A numerical investigation of the disturbance amplification in a Mach 4.8 flat-plate boundary layer with a localized two-dimensional roughness element is presented. The height of the roughness is varied and reaches up to approximately 70% of the boundary-layer thickness. Simulations are based on a time-accurate integration of the compressible Navier–Stokes equations, with a small disturbance of fixed frequency being triggered via blowing and suction upstream of the roughness element. The roughness element considerably alters the instability of the boundary layer, leading to increased amplification or damping of a modal wave depending on the frequency range. The roughness is also the source of an additional perturbation. Even though this additional mode is stable, the interaction with the unstable mode in the form of constructive and destructive interference behind the roughness element leads to a beating and therefore transiently increased disturbance amplitude. Far downstream of the roughness, the amplification rate of a flat-plate boundary layer is recovered. Overall, the two-dimensional roughness element behaves as disturbance amplifier with a limited bandwidth capable of filtering a range of frequencies and strongly amplifying only a selected range.

Laboratory and field measurements of the generation of gravity waves by turbulent winds imply that a theoretical model based on laminar flow may be adequate on a laboratory, but not an oceanographic, scale. This suggests that the significance of wave-induced perturbations in the turbulent Reynolds stresses for momentum transfer from wind to waves must increase with an appropriate scale parameter. A generalization of the laminar model is constructed by averaging the linearized equations of motion for a turbulent shear flow in a direction (say y) parallel to the wave crests of a particular Fourier component of the surface-wave field. It is shown that the resulting, mean momentum transfer to this component comprises: (i) a singular part, which is proportional to the product of the velocity-profile curvature and the mean square of the wave-induced vertical velocity in the critical layer, where the mean wind speed is equal to the wave speed; (ii) a vertical integral of the mean product of the vertical velocity and the vorticity ω, where ω is the wave-induced perturbation in the total vorticity along a streamline of the y-averaged motion; (iii) the perturbation in the mean turbulent shear stress at the air-water interface. The equation that governs the advection of the vorticity ω under the action of the perturbations in the turbulent Reynolds stresses is derived. Further theoretical progress appears to demand some ad hoc hypothesis for the specification of these turbulent Reynolds stresses. Two such hypotheses are discussed briefly, but it does not appear worth while, in the absence of more detailed experimental data, to carry out elaborate numerical calculations at this time.

We present a calculation of the hydrodynamic self-diffusion coefficient of a tagged particle in a dilute mono-dispersed suspension of small neutrally buoyant spheres undergoing a steady simple shearing motion. The displacement of the tagged particle parallel to the longitudinal or streamwise direction resulting from a ‘collision’ with one other particle is calculated on the assumption that inertia and Brownian motion effects are negligible. Summing over different pairs leads to a logarithmically divergent integral for the diffusivity which is rendered finite by allowing for the cutoff due to the occasional presence of another pair of particles. The longitudinal shearinduced self-diffusion coefficient is thus found to be 0.267a2γ{cln c−1 + O(c)], where γ denotes the applied shear rate, a is the radius of the spheres and c their volume concentration.

The instability of rectangular jets is investigated using a vortex-sheet model. It is shown that such jets support four linearly independent families of instability waves. Within each family there are infinitely many modes. A way to classify these modes according to the characteristics of their mode shapes or eigenfunctions is proposed. The stability equation for jets of this geometry is non-separable so that the traditional methods of analysis are not applicable. It is demonstrated that the boundary element method can be used to calculate the dispersion relations and eigenfunctions of these instability wave modes. The method is robust and efficient. A parametric study of the instability wave characteristics has been carried out. A sample of the numerical results is reported here. It is found that the first and third modes of each instability wave family are corner modes. The pressure fluctuations associated with these instability waves are localized near the corners of the jet. The second mode, however, is a centre mode with maximum fluctuations concentrated in the central portion of the jet flow. The centre mode has the largest spatial growth rate. It is anticipated that as the instability waves propagate downstream the centre mode would emerge as the dominant instability of the jet.

