A stability analysis of meandering and braiding perturbations in a model alluvial river is described. A perturbation technique, involving a small parameter representing the ratio of sediment transport to water transport, is used to obtain the following results.

Under appropriate conditions, the existence of sediment transport and friction are necessary conditions for the occurrence of instability in the flow and on the bed; thus instability is not inherent in the flow alone. An Anderson-type scale relation for longitudinal instability is obtained for meandering. A relation estimating the number of braids and differentiating between meandering and braided regimes is derived. These relations are independent of sediment transport.

The axial and radial velocity components w and u, and the concentration c of a Rhodamine 6G dye were measured simultaneously in a turbulent buoyant jet, using laser-Doppler anemometry combined with a recently developed laser-induced-fluorescence concentration measurement technique. These non-intrusive techniques enable measurements in a region of plume motion where conventional probe-based techniques have had difficulties. The results of the study show that the asymptotic decay laws for velocity and concentration of a tracer transported by the flow are verified experimentally in both jets and plumes. The momentum and volume fluxes and the mean dilution factor are determined in dimensionless form as a function of the normalized distance from the flow source. Contradictory results from earlier experimental plume investigations concerning the decay laws of w and c and the plume width ratio bc/bw are discussed. The turbulence properties and the transition from momentum-driven jets to buoyancy-driven plumes are presented. The turbulence is found to scale with the mean flow as predicted by dimensional analysis and self-similarity. Buoyancy-produced turbulence is found to transport twice as much tracer as jet turbulence. Although velocity statistics in jets and plumes are found to be highly self-similar there is a strong disparity in the distribution of tracer concentration in the two flows. This occurs in the time-average mean flows as well as the r.m.s. turbulent quantities. Instantaneous concentration fluctuations are found to exceed time averages by as much as a factor of 3. The experimental results should provide a reasonable basis for validation of computer models of axisymmetric plumes.

Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.

Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

Six subgrid models for the turbulent stress tensor are tested by conducting large-eddy simulations (LES) of the weakly compressible temporal mixing layer: the Smagorinsky, similarity, gradient, dynamic eddy-viscosity, dynamic mixed and dynamic Clark models. The last three models are variations of the first three models using the dynamic approach. Two sets of simulations are performed in order to assess the quality of the six models. The LES results corresponding to the first set are compared with filtered results obtained from a direct numerical simulation (DNS). It appears that the dynamic models lead to more accurate results than the non-dynamic models tested. An adequate mechanism to dissipate energy from resolved to subgrid scales is essential. The dynamic models have this property, but the Smagorinsky model is too dissipative during transition, whereas the similarity and gradient models are not sufficiently dissipative for the smallest resolved scales. In this set of simulations, at moderate Reynolds number, the dynamic mixed and Clark models are found to be slightly more accurate than the dynamic eddy-viscosity model. The second set of LES concerns the mixing layer at a considerably higher Reynolds number and in a larger computational domain. An accurate DNS for this mixing layer can currently not be performed, thus in this case the LES are tested by investigating whether they resemble a self-similar turbulent flow. It is found that the dynamic models generate better results than the non-dynamic models. The closest approximation to a self-similar state was obtained using the dynamic eddy-viscosity model.

This paper refers to the work of Moeckel (1952) on the interaction of an oblique shock wave with a shear layer in steady supersonic flow and the work of Chester (1955) and Chisnell (1957) on the propagation of a shock wave down a non-uniform tube. It is shown that their basic results can be obtained by the application of the following simple rule. The relevant equations of motion are first written in characteristic form. Then the rule is to apply the differential relation which must be satisfied by the flow quantities along a characteristic to the flow quantities just behind the shock wave. Together with the shock relations this rule determines the motion of the shock wave. The accuracy of the results for a wide range of problems and for all shock strengths is truly surprising.

The results are exactly the same as were found by the authors cited above. The derivation given here is simpler to perfom (although the original methods were by no means involved) and of somewhat wider application, but the main reason for presenting this discussion is to try to throw further light on these remarkable results.

In discussing the underlying reasons for this rule, it is convenient to use the propagation in a non-uniform tube as a typical example, but applications to a number of problems are given later. A list of some of these appears at the beginning of the introductory section.

