The evolution of a stratified shear layer with mean shear in the horizontal direction, orthogonal to gravity, is numerically investigated with focus on the structural organization of the vorticity and density fields. Although the Reynolds number of the flow increases with time, facilitating instabilities and turbulence, the bulk Richardson number signifying the level of stratification also increases. Remarkably rich dynamics is found: turbulence; the emergence of coherent core/braid regions from turbulence; formation of a lattice of dislocated vortex cores connected by thin horizontal sheets of collapsed density and vorticity; density-driven intrusions at the edges of the shear layer; and internal wave generation and propagation. Stratification introduces significant vertical variability although it inhibits the vertical velocity. The molecular dissipation of turbulent kinetic energy and of turbulent potential energy are both found to be substantial even in the case with highest stratification, and primarily concentrated in thin horizontal sheets. The simulation data are used to help explain how buoyancy induces the emergence of columnar vortex cores from turbulence and then dislocates these cores to eventually form a lattice of ‘pancake’ eddies connected by thin sheets with large vertical shear (horizontal vorticity) and density gradient.

Unsteady turbulence in uniformly stratified unsheared flow is analysed using rapid distortion theory (RDT). For inviscid flow with no molecular diffusion the theory shows how the initial conditions, such as the initial turbulent kinetic energy KE0 and potential energy PE0, determine the partition of energy between the potential energy associated with density fluctuation and the kinetic energy associated with each of the velocity components during the subsequent development of the turbulence. One parameter is an exception to this sensitivity to initial conditions, namely the limit at large time of the ratio of potential energy to vertical kinetic energy. In the linear theory, this ratio depends neither on the Reynolds number Re, nor the Prandtl number Pr nor the Froude number Fr. This is consistent with turbulence measurements in the atmosphere, wind tunnel and water tank experiments, and with large-eddy simulations, where similar values of the ratio are found. The RDT results are extended to show the effects of viscosity and diffusion where Re is not very large, explaining the sensitivity of the spectra and the fluxes to the value of the Prandtl number Pr. When Pr is larger than 1, the high-wavenumber components of the three-dimensional spectra induce a vertical flux of temperature (density) that is positive (negative), and therefore ‘countergradient.’ On the other hand, when the thermal diffusivity is stronger and Pr is less than 1, lower-wavenumber components become countergradient sooner since the high-wavenumber components are prevented from becoming countergradient. When all the wavenumber components are integrated to derive the total vertical density flux, it becomes countergradient more quickly and more strongly in high-Pr than in low-Pr turbulence. All these theoretically derived differences between high-Pr and low-Pr turbulence are consistent with the experimental measurements in water tank and wind tunnel experiments and numerical simulations. It is shown that the initial kinetic and potential energy spectrum forms E(k) and S(k) near k = 0 determine the long-time limit values of the variances and the covariances, including their decay rate with time. In the special case of Pr = 1, the oscillation time period of the three-dimensional spectrum function is independent of the wavenumber and is the same as that of an inviscid fluid with the effect of viscosity/diffusion being limited to the damping of all the wavenumber components in-phase with each other. Furthermore, the non-dimensional ratios of the covariances, including the normalized vertical density flux and the anisotropy tensor, agree with the inviscid results if S(k) is proportional to E(k), or if either S(k) or E(k) is identically zero. However, even when Pr = 1, in the ‘one-dimensional spectrum’ in the x-direction, there is a transitory countergradient flux for high wavenumbers; only in this case is there a qualitative difference with the three-dimensioanl spectrum. This paper shows that the characteristic differences in the behaviour of stably stratified turbulence reported in previous DNS experiments at moderate Reynolds numbers can largely be explained by linear oscillations and simple molecular or eddy diffusion rather than by any new kinds of nonlinear mixing processes.

We consider the effects of strong critical-layer nonlinearity on the spatially growing instabilities of a shear layer between two parallel streams. A composite expansion technique is used to obtain a single formula that accounts for both shear-layer spreading and nonlinear critical-layer effects. Nonlinearity causes the instability to saturate well upstream of the linear neutral stability point. It also produces vorticity roll-up that cannot be predicted by linear theory.

