The propagation characteristics of inviscid axisymmetric linearized disturbances to swirling jets are investigated for two families of model velocity profiles. Briggs' method is applied to dispersion relations to determine when the basic swirling jets are absolutely or convectively unstable. The method is also applied to the neutral inertial waves used by Benjamin to characterize the subcritical or supercritical nature of the flow. Although these waves are neutral, Briggs' method nonetheless indicates a qualitative change in behaviour at Benjamin's criticality condition. The first model jet has uniform axial velocity, rigid-body rotation and issues into still fluid. A known difficulty in the application of Briggs' method to the analytical dispersion relation for inviscid waves in this flow is resolved. The difficulty is that the pinch point can cross into the left half of the complex-wavenumber plane, where waves grow exponentially with radius and fail to satisfy homogeneous boundary conditions. In this paper the physical consequences of this behaviour are explained. It is shown that if the still fluid is of infinite extent in the radial direction, then the jet is convectively unstable to axisymmetric waves, but if the jet is confined by an outer cylinder concentric with the jet axis, then it becomes absolutely unstable to axisymmetric waves provided that the swirl (ratio of azimuthal to axial velocity) is large enough. This destabilizing effect of confinement occurs however large the radius of the outer cylinder. A second family of model swirling jets with smooth profiles and a finite thickness shear layer at the jet edge is also studied. The inviscid stability equations are solved numerically in this case. The results from the analytical dispersion relations are reproduced as the shear layer thickness tends to zero. However, increasing this thickness acts to destabilize the absolute instability of axisymmetric waves, apparently due to the centrifugal instability present in the shear layer. It is suggested that the transition from convective to absolute instability could be associated with the onset of an unsteady vortex breakdown. The swirl required to produce this transition can be either greater, or less, than the swirl required to produce the transition from supercritical to subcritical flow, depending on the details of the basic velocity profiles. A codimension-two point in parameter space can exist where the unsteady bifurcation associated with the convective–absolute transition coincides with the steady bifurcation associated with the supercritical–subcritical transition. Such codimension-two points can control a rich variety of nonlinear dynamical behaviour.

In this study, it is shown how the equivalent sand roughness required in the Moody chart can be calculated for arbitrarily shaped wall roughnesses. After a discussion of how to define the wall location and roughness height in the most reasonable way, a numerical approach based on the determination of entropy production in rough pipes and channels is presented. As test cases, three different two-dimensional roughness types have been chosen which are representative of regular roughnesses on machined surfaces. In the turbulent range, skin friction results with these test roughnesses can be linked to Nikuradse's sand roughness results by a constant factor. For laminar flows, a significant effect of wall roughness is identified which in most other studies is neglected completely. The dissipation model of this study is validated with experimental data for laminar and turbulent flows.

Supersonic impinging jet(s) inherently produce a highly unsteady flow field. The occurrence of such flows leads to many adverse effects for short take-off and vertical landing (STOVL) aircraft such as: a significant increase in the noise level, very high unsteady loads on nearby structures and an appreciable loss in lift during hover. In prior studies, we have demonstrated that arrays of microjets, appropriately placed near the nozzle exit, effectively disrupt the feedback loop inherent in impinging jet flows. In these studies, the effectiveness of the control was found to be strongly dependent on a number of geometric and flow parameters, such as the impingement plane distance, microjet orientation and jet operating conditions. In this paper, the effects of some of these parameters that appear to determine control efficiency are examined and some of the fundamental mechanisms behind this control approach are explored. Through comprehensive two- and three-component velocity (and vorticity) field measurements it has been clearly demonstrated that the activation of microjets leads to a local thickening of the jet shear layer, near the nozzle exit, making it more stable and less receptive to disturbances. Furthermore, microjets generate strong streamwise vorticity in the form of well-organized, counter-rotating vortex pairs. This increase in streamwise vorticity is concomitant with a reduction in the azimuthal vorticity of the primary jet. Based on these results and a simplified analysis of vorticity transport, it is suggested that the generation of these streamwise vortices is mainly a result of the redirection of the azimuthal vorticity by vorticity tilting and stretching mechanisms. The emergence of these longitudinal structures weakens the large-scale axisymmetric structures in the jet shear layer while introducing substantial three-dimensionality into the flow. Together, these factors lead to the attenuation of the feedback loop and a significant reduction of flow unsteadiness.

Slender-body asymptotic theory is used to evaluate the translational and rotational electrophoretic velocities of initially uncharged polarizable bodies of revolution. These velocities are obtained as asymptotic expansions in the small particle slenderness. Conducting particles which lack fore–aft symmetry translate parallel to the applied field direction, regardless of their orientation relative to it. Both conducting and dielectric particles tend to align with the field. The translational and rotational velocities of dielectric particles are asymptotically smaller than those of comparable conducting particles.

