This is a theoretical and experimental study of the basic hydraulics of two flowing layers. Unlike single-layer flows, two-layer flows respond quite differently to bottom depth as opposed to width variations. Bottom-depth changes affect the lower layer directly and the upper layer only indirectly. Changes in width can affect both layers. In fact for flows through a contraction control two distinct flow configurations are possible; which one actually occurs depends on the requirements of matching a downstream flow. Two-layer flows can pass through internally critical conditions at other than the narrowest section. When the two layers are flowing in the same direction, the result is a strong coupling between the two layers in the neighbourhood of the control. For contractions a particularly simple flow then exists upstream in which there is no longer any significant interfacial dynamics; downstream in the divergent section the flow remains internally supercritical, causing one of the layers to be rapidly accelerated with a resulting instability at the interface. A brief discussion of internal hydraulic jumps based upon the energy equations as opposed to the more traditional momentum equations is included. Previous uniqueness problems are thereby avoided.

The adequacy of direct one-step chemical kinetics for describing ignition and extinction in initially unmixed gases is studied through the particular case of inviscid axisymmetric stagnation-point flow. Oxidant is assumed to blow from upstream infinity at a non-gaseous reservoir of pure fuel at its boiling (or sublimating) temperature. Before reaching the reservoir the oxidant reacts with gaseous fuel flowing in the opposite direction to form product and release heat. This heat is in part conducted and diffused to the reservoir interface to transform more fuel into the gaseous state and continue the steady-state burning. Second-order Arrhenius kinetics for Lewis-number unity is examined. A critical parameter characterizing the phenomenon is shown to be the first Damkohler similarity group D1, the ratio of a time characterizing the flow to a time characterizing the chemical activity.

For small D1 the reactants convect away heat without releasing the energy stored in their chemical bonds. Regular perturbation about chemically frozen flow establishes this condition as the weak burning limit. For large D1 singular perturbation describes a narrow region of intense chemical activity. For infinite D1 (indefinitely fast rate of reaction) the region is reduced to a surface of discontinuity (the thin-flame kinetics of Burke & Schumann).

For intermediate D1 numerical techniques establish that a solution describing burning of moderate intensity joins the two previously mentioned asymptotic limits. It is suggested that sudden transition of the system between the various branches in this domain of intermediate D1 accounts for the phenomena of ignition and extinction of burning.

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid–gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman–Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

This paper is concerned with small amplitude vortical and entropic unsteady motions imposed on steady potential flows. Its main purpose is to show that, even in this unsteady compressible and vortical flow, the perturbations in pressure p’ and velocity u can be written as p’ = ρ0D0ϕ/Dt and u = ϕ + u(I) respectively, where D0/Dt is the convective derivative relative to the mean potential flow, u(I) is a known function of the imposed upstream disturbance and ϕ is a solution to the linear inhomogeneous wave equation \[ \frac{D_0}{Dt}\bigg(\frac{1}{c^2_0}\frac{D_0\phi}{Dt}\bigg)-\frac{1}{\rho_0}\nabla\cdot(\rho_0\nabla\phi)=\frac{1}{\rho_0}\nabla\cdot\rho_0{\bf u}^{(I)} \] with a dipole source term ρ0−1 [xdtri ]ρ0u(I) whose strength ρ0u(I) is a known function of the imposed upstream distortion field. (Here c0 and ρ0 denote the speed of sound and density of the background potential flow.) This equation is used to extend Hunt's (1973) generalization of the ‘rapid-distortion’ theory of turbulence developed by Batchelor & Proudman (1954) and Ribner & Tucker (1953). These theories predict changes occurring in weakly turbulent flows that are distorted (by solid obstacles and other external influences) in a time short relative to the Lagrangian integral scale.

The theory is applied to the unsteady supersonic flow around a corner and a closed-form analytical solution is obtained. Detailed calculations are carried out to show how the expansion at the corner affects a turbulent incident stream.

The nonlinear stability of a viscous film flowing steadily down an inclined plane is investigated by the method of multiple scales. It is shown that the super-critically stable, finite amplitude, long, monochromatic wave obtained by Lin (1969, 1970, 1971) is stable to side-band disturbances under modal interaction if the bandwidth is less in magnitude than to the ratio of the amplitude to the film thickness. Near the upper branch of the linear neutral-stability curve where the amplification rate ci is O(ε2), the nonlinear evolution of initially infinitesimal waves of a finite bandwidth is shown to obey the Landau-Stuart equation, Near the lower branch of the neutral curve, the nonlinear evolution is stronger. An equation is derived for describing this strong nonlinear development of relatively long waves. In practice, disturbance of this type clusters in the form of a hump which cannot be constructed only by the first few harmonics.