There has been some debate recently on whether the far-wake structure downstream of a cylinder is dependent on, or ‘connected’ with, the precise details of the near-wake structure. Indeed, it has previously been suggested that the far-wake scale and frequency are unconnected with those of the near wake. In the present paper, we demonstrate that both the far-wake scale and frequency are dependent on the near wake. Surprisingly, the characteristic that actually forges a ‘connection’ between the near and far wakes is the sensitivity to free-stream disturbances. It is these disturbances that are also responsible for the regular three-dimensional patterns that may be visualized. Observations of a regular ‘honeycomb’-like three-dimensional pattern in the far wake is found to be caused by an interaction between oblique shedding waves from upstream and large-scale two-dimensional waves, amplified from the free-stream disturbances. The symmetry and spanwise wavelength of Cimbala, Nagib & Roshko's (1988) three-dimensional pattern are precisely consistent with such wave interactions. In the presence of parallel shedding, the lack of a honeycomb pattern shows that such a three-dimensional pattern is clearly dependent on upstream oblique vortex shedding.

With the deductions above as a starting point, we describe a new mechanism for the resonance of oblique waves, as follows. In the case of two-dimensional waves, corresponding to a very small spectral peak in the free stream (fT) interacting (quadratically) with the oblique shedding waves frequency (fK), it appears that the most amplified or resonant frequency in the far wake is a combination frequency fFW = (fK–fT), which corresponds physically with ‘oblique resonance waves’ at a large oblique angle. The large scatter in (fFW/fK) from previous studies is principally caused by the broad response of the far wake to a range of free-stream spectral peaks (fT). We present clear visualization of the oblique wave phenomenon, coupled with velocity measurements which demonstrate that the secondary oblique wave energy can far exceed the secondary two-dimensional wave energy by up to two orders of magnitude. Further experiments show that, in the absence of an influential free-stream spectral peak, the far wake does not resonate, but instead has a low-amplitude broad spectral response. The present phenomena are due to nonlinear instabilities in the far wake, and are not related to vortex pairing. There would appear to be distinct differences between this oblique wave resonance and the subharmonic resonances that have been previously studied in channel flow, boundary layers, mixing layers and airfoil wakes.

We present a theoretical investigation of the self-diffusiophoresis of a class of two-faced, two-dimensional Janus particles propelled by the production of gradients in the concentration of a solute diffusing into a surrounding fluid at zero Reynolds and Péclet numbers. Those concentration gradients produce a tangential boundary slip resulting in translation and rotation of the particle, as a consequence of the fact that it is free of both force and torque. The model Janus particles studied here have piecewise constant surface mobilities and surface activities over two faces. An isolated circular Janus particle is studied first and its speed of locomotion in free space is found analytically. Confinement effects are then investigated by placing such a Janus particle near a straight no-slip wall. The governing nonlinear dynamical system is found in explicit form. It is used to study how the geometry, location and orientation of the particle relative to the wall affect its motion. It is found that if the particles do not hit the wall in finite time they are eventually repelled away from it.

This paper is concerned with forced oscillations of fluid in a rectangular container. From the linearized approximation of the equations governing these oscillations, resonance frequencies are obtained for which the amplitude of the oscillations becomes infinite. Observation shows that under these circumstances a hydraulic jump is formed, which travels periodically back and forth between the walls of the container. This hydraulic jump is a non-linear phenomenon, analogous to the shock wave appearing in one-dimensional gas flow under similar resonance conditions.

A theory developed by previous authors for one-dimensional gas flow is applied to the fluid oscillations in order to calculate the strength and the phase of the jump. The moment exerted on the container is also calculated. These quantities were measured experimentally at the lowest resonance frequency and the results are in good agreement with the theoretical values.

