When a small air bubble bursts from an equilibrium position at an air/water interface, a complex motion ensues resulting in the production of a high-speed liquid jet. This free-surface motion following the burst is modelled numerically using a boundary integral method. Jet formation and liquid entrainment rates from jet breakup into drops are calculated and compared with existing experimental evidence. In order to investigate viscous effects, a boundary layer is included in the calculations by employing a time-stepping technique which allows the boundary mesh to remain orthogonal to the surface. This allows an approximation of the vorticity development in the region of boundary-layer separation during jet formation. Calculated values of pressure and energy dissipation rates in the fluid indicate a violent motion, particularly for smaller bubbles. This has important implications for the biological industry where animal cells in bioreactors have been found to be killed by the presence of small bubbles.

A theoretical method is presented for predicting the deformation and the conditions for breakup of a liquid droplet freely suspended in a general linear shear field. This is achieved by expanding the solution to the creeping-flow equations in powers of the deformation parameter ε and using linear stability theory to determine the onset of bursting. When compared with numerical solutions and with the available experimental data, the theoretical results are generally found to be of acceptable accuracy although, in some cases, the agreement is only qualitative.

The bursting frequency in turbulent boundary layers has been measured over the Reynolds-number range 103 < U∞/ν < 104. When scaled with the variables appropriate for the wall region, the non-dimensional frequency was constant independent of Reynolds number. A strong effect of the sensor size was noted on the measured bursting frequency. Only sensors having a spatial scale less than twenty viscous lengthscales were free from spatial-averaging effects and yielded consistent results. The spatial-resolution problem was apparently the reason for erroneous results reported in the past.

The ‘preferred mode’ of an incompressible axisymmetric free jet has been organized through controlled perturbation, and spatial distributions of time-average as well as phase-average flow properties in the near field are documented. The excitation produces noticeable changes in the time-average measures of the jet, although these changes are less dramatic than those for the excitation producing stable vortex pairing. For different stages in the evolution of the preferred-mode coherent structure, the phase-average vorticity, coherent Reynolds stress, and incoherent turbulence intensities and Reynolds stress have been educed through phase-locked hot-wire measurements, over the spatial extent of the structure and without invoking the Taylor hypothesis. For a particular stage of the evolution (i.e. when the structure is centred at x/D ≃ 3) the distributions of these quantities have been compared for both initially laminar and fully turbulent exit boundary layers, and for four jet Reynolds numbers. The relative merits of the coherent structure streamline and pseudo-stream-function patterns, as compared with phase-average velocity contours, for structure boundary identification have been discussed. The structure shape and size agree closely with those inferred from the average streamline pattern of the natural structure educed by Yule (1978).

These data as well as τ-spectra show that even excitation at the preferred mode cannot sustain the initially organized large-scale coherent structure beyond eight diameters from the jet exit. The background turbulence is organized by the coherent motions in such a way that the maximum rate of decrease of the coherent vorticity occurs at the structure centres which are the saddle points of the background-turbulence Reynolds-stress distributions. The structure centres are also the locations of peak phase-average turbulence intensities. The evolving shape of the structure as it travels downstream helps explain the transverse variations of the wavelength and convection velocity across the mixing layer. The coherent structure characteristics are found to be independent of whether the initial boundary layer is laminar or turbulent, but depend somewhat on the jet Reynolds number. With increasing Reynolds number, the structure decreases in the streamwise length and increases in the radial width and becomes relatively more energetic, and more efficient in the production of coherent Reynolds stress.

Numerical experiments are conducted to study high-Rayleigh-number convective turbulence ($Ra$ ranging from $2\times 10^6$ up to $2\times 10^{11}$) in a $\Gamma=1/2$ aspect-ratio cylindrical cell heated from below and cooled from above and filled with gaseous helium ($Pr=0.7$). The numerical approach allows three-dimensional velocity, vorticity and temperature fields to be analysed. Furthermore, several numerical probes are placed within the fluid volume, permitting point-wise velocity and temperature time series to be extracted. Taking advantage of the data accessibility provided by the direct numerical simulation the flow dynamics has been explored and separated into its mean large-scale and fluctuating components, both in the bulk and in the boundary layer regions. The existence of large-scale structures creating a mean flow sweeping the horizontal walls has been confirmed. However, the presence of a single recirculation cell filling the whole volume was observed only for $Ra < 10^9 - 10^{10}$ and with reduced intensity compared to axisymmetric toroidal vortices attached to the horizontal plates. At larger $Ra$ the single cell is no longer observed, and the bulk recirculation breaks up into two counter-rotating asymmetric unity-aspect-ratio rolls. This transition has an appreciable impact on the boundary layer structure and on the global heat transfer properties. The large-scale structure signature is reflected in the statistics of the bulk turbulence as well, which, taking advantage of the large number of numerical probes available, is examined both in terms of frequency spectra and of temperature structure functions. The present results are also discussed within the framework of recent theoretical developments showing that the effect of the aspect ratio on the global heat transfer properties at large $Ra$ still remains an open question.