The structure of turbulent flow over large roughness consisting of regular arrays of cubical obstacles is investigated numerically under constant pressure gradient conditions. Results are analysed in terms of first- and second-order statistics, by visualization of instantaneous flow fields and by conditional averaging. The accuracy of the simulations is established by detailed comparisons of first- and second-order statistics with wind-tunnel measurements. Coherent structures in the log region are investigated. Structure angles are computed from two-point correlations, and quadrant analysis is performed to determine the relative importance of Q2 and Q4 events (ejections and sweeps) as a function of height above the roughness. Flow visualization shows the existence of low-momentum regions (LMRs) as well as vortical structures throughout the log layer. Filtering techniques are used to reveal instantaneous examples of the association of the vortices with the LMRs, and linear stochastic estimation and conditional averaging are employed to deduce their statistical properties. The conditional averaging results reveal the presence of LMRs and regions of Q2 and Q4 events that appear to be associated with hairpin-like vortices, but a quantitative correspondence between the sizes of the vortices and those of the LMRs is difficult to establish; a simple estimate of the ratio of the vortex width to the LMR width gives a value that is several times larger than the corresponding ratio over smooth walls. The shape and inclination of the vortices and their spatial organization are compared to recent findings over smooth walls. Characteristic length scales are shown to scale linearly with height in the log region. Whilst there are striking qualitative similarities with smooth walls, there are also important differences in detail regarding: (i) structure angles and sizes and their dependence on distance from the rough surface; (ii) the flow structure close to the roughness; (iii) the roles of inflows into and outflows from cavities within the roughness; (iv) larger vortices on the rough wall compared to the smooth wall; (v) the effect of the different generation mechanism at the wall in setting the scales of structures.

The particle tracking (PT) technique is used to study turbulent diffusion of particle pairs in a three-dimensional turbulent flow generated by two oscillating grids. The experimental data show a range where the Richardson–Obukhov law 〈r2〉 = Cεt3 is satisfied, and the Richardson–Obukhov constant is found to be C = 0.5. A number of models predict much larger values. Furthermore, the distance–neighbour function is studied in detail in order to determine its general shape. The results are compared with the predictions of three models: Richardson (1926), Batchelor (1952) and Kraichnan (1966a). These three models predict different behaviours of the distance–neighbour function, and of the three, only Richardson's model is found to be consistent with the measurements. We have corrected a minor error in Kraichnan's (1996a) Lagrangian history direct interaction calculations with the result that we had to increase his theoretical value from C = 2.42 to C = 5.5.

The problem of determining the steady axially symmetrical motion induced by a sphere rotating with constant angular velocity about a diameter in an incompressible viscous fluid which is at rest at large distances from it is considered. The basic independent variables are the polar co-ordinates (r, θ) in a plane through the axis of rotation and with origin at the centre of the sphere. The equations of motion are reduced to three sets of nonlinear second-order ordinary differential equations in the radial variable by expanding the flow variables as series of orthogonal Gegenbauer functions with argument μ = cosθ. Numerical solutions of the finite set of equations obtained by truncating the series after a given number of terms are obtained. The calculations are carried out for Reynolds numbers in the range R = 1 to R = 100, and the results are compared with various other theoretical results and with experimental observations.

The torque exerted by the fluid on the sphere is found to be in good agreement with theory at low Reynolds numbers and appears to tend towards the results of steady boundary-layer theory for increasing Reynolds number. There is excellent agreement with experimental results over the range considered. A region of inflow to the sphere near the poles is balanced by a region of outflow near the equator and as the Reynolds number increases the inflow region increases and the region of outflow becomes narrower. The radial velocity increases with Reynolds number at the equator, indicating the formation of a radial jet over the narrowing region of outflow. There is no evidence of any separation of the flow from the surface of the sphere near the equator over the range of Reynolds numbers considered.

Drag reduction strategies for the turbulent flow around a D-shaped body are examined experimentally and theoretically. A reduced-order vortex model describes the interaction between the shear layer and wake dynamics and guides a path to an efficient feedback control design. The derived feedback controller desynchronizes shear-layer and wake dynamics, thus postponing vortex formation. This actuation is tested in a wind tunnel. The Reynolds number based on the height of the body ranges from 23000 to 70000. We achieve a 40% increase in base pressure associated with a 15% drag reduction employing zero-net-mass-flux actuation. Our controller outperforms other approaches based on open-loop forcing and extremum-seeking feedback strategies in terms of drag reduction, adaptivity, and the required actuation energy.