We perform numerical box simulations of strongly stratified turbulence. The equations solved are the Boussinesq equations with constant Brunt–Väisälä frequency and forcing either in rotational or divergent modes, or, with another terminology, in vortical or wave modes. In both cases, we observe a forward energy cascade and inertial-range scaling of the horizontal kinetic and potential energy spectra. With forcing in rotational modes, there is approximate equipartition of kinetic energy between rotational and divergent modes in the inertial range. With forcing in divergent modes the results are sensitive to the vertical forcing wavenumber kfv. If kfv is sufficiently large the dynamics is very similar to the dynamics of the simulations which are forced in rotational modes, with approximate equipartition of kinetic energy in rotational and divergent modes in the inertial range. Frequency spectra of rotational, divergent and potential energy are calculated for individual Fourier modes. Waves are present at low horizontal wavenumbers corresponding to the largest scales in the boxes. In the inertial range, the frequency spectra exhibit no distinctive peaks in the internal wave frequency. In modes for which the vertical wavenumber is considerably larger than the horizontal wavenumber, the frequency spectra of rotational and divergent modes fall on top of each other. The simulation results indicate that the dynamics of rotational and divergent modes develop on the same time scale in stratified turbulence. We discuss the relevance of our results to atmospheric and oceanic dynamics. In particular, we review a number of observational reports indicating that stratified turbulence may be a prevalent dynamic process in the ocean at horizontal scales of the order of 10 or 100 m up to several kilometres.

Characteristics of laboratory-scale bubble-driven buoyant plumes in a stably stratified quiescent fluid are studied using large-eddy simulation (LES). As a bubble plume entrains stratified ambient water, its net buoyancy decreases due to the increasing density difference between the entrained and ambient fluids. A large fraction of the entrained fluid eventually detrains and falls along an annular outer plume from a height of maximum rise (peel height) to a neutral buoyancy level (trap height), during which less buoyant scalars (e.g. small droplets) are trapped and dispersed horizontally, forming quasi-horizontal intrusion layers. The inner/outer double-plume structure and the peel/intrusion process are found to be more distinct for cases with small bubble rise velocity, while weak and unstable when the slip velocity is large. LES results are averaged to generate distributions of mean velocity and turbulent fluxes. These distributions provide data for assessing the performance of previously developed closures used in one-dimensional integral plume models. In particular, the various LES cases considered in this study yield consistent behaviour for the entrainment coefficients for various plume cases. Furthermore, a new continuous peeling model is derived based on the insights obtained from LES results. Comparing to previous peeling models, the new model behaves in a more self-consistent manner, and it is expected to provide more reliable performance when applied in integral plume models.

Transition to uniform turbulence in cylindrical pipe flow occurs experimentally via the spatial expansion of isolated coherent structures called ‘slugs’, triggered by localized finite-amplitude disturbances. We study this process numerically by examining the preferred route in phase space through which a critical disturbance initiates a ‘slug’. This entails first identifying the relative attractor – ‘edge state’ – on the laminar–turbulent boundary in a long pipe and then studying the dynamics along its low-dimensional unstable manifold, leading to the turbulent state. Even though the fully turbulent state delocalizes at Re ≈ 2300, the edge state is found to be localized over the range Re = 2000–6000, and progressively reduces in both energy and spatial extent as Re is increased. A key process in the genesis of a slug is found to be vortex shedding via a Kelvin–Helmholtz mechanism from wall-attached shear layers quickly formed at the edge state's upstream boundary. Whether these shedded vortices travel on average faster or slower downstream than the developing turbulence determines whether a puff or a slug (respectively) is formed. This observation suggests that slugs are out-of-equilibrium puffs which therefore do not co-exist with stable puffs.