A decomposition of the energy and helicity fluxes is used to analyse turbulent hydrodynamic flows. The decomposition is based on the projection of the flow to a helical basis. This allows the roles of interactions among modes of different helicities to be investigated separately. The proposed formalism is applied to large-scale numerical simulations of non-helical and helical flows, where the decomposed fluxes are explicitly calculated. It is shown that the total energy flux can be split into three fluxes that independently remain constant in the inertial range. One of these fluxes which corresponds to the interactions of fields with the same helicity is negative, implying the presence of an inverse cascade that is ‘hidden’ inside the forward cascade. Similarly to the energy flux, it is also shown that the helicity flux can be decomposed into two fluxes that remain constant in the inertial range. Implications of these results as well as possible new directions for investigations are discussed.

An open system or open resonator is a domain of wave activity separated from the exterior by a partly open or partly transparent surface. Such open resonators lose energy to infinity through radiation. The numerical computation of the corresponding resonances is complicated by spurious reflections of the outgoing waves at the necessarily finite grid boundaries. These reflections can be reduced to extremely low levels by applying perfectly matched layer (PML) absorbing boundary conditions, which separate the discrete resonances from the continuous spectrum. Using a simple one-dimensional model problem, the influence of the various PML parameters is determined by a numerical error analysis. In addition to one-dimensional open resonators, two-dimensional open resonators as well as various resonating structures in waveguides are considered, and the resonant spectra and selected modes are evaluated. For the first time, leaky modes are computed for several resonating structures in a waveguide in addition to the trapped modes published in the literature. In applications, leaky mode resonances are often more important than trapped mode resonances. Gap tones, observed in a model problem of high-lift configurations, are identified as transversal resonant modes with the lowest radiation losses.

A linear stability analysis is presented for flow between concentric cylinders when a fully developed axial flow is present. Small perturbations are assumed to be nonaxisymmetric. This leads to an eigenvalue problem with four eigenvalues: the critical Taylor number, an amplification factor and two wavenumbers. The presence of the tangential wavenumber permits prediction of the stability of spiral flow. This made it possible to model the flow more accurately and to extend the range of calculations to higher axial Reynolds numbers than had previously been attainable. Calculations were carried out for radius ratios from 0·95 to 0·1, Reynolds numbers as large as 300 and cases with co-rotation and counter-rotation of the cylinders.

Measurements obtained in boundary layers developing downstream of the highly turbulent, separated flow generated at the leading edge of a blunt flat plate are presented. Two cases are considered: first, when there is only very low (wind tunnel) turbulence present in the free-stream flow and, second, when roughly isotropic, homogeneous turbulence is introduced. With conditions adjusted to ensure that the separated region was of the same length in both cases, the flow around reattachment was significantly different and subsequent differences in the development rate of the two boundary layers are identified. The paper complements, but is much more extensive than, the earlier presentation of some of the basic data (Castro & Epik 1996), confirming not only that the development process is very slow, but also that it is non-monotonic. Turbulence stress levels fall below those typical of zero-pressure-gradient boundary layers and, in many ways, the boundary layer has features similar to those found in standard boundary layers perturbed by free-stream turbulence. It is argued that, at least as far as the turbulence structure is concerned, the inner layer region develops no more quickly than does the outer flow and it is the latter which essentially determines the overall rate of development of the whole flow. Some numerical computations are used to assess the extent to which current turbulence models are adequate for such flows.

This paper presents an analytic solution for aerodynamic noise generated by an unsteady wall pressure gust interacting with a spanwise-variable trailing edge in a background steady uniform flow. Viscous and nonlinear effects are neglected. The Wiener–Hopf method is used in conjunction with a non-orthogonal coordinate transformation and separation of variables to permit analytical progress. The solution is obtained in terms of a tailored modal expansion in the spanwise coordinate; however, only finitely many modes are cut-on, therefore the far-field noise can be quickly evaluated. The solution gives insight into the potential mechanisms behind the reduction of noise for plates with serrated trailing edges compared to those with straight edges. The two mechanisms behind the noise reduction are an increased destructive interference in the far field, and a redistribution of acoustic energy from low cut-on modes to higher cut-off modes. Five different test-case trailing-edge geometries are considered. The analytic solution identifies which geometries are most effective in different frequency ranges: geometries which promote destructive interference are best at low frequencies, whilst geometries which promote a redistribution of energy are better at high frequencies.