An investigation of the interfacial-tension-driven fragmentation of a very long fluid filament in a quiescent viscous fluid is presented. Experiments covering almost three orders of magnitude in viscosity ratio reveal as many as 19 satellite droplets in between the largest droplets; complementary boundary-integral calculations are used to study numerically the evolution of the filament as a function of the viscosity ratio of the fluids and the initial wavenumber of the interface perturbation. Satellite drops are generated owing to multiple breakup sequences around the neck region of a highly deformed filament. In low-viscosity ratio systems, p < O(0.1), the breakup mechanism is self-repeating in the sense that every pinch-off is always associated with the formation of a neck, the neck undergoes pinch-off, and the process repeats. In general the agreement between computations and experiments is excellent; both indicate that the initial wavenumber of the disturbance is important in the quantitative details of the generated drop size distributions. However, these details are insignificant when compared with the large variations produced in the drop size distributions owing to variation in the viscosity ratio.

Several models are developed for the high-wavenumber portion of the spectral transfer function of scalar quantities advected by high-Reynolds-number, locally isotropic turbulent flow. These models are applicable for arbitrary Prandtl or Schmidt number, v/D, and the resultant scalar spectra are compared with several experiments having different v/D. The ‘bump’ in the temperature spectrum of air observed over land is shown to be due to a tendency toward a viscous-convective range and the presence of this bump is consistent with experiments for large v/D. The wavenumbers defining the transition between the inertial-convective range and viscous-convective range for asymptotically large v/D (denoted k* and k1* for the three- and one-dimensional spectra) are determined by comparison of the models with experiments. A measurement of the transitional wavenumber k1* [denoted (k1*)s] is found to depend on v/D and on any filter cut-off. On the basis of the k* values it is shown that measurements of β1 from temperature spectra in moderate Reynolds number turbulence in air (v/D = 0·72) maybe over-estimates and that the inertial-diffusive range of temperature fluctuations in mercury (v/D ≃ 0·02) is of very limited extent.

We study the existence and the role of solitary waves in the instability of a fluid layer flowing down an inclined plane. The approach presented is fully nonlinear. Solitary waves steady in a moving frame are described by homoclinic trajectories of an associated ordinary differential equation. They are searched numerically for a given value of viscosity and surface tension. Several kinds of solitary waves can exist, characterized by their number n of humps. We investigate the stability of these waves by integrating the initial-value problem directly. Solitary waves with more than 1 hump did not appear in the simulation, and moreover a catastrophic behaviour took place for too large a Reynolds number (R [gsim ] R*1) or too large an amplitude, suggesting a finite-time singularity. The long-term evolution is shown to be a very slow relaxation to a steady state in a moving frame. The relation to the experimental observation of localized wavetrains is also discussed.

Exact coherent states in no-slip plane Poiseuille flow are calculated by homotopy from free-slip to no-slip boundary conditions. These coherent states are unstable travelling waves. They consist of wavy low-speed streaks flanked by staggered streamwise vortices closely resembling the asymmetric coherent structures observed in the near-wall region of turbulent flows. The travelling waves arise from a saddle-node bifurcation at a sub-turbulent Reynolds number with wall-normal, spanwise and streamwise dimensions smaller than but comparable to 50+, 100+ and 250+, respectively. These coherent solutions come in pairs with distinct structure and instabilities. There is a three-dimensional continuum of such exact coherent states.

The low-frequency unsteadiness is characterized in the direct numerical simulation of a shock wave–turbulent boundary layer interaction generated by a compression ramp in Mach 2.9 flow. Consistent with experimental observations, the shock wave in the simulation undergoes a broadband streamwise oscillation at frequencies approximately two orders of magnitude lower than the characteristic frequency of the energetic turbulent scales in the incoming boundary layer. The statistical relation between the low-frequency shock motion and the upstream and downstream flow is investigated. The shock motion is found to be related to a breathing of the separation bubble and an associated flapping of the separated shear layer. A much weaker statistical relation is found with the incoming boundary layer. In order to further characterize the low-frequency mode in the downstream separated flow, the temporal evolution of the low-pass filtered flow field is investigated. The nature of the velocity and vorticity profiles in the initial part of the interaction is found to change considerably depending on the phase of the low-frequency motion. It is conjectured that these changes are due to an inherent instability in the downstream separated flow, and that this instability is the physical origin of the low-frequency unsteadiness. The low-frequency mode observed here is, in certain aspects, reminiscent of an unstable global mode obtained by linear stability analysis of the mean flow in a reflected shock interaction (Touber & Sandham, Theor. Comput. Fluid Dyn., vol. 23, 2009, pp. 79–107).