The motion of the head of a gravity current travelling down a slope of angle θ to the horizontal is investigated in the laboratory. The head is produced by suddenly initiating a buoyancy flux from a line source at the top of the slope. It is found that for very small slopes (θ [les ] 0.5°) the head decelerates with distance from the source, but at greater slopes the buoyancy force is large enough to overcome frictional effects and a steady head velocity results. Over a wide range of slope angles the front velocity Uf, non-dimensionalized by the cube root of the buoyancy flux (g′0Q)1/3, is almost independent of the slope angle and Uf/(g’0Q)1/3 = 1.5 ± 0.2 for 5° [les ] θ [les ] 90°. This result is shown to follow from some simple analysis which relates the velocity of the front to the following flow. For a Boussinesq plume the front velocity is found to be approximately 60% of the mean velocity of the following flow. This means that the head increases in size as it travels down the slope, both by direct entrainment into the head itself and by addition of fluid from the following flow. We find that direct entrainment increases with increasing slope and accounts for one-tenth of the growth of the head at 10° and about two-thirds at 90°.

A free surface may be deformed by fluid motions; such deformation may lead to surface roughness, breakup, or disintegration. This paper describes the wide range of free-surface deformations that occur when there is turbulence at the surface, and focuses on turbulence in the denser, liquid, medium. This turbulence may be generated at the surface as in breaking water waves, or may reach the surface from other sources such as bed boundary layers or submerged jets. The discussion is structured by consideration of the stabilizing influences of gravity and surface tension against the disrupting effect of the turbulent kinetic energy. This leads to a two-parameter description of the surface behaviour which gives a framework for further experimental and theoretical studies. Much of the discussion is necessarily heuristic, and is often limited by a lack of appropriate experimental observations. It is intended that such experiments be stimulated, to test the value or otherwise of our two-parameter description.

An analysis is presented of the dynamics of a small deformable capsule freely suspended in a viscous fluid undergoing shear. The capsule consists of an elastic membrane which encloses another viscous fluid, and it deforms in response to the applied external stresses and the elastic forces generated within the membrane. Equations are derived which give its time-dependent deformation in the limit that the departure of the shape from sphericity is small. The form of the shear flow is arbitrary and a general (two-dimensional) elastic material is considered. Limiting forms are obtained for highly viscous capsules and for membranes which are area-preserving, and earlier results for surface tension droplets and incompressible isotropic membranes are derived as particular cases. Results for the viscosity of a dilute suspension of capsules are also given.

The theoretical prediction for the relaxation rate of the shape is derived for an interface which has elastic properties appropriate for a red-blood-cell membrane, and is compared with experimental observations of erythrocytes.

Effects of chordwise, spanwise, and isotropic flexibility on the force generation and propulsive efficiency of flapping wings are elucidated. For a moving body immersed in viscous fluid, different types of forces, as a function of the Reynolds number, reduced frequency (k), and Strouhal number (St), acting on the moving body are identified based on a scaling argument. In particular, at the Reynolds number regime of and the reduced frequency of , the added mass force, related to the acceleration of the wing, is important. Based on the order of magnitude and energy balance arguments, a relationship between the propulsive force and the maximum relative wing-tip deformation parameter () is established. The parameter depends on the density ratio, St, k, natural and flapping frequency ratio, and flapping amplitude. The lift generation, and the propulsive efficiency can be deduced by the same scaling procedures. It seems that the maximum propulsive force is obtained when flapping near the resonance, whereas the optimal propulsive efficiency is reached when flapping at about half of the natural frequency; both are supported by the reported studies. The established scaling relationships can offer direct guidance for micro air vehicle design and performance analysis.

