The flow around two stationary cylinders in tandem arrangement at the laminar and early turbulent regime, ($\hbox{\it Re}\,{=}\,10^2$–$10^3$), is studied using two- and three-dimensional direct numerical simulations. A range of spacings between the cylinders from 1.1 to 5.0 diameters is considered with emphasis on identifying the effects of three-dimensionality and cylinder spacing as well as their coupling. To achieve this, we compare the two-dimensional with corresponding three-dimensional results as well as the tandem cylinder system results with those of a single cylinder. The critical spacing for vortex formation and shedding in the gap region depends on the Reynolds number. This dependence is associated with the formation length and base pressure suction variations of a single cylinder with Reynolds number. This association is useful in explaining some of the discrepancies between the two-dimensional and three-dimensional results. A major effect of three-dimensionality is in the exact value of the critical spacing, resulting in deviations from the two-dimensional predictions for the vorticity fields, the forces on the downstream cylinder, and the shedding frequency of the tandem system. Two-dimensional simulations under-predict the critical spacing, leading to erroneous results for the forces and shedding frequencies over a range of spacings where the flow is qualitatively different. To quantify the three-dimensional effects we first employ enstrophy, decomposed into a primary and a secondary component. The primary component involves the vorticity parallel to the cylinder axis, while the secondary component incorporates the streamwise and transverse components of the vorticity vector. Comparison with the single cylinder case reveals that the presence of the downstream cylinder at spacings lower than the critical value has a stabilizing effect on both the primary and secondary enstrophy. Systematic quantification of three-dimensionalities involves finding measures for the intensity of the spanwise fluctuations of the forces. This also verifies the stabilization scenario, suggesting that when the second cylinder is placed at a distance smaller than the critical one, three-dimensional effects are suppressed compared to the single-cylinder case. However, when the spacing exceeds the critical value, the upstream cylinder tends to behave like a single cylinder, but three-dimensionality in the flow generally increases.

A set of complete two- and three-dimensional direct numerical simulations (DNS) in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is presented in this paper. Although the physical phenomenon is three-dimensional, owing to its prohibitive computational costs the majority of the previous DNS of turbulent and transition natural convection flows in enclosed cavities assumed a two-dimensional behaviour. The configurations selected here (Rayleigh number based on the cavity height 6.4 × 108, 2 × 109 and 1010, Pr = 0.71) are an extension to three dimensions of previous two-dimensional problems.

An overview of the numerical algorithm and the methodology used to verify the code and the simulations is presented. The main features of the flow, including the time-averaged flow structure, the power spectra and probability density distributions of a set of selected monitoring points, the turbulent statistics, the global kinetic energy balances and the internal waves motion phenomenon are described and discussed.

As expected, significant differences are observed between two- and three-dimensional results. For two-dimensional simulations the oscillations at the downstream part of the vertical boundary layer are clearly stronger, ejecting large eddies to the cavity core. In the three-dimensional simulations these large eddies do not persist and their energy is rapidly passed down to smaller scales of motion. It yields on a reduction of the large-scale mixing effect at the hot upper and cold lower regions and consequently the cavity core still remains almost motionless even for the highest Rayleigh number. The boundary layers remain laminar in their upstream parts up to the point where these eddies are ejected. The point where this phenomenon occurs clearly moves upstream for the three-dimensional simulations. It is also shown that, even for the three-dimensional simulations, these eddies are large enough to permanently excite an internal wave motion in the stratified core region. All these differences become more marked for the highest Rayleigh number.