This study deals with the flow around a circular cylinder placed near a plane wall and exposed to an oscillatory flow. The study comprises instantaneous pressure distribution measurements around the cylinder at high Reynolds numbers (mostly at Re ∼ 105) and a flow visualization study of vortex motions at relatively smaller Reynolds numbers (Re ∼ 103–104). The range of the gap-to-diameter ratio is from 0 to 2 for the pressure measurements and from 0 to 25 for the flow visualization experiments. The range of the Keulegan–Carpenter number KC is from 4 to 65 for the pressure measurements and from 0 to 60 for the flow visualization tests. The details of vortex motions around the cylinder are identified for specific values of the gap-to-diameter ratio and for the KC regimes known from research on wall-free cylinders. The findings of the flow visualization study are used to interpret the variations in pressure with time around the pipe. The results indicate that the flow pattern and the pressure distribution change significantly because of the close proximity of the boundary where the symmetry in the formation of vortices breaks down, and also the characteristic transverse vortex street observed for wall-free cylinders for 7 < KC < 13 disappears. The results further indicate that the vortex shedding persists for smaller and smaller values of the gap-to-diameter ratio, as KC is decreased. The Strouhal frequency increases with decreasing gap-to-diameter ratio. The increase in the Strouhal frequency with respect to its wall-free-cylinder value can be as much as 50% when the cylinder is placed very close to the wall with a gap-to-diameter ratio of O(0.1).

A resonant subharmonic interaction between two axisymmetric travelling waves was induced in the shear layer of an axisymmetric jet by controlled sinusoidal perturbations with two frequencies separated by one octave. Wherever the two excited waves are non-dispersive and the fundamental is close to its linear neutral point the two waves may interact in a manner that enhances the amplification rate of the subharmonic wave train. The amplified subharmonic will exceed the fundamental's level to become the dominant instability component. The initial phase difference between the subharmonic and the fundamental plays an important role in the amplification of the subharmonic. For specific phase angles between the two excited waves a suppression of the subharmonic may be observed. The influence of other initial parameters such as amplitude ratio, overall forcing level, excitation frequency and flow conditions at the nozzle (i.e. the initial turbulence level and the initial momentum thickness) was also investigated. An increase in the combined forcing level reduces the effect of the initial phase difference on the amplification of the subharmonic. Stronger excitation moves the location at which the two waves are locked in space further upstream while the effect of the initial phase difference decreases. The energy transfer to the subharmonic wave has been analysed by estimating the production terms. The results clearly indicate that most of the energy for the resonant growth of the subharmonic comes directly from the mean flow. The fundamental wave acts as a catalyst, as long as the resonance conditions are satisfied, enhancing the rate of energy transfer from the mean flow to the subharmonic.

This work reports an experimental characterisation of the flow properties in a homogeneous bubble swarm rising at high Reynolds numbers within a homogeneous and isotropic turbulent flow. Both the gas volume fraction $\unicode[STIX]{x1D6FC}$ and the velocity fluctuations $u_{0}^{\prime }$ of the carrier flow before bubble injection are varied, respectively, in the ranges $0\,\%<\unicode[STIX]{x1D6FC}<0.93\,\%$ and $2.3~\text{cm}~\text{s}^{-1}<u_{0}^{\prime }<5.5~\text{cm}~\text{s}^{-1}$. The so-called bubblance parameter ($b=V_{r}^{2}\unicode[STIX]{x1D6FC}/u_{0}^{\prime 2}$, where $V_{r}$ is the bubble relative rise velocity) is used to compare the ratio of the kinetic energy generated by the bubbles to the one produced by the incident turbulence, and is varied from 0 to 1.3. Conditional measurements of the velocity field downstream of the bubbles in the vertical direction allow us to disentangle three regions that have specific statistical properties, namely the primary wake, the secondary wake and the far field. While the fluctuations in the primary wake are similar to that of a single bubble rising in a liquid at rest, the statistics of the velocity fluctuations in the far field follow a Gaussian distribution, similar to that produced by the homogenous and isotropic turbulence at the largest scales. In the secondary wake region, the conditional probability density function of the velocity fluctuations is asymmetric and shows an exponential tail for the positive fluctuations and a Gaussian one for the negative fluctuations. The overall agitation thus results from the combination of these three contributions and depends mainly on the bubblance parameter. For $0<b<0.7$, the overall velocity fluctuations in the vertical direction evolve as $b^{0.4}$ and are mostly driven by the far-field agitation, whereas the fluctuations increase as $b^{1.3}$ for larger values of the bubblance parameter ($b>0.7$), in which the significant contributions come both from the secondary wake and the far field. Thus, the bubblance parameter is a suitable parameter to characterise the evolution of liquid agitation in bubbly turbulent flows.