Flying insects and robots that mimic them flap and rotate (or ‘pitch’) their wings with large angular amplitudes. The reciprocating nature of flapping requires rotation of the wing at the end of each stroke. Insects or flapping-wing robots could achieve this by directly exerting moments about the axis of rotation using auxiliary muscles or actuators. However, completely passive rotational dynamics might be preferred for efficiency purposes, or, in the case of a robot, decreased mechanical complexity and reduced system mass. Herein, the detailed equations of motion are derived for wing rotational dynamics, and a blade-element model is used to supply aerodynamic force and moment estimates. Passive-rotation flapping experiments with insect-scale mechanically driven artificial wings are conducted to simultaneously measure aerodynamic forces and three-degree-of-freedom kinematics (flapping, rotation and out-of-plane deviation), allowing a detailed evaluation of the blade-element model and the derived equations of motion. Variations in flapping kinematics, wing-beat frequency, stroke amplitude and torsional compliance are made to test the generality of the model. All experiments showed strong agreement with predicted forces and kinematics, without variation or fitting of model parameters.

Transition arising from a separated region of flow is quite common and plays an important role in engineering. It is difficult to predict using conventional models and the transition mechanism is still not fully understood. We report the results of a numerical simulation to study the physics of separated boundary-layer transition induced by a change of curvature of the surface. The geometry is a flat plate with a semicircular leading edge. The Reynolds number based on the uniform inlet velocity and the leading-edge diameter is 3450. The simulated mean and turbulence quantities compare well with the available experimental data.

The numerical data have been comprehensively analysed to elucidate the entire transition process leading to breakdown to turbulence. It is evident from the simulation that the primary two-dimensional instability originates from the free shear in the bubble as the free shear layer is inviscidly unstable via the Kelvin–Helmholtz mechanism. These initial two-dimensional instability waves grow downstream with a amplification rate usually larger than that of Tollmien–Schlichting waves. Three-dimensional motions start to develop slowly under any small spanwise disturbance via a secondary instability mechanism associated with distortion of two-dimensional spanwise vortices and the formation of a spanwise peak–valley wave structure. Further downstream the distorted spanwise two-dimensional vortices roll up, leading to streamwise vorticity formation. Significant growth of three-dimensional motions occurs at about half the mean bubble length with hairpin vortices appearing at this stage, leading eventually to full breakdown to turbulence around the mean reattachment point. Vortex shedding from the separated shear layer is also observed and the ‘instantaneous reattachment’ position moves over a distance up to 50% of the mean reattachment length. Following reattachment, a turbulent boundary layer is established very quickly, but it is different from an equilibrium boundary layer.

We present here a brief description of a numerical technique suitable for solving axisymmetric (or two-dimensional) free-boundary problems of fluid mechanics. The technique is based on a finite-difference solution of the equations of motion on an orthogonal curvilinear coordinate system, which is also constructed numerically and always adjusted so as to fit the current boundary shape. The overall solution is achieved via a global iterative process, with the condition of balance between total normal stress and the capillary pressure at the free boundary being used to drive the boundary shape to its ultimate equilibrium position.

Unsteady flow fields over a circular cylinder are used for training and then prediction using four different deep learning networks: generative adversarial networks with and without consideration of conservation laws; and convolutional neural networks with and without consideration of conservation laws. Flow fields at future occasions are predicted based on information on flow fields at previous occasions. Predictions of deep learning networks are made for flow fields at Reynolds numbers that were not used during training. Physical loss functions are proposed to explicitly provide information on conservation of mass and momentum to deep learning networks. An adversarial training is applied to extract features of flow dynamics in an unsupervised manner. Effects of the proposed physical loss functions and adversarial training on predicted results are analysed. Captured and missed flow physics from predictions are also analysed. Predicted flow fields using deep learning networks are in good agreement with flow fields computed by numerical simulations.

A converging channel of area ratio 34 : 1 has been developed to produce strong shock waves in a tube. Shock waves of speeds M1=3−7·5 in oxygen and M1 = 5−10 in nitrogen have now been studied with an interferometer, and values of the relaxation time τ for the approach to vibrational equilibrium behind the shocks have been measured. The value of τ (atmospheric) varies from 54μsec at 800° K to 1·3 μsec at 3000° K for oxygen and from 19 μsec at 3500° K to 5 μsec at 5500° K for nitrogen. For oxygen, the graph of log τ against T−1/3 is not quite the straight line predicted by the Landau-Teller theory. The density ratios across the shocks were measured and compared with values calculated by the Bethe-Teller method for variable specific heats. Agreement between the measured and calculated values is satisfactory. Experiments were also performed on oxygen-nitrogen mixtures to determine the effect of nitrogen on the approach to equilibrium of the oxygen. It was found that O2 and N2 collisions at approximately 2000° K are 40% as effective in transferring energy to the oxygen as O2 and O2 collisions. A device that detects a shock with a time lag of less than 1μsec, consisting of an evaporated gold film which changes its resistance when heated by the shock, was also developed.