A laminar Mach 6 flat plate boundary layer is perturbed using three different types of disturbances introduced through blowing and suction. The linear and nonlinear development and eventual breakdown to turbulence are investigated using direct numerical simulation. The three different transition mechanisms compared are first mode oblique breakdown, second mode oblique breakdown and second mode fundamental resonance. The focus of the present work is to compare the nonlinear development and breakdown to turbulence for the different transition mechanisms and explain the heat transfer overshoot observed in experiments. First mode oblique breakdown leads to the shortest transition length and a clear peak in wall heat transfer in the transitional region. For all three transition mechanisms, the development of streamwise streaks precedes the breakdown to fully turbulent flow. The modal linear and nonlinear development are analysed including the breakdown of the streaks. The effect of wall cooling is investigated for second mode fundamental resonance and no qualitative differences in the nonlinear processes are observed. Finally, the development towards fully turbulent flow including mean flow, turbulent spectra, and turbulent fluctuations is shown and the first mode oblique breakdown simulation shows the furthest development towards a fully turbulent flow.

Forcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier–Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifth- and sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. Fluid Mech., vol. 289, 1995, pp. 179–198) and Malenica & Molin (J. Fluid Mech., vol. 302, 1995, pp. 203–229) was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.

Air flowing over a liquid surface encounters an increased resistance if waves are present. The relation of this increased resistance to the properties of the waves has been studied. Air and a liquid flowed co-currently in an enclosed channel which is 12 in. wide and 1 in. high and which is long enough so that flow in the air and the liquid and the interfacial structure are fully developed. The drag on interfaces with three-dimensional wave structures was found to increase with the square of the gas velocity and to depend more on the height of the waves than on other parameters characterizing the interface. The ratio of the equivalent sand roughness to the root-mean-square of the fluctuations in the height of the liquid film is approximately equal to 3 √2. The velocity profiles in the gas were found to be different from what has been reported for flows over sand roughened surfaces.

The equations governing the free-surface flow of a tubular film of liquid are derived from physical arguments, which throw some light on the formal process described in part 1. The solutions of the equations are discussed, in particular with reference to the film-blowing process for the manufacture of thin sheets of thermoplastic material. The qualitative adequacy of a model based on the dominance of viscous forces is demonstrated, and the effect of surface tension, air drag and non-isothermal flow is discussed briefly.

In order to elucidate the turbulent structure below a shear-free gas-liquid interface, turbulence measurements were made in a 50 cm square by 40 cm deep tank stirred by a vertically oscillating grid well below the surface, using a split-film anemometer probe rotating in a horizontal circle. This instrument is able to measure both vertical and horizontal velocity fluctuations to within 0.4 mm of the surface, from which spatial spectra and profiles of r.m.s. velocity fluctuations and integral lengthscales can be calculated. The turbulent structure is affected by the presence of the surface within a ‘surface-influenced layer’ roughly one integral scale, or ten per cent of the distance from the surface to the centre of the grid stroke, in thickness. The shapes of the spectra and profiles within the surface-influenced layer are predicted to a good first approximation by the source theory of Hunt & Graham (1978), which treats the turbulent structure as the superposition of homogeneous turbulence with an irrotational velocity field driven by a source distribution at the surface which cancels the vertical velocity fluctuations there. The magnitudes (as opposed to the shapes) of the profiles scale according to the values that would otherwise occur in the vicinity of the surface-influenced layer were the surface not present. These magnitudes are adequately predicted by the bulk relations determined by Hopfinger & Toly (1976) and Thompson & Turner (1975), with no apparent dependence on turbulent Reynolds number. There are some minor discrepancies between the measured profiles and those of Hunt & Graham. A thin layer of reduced velocity fluctuations below what would be expected from the theory was observed near the surface. Also, anisotropy in the velocity spectra at depths within the surface-influenced layer extended well into the inertial subrange, whereas the Hunt & Graham theory predicts no anisotropy at high wavenumbers.

