This paper presents 5 kHz stereo particle image velocimetry and OH planar laser induced fluorescence measurements of transversely forced swirl flames. The presence of transverse forcing on this naturally unstable flow both influences the natural instabilities, as well as amplifies disturbances that may not necessarily manifest themselves during natural oscillations. By manipulating the structure of the acoustic forcing field, both axisymmetric and helical modes are preferentially excited away from the frequency of natural instability. The paper presents a method for spatially interpolating the phase locked $r{-}z$ and $r{-}\unicode[STIX]{x1D703}$ planar velocity and flame position data, extracting the full three-dimensional structure of the helical disturbances. These helical disturbances are also decomposed into symmetric and anti-symmetric disturbances about the jet core, showing the subsequent axial evolution (in magnitude and phase) of each of these underlying disturbances. It is shown that out-of-phase acoustic forcing excites $m=\pm 1$ modes, but the flow field preferentially amplifies the counter-winding, co-rotating helical disturbance over the co-winding, counter-rotating helical disturbance. This causes the flow and flame to transition from a transverse flapping near the jet exit to a precessing motion further downstream. In contrast, in-phase forcing promotes axisymmetric $m=0$ disturbances which dominate the flow field over the entire axial domain. In both cases, the amplitudes of the anti-symmetric disturbances about the jet core grow with downstream distance before saturating and decaying, while the symmetric disturbances appear nearly negligible. It is suggested that this saturation and decay is due to linear effects (e.g. a negative spatial growth rate), rather than nonlinear interactions.

An analysis is made of the free outflow of fluid from those underflow gates known as radial or Tainter gates. Attention is focused on a correct treatment of the effects of gate curvature and of gravity on the flow. The rapidly convergent, iterative solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. A limited comparison with some experimental results shows agreement to be good.

We consider the role of heating and cooling in the steady drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape. The internal holes and the external boundary evolve as a result of the axial drawing and surface-tension effects. The heating and cooling affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We use asymptotic techniques to show that, under a suitable transformation, this complicated three-dimensional free boundary problem can be formulated in such a way that the transverse aspect of the flow can be reduced to the solution of a standard Stokes flow problem in the absence of axial stretching. The solution of this standard problem can then be substituted into a system of three ordinary differential equations that completely determine the flow. We use this approach to develop a very simple numerical method that can determine the way that thermal effects impact on the drawing of threads by devices that either specify the fibre tension or the draw ratio. We also develop a numerical method to solve the inverse problem of determining the initial cross-sectional geometry, draw tension and, importantly, heater temperature to obtain a desired cross-sectional shape and change in cross-sectional area at the device exit. This precisely allows one to determine the pattern of air holes in the preform that will achieve the desired hole pattern in the stretched fibre.

When a liquid metal flows around a truncated cylinder in the presence of a magnetic field which is parallel to the axis of the cylinder, a stagnant region develops above the cylinder. We call this region a Ludford column. The Ludford column represents the magnetohydrodynamics (MHD) analogue of the well-known Taylor columns in rotating flows. Whereas Taylor columns can be easily visualized using dye, the visualization of Ludford columns has remained elusive up to now because liquid metals are opaque. We demonstrate that this fundamental limitation of experimental MHD can be overcome by using a superconducting 5 T magnet. This facility permits us to perform MHD experiments in which the opaque liquid metals are replaced with a transparent electrolyte while maintaining the key MHD effects. We report results of a series of flow experiments in which an aqueous solution of sulphuric acid flows around a bar with square cross-section (which for simplicity shall be referred to as a cylinder). We vary the Reynolds number in the range $5\lt Re\lt 100$ and the Hartmann number in the range $0\lt Ha\lt 14$. The experimental procedure involves flow visualizations using tracer particles as well as velocity measurements using particle image velocimetry (PIV). Our experiments provide direct access to the Ludford column for the first time and reveal the spatial structure of this basic feature of MHD flows.

An asymptotic method for analysing slender non-axisymmetric drops, bubbles and jets in a general straining flow is developed. The method relies on the slenderness of the geometry to reduce the three-dimensional equations to a sequence of weakly coupled, quasi-two-dimensional Stokes flow problems for the cross-sectional evolution. Exact solution techniques for the flow outside a bubble in two-dimensional Stokes flow are generalized to solve for the transverse flow field, allowing large non-axisymmetric deformations to be described. A generalization to the case where the interior of the bubble contains a slightly viscous fluid is also presented.

Our method is used to compute steady non-axisymmetric solution branches for inviscid bubbles and slightly viscous drops. We also present unsteady numerical solutions showing how the eccentricity of the cross-section adjusts to a non-axisymmetric external flow. Finally, we use our theory to investigate how the pinch-off of a jet of relatively inviscid fluid is affected by a two-dimensional straining cross-flow.