We consider a nominally uniform flow over a semi-infinite flat plate and show how a small slowly modulated (predominantly streamwise) disturbance of the upstream flow is amplified by leading-edge bluntness effects and eventually develops into a small-amplitude but nonlinear spanwise motion far downstream from the edge. This motion is then imposed on the viscous boundary layer at the surface of the plate – causing an order-one change in its profile shape, which can reduce the wall shear to zero and thereby causes the boundary layer to separate. The present study is similar to an earlier steady flow analysis, but the unsteady effects now cause the upstream boundary layer to develop inflectional profiles which can support rapidly growing inviscid instabilities that give rise to transition before the separation can occur.

We describe a tensorial generalization of the Navier slip boundary condition and illustrate its use in solving for flows around anisotropic textured surfaces. Tensorial slip can be derived from molecular or microstructural theories or simply postulated as a constitutive relation, subject to certain general constraints on the interfacial mobility. The power of the tensor formalism is to capture complicated effects of surface anisotropy, while preserving a simple fluid domain. This is demonstrated by exact solutions for laminar shear flow and pressure-driven flow between parallel plates of arbitrary and different textures. From such solutions, the effects of rotating a texture follow from simple matrix algebra. Our results may be useful for extracting local slip tensors from global measurements, such as the permeability of a textured channel or the force required to move a patterned surface, in experiments or simulations.

Direct numerical simulations (DNS) of bypass transition due to high-amplitude free-stream turbulence (FST) are carried out for a flat-plate boundary layer. The computational domain begins upstream of the plate leading edge and extends into the fully turbulent region. Thus, there is no ad hoc treatment to account for the initial ingestion of FST into the laminar boundary layer. We study the effects of both the FST length scale and the disturbance behaviour near the plate leading edge on the details of bypass transition farther downstream. In one set of simulations, the FST parameters are chosen to match the ERCOFTAC benchmark case T3B. The inferred FST integral length scale L11 is significantly larger (RL = UL11/ν = 6580) than that employed in previous simulations of bypass transition (RL ≃ 1000). An additional set of simulations was performed at RL = 1081 to compare the transition behaviour in the T3B case with that of a smaller value of FST length scale. The FST length scale is found to have a profound impact on the mechanism of transition. While streamwise streaks (Klebanoff modes) are observed at both values of the FST length scale, they appear to have clear dynamical significance only at the smaller value of RL, where transition is concomitant with streak breakdown. For the T3B case, turbulent spots form upstream of the region where streaks could be detected. Spot precursors are traced to quasi-periodic spanwise structures, first observed as short wavepackets in the wall-normal velocity component inside the boundary layer. These structures are reoriented to become horseshoe vortices, which break down into young turbulent spots. Two of the four spots examined for this case had a downstream-pointing shape, similar to those found in experimental studies of transitional boundary layers. Additionally, our simulations indicate the importance of leading-edge receptivity for the onset of transition. Specifically, higher fluctuations of the vertical velocity at the leading edge of the plate result in higher levels of streamwise Reynolds stress inside the developing boundary layer, facilitating breakdown to turbulence.

This paper presents a numerical analysis of the transition from selective withdrawal to viscous entrainment. In our model problem, an interface between two immiscible layers of equal viscosity is deformed by an axisymmetric withdrawal flow, which is driven by a point sink located some distance above the interface in the upper layer. We find that steady-state hump solutions, corresponding to selective withdrawal of liquid from the upper layer, cease to exist above a threshold withdrawal flux, and that this transition corresponds to a saddle-node bifurcation for the hump solutions. Numerical results on the shape evolution of the steady-state interface are compared against previous experimental measurements. We find good agreement where the data overlap. However, the larger dynamic range of the numerical results allows us to show that the large increase in the curvature of the hump tip near transition is not consistent with an approach towards a power-law cusp shape, an interpretation previously suggested from inspection of the experimental measurements alone. Instead, the large increase in the curvature at the hump tip reflects a robust trend in the steady-state interface evolution. For large deflections, the hump height is proportional to the logarithm of the curvature at the hump tip; thus small changes in hump height correspond to large changes in the value of the hump curvature.

A spatially developing supersonic boundary layer at Mach 2 is analysed by means of direct numerical simulation of the compressible Navier--Stokes equations, with the objective of quantitatively characterizing the coherent vortical structures. The study shows structural similarities with the incompressible case. In particular, the inner layer is mainly populated by quasi-streamwise vortices, while in the outer layer we observe a large variety of structures, including hairpin vortices and hairpin packets. The characteristic properties of the educed structures are found to be nearly uniform throughout the outer layer, and to be weakly affected by the local vortex orientation. In the outer layer, typical core radii vary in the range of 5–6 dissipative length scales, and the associated circulation is approximately constant, and of the order of 180 wall units. The statistical properties of the vortical structures in the outer layer are similar to those of an ensemble of non-interacting closed-loop vortices with a nearly planar head inclined at an angle of approximately 20° with respect to the wall, and with an overall size of approximately 30 dissipative length scales.