In this paper, we consider the evolution of an initially small voidage disturbance in a gas-fluidized bed. Using a one-dimensional model proposed by Needham & Merkin (1983), Crighton (1991) has shown that weakly nonlinear waves of voidage propagate according to the Korteweg–de Vries equation with perturbation terms which can be either amplifying or dissipative, depending on the sign of a coefficient. Here, we investigate the unstable side of the threshold and examine the growth of a single KdV voidage soliton, following its development through several different regimes. As the size of the soliton increases, KdV remains the leading-order equation for some time, but the perturbation terms change, thereby altering the dependence of the amplitude on time. Eventually the disturbance attains a finite amplitude and corresponds to a fully nonlinear solitary wave solution. This matches back directly onto the KdV soliton and tends exponentially to a limiting size. We interpret the series of large-amplitude localized pulses of voidage formed in this way from initial disturbances as corresponding to the ‘voidage slugs’ observed in gas fluidization in narrow tubes.

The connection between fluid flow and optical flow is explored in typical flow visualizations to provide a rational foundation for application of the optical flow method to image-based fluid velocity measurements. The projected-motion equations are derived, and the physics-based optical flow equation is given. In general, the optical flow is proportional to the path-averaged velocity of fluid or particles weighted with a relevant field quantity. The variational formulation and the corresponding Euler–Lagrange equation are given for optical flow computation. An error analysis for optical flow computation is provided, which is quantitatively examined by simulations on synthetic grid images. Direct comparisons between the optical flow method and the correlation-based method are made in simulations on synthetic particle images and experiments in a strongly excited turbulent jet.

This paper reports an experimental investigation of the near wake of a finite-length square cylinder, with one end mounted on a flat plate and the other free. The cylinder aspect ratio or height-to-width ratio H/d ranges from 3 to 7. Measurements were carried out mainly in a closed-loop low-speed wind tunnel at a Reynolds number Red, based on d and the free-stream velocity of 9300 using hot-wire anemometry, laser Doppler anemometry and particle image velocimetry (PIV). The planar PIV measurements were performed in the three orthogonal planes of the three-dimensional cylinder wake, along with flow visualization conducted simultaneously in two orthogonal planes (Red = 221). Three types of vortices, i.e. the tip, base and spanwise vortices were observed and the near wake is characterized by the interactions of these vortices. Both flow visualization and two-point correlation point to an inherent connection between the three types of vortices. A model is proposed for the three-dimensional flow structure based on the present measurements, which is distinct from previously proposed models. The instantaneous flow structure around the cylinder is arch-type, regardless of H/d, consisting of two spanwise vortical ‘legs’, one on each side of the cylinder, and their connection or ‘bridge’ near the free end. Both tip and base vortices are the streamwise projections of the arch-type structure in the (y, z) plane, associated with the free-end downwash flow and upwash flow from the wall, respectively. Other issues such as the topological characteristics, spatial arrangement and interactions among the vortical structures are also addressed.

We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldy's approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties.

Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation.

An evaluation of the effectiveness of the VITA, Quadrant, TPAV, U -level, Positive slope, and VITA with slope burst-detection algorithms has been done by making direct comparisons with flow visualization. Measurements were made in a water channel using an X-type hot-film probe located in the near-wall region. Individual ejections from bursts which contacted the probe were identified using dye flow visualization. The effectiveness of each of the detection algorithms was found to be highly dependent on the operational parameters, i.e. threshold levels and averaging or window times. These parameters were adjusted so that the number of events detected by each of the algorithms corresponded to the number of ejections identified by flow visualization, while the probability of a false detection was minimized. Comparing the detection algorithm using these optimum parameter settings, the Quadrant technique was found to have the greatest reliability with a high probability of detecting the ejections and a low probability of false detections. Furthermore, it was found that the ejections detected by the Quadrant technique could be grouped into bursts by analysing the probability distribution of the time between ejections.