We use linear proportional control for the suppression of the Marangoni instability in a thin film heated from below. Our keen interest is focused on the recently revealed oscillatory mode caused by a coupling of two long-wave monotonic instabilities, the Pearson and deformational ones. Shklyaev et al. (Phys. Rev. E, vol. 85, 2012, 016328) showed that the oscillatory mode is critical in the case of a substrate of very low conductivity. To stabilize the no-motion state of the film, we apply two linear feedback control strategies based on the heat flux variation at the substrate. Strategy (I) uses the interfacial deflection from the mean position as the criterion of instability onset. Within strategy (II) the variable that describes the instability is the deviation of the measured temperatures from the desired, conductive values. We perform two types of calculations. The first one is the linear stability analysis of the nonlinear amplitude equations that are derived within the lubrication approximation. The second one is the linear stability analysis that is carried out within the Bénard–Marangoni problem for arbitrary wavelengths. Comparison of different control strategies reveals feedback control by the deviation of the free surface temperature as the most effective way to suppress the Marangoni instability.

In elastic macromolecules, the value of the short-time diffusion coefficient depends on the choice of the point the displacement of which is tracked. On the other hand, the experimentally more relevant long-time diffusion coefficient is independent of the reference point, but its estimation usually requires computationally expensive Brownian dynamics simulations. Here we show how to obtain a precise estimate of the long-time diffusion coefficient of elastic macromolecules in a fast and robust manner, without invoking Brownian dynamics.

The interaction of a gas cloud of specified mass, momentum and energy with an atmosphere in uniform motion is studied using the kinetic theory of gases. The Krook collision model is used to obtain analytically interpretable and numerically tractable results describing the spatial and temporal evolution of the cloud. The cloud originates from a continuum source and expands to a free molecular flow. It then begins to collide with the atmosphere, and is gradually transformed into a continuum flow diffusing through the atmosphere while being convected by the uniform motion.

Unsteady heat or mass transport from a particle with an arbitrary shape suspended in a fluid of infinite expanse is considered in the limit of small Péclet numbers where diffusion is dominant. In a frame of reference in which the particle appears to be stationary, the velocity of the fluid is uniform or varies in a linear manner with respect to the spatial coordinates, with an arbitrary time dependence. The temperature or concentration of a species at the surface of the particle is held at a certain constant value, whereas that at infinity is held at another constant value. Two particular problems are considered, both to leading order with respect to the Péclet number: (a) the rate of transport from a particle that is introduced suddenly into a steady flow near the steady state; and (b) the average rate of transport from a particle that is suspended in a time-periodic flow. The theory uses the method of matched asymptotic expansions and employs the Green's function of the convection–diffusion equation for a generally unsteady uniform or linear flow. The Green's function is derived in closed form by first performing a transformation to a Lagrangian framework. In the first problem of transient transport, it is found that the functional form of the rate of transport near the steady state is affected strongly by the structure of the incident flow: the decay in uniform or elongational flow is exponential, whereas the decay in simple shear flow is algebraic. In the second problem of transport in a periodic flow, it is found that the value of a properly defined frequency parameter has a strong influence on the mean rate of transport, for all types of flow. The oscillation induces convective mixing and thereby reduces the mean rate of transport by a substantial factor. The ability of the theory to describe another situation of heat or mass transport considered by Pedley is also discussed.

Motivated by the success of wavepackets in modelling the noise from subsonic and perfectly expanded supersonic jets, we apply the wavepacket model to imperfectly expanded supersonic jets. Recent studies with subsonic jets have demonstrated the importance of capturing the ‘jitter’ of wavepackets in order to correctly predict the intensity of far-field sound. Wavepacket jitter may be statistically represented using a two-point coherence function; accurate prediction of noise requires identification of this coherence function. Following the analysis of Cavalieri & Agarwal (J. Fluid Mech., vol. 748, 2014. pp. 399–415), we extend their methodology to model the acoustic sources of broadband shock-associated noise in imperfectly expanded supersonic jets using cross-spectral densities of the turbulent and shock-cell quantities. The aim is to determine the relationship between wavepacket coherence-decay and far-field broadband shock-associated noise, using the model as a vehicle to explore the flow mechanisms at work. Unlike the subsonic case where inclusion of coherence decay amplifies the sound pressure level over the whole acoustic spectrum, we find that it does not play such a critical role in determining the peak sound amplitude for shock-cell noise. When higher-order shock-cell modes are used to reconstruct the acoustic spectrum at higher frequencies, however, the inclusion of a jittering wavepacket is necessary. These results suggest that the requirement for coherence decay identified in prior broadband shock-associated noise (BBSAN) models is in reality the statistical signature of jittering wavepackets. The results from this modelling approach suggest that nonlinear jittering effects of wavepackets need to be included in dynamic models for broadband shock-associated noise.