The aim of the present paper is to analyse the dynamics of the Lamb–Oseen vortex when continuously forced by a random excitation. Stochastic forcing is classically used to mimic external perturbations in realistic configurations, such as variations of atmospheric conditions, weak compressibility effects, wing-generated turbulence injected into aircraft wakes, or free-stream turbulence in wind tunnel experiments. The linear response of the Lamb–Oseen vortex to stochastic forcing can be decomposed in relation to the azimuthal symmetry of the perturbation given by the azimuthal wavenumber m. In the axisymmetric case m = 0, we find that the response is characterized by the generation of vortex rings at the outer periphery of the vortex core. This result is consistent with recurrent observations of such dynamics in the study of vortex–turbulence interaction. When considering helical perturbations m = 1, the response at large axial wavelengths consists of a global translation of the vortex, a feature very similar to the phenomenon of vortex meandering (or wandering) observed experimentally, corresponding to an erratic displacement of the vortex core. At smaller wavelengths, we find that stochastic forcing can excite specific oscillating modes of the Lamb–Oseen vortex. More precisely, damped critical-layer modes can emerge via a resonance mechanism. For perturbations with higher azimuthal wavenumber m ≥ 2, we find no structure that clearly dominates the response of the vortex.

The laminar–turbulent boundary Σ is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognizing approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on Σ are organized around only a few travelling waves and their heteroclinic connections. If the flow is suitably constrained to a subspace with a discrete rotational symmetry, it is possible to find locally attracting travelling waves embedded within Σ. Four new types of travelling waves were found using this approach.

In this paper we present an analytical solution to the problem of sound radiation from semi-infinite coaxial cylinders, as a model for rearward noise emission by aeroengines. The cylinders carry uniform subsonic flows, whose Mach numbers may differ from each other and from that of the external flow. The incident field takes the form of a downstream-going acoustic mode in either the outer cylinder (the bypass flow) or the inner cylinder (the jet). The key geometrical ingredient of our problem is that the two open ends are staggered by a finite axial distance, so that the inner cylinder can be either buried upstream inside the outer cylinder, or can protrude downstream beyond the end of the outer cylinder (sometimes called the ‘half-cowl’ configuration). The solution is found by solving a matrix Wiener–Hopf equation, which involves the factorization of a certain matrix in the form − = +, with ± analytic, invertible and with algebraic behaviour at infinity in the upper and lower halves of the complex Fourier plane respectively. It turns out that the method of solution is different for the buried and protruding cases. In the buried case the well-known pole removal technique can be applied to a certain meromorphic function (denoted k11), but in the protruding case the corresponding function k22 is no longer meromorphic. Progress is made, however, by using a Padé representation of k22 to yield a meromorphic problem which can then be solved using the pole removal technique as before. A range of results is presented, for both buried and protruding systems and with and without mean flow, and it becomes clear that the stagger of the two open ends can have a very significant effect on the far-field noise. We also obtain reasonable agreement between our predictions and some experimental results. One particular noise mechanism we identify in the presence of mean shear is the way in which a Kelvin–Helmholtz instability mode launched from the upstream trailing edge can be scattered into sound by its interaction with the downstream edge, provided that the separation between the edges is sufficiently large in a way which we identify.

Instability of coupled density fronts, and its fully nonlinear evolution are studied within the idealized reduced-gravity rotating shallow-water model. By using the collocation method, we benchmark the classical stability results on zero potential vorticity (PV) fronts and generalize them to non-zero PV fronts. In both cases, we find a series of instability zones intertwined with the stability regions along the along-front wavenumber axis, the most unstable modes being long wave. We then study the nonlinear evolution of the unstable modes with the help of a high-resolution well-balanced finite-volume numerical scheme by initializing it with the unstable modes found from the linear stability analysis. The most unstable long-wave mode evolves as follows: after a couple of inertial periods, the coupled fronts are pinched at some location and a series of weakly connected co-rotating elliptic anticyclonic vortices is formed, thus totally changing the character of the flow. The characteristics of these vortices are close to known rodon lens solutions. The shorter-wave unstable modes from the next instability zones are strongly concentrated in the frontal regions, have sharp gradients, and are saturated owing to dissipation without qualitatively changing the flow pattern.