The evolution of three-dimensional temporally evolving plane mixing layers through as many as three pairings has been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity. Three-dimensional perturbations were used with amplitudes ranging from infinitesimal to large enough to trigger a rapid transition to turbulence. Pairing is found to inhibit the growth of infinitesimal three-dimensional disturbances, and to trigger the transition to turbulence in highly three-dimensional flows. The mechanisms responsible for the growth of three-dimensionality and onset of transition to turbulence are described. The transition to turbulence is accompanied by the formation of thin sheets of spanwise vorticity, which undergo secondary rollups. The post-transitional simulated flow fields exhibit many properties characteristic of turbulent flows.

Direct numerical simulations are performed of a confined three-dimensional, temporally developing, initially isothermal gas mixing layer with one stream laden with as many as 7.3×105 evaporating hydrocarbon droplets, at moderate gas temperature and subsonic Mach number. Complete two-way phase couplings of mass, momentum and energy are incorporated which are based on a thermodynamically self-consistent specification of the vapour enthalpy, internal energy and latent heat of vaporization. Effects of the initial liquid mass loading ratio (ML), initial Stokes number (St0), initial droplet temperature and flow three-dimensionality on the mixing layer growth and development are discussed. The dominant parameter governing flow modulation is found to be the liquid mass loading ratio. Variations in the initial Stokes number over the range 0.5[les ]St0[les ]2.0 do not cause significant modulations of either first- or second-order gas phase statistics. The mixing layer growth rate and kinetic energy are increasingly attenuated for increasing liquid loadings in the range 0[les ]ML[les ]0.35. The laden stream becomes saturated before evaporation is completed for all but the smallest liquid loadings owing to: (i) latent heat effects which reduce the gas temperature, and (ii) build up of the evaporated vapour mass fraction. However, droplets continue to be entrained into the layer where they evaporate owing to contact with the relatively higher-temperature vapour-free gas stream. The droplets within the layer are observed to be centrifuged out of high-vorticity regions and to migrate towards high-strain regions of the flow. This results in the formation of concentration streaks in spanwise braid regions which are wrapped around the periphery of secondary streamwise vortices. Persistent regions of positive and negative slip velocity and slip temperature are identified. The velocity component variances in both the streamwise and spanwise directions are found to be larger for the droplets than for the gas phase on the unladen stream side of the layer; however, the cross-stream velocity and temperature variances are larger for the gas. Finally, both the mean streamwise gas velocity and droplet number density profiles are observed to coincide for all ML when the cross-stream coordinate is normalized by the instantaneous vorticity thickness; however, first-order thermodynamic profiles do not coincide.

It is well known that when a high subsonic (Mach number > 0.7) high Reynolds number (Re > 2 × 105) jet is directed normal to a wall intense discrete frequency sound waves called impingement tones are emitted. This phenomenon has been studied by a number of investigators in the past. It is generally accepted that the tones are generated by a feedback loop. Despite this general agreement critical difference in opinion as to how the feedback is achieved remains unresolved. Early investigators (e.g. Wagner 1971; Neuwerth 1973, 1974) proposed that the feedback loop is closed by acoustic disturbances which propagate from the wall to the nozzle exit inside the jet. Recent investigators (e.g. Ho & Nosseir 1981; Umeda et al. 1987), However, believed that the feedback is achieved by sound waves propagating outside the jet. In this paper a new feedback mechanism is proposed. It is suggested that the feedback is achieved by upstream-propagating waves associated with the lowest-order intrinsic neutral wave modes of the jet flow. These wave modes have well-defined radial and azimuthal pressure and velocity distributions. These distributions are dictated by the mean flow of the jet exactly as in the case of the well-known Kelvin-Helmholtz instability waves. The characteristics of these waves are calculated and studied. These characteristics provide a natural explanation of why the unsteady flow fields of subsonic impinging jets must be axisymmetric, whereas those for supersonic jets may be either axisymmetric or helical (flapping). In addition they also offer, for the first time, an explanation as to why no stable impingement tones have been observed for (cold) subsonic jets with Mach number less than 0.6. Furthermore, the new model allows the prediction of the average Strouhal number of impingement tones as a function of jet Mach number. The predicted results compare very favourably with measurements. For subsonic jets the pressure and velocity field of these upstream-propagating neutral waves are found to be confined primarily inside the jet. This is in agreement with the observations of Wagner (1971) and Neuwerth (1973, 1974) and their contention that the feedback disturbances actually propagate upstream inside the jet.