The aerodynamic performance of lifting surfaces operating at low Reynolds number conditions is impaired by laminar separation. In most cases, transition to turbulence occurs in the separated shear layer as a result of a series of strong hydrodynamic instability mechanisms. Although the understanding of these mechanisms has been significantly advanced over the past decades, key questions remain unanswered about the influence of external factors such as free-stream turbulence (FST) and others on transition and separation. The present study is driven by the need for more accurate predictions of separation and transition phenomena in ‘real world’ applications, where elevated levels of FST can play a significant role (e.g. turbomachinery). Numerical investigations have become an integral part in the effort to enhance our understanding of the intricate interactions between separation and transition. Due to the development of advanced numerical methods and the increase in the performance of supercomputers with parallel architecture, it has become feasible for low Reynolds number application ($O(10^{5})$) to carry out direct numerical simulations (DNS) such that all relevant spatial and temporal scales are resolved without the use of turbulence modelling. Because the employed high-order accurate DNS are characterized by very low levels of background noise, they lend themselves to transition research where the amplification of small disturbances, sometimes even growing from numerical round-off, can be examined in great detail. When comparing results from DNS and experiment, however, it is beneficial, if not necessary, to increase the background disturbance levels in the DNS to levels that are typical for the experiment. For the current work, a numerical model that emulates a realistic free-stream turbulent environment was adapted and implemented into an existing Navier–Stokes code based on a vorticity–velocity formulation. The role FST plays in the transition process was then investigated for a laminar separation bubble forming on a flat plate. FST was shown to cause the formation of the well-known Klebanoff mode that is represented by streamwise-elongated streaks inside the boundary layer. Increasing the FST levels led to accelerated transition, a reduction in bubble size and better agreement with the experiments. Moreover, the stage of linear disturbance growth due to the inviscid shear-layer instability was found to not be ‘bypassed’.

We examine the effects of the variation in dissipation-range resolution on the accuracy of inertial-range statistics and intermittency in terms of the direct numerical simulations of homogeneous turbulence and passive-scalar turbulence by changing the spatial resolution up to 20483 grid points while maintaining a constant Reynolds number at Rλ ≃ 180 or ≃ 420 and Schmidt number at Sc = 1. Although large fluctuations of the derivative fields depended strongly on Kmaxη and were underestimated when Kmaxη≃1, where Kmax is the maximum wavenumber in the computations and η is the mean Kolmogorov length, the behaviour of the spectra and the scaling exponents of the structure functions up to the eighth order in the range of scales greater than 10η was insensitive to variations in Kmaxη, even when Kmaxη≃1. The relationship between the spatial resolution and asymptotic tail of the probability density functions of the energy dissipation fields was studied using the multifractal model for dissipation, and the results were confirmed by comparison to the simulation data. Degradation of the statistics arises from modifications to the flow dynamics due to the finite wavenumber cutoff and the use of a coarser filter width for the data, which is obtained using a reasonable accuracy criterion for the flow dynamics. The effect of the former was less than that of the latter for the low-to-moderate-order statistics when Kmaxη≥1. We also discuss the universality of the inertial-range statistics with respect to variations in the dissipation-range characteristics.

In this paper, we use numerical simulation and laboratory experimental observation to show that fire whirls can be generated spontaneously through the interaction between a central flame and surrounding organized or randomly distributed flames. The momentum of the air stream entrained by the main flame decreases as it crosses a surrounding flame, so that the main flame rotates if surrounding flames are arranged in such a way as to block the passage of the air stream directed towards the centre of the main flame and to favour flows in a particular circumferential direction. An analysis is performed to study the role of the rotation speed in the flame height. It is found that the flame height initially decreases to a minimum owing to the inflow boundary layer wind reducing the initial vertical velocity of gas for low rotation speed and to entrainment enhancement reducing the rising time, and then it increases owing to the pressure reduction at the centre of the rotating vortex and entrainment suppression extending the rising time.