This paper presents an experimental study of internal waves generated by circular and rectilinear oscillations of a circular cylinder in a uniformly stratified fluid. The synthetic schlieren technique is used for quantitative analysis of the internal-wave parameters. It is shown that at small oscillation amplitudes, the wave pattern observed for circular oscillations is in good agreement with linear theory: internal waves are radiated in the wave beams passing through the first and third quadrants of a Cartesian coordinate system for the clockwise direction of the cylinder motion, and the intensity of these waves is twice the intensity measured for ‘St Andrew's cross’ waves generated by purely horizontal or vertical oscillations of the same frequency and amplitude. As the amplitude of circular oscillations increases, significant nonlinear effects are observed: (i) a strong density-gradient ‘zero-frequency’ disturbance is generated, and (ii) a region of intense fluid stirring is formed around the cylinder serving as an additional dissipative mechanism that changes the shape of wave envelopes and decreases the intensity of wave motions. In the same range of oscillation amplitudes, the wave generation by rectilinear (horizontal and vertical) oscillations is shown to be by and large a linear process, with moderate manifestations of nonlinearity such as weak ‘zero-frequency’ disturbance and weak variation of the shape of wave envelopes with the oscillation amplitude. Analysis of spatiotemporal images reveals different scenarios of transient effects in the cases of circular and rectilinear oscillations. In general, circular oscillations tend to generate disturbances evolving at longer time scales.

Transitions between regular oscillations and bursting oscillations that involve a bifurcational process which culminates in the creation of a relative periodic orbit of infinite period and infinite length are investigated both experimentally and numerically in a short-aspect-ratio Taylor–Couette flow. This bifurcational process is novel in that it is the accumulation point of a period-adding cascade at which the mid-height reflection symmetry is broken. It is very rich and complex, involving very-low-frequency states arising via homoclinic and heteroclinic dynamics, providing the required patching between states with very different dynamics in neighbouring regions of parameter space. The use of nonlinear dynamical systems theory together with symmetry considerations has been crucial in interpreting the laboratory experimental data as well as the results from the direct numerical simulations. The phenomenon corresponds to dynamics well beyond the first few bifurcations from the basic state and so is beyond the reach of traditional hydrodynamic stability analysis, but it is not fully developed turbulence where a statistical or asymptotic approach could be employed. It is a transitional phenomenon, where the phase dynamics of the large-scale structures (jets of angular momentum emanating from the boundary layer on the rotating inner cylinder) becomes complicated. Yet the complicated phase dynamics remains accessible to an analysis of its space–time characteristics and a comprehensive mechanical characterization emerges. The excellent agreement between the experiments and the numerical simulations demonstrates the robustness of this complex bifurcation phenomenon in a physically realized system with its inherent imperfections and noise. Movies are available with the online version of the paper.

Wall roughness produces a downward shift of the mean streamwise velocity profile in the log region, known as the roughness function. The dependence of the roughness function on the height and arrangement of roughness elements has been confirmed in several studies where regular rough walls were analysed; less attention has been paid to non-regular rough walls. Here, a numerical analysis of turbulent flows over irregularly shaped rough walls is performed, clearly identifying the importance of a parameter, called the effective slope (ES) of the wall corrugations, in characterizing the geometry of non-smooth irregular walls. The effective slope proves to be one of the fundamental geometric parameters for scaling the roughness function. Specifically, for a moderate range of roughness heights, both in the transitionally and in the fully rough regime, ES appears to scale the roughness function for a wide range of irregular rough geometric configurations. The effective slope determines the relative importance of friction drag and pressure drag. For ES ~ 0.15 we find that the friction contribution to the total wall stress is nearly in balance with the pressure-drag contribution. This value separates the region where the roughness function ΔU+ = f(ES) is linear from that where a smooth nonlinear behaviour is observed. In the cases investigated, value ES ~ 0.15 also separates the transitionally rough regime from the fully rough regime.

A general procedure is presented for computing axisymmetric swirling vortices which are steady with respect to an inviscid flow that is either uniform at infinity or includes shear. We consider cases both with and without a spherical obstacle. Choices of numerical parameters are given which yield vortex rings with swirl, attached vortices with swirl analogous to spherical vortices found by Moffatt, tubes of vorticity extending to infinity and Beltrami flows. When there is a spherical obstacle we have found multiple solutions for each set of parameters. Flows are found by numerically solving the Bragg–Hawthorne equation using a non-Newton-based iterative procedure which is robust in its dependence on an initial guess.

The flow of an impinging non-Newtonian jet onto a solid flat plate is examined theoretically in this study. Similarity solutions are sought for both shear-thinning and shear-thickening fluids of the power-law type. The jet is assumed to spread out in a thin layer bounded by a hydraulic jump. In addition to the stagnation-flow region, the flow domain is divided into three main regions: a developing boundary layer, fully viscous boundary layer and hydraulic jump. The anomalous behaviour of power-law fluids at small shear rate is remedied by seeking a two-layer solution in each domain. Such anomalies include the singularity of viscosity for shear-thinning fluids, and the vanishing of viscosity as well the overshoot in velocity for shear-thickening fluids. Although the rate of shear-thinning appears to affect significantly the film profile and velocity, only the overall viscosity influences the position of the hydraulic jump.