It is known that perfectly trapped surface waves exist at certain eigenfrequencies near a vertical cylinder in a long channel, or an infinite and periodic array of vertical cylinders, and excitation by incident waves of the same frequency is not possible according to the linear theory. We present a nonlinear theory whereby a trapped wave near a cylinder in a channel is excited subharmonically by an incident wave of twice the eigenfrequency. The effects of geometrical parameters on the initial growth of resonance and the final amplification are studied in detail.

The canonical interaction between a two-dimensional weak Gaussian disturbance (entropy spot, density spot, weak vortex) with an exothermic/endothermic planar shock wave is studied via the linear interaction approximation. To this end, a unified framework based on an extended Kovásznay decomposition that simultaneously accounts for non-acoustic density disturbances along with a poloidal–toroidal splitting of the vorticity mode and for heat release is proposed. An extended version of Chu’s definition for the energy of disturbances in compressible flows encompassing multi-component mixtures of gases is also proposed. This new definition precludes spurious non-normal phenomena when computing the total energy of extended Kovásznay modes. Detailed results are provided for three cases, along with fully general expressions for mixed solutions that combine incoming vortical, entropy and density disturbances.

Mixing enhancement in a mixing layer is considered in terms of a ‘vortex generator’ that uses fluid dynamically generated counter-rotating longitudinal vortices rather than explicit winglets or similar devices. This view is reached through considering the centrifugal instability of weak initial Görtler vortices on a slightly concave wall that are allowed to develop to their various nonlinear stages through selecting the cutoff lengths of the trailing edge prior to their release into the mixing region. These vortices are released from one side of the (say, upper) stream in the present work. The quantitative entrainment properties of the longitudinal vortices are studied to select an optimal trailing-edge cutoff for fixed upstream conditions. As the vortices develop along the wall, they are intensified because of the centrifugal instability mechanism and because of the work done by the Reynolds stress of the vortices against the local mean flow rate of strain; simultaneously, the region of strong streamwise vorticity moves away from the wall. This selection process is explained through a balance between the vorticity strength and proximity to the lower stream when the trailing edge is cut off: it is shown, therefore, that vortices of relatively modest strength and kinetic energy that are close to the interface separating the two streams provide mixing properties superior to stronger vortices located too far from the interface. Energy-balancing mechanisms and the stretching of the initial interface are studied, as are the effects of the velocity ratio and the spanwise wavelengths other than the fundamental. In order further to enhance mixing by exploiting the inherent secondary instability of primary steady longitudinal vortices, the most amplified secondary instability of the optimal-trailing-edge cutoff situation, which is the sinuous mode, is studied in detail in terms of the nonlinear development and modification of the steady vortical flow. Local energy-exchange mechanisms are studied, as are the mixing properties of the modified steady flow, which are shown to be significantly improved compared to the unmodified steady flow. Though the initiation of steady longitudinal vortices relies on centrifugal instability upstream, such vortices are able to develop self-sustaining and amplifying properties through the Reynolds stresses in the mixing region even without centrifugal instability reinforcement. The secondary instability is initiated and sustained entirely through its own three-dimensional Reynolds stress properties, which work against the three-dimensional rates of strain in the entire steady flow. This contrasts with initially generated potential-like vortices that decay downstream in the presence of dissipative mechanisms without the production mechanisms due to the Reynolds stresses.

We consider theoretically and experimentally the propagation in porous media of variable-density gravity currents containing a stably stratified density field, with most previous studies of gravity currents having focused on cases of uniform density. New thin-layer equations are developed to describe stably stratified fluid flows in which the density field is materially advected with the flow. Similarity solutions describing both the fixed-volume release of a distributed density stratification and the continuous input of fluid containing a distribution of densities are obtained. The results indicate that the density distribution of the stratification significantly influences the vertical structure of the gravity current. When more mass is distributed into lighter densities, it is found that the shape of the current changes from the convex shape familiar from studies of the uniform-density case to a concave shape in which lighter fluid accumulates primarily vertically above the origin of the current. For a constant-volume release, the density contours stratify horizontally, a simplification which is used to develop analytical solutions. For currents introduced continuously, the horizontal velocity varies with vertical position, a feature which does not apply to uniform-density gravity currents in porous media. Despite significant effects on vertical structure, the density distribution has almost no effect on overall horizontal propagation, for a given total mass. Good agreement with data from a laboratory study confirms the predictions of the model.

The lift-generating capabilities of a translating and passively pitching rectangular plate are assessed experimentally. The plate pitch dynamics are generated by a rotational spring, and a genetic algorithm isolates a set of spring parameters maximizing the average lift. Our experiments identified a range of parameters that produce kinematic trajectories associated with optimal lift production. The stroke length and the dynamic response of the spring at the driving frequency are revealed to play crucial roles in the generation of such trajectories. Measurements taken with digital particle image velocimetry are used to analyse the results.