Direct numerical simulations have been performed to study the dynamics of isolated turbulent spots in compressible isothermal-wall boundary layers. Results of a bypass transition scenario at Mach 2, 4 and 6 are presented. At all Mach numbers the evolved spots have a leading-edge overhang, followed by a turbulent core and a calmed region at the rear interface. The spots have an upstream-pointing arrowhead shape when visualized by near-wall slices, but a downstream-pointing arrowhead in slices away from the wall. The lateral spreading of the spot decreases substantially with the Mach number, consistent with a growth mechanism based on the instability of lateral shear layers. Evidence for a supersonic (Mack) mode substructure is found in the Mach 6 case, where coherent spanwise structures are observed under the spot overhang region.

A field experiment – the Snow Horizontal Array Turbulence Study (SnoHATS) – has been performed over an extensive glacier in Switzerland in order to study small-scale turbulence in the stable atmospheric surface layer, and to investigate the role, dynamics and modelling of the subgrid scales (SGSs) in the context of large-eddy simulations. The a priori data analysis aims at comparing the role and behaviour of the SGSs under stable conditions with previous studies under neutral or unstable conditions. It is found that the SGSs in a stable surface layer remain an important sink of temperature variance and turbulent kinetic energy from the resolved scales and carry a significant portion of the fluxes when the filter scale is larger than the distance to the wall. The fraction of SGS fluxes (out of the total fluxes) is found to be independent of stability. In addition, the stress–strain alignment is similar to the alignment under neutral and unstable conditions. The model coefficients vary considerably with stability but in a manner consistent with previous findings, which also showed that scale-dependent dynamic models can capture this variation. Furthermore, the variation of the coefficients for both momentum and heat SGS fluxes can be shown to be better explained by stability parameters based on vertical gradients, rather than vertical fluxes. These findings suggest that small-scale turbulence dynamics and SGS modelling under stable conditions share many important properties with neutral and convective conditions, and that a unified approach is thus possible. This paper concludes with a discussion of some other challenges for stable boundary-layer simulations that are not encountered in the neutral or unstable cases.

The effects of small-amplitude, two-dimensional grooves on pressure losses in a laminar channel flow have been analysed. Grooves with an arbitrary shape and an arbitrary orientation with respect to the flow direction have been considered. It has been demonstrated that losses can be expressed as a superposition of two parts, one associated with change in the mean positions of the walls and one induced by flow modulations associated with the geometry of the grooves. The former effect can be determined analytically, while the latter has to be determined numerically and can be captured with an acceptable accuracy using reduced-order geometry models. Projection of the wall shape onto a Fourier space has been used to generate such a model. It has been found that in most cases replacement of the actual wall geometry with the leading mode of the relevant Fourier expansion permits determination of pressure losses with an error of less than 10 %. Detailed results are given for sinusoidal grooves for the range of parameters of practical interest. These results describe the performance of arbitrary grooves with the accuracy set by the properties of the reduced-order geometry model and are exact for sinusoidal grooves. The results show a strong dependence of the pressure losses on the groove orientation. Longitudinal grooves produce the smallest drag, and oblique grooves with an inclination angle of ${\sim }42\textdegree $ exhibit the largest flow turning potential. Detailed analyses of the extreme cases, i.e. transverse and longitudinal grooves, have been carried out. For transverse grooves with small wavenumbers, the dominant part of the drag is produced by shear, while the pressure form drag and the pressure interaction drag provide minor contributions. For the same grooves with large wavenumbers, the stream lifts up above the grooves due to their blocking effect, resulting in a change in the mechanics of drag formation: the contributions of shear decrease while the contributions of the pressure interaction drag increase, leading to an overall drag increase. In the case of longitudinal grooves, drag is produced by shear, and its rearrangement results in a drag decrease for long-wavelength grooves in spite of an increase of the wetted surface area. An increase of the wavenumber leads to the fluid being squeezed from the troughs and the stream being forced to lift up above the grooves. The shear is nearly eliminated from a large fraction of the wall but the overall drag increases due to reduction of the effective channel opening. It is shown that properly structured grooves are able to eliminate wall shear from the majority of the wetted surface area regardless of the groove orientation, thus exhibiting the potential for the creation of drag-reducing surfaces. Such surfaces can become practicable if a method for elimination of the undesired pressure and shear peaks through proper groove shaping can be found.

