of steady and unsteady ship waves propagating on a free surface discretized with panels are studied. The wave propagation is characterized by an explicit discrete dispersion relation which allows the systematic analysis of the distortion of the wave pattern due to discretization and the derivation of a stability criterion to be met by the numerical algorithm. The conclusions of the study are applied to a panel method used for the computation of steady and time-harmonic free-surface flows past elementary singularities and a ship hull.

Two-point hot-wire measurements of streamwise velocity were performed in the logarithmic and wake regions of turbulent pipe flow for Reynolds numbers, based on pipe diameter, ranging from 7.6 × 104 to 8.3 × 106 at four wall-normal positions with azimuthal probe separation. The azimuthal correlations were found to be consistent with the presence of very large-scale coherent regions of low-wavenumber, low-momentum fluid observed in previous studies of wall-bounded flows and were found to be independent of changing Reynolds number and surface roughness effects. At the edge of the logarithmic layer the azimuthal scale determined from the correlations was found to be similar to that observed for channel flows but larger than that observed for boundary layers, inconsistent with the concept of a universal logarithmic region. As the wall-normal position increased outside the logarithmic layer, there was a decrease in azimuthal scale relative to that of channel flow. Using cross-spectral analysis, high-wavenumber motion was found to grow azimuthally with wall-normal distance at a faster rate than the low-wavenumber motions.

This paper investigates the non-uniform motion of a thin plate of finite aspect ratio, with a rounded leading edge and sharp trailing edge, executing heaving and pitching oscillations at zero mean lift. Such vertical motions characterize the horizontal lunate tails with which cetacean mammals propel themselves, and the same motions, turned through 90° to become horizontal motions of sideslip and yaw, characterize the vertical lunate tails of certain fast-swimming fishes. An oscillating vortex sheet consisting of streamwise and spanwise components is shed to trail behind the body and it is this additional feature of the streamwise component resulting from the finiteness of the plate that makes this study a generalization of the two-dimensional treatment of lunate-tail propulsion by Lighthill (1970). The forward thrust, the power required, the energy imparted to the wake and the hydromechanical propulsive efficiency are determined for this general motion as functions of the physical parameters defining the problem: namely the aspect ratio, the reduced frequency, the feathering parameter and the position of the pitching axis. The dependence of the thrust coefficient and propulsive efficiency on these physical parameters, for the complete range of variation consistent with the assumptions of the problem, has been depicted graphically.