We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as $k^{-\frac{5}{3}}$ at the cutoff is studied. Then, a spectral large-eddy simulation of decaying isotropic turbulence convecting a passive temperature is performed, at a resolution of 1283 collocation points. It is shown that the temperature spectrum tends to follow in the energetic scales a k−1 range, followed by a $k^{-\frac{5}{3}}$ inertial–convective range at higher wavenumbers. This is in agreement with previous independent calculations (Lesieur & Rogallo 1989). When self-similar spectra have developed, the temperature variance and kinetic energy decay respectively like t−1.37 and t−1.85, with identical initial spectra peaking at ki = 20 and ∝ k8 for k → 0. In the k−1 range, the temperature spectrum is found to collapse according to the law ET(k, t) = 0.1η(〈u2〉/ε) k−1, where ε and η are the kinetic energy and temperature variance dissipation rates. The spectral eddy viscosity and diffusivity are recalculated explicitly from the large-eddy simulation: the anomalous ∝ ln k behaviour of the eddy diffusivity in the eddy-viscosity plateau is shown to be associated with the large-scale intermittency of the passive temperature: the p.d.f. of the velocity component u is Gaussian (∼ exp − X2), while the scalar T, the velocity derivatives ∂u/∂x and ∂u/∂z, and the temperature derivative ∂T/∂z are all close to exponential exp - |X| at high |X|. The pressure distribution is exponential at low pressure and Gaussian at high.

For stably stratified Boussinesq turbulence, the coupling between the temperature and the velocity fields leads to the disappearance of the ‘anomalous’ temperature behaviour (k−1 range, logarithmic eddy diffusivity and exponential probability density function for T). These are the highest-resolution calculations ever performed for this problem. We also split the eddy viscous coefficients into a vortex and a wave component. In both cases (unstratified and stratified), comparisons with direct numerical simulations are performed.

Finally we propose a generalization of the spectral eddy viscosity to highly intermittent situations in physical space: in this structure-function model, the spectral eddy viscosity is based upon a kinetic energy spectrum local in space. The latter is calculated with the aid of a local second-order velocity structure function. This structure function model is compared with other models, including Smagorinsky's, for isotropic decaying turbulence, and with high-resolution direct simulations. It is shown that low-pressure regions mark coherent structures of high vorticity. The pressure spectra are shown to follow Batchelor's quasi-normal law: $\alpha C^2_{\rm k}\epsilon^{\frac{4}{3}}k^{-\frac{7}{3}}$ (Ck is Kolmogorov's constant), with α ≈ 1.32.

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, α (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for α < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < α < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of α a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

An experiment is described in which a constant stress is applied to the surface of an initially quiescent tank of fluid with a uniform density gradient. The development of the turbulent layer by entrainment of the underlying fluid is described and it is found that the entrainment coefficient E, the ratio of the entrainment velocity ue to the friction velocity u* is given in terms of the depth D of the mixed layer and the density jump δρ across the entrainment interface by the relation \[ E = \frac{u_e}{U_{*}} = 2.5\frac{\rho_0u^2_{*}}{g\delta\rho D}. \] The rate of increase of potential energy of the stratified fluid was found to be proportional to the rate of dissipation of kinetic energy per unit area in the turbulent layer. The form of these results is consistent with those found by Turner with an agitation tank, but the parameters used here allow direct application to entrainment in the ocean.

The swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

In a high-speed subsonic jet impinging on a flat plate, the surface pressure fluctuations have a broad spectrum due to the turbulent nature of the high-Reynolds-number jet. However, these pressure fluctuations dramatically change their pattern into almost periodic waves, if the plate is placed close to the nozzle (x0/d < 7·5). In the present study extensive measurements of the near-field pressure provide solid support for the hypothesis that a feedback mechanism is responsible for the sudden change observed in the pressure fluctuations at the onset of resonance. The feedback loop consists of two elements: the downstream-convected coherent structures and upstream-propagating pressure waves generated by the impingement of the coherent structures on the plate. The upstream-propagating waves and the coherent structures are phase-locked at the nozzle exit. The upstream-propagating waves excite the thin shear layer near the nozzle lip and produce periodic coherent structures. The period is determined by the convection speed of the coherent structures, the speed of the upstream-propagating waves as well as the distance between the nozzle and the plate. An instability process, herein referred to as the ‘collective interaction’, was found to be critical in closing the feedback loop near the nozzle lip.

This paper aims at a description of boundary-layer flow which is subjected to free-stream turbulence in the range from 1–6% and is based on both flow visualization results and extensive hot-wire measurements. Such flows develop streamwise elongated regions of high and low streamwise velocity which seem to lead to secondary instability and breakdown to turbulence. The initial growth of the streaky structures is found to be closely related to algebraic or transient growth theory. The data have been used to determine streamwise and spanwise scales of the streaky structures. Both the flow visualization and the hot-wire measurements show that close to the leading edge the spanwise scale is large as compared to the boundary-layer thickness, but further downstream the spanwise scale approaches the boundary-layer thickness. Wavenumber spectra in both the streamwise and the spanwise directions were calculated. A scaling for the streamwise structure of the disturbance was found, which allows us to collapse the spectra from different downstream positions. The scaling combines the facts that the streaky structures increase their streamwise length in the downstream direction which becomes proportional to the boundary-layer thickness and that the energy growth is algebraic, close to proportional to the downstream distance.