Explicit analytical expressions for the drag and diffusion coefficients of a spherical particle attached to the flat interface between two immiscible fluids are constructed for the case of a vanishing viscosity ratio between the fluid phases. The model is designed to account explicitly for the dependence on the contact angle between the two fluids and the solid surface. The Lorentz reciprocal theorem is applied in the context of geometric perturbations from the limiting cases of $90^{\circ }$ and $180^{\circ }$ contact angles. The model agrees well with the experimental and numerical data from the literature. Also, an advantage of the method utilized is that the drag and diffusion coefficients can be calculated up to one order higher in the perturbation parameter than the known velocity and pressure fields. Extensions to other particle shapes with known velocity and pressure fields are straightforward.

For small-amplitude vortical and entropic unsteady disturbances of potential flows, Goldstein proposed a partial splitting of the velocity field into a vortical part u(I) that is a known function of the imposed upstream disturbance and a potential part ∇ϕ satisfying a linear inhomogeneous wave equation with a dipole-type source term. The present paper deals with flows around bodies with a stagnation point. It is shown that for such flows u(I) becomes singular along the entire body surface and its wake and as a result ∇ϕ will also be singular along the entire body surface. The paper proposes a modified splitting of the velocity field into a vortical part u(R) that has zero streamwise and normal components along the body surface, an entropy-dependent part and a regular part ∇ϕ* that satisfies a linear inhomogeneous wave equation with a modified source term.

For periodic disturbances, explicit expressions for u(R) are given for three-dimensional flows past a single obstacle and for two-dimensional mean flows past a linear cascade. For weakly sheared flows, it is shown that if the mean flow has only a finite number of isolated stagnation points, u(R) will be finite along the body surface. On the other hand, if the mean flow has a stagnation line along the body surface such as in two-dimensional flows then the component of u(R) in this direction will have a logarithmic singularity.

For incompressible flows, the boundary-value problem for ϕ* is formulated in terms of an integral equation of the Fredholm type. The theory is applied to a typical bluff body. Detailed calculations are carried out to show the velocity and pressure fields in response to incident harmonic disturbances.

Many growing plant cells undergo rapid axial elongation with negligible radial expansion. Growth is driven by high internal turgor pressure causing viscous stretching of the cell wall, with embedded cellulose microfibrils providing the wall with strongly anisotropic properties. We present a theoretical model of a growing cell, representing the primary cell wall as a thin axisymmetric fibre-reinforced viscous sheet supported between rigid end plates. Asymptotic reduction of the governing equations, under simple sets of assumptions about the fibre and wall properties, yields variants of the traditional Lockhart equation, which relates the axial cell growth rate to the internal pressure. The model provides insights into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility, and shows how either dynamical changes in wall material properties or passive fibre reorientation may suppress cell elongation.

Receptivity, disturbance growth and breakdown to turbulence in Görtler flow are studied by spatial direct numerical simulation (DNS). The boundary layer is exposed to free-stream vortical modes and localized wall roughness. We propose a normalization of the roughness-induced receptivity coefficient by the square root of the Görtler number. This scaling removes the dependence of the receptivity coefficient on wall curvature. It is found that vortical modes are more efficient at generating Görtler vortices than localized roughness. The boundary layer is most receptive to zero- and low-frequency free-stream vortices, exciting steady and slowly travelling Görtler modes. The associated receptivity mechanism is linear and involves the generation of boundary-layer streaks, which soon evolve into unstable Görtler vortices. This connection between transient and exponential amplification is absent on flat plates and promotes transition to turbulence on curved walls. We demonstrate that the Görtler boundary layer is also receptive to high-frequency free-stream vorticity, which triggers steady Görtler rolls via a nonlinear receptivity mechanism. In addition to the receptivity study, we have carried out DNS of boundary-layer transition due to broadband free-stream turbulence with different intensities and frequency spectra. It is found that nonlinear receptivity dominates over the linear mechanism unless the free-stream fluctuations are concentrated in the low-frequency range. In the latter case, transition is accelerated due to the presence of travelling Görtler modes.