Work recently reported by the authors (Perry & Abell 1975) on smooth-walled pipe flow showed support for the Townsend (1976) structural similarity principle as regards viscosity not being directly relevant in controlling the mean relative motions and the energy-containing turbulent motions. The work also supported a universal spectral behaviour in the wall region of the flow. In many hypotheses for rough-walled pipe flow, surface roughness, like viscosity, enters the problem only via external boundary conditions. Data obtained in a rough pipe are reported here and on first appearance the results seem to contradict the Townsend hypothesis and to threaten the very foundation upon which many similarity laws for rough-walled flows are based. However, on closer examination of the spectrum scaling of smooth-walled pipe flow the low and high wavenumber energy not necessarily associated with the universal similarity range can be accounted for. The broad-band longitudinal turbulence results for a rough-walled pipe can then be predicted from the smooth-wall scaling. The conclusion is that, despite the apparent anomalies, the turbulence structure in a rough pipe appears to follow the same scaling laws as for a smooth pipe, given a sufficient length of flow development in both cases. The deduced functional forms are consistent with Townsend's (1976) attached-eddy hypothesis.

Sir Charles Darwin has advocated a study of the ‘drift’ of material surfaces in the classically investigated irrotational flows past bodies. This suggestion is followed up and given further support in the present paper. In particular, it is shown how secondary flows can be evaluated by use of the ‘drift function’ t for the primary flow. This is a function (§1) such that material surfaces initially at right angles to the stream drift into shapes expressible by equations t=constant.

The analysis leads to a simple expression (§4) for the secondary velocity field in the flow past an infinite cylinder of any cross-section, with the upstream velocity normal to its axis and increasing linearly with distance along the axis-a problem in which only the secondary vorticity field was previously known. The drift past a sphere is computed and illustrated (§6), and the secondary vorticity field in shear flow past a sphere is tabulated (§7). There is also a detailed study (§3) of the asymptotic form of the secondary velocity field in flow past any body, based on a result of Darwin concerning ‘hydrodynamic mass’.

Premixed turbulent combustion in the flamelet regime is analysed on the basis of a field equation. This equation describes the instantaneous flame contour as an isoscalar surface of the scalar field G(x,t). The field equation contains the laminar burning velocity sL as velocity scale and its extension includes the effect of flame stretch involving the Markstein length [Lscr ] as a characteristic lengthscale of the order of the flame thickness. The scalar G(x,t) plays a similar role for premixed flamelet combustion as the mixture fraction Z(x,t) in the theory of non-premixed flamelet combustion.

Equations for the mean $\overline{G}$ and variance $\overline{G^{\prime 2}}$ are derived. Additional closure problems arise for the mean source terms in these equations. In order to understand the nature of these terms an ensemble of premixed flamelets with arbitrary initial conditions in constant-density homogeneous isotropic turbulence is considered. An equation for the two-point correlation $\overline{G^{\prime}({\boldmath x},t)G^{\prime}({\boldmath x}+{\boldmath r},t)}$ is derived. When this equation is transformed into spectral space, closure approximations based on the assumption of locality and on dimensional analysis are introduced. This leads to a linear equation for the scalar spectrum function Γ(k,t), which can be solved analytically. The solution Γ(k,t) is analysed by assuming a small-wavenumber cutoff at k0 = lT−1, where lT is the integral lengthscale of turbulence. There exists a $k^{-\frac{5}{3}}$ spectrum between lT and LG, where LG is the Gibson scale. At this scale turbulent fluctuations of the scalar field G(x,t) are kinematically restored by the smoothing effect of laminar flame propagation. A quantity called kinematic restoration ω is introduced, which plays a role similar to the scalar dissipation χ for diffusive scalars.

By calculating the appropriate moments of Γ(k,t), an algebraic relation between ω, $\omega,\overline{G^{\prime}({\boldmath x},t)^2}$, the integral lengthscale lT and the viscous dissipation ε is derived. Furthermore, the scalar dissipation χ[Lscr ], based on the Markstein diffusivity [Dscr ][Lscr ] = sL [Lscr ], and the scalar-strain co-variance Σ[Lscr ] are related to ω. Dimensional analysis, again, leads to a closure of the main source term in the equation for the mean scalar $\overline{G}$. For the case of plane normal and oblique turbulent flames the turbulent burning velocity sT and the flame shape is calculated. In the absence of flame stretch the linear relation sT ∼ u′ is recovered. The flame brush thickness is of the order of the integral lengthscale. In the case of a V-shaped flame its increase with downstream position is calculated.