Numerical models of the geodynamo are usually classified into two categories: dipolar modes, observed when the inertial term is small enough; and multipolar fluctuating dynamos, for stronger forcing. We show that a third dynamo branch corresponding to a dominant force balance between the Coriolis force and the Lorentz force can be produced numerically. This force balance is usually referred to as the strong-field limit. This solution coexists with the often described viscous branch. Direct numerical simulations exhibit a transition from a weak-field dynamo branch, in which viscous effects set the dominant length scale, and the strong-field branch, in which viscous and inertial effects are largely negligible. These results indicate that a distinguished limit needs to be sought to produce numerical models relevant to the geodynamo and that the usual approach of minimising the magnetic Prandtl number (ratio of the fluid kinematic viscosity to its magnetic diffusivity) at a given Ekman number is misleading.

Large-eddy simulations of single-shock-driven mixing suggest that, for sufficiently high incident Mach numbers, a two-gas mixing layer ultimately evolves to a late-time, fully developed turbulent flow, with Kolmogorov-like inertial subrange following a power law. After estimating the kinetic energy injected into the diffuse density layer during the initial shock–interface interaction, we propose a semi-empirical characterization of fully developed turbulence in such flows, based on scale separation, as a function of the initial parameter space, as , which corresponds to late-time Taylor-scale Reynolds numbers . In this expression, represents the post-shock perturbation amplitude, the change in interface velocity induced by the shock refraction, the characteristic kinematic viscosity of the mixture, the inner diffuse thickness of the initial density profile, the post-shock Atwood ratio, and for the gas combination and post-shock perturbation amplitude considered. The initially perturbed interface separating air and SF6 (pre-shock Atwood ratio ) was impacted in a heavy–light configuration by a shock wave of Mach number , 1.25, 1.56, 3.0 or 5.0, for which is fixed at about 25 % of the dominant wavelength of an initial, Gaussian perturbation spectrum. Only partial isotropization of the flow (in the sense of turbulent kinetic energy and dissipation) is observed during the late-time evolution of the mixing zone. For all Mach numbers considered, the late-time flow resembles homogeneous decaying turbulence of Batchelor type, with a turbulent kinetic energy decay exponent and large-scale () energy spectrum , and a molecular mixing fraction parameter, . An appropriate time scale characterizing the Taylor-scale Reynolds number decay, as well as the evolution of mixing parameters such as and the effective Atwood ratio , seem to indicate the existence of low- and high-Mach-number regimes.

A smoke visualization technique was used to study the laminar wake behind a single cylinder. Observations of the wake revealed the nature of the process of formation of a Kármán vortex street: instability of the shear layers was coupled with the rolling up process induced by the concentration of vorticity.

An unexpected mass transfer was observed through the Kármán vortex street which cause mixing of the fluid surrounding the wake.

An alternative foundation is developed for the large-eddy simulation (LES) of turbulent flows. It is based on ‘self-conditioned fields’, for example, the mean velocity field conditional on a discrete representation of the filtered velocity field. It is shown that the self-conditioned velocity field minimizes the residual kinetic energy, and that, with the ideal model, the method yields the correct one-time behaviour as determined by the Navier–Stokes equations. The approach is extended to the self-conditioned probability density function (PDF) of compositions. Compared to LES formulations based on the filtered velocity and the filtered density function, the self-conditioned field approach has several advantages: for laminar flow, and in the direct-numerical-simulation limit, the residual fluctuations are zero or exponentially small; full account is taken of the probability distribution of turbulent fields; there are no commutation issues; and there are no issues with filtering at walls, where the self-conditioned velocity is zero. The exact evolution equations for the self-conditioned velocity and composition PDF are derived. Basic models are presented, and the development of improved models is discussed.