A theory is given for predicting the absorption of the power in an incident sinusoidal wave train by means of a damped, oscillating, partly or completely submerged body. General expressions for the efficiency of wave absorption when the body oscillates in one or, in some cases, two modes are given. It is shown that 100% efficiency is possible in some cases. Curves describing the variation of efficiency and amplitude of the body with wavenumber for various bodies are presented.

A theoretical study is made of shearing flows bounded by a simple-harmonic wavy surface, the main object being to calculate the normal and tangential stresses on the boundary. The type of flow considered is approximately parallel in the absence of the waves, being exemplified by two-dimensional boundary layers over a plane. Account is taken of viscosity; but, as the Reynolds number is assumed to be large, its effects are seen to be confined within narrow ‘friction layers’, one of which adjoins the wave and another surrounds the ‘critical point’ where the velocity of flow equals the wave velocity. The boundary conditions are made as general as possible by including the three cases where respectively the boundary is rigid, flexible yet still solid, or completely mobile as if it were the interface with a second fluid.

The theory is developed on the model of stable laminar flow, although it is proposed that the same theory may usefully be applied also to examples of turbulent flow considered as ‘pseudo-laminar’ with velocity profiles corresponding to the mean-velocity distribution. Use is made of curvilinear co-ordinates which follow the contour of the wave-train. This admits a linearized form of the problem whose validity requires only that the wave amplitude be small in comparison with the wavelength, even when large velocity gradients exist close to the boundary. The analysis is made largely without restriction to particular forms of the velocity profile; but eventually consideration is given to the example of a linear profile and the example of a boundary-layer profile approximated by a quarter-period sinusoid. In § 7 some general methods are set out for the treatment of disturbed boundary-layer proses: these apply with greatest precision to thin boundary layers, but are also useful for the initially very steep but on the whole fairly diffuse profiles which occur in most practical instances of turbulent flow over waves.

The phase relationships found between the stresses and the wave elevation are discussed for several examples, and their interest in connexion with problems of wave generation by wind is pointed out. It is shown that in most circumstances the stresses are distributed in much the same way as if the leeward slopes of the waves were sheltered. For instance, the pressure distribution often has a substantial component in phase with the wave slope, just as if a wake were formed behind each wave crest—although of course actual separation effects are outside the scope of the present theory. In this aspect, the analysis amplifies the work of Miles (1957).

Numerical simulations are used to investigate the resonant instabilities in two-dimensional flow past an open cavity. The compressible Navier–Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field, which is shown to be in good visual agreement with schlieren photographs from experiments at several different Mach numbers. The results show a transition from a shear-layer mode, primarily for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear-layer mode is characterized well by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare well with predictions based on linear stability analysis of the Kelvin–Helmholtz mode. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib & Roshko (1987) for incompressible flow with a laminar upstream boundary layer. Transition to wake mode occurs as the length and/or depth of the cavity becomes large compared to the upstream boundary-layer thickness, or as the Mach and/or Reynolds numbers are raised. Under these conditions, it is shown that the Kelvin–Helmholtz instability grows to sufficient strength that a strong recirculating flow is induced in the cavity. The resulting mean flow is similar to wake profiles that are absolutely unstable, and absolute instability may provide an explanation of the hydrodynamic feedback mechanism that leads to wake mode. Predictive criteria for the onset of shear-layer oscillations (from steady flow) and for the transition to wake mode are developed based on linear theory for amplification rates in the shear layer, and a simple model for the acoustic efficiency of edge scattering.

Flow in the separation bubble formed along the sides of a blunt flat plate with right-angled corners has been studied in terms of extensive single- and two-point measurements of velocity and surface-pressure fluctuations. The cross-correlations between the surface-pressure and velocity fluctuations are found to be useful for the study of large-scale vortex structure in the bubble. Large-scale vortices are shed downstream from the separation bubble with a frequency of about 0.6U∞/xR, where U∞ is the approaching velocity and xR is the time-mean length of the bubble. On top of this regular vortex shedding, there exists a large-scale unsteadiness in the bubble. Vortices which are much larger than the regular vortices are shed with frequencies less than about 0.2U∞/xR. The large-scale unsteadiness is accompanied by enlargement and shrinkage of the bubble and also by a flapping motion of the shear layer near the separation line. The intermittent nature of the flow in the bubble is clarified in some detail. The distributions of the cross-correlations between the pressure and velocity fluctuations demonstrate the vortex structure in the reattaching zone. The longitudinal distance between the vortices is estimated to be (0.7–0.8) xR and their convection velocity is about 0.5U∞ near the reattachment line. The cross-correlations also suggest the existence of a longitudinal counter-rotating system in the bubble. The distance between the axes of the rotation is of the order of 0.6xR. Variations of timescales, lengthscales and phase velocities of the vortices are presented and discussed.