Steady-state two-dimensional results obtained from numerical solutions to the transient Navier-Stokes equations are given for laminar convective motion of a gas in an enclosed vertical slot with large horizontal temperature differences. We present results for air using the ideal-gas law and Sutherland-law transport properties, although the results are also valid for hydrogen. Wide ranges of aspect-ratio, Rayleigh-number and temperature-difference parameters are examined. The results are compared in detail with the exact solution in the conduction and fully developed merged boundary-layer limits for arbitrary temperature difference, and to the well-established Boussinesq limit for small temperature difference. It is found that the static pressure, and temperature and velocity distributions are very sensitive to property variations, even though the average heat flux is not. In addition we observe a net vertical heat flux to be the same as that obtained from the Boussinesq equations. We concentrate on the boundary-layer regime, but we present a rather complete picture of different flow regimes in Rayleigh-number, aspect-ratio and temperature-difference parameter space. We observe that, with increasing temperature difference, lower critical Rayleigh numbers for stationary and oscillatory instabilities are obtained. In addition we observe that in some cases the physical nature of the instability changes with increasing temperature difference.

Nitric oxide formation in gas turbine combustion depends on four key factors: flame stabilization, heat transfer, fuel–air mixing and combustion instability. The design of modern gas turbine burners requires delicate compromises between fuel efficiency, emissions of oxides of nitrogen (NOx) and combustion stability. Burner designs allowing substantial NOx reduction are often prone to combustion oscillations. These oscillations also change the NOx fields. Being able to predict not only the main species field in a burner but also the pollutant and the oscillation levels is now a major challenge for combustion modelling. This must include a realistic treatment of unsteady acoustic phenomena (which create instabilities) and also of heat transfer mechanisms (convection and radiation) which control NOx generation.

In this work, large-eddy simulation (LES) is applied to a realistic gas turbine combustion chamber configuration where pure methane is injected through multiple holes in a cone-shaped burner. In addition to a non-reactive simulation, this article presents three reactive simulations and compares them to experimental results. The first reactive simulation neglects effects of cooling air on flame stabilization and heat losses by radiation and convection. The second reactive simulation shows how cooling air and heat transfer affect nitric oxide emissions. Finally, the third reactive simulation shows the effects of combustion instability on nitric oxide emissions. Additionally, the combustion instability is analysed in detail, including the evaluation of the terms in the acoustic energy equation and the identification of the mechanism driving the oscillation.

Results confirm that LES of gas turbine combustion requires not only an accurate chemical scheme and realistic heat transfer models but also a proper description of the acoustics in order to predict nitric oxide emissions and pressure oscillation levels simultaneously.

Experimental data are presented from 138 synchronized channels of hot-wire anemometry in an investigation of the large-scale, or coherent, structures in a high Reynolds number fully developed, turbulent axisymmetric shear layer. The dynamics of the structures are obtained from instantaneous realizations of the streamwise velocity field in a single plane, x/D = 3, downstream of a round jet nozzle. The Proper Orthogonal Decomposition (POD) technique is applied to an ensemble of these realizations to determine optimal representations of the velocity field, in a mean-square sense, in terms of an orthogonal basis. The coefficients of the orthogonal functions, which describe the temporal evolution of the POD eigenfunctions, are determined by projecting instantaneous realizations of the velocity field onto the basis.

Evidence is presented to show that with a partial reconstruction of the velocity field, using only the first radial POD mode, the large-scale structure is objectively educed from the turbulent field. Further, it is shown that only five azimuthal Fourier modes (0,3,4,5,6) are necessary to represent the evolution of the large-scale structure. The results of the velocity reconstruction using the POD provide evidence for azimuthally coherent structures that exist near the potential core. In addition to the azimuthal structures near the potential core, evidence is also found for the presence of counter-rotating, streamwise vortex pairs (or ribs) in the region between successive azimuthally coherent structures as well as coexisting for short periods with them. The large-scale structure cycle, which includes the appearance of the ring structure, the advection of fluid by the ribs in the braid region and their advection toward the outside of the layer by a following ring structure, repeats approximately every one integral time scale. One surprising result was that the most spatially correlated structure in the flow, the coherent ring near the potential core which ejects fluid in the streamwise direction in a volcano-like eruption, is also the one with the shortest time scale.