The Navier–Stokes equations are solved numerically for two-dimensional peristaltic flows by using the finite-difference technique employing the upwind SOR method, and the velocity, pressure and stress fields for various peristaltic flows are obtained. The influences of the magnitudes of wave amplitude, wavelength and Reynolds number on the flow are investigated through numerical calculations, and the results are compared with those of the perturbation analysis. The paper is mainly concerned with elucidating the characteristics of the peristaltic flow at moderate Reynolds numbers where peristaltic pumping has a possibility of engineering application. As a result, it is found that the validity of the perturbation solutions by Jaffrin (1973) and Zien & Ostrach (1970) are restricted within a narrower range than that which they predicted, and that the reflux phenomenon in the flow does change the whole situation according to Reynolds number.

A study is made of hydromagnetic oscillations in a rotating fluid sphere. The basic state is chosen as a uniform current parallel to the axis of rotation. It is found that the non-dissipative normal modes are described by a modified form of the Poincaré eigenvalue problem. For small rotation rates, the lowest non-axisymmetric modes are unstable. For rotation rates of geophysical interest all normal modes are stable. The introduction of ohmic dissipation leads to a hydro-magnetic boundary-layer problem. Solutions for the boundary layer are outlined indicating its role in altering the free periods, damping the oscillations and producing external magnetic fields. Dispersion relations are derived which establish that the zonal phase velocities of both ‘fast’ hydrodynamic and ‘slow’ hydro-magnetic waves can be of either sign. Observations of the secular variations of the earth's magnetic field indicate motion primarily towards the west. A mechanism for the selective excitation of the observed motion is discussed.

Water drops falling on a deep pool can either coalesce to form a vortex ring or splash, depending on the impact conditions. The transition between coalescence and splashing proceeds via a number of intermediate steps, such as thick and thin jet formation and gas-bubble entrapment. We perform simulations to determine the conditions under which bubble entrapment and jet formation occur. A regime map is established for Weber numbers ranging from 50 to 300 and Froude numbers from 25 to 600. Vortex ring formation is seen for all of the regimes; it is greater for the coalescence regime and less in the case of the thin jet regime.

In the present paper we formulate a predictive theory of the formation of sand ripples under sea waves. The theory is based on a linear stability analysis of a flat sandy bottom subject to a viscous oscillatory flow. The conditions for decay or amplification of a bottom perturbation are determined along with the wavelength of the most unstable component as a function of the Reynolds number of the flow and of the Froude and Reynolds numbers of the sediments. A comparison between theoretical findings and experimental data supports the validity of the present theory. An analytical solution for viscous oscillatory flow over a small-amplitude wavy bottom is determined for arbitrary values of the ratio r between the amplitude of fluid displacement and the wavelength of bottom waviness. Previous works by Lyne (1971) and Sleath (1976), who considered small or large values of r, are thus extended.

In the presence of an air stream, a uniform liquid film on a horizontal flat plate may be unstable to small disturbances, and waves may arise. In this paper the hydrodynamic stability of thin liquid films is examined both experimentally and theoretically.

The experiments concern water films thinner than those which have been examined in the past. It is found that, when the film thickness is sufficiently small, a previously unknown type of instability occurs. The theoretical analysis explains this surprising phenomenon.

Due to interaction of the mean airflow and small disturbances of the liquid-air interface, normal and tangential stress perturbations are produced at the liquid surface. It is shown that small wave-like disturbances become unstable when the joint influence of the component of normal stress in phase with the wave elevation and the component of tangential stress in phase with the wave slope is sufficient to overcome the ‘stiffness’ of the liquid surface due to gravity and surface tension. It is found that the destabilizing role of the tangential stress component is dominant for very thin films, and that instability may occur whatever the velocity of the air stream, provided the film is made sufficiently thin.