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we restrict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non-dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena.

We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued.

A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier–Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.

A generalization of criticality – called secondary criticality – is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves – from Stokes gravity waves in shallow water – is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then ‘critical’ if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin–Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

The oxygen vibrational and dissociation relaxation behind regular reflected shocks has been calculated and measured. Numerical calculations using published rate coefficients supplied the relaxation-zone data needed to estimate the range of most useful experimental conditions. Then photographs of the shock reflexion were taken using a complementary double-exposure interferometer. The density profiles in the relaxation zones behind the reflected shocks were measured by means of a multibeam laser-differential interferometer. The results of these experiments confirmed the theoretical model adopted for the calculations within a certain range of experimental conditions, but clearly revealed the need for revising the rate coefficients. New calculations with different vibrational relaxation times and dissociation rate coefficients then had the result that the best fit of calculated to measured profiles was obtained when the following values were inserted.

Vibration\begin{eqnarray*} & p\tau_v = A_v\exp(B_vT^{-\frac{1}{3}}),\\ & A_v = (2.1\pm 0.2)\times 10^{-5}\,{\rm kg/ms},\quad B_v = 129\,{}^{\circ}{\rm K}^{\frac{1}{3}}. \end{eqnarray*}

Dissociation: O2 + O_2[rlarr ] 2O + O_2 \begin{eqnarray*} & {\mathop {k_1}\limits^{\rightharpoonup}} = A_1T^{-2.5}\exp (-\theta_D/T),\\ & A_1 = (6.2 \pm 0.5)\times 10^{18}\,{\rm m}^3\,{}^{\circ}{\rm K}^{2.5}/{\rm mol}\,{\rm s},\quad\theta_D = 59\,136\,{}^{\circ}{\rm K}. \end{eqnarray*}

Dissociation: O2 + O[rlarr ]3O \begin{eqnarray*} & {\mathop {k_1}\limits^{\rightharpoonup}} = A_2T^{-1.0}\exp (-\theta_D/T),\\ & A_2 = (4.0 \mp 0.5)\times 10^{-13}\,{\rm m}^3\,{}^{\circ}{\rm K}/{\rm mol}\,{\rm s}. \end{eqnarray*}

Benjamin–Feir instability of nonlinear gravity–capillary waves is studied experimentally. The experimental results are compared with computations performed for values of wavelength and steepness identical to those employed in the experiments. The theoretical approach is based on the Zakharov nonlinear equation which is modified here to incorporate weak viscous dissipation. Experiments are performed in a wave ume which has an accurately controlled wavemaker for generation of the carrier wave, as well as an additional independent conical wavemaker for generation of controlled three-dimensional disturbances. The approach adopted in the present experimental investigation allows therefore the determination of the actual boundaries of the instability domain, and not just the most unstable disturbances. Instantaneous surface elevation measurements are performed with capacitance-type wave gauges. Multipoint measurements make it possible to determine the angular dependence of the amplitude of the forced and unforced disturbances, as well as their variation along the tank. The limits of the instability domains obtained experimentally for each set of carrier wave parameters agree favourably with those computed numerically using the model equation. The numerical study shows that application of the Zakharov equation, which is free of the narrow-band approximation adopted in the derivation of the nonlinear Schrödinger (NLS) equation, may lead to qualitatively different results regarding the stability of nonlinear gravity–capillary waves. The present experiments support the results of the numerical investigation.

The wall-jetting effect in Mach reflections in viscous pseudo-steady flows (as obtained in shock tubes) is investigated numerically. The W-modification of Godunov's scheme has been modified to solve the Navier–Stokes equations using a splitting into physical processes. The viscous terms are approximated using an explicit scheme with central differences in space and a two-step Runge–Kutta method in time. Two analytical models are considered. The first is a self-similar viscous flow model in which we consider a flow field with characteristic size $L$, and assume that as the characteristic size grows from 0 to $L$, the viscosity of the gas ahead of the shock wave varies from 0 to $\mu_{0}$. Consequently, the flow can be made self-similar by using the parameter $\textit{Re}\,{=}\,\rho_{0}a_{0}L/\mu_{0}$. The second is a real non-stationary viscous flow, in which the molecular viscosity during the growth of a characteristic size from 0 to $L$ remains constant and is equal $\mu_{0}$. As a result the viscous effects are only partially accounted for in the self-similar viscous flow model in comparison to a real non-stationary viscous flow model, since they are smaller in the former case. The present investigation complements our previous investigation of the wall-jetting effect in Mach reflection in inviscid pseudo-steady flows (Henderson et al. 2003).