Late transitional and turbulent flows in the mixing-layer region of a round jet are investigated for a range of Reynolds numbers by using flow-visualization and hotwire techniques. Attention is focused on the vortices in the transition region and the large eddies in the turbulent region. The interaction and coalescence of vortex rings in the transition region are described. The transition region is characterized by a growth of three-dimensional flow due to a wave instability of the cores of the vortex rings. The merging of these distorted vortices produces large eddies which can remain coherent up to the end of the potential-core region of the jet. A conditional sampling technique is used to measure eddies moving near the jet centre-line. These eddies differ significantly from the ring vortices as they are three-dimensional and contain irregular small-scale turbulence. However, when averaged, their structure is similar in cross-section to that of a vortex ring. These sampled eddies contribute greatly to local velocity fluctuations and statistical correlations. The experiments indicate a need for careful consideration of the meanings of terms such as ‘vortex’, ‘eddy’ and ‘turbulent flow’. In particular care must be taken to discriminate between the orderly, easily visualized, vortices in the transition regions of free shear flows and the less clearly visualized, but strong, large eddies in the fully developed turbulent regions.

The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate γ/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, γ/D [Lt ] 1, has been known for many years. After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r−3) [Lt ] γ/D, and for an intermediate regime relevant for extreme aspect ratios where 1 [Lt ] γ/D [Lt ] (r3 + r−3). The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r ∼ 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior continuum mechanical and experimental results.

A model to explain the low-frequency unsteadiness found in shock-induced separation is proposed for cases in which the flow is reattaching downstream. It is based on the properties of fluid entrainment in the mixing layer generated downstream of the separation shock whose low-frequency motions are related to successive contractions and dilatations of the separated bubble. The main aerodynamic parameters on which the process depends are presented. This model is consistent with experimental observations obtained by particle image velocimetry (PIV) in a Mach 2.3 oblique shock wave/turbulent boundary layer interaction, as well as with several different configurations reported in the literature for Mach numbers ranging from 0 to 5.

A rapid method for the prediction of flow separation results from an approximate solution of the equations of motion; a single empirical factor is required. The equations are integrated by a modified ‘inner and outer solutions’ technique developed recently for laminar boundary layers, the criterion for separation being obtained as a simple formula applying directly to the separation position. At Reynolds numbers of the order of 106, the criterion is $C_p(xdC_p|dx)^{\frac {1}{2}} = 0 \cdot 39 (10^{-6}R)^{\frac {1}{10}},$ when d2p/dx2 [ges ] 0 and Cp [les ] 4/7; the coefficient 0·39 is replaced by 0·35 when d2p/dx2 < 0.

The prediction of the pressure rise to separation is likely to be from 0 to 10% too low, which puts it second in accuracy to those methods, such as Maskell's (1951), which utilize the Ludweig-Tillmann skin friction law. However, the convenience of the method makes the present error acceptable for many applications, while a greater accuracy should be attainable from an improved allowance for the quantity d2p/dx2.

The main derivation is for arbitrary pressure distributions, while an extension leads to the pressure distribution which just maintains zero skin friction throughout the region of pressure rise.

The concept of a turbulent inner layer with zero wall stress is put forward, and it is deduced that in the neighbourhood of the wall the velocity is proportional to the square root of the distance from the wall.

In order to understand the rheological and transport properties of a suspension of swimming micro-organisms, it is necessary to analyse the fluid-dynamical interaction of pairs of such swimming cells. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). The effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The interaction of two squirmers is calculated analytically for the limits of small and large separations and is also calculated numerically using a boundary-element method. The analytical and the numerical results for the translational–rotational velocities and for the stresslet of two squirmers correspond very well. We sought to generate a database for an interacting pair of squirmers from which one can easily predict the motion of a collection of squirmers. The behaviour of two interacting squirmers is discussed phenomenologically, too. The results for the trajectories of two squirmers show that first the squirmers attract each other, then they change their orientation dramatically when they are in near contact and finally they separate from each other. The effect of bottom-heaviness is considerable. Restricting the trajectories to two dimensions is shown to give misleading results. Some movies of interacting squirmers are available with the online version of the paper.