Red blood cells or artificial vesicles may be conveniently represented by capsules, i.e. liquid droplets surrounded by deformable membranes. The aim of this paper is to assess the importance of viscoelastic properties of the membrane on the motion of a capsule freely suspended in a viscous liquid subjected to shear flow. A regular perturbation solution of the general problem is obtained when the particle is initially spherical and undergoing small deformations. With a purely viscous membrane (infinite relaxation time) the capsule deforms into an ellipsoid and has a continuous flipping motion. When the membrane relaxation time is of the same order as the shear time, the particle reaches a steady ellipsoidal shape which is oriented with respect to streamlines at an angle that varies between 45° and 0°, and decreases with increasing shear rates. Furthermore it is predicted that the deformation reaches a maximum value, which is consistent with experimental observations of red blood cells.

Small spheroidal particles suspended in a sheared viscous liquid are sometimes observed to take up slowly preferred orientations, relative to the motion of the undisturbed liquid, which are independent of the initial conditions of release. These obsevations cannot be accounted for by the solution, obtained by Jeffery (1922), of the linearized Navier-Stokes equations. It is shown in this paper that the effect of the inertia of the liquid is to alter slowly the orbit of the particle in accordance with Jeffery's hypothesis that the particle ultimately moves in such a way that the dissipation of energy is a minimum, but that this effect is orders of magnitude too small to account for any of the experimental observations.

It is suggested that non-Newtonian properties of the liquid account for the observations. It is shown that the rate of orientation of a particle would then be independent of its size, and this prediction is verified experimentally. Other experimental evidence in support of this suggestion is also described.

Some remarks are also made about the possible effect of collisions between the particles when more than one particle is present.

The inviscid instability of a longitudinal vortex structure within a steady boundary layer is investigated. The instability has wavelength comparable with the boundary-layer thickness so that a quasi-parallel approach to the instability problem can be justified. The generalization of the Rayleigh equation to such a flow is obtained and solved for the case when the vortex structure is induced by curvature. Two distinct modes of instability are found; these modes correspond with experimental observations on the breakdown process for Görtler vortices.

The first and higher harmonic components of the resonant fluid response in the gap between two identical fixed rectangular boxes are experimentally investigated in a wave basin. Gap response is excited by transient wave groups (being based on scaled versions of the autocorrelation function of sea-state spectra, representing NewWaves, the average shape of large waves in a sea state). Several different wave groups with different maximum surface elevations, spectral peak frequencies and bandwidths are used, while the bilge shape of the boxes and approach angle of the waves are also varied. Unlike a simple regular wave, it is complicated to separate the harmonic components for a transient wave group due to nonlinear wave–wave and wave–structure interactions. A four-phase combination methodology is used to separate the first four harmonic components, and this also allows higher harmonic components to be isolated with simple digital frequency filtering. Harmonic components up to 14th order in the incident wave amplitude have been extracted. It is shown that for an incident group with appropriate frequency content, the linear gap response may be substantially smaller than the second harmonic component, which is strongly driven via quadratic coupling of the linear terms from the incident wave and occurs in the gap resonant modes. Double frequency excitation may have important practical implications for offshore operations. Fourth and zeroth (long-wave) harmonics in the gap are further driven via quadratic coupling of the second harmonic itself. Linear damping coefficients for the first few modes of the gap resonant response are derived from measured time series using a numerical fit and shown to be higher than those from linear diffraction calculations.

It is shown that the dispersion of a substance, with molecular diffusivity κ, in a stationary, homogeneous, turbulent velocity field can be formulated in terms of a ‘substance auto-correlation function’, this being a generalization of the well-known Lagrangian correlation between the velocity of a fluid particle at different times. It is found that the interaction between the molecular diffusion and the turbulent motion reduces the dispersion from the value it would have if the processes of molecular and turbulent diffusion were independent and additive. The conflict, between the results obtained in this paper and previous results which implied that the interaction increases the dispersion, is resolved. The ratio, of the contributions to the dispersion from the interaction term and the turbulent diffusion term, is obtained for comparatively large times by the use of intuitive arguments, and is found to be inversely proportional to the Prandtl number and a Reynolds number of the turbulence.