Quasi-steady aerodynamic models play an important role in evaluating aerodynamic performance and conducting design and optimization of flapping wings. The kinematics of flapping wings is generally a resultant motion of wing translation (yaw) and rotation (pitch and roll). Most quasi-steady models are aimed at predicting the lift and thrust generation of flapping wings with prescribed kinematics. Nevertheless, it is insufficient to limit flapping wings to prescribed kinematics only since passive pitching motion is widely observed in natural flapping flights and preferred for the wing design of flapping wing micro air vehicles (FWMAVs). In addition to the aerodynamic forces, an accurate estimation of the aerodynamic torque about the pitching axis is required to study the passive pitching motion of flapping flights. The unsteadiness arising from the wing’s rotation complicates the estimation of the centre of pressure (CP) and the aerodynamic torque within the context of quasi-steady analysis. Although there are a few attempts in literature to model the torque analytically, the involved problems are still not completely solved. In this work, we present an analytical quasi-steady model by including four aerodynamic loading terms. The loads result from the wings translation, rotation, their coupling as well as the added-mass effect. The necessity of including all the four terms in a quasi-steady model in order to predict both the aerodynamic force and torque is demonstrated. Validations indicate a good accuracy of predicting the CP, the aerodynamic loads and the passive pitching motion for various Reynolds numbers. Moreover, compared to the existing quasi-steady models, the presented model does not rely on any empirical parameters and thus is more predictive, which enables application to the shape and kinematics optimization of flapping wings.

We conducted direct numerical simulations of turbulent flow over three-dimensional sinusoidal roughness in a channel. A passive scalar is present in the flow with Prandtl number $Pr=0.7$, to study heat transfer by forced convection over this rough surface. The minimal-span channel is used to circumvent the high cost of simulating high-Reynolds-number flows, which enables a range of rough surfaces to be efficiently simulated. The near-wall temperature profile in the minimal-span channel agrees well with that of the conventional full-span channel, indicating that it can be readily used for heat-transfer studies at a much reduced cost compared to conventional direct numerical simulation. As the roughness Reynolds number, $k^{+}$, is increased, the Hama roughness function, $\unicode[STIX]{x0394}U^{+}$, increases in the transitionally rough regime before tending towards the fully rough asymptote of $\unicode[STIX]{x1D705}_{m}^{-1}\log (k^{+})+C$, where $C$ is a constant that depends on the particular roughness geometry and $\unicode[STIX]{x1D705}_{m}\approx 0.4$ is the von Kármán constant. In this fully rough regime, the skin-friction coefficient is constant with bulk Reynolds number, $Re_{b}$. Meanwhile, the temperature difference between smooth- and rough-wall flows, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}^{+}$, appears to tend towards a constant value, $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$. This corresponds to the Stanton number (the temperature analogue of the skin-friction coefficient) monotonically decreasing with $Re_{b}$ in the fully rough regime. Using shifted logarithmic velocity and temperature profiles, the heat-transfer law as described by the Stanton number in the fully rough regime can be derived once both the equivalent sand-grain roughness $k_{s}/k$ and the temperature difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$ are known. In meteorology, this corresponds to the ratio of momentum and heat-transfer roughness lengths, $z_{0m}/z_{0h}$, being linearly proportional to the inner-normalised momentum roughness length, $z_{0m}^{+}$, where the constant of proportionality is related to $\unicode[STIX]{x0394}\unicode[STIX]{x1D6E9}_{FR}^{+}$. While Reynolds analogy, or similarity between momentum and heat transfer, breaks down for the bulk skin-friction and heat-transfer coefficients, similar distribution patterns between the heat flux and viscous component of the wall shear stress are observed. Instantaneous visualisations of the temperature field show a thin thermal diffusive sublayer following the roughness geometry in the fully rough regime, resembling the viscous sublayer of a contorted smooth wall.

The natural frequencies and damping rates of surface waves in a circular cylinder with pinned-end boundary conditions are calculated in terms of the gravitational Reynolds and Bond numbers, C−1 and B, and the slenderness of the cylinder Λ, in the limit C→0. We consider higher-order approximations that include the effect of viscous dissipation in the Stokes boundary layers and the bulk. A comparison with clean-surface experiments by Henderson & Miles (1994) shows a satisfactory agreement except for the first axisymmetric mode, which exhibits a 26% discrepancy. The much larger dramatic discrepancy of former theoretical predictions is hereby improved and explained.

In this paper, we revisit Janssen’s (J. Phys. Oceanogr., vol. 33 (4), 2003, pp. 863–884) formulation for the dynamic excess kurtosis of weakly nonlinear gravity waves in deep water. For narrowband directional spectra, the formulation is given by a sixfold integral that depends upon the Benjamin–Feir index and the parameter $R={\it\sigma}_{{\it\theta}}^{2}/2{\it\nu}^{2}$, a measure of short-crestedness for the dominant waves, with ${\it\nu}$ and ${\it\sigma}_{{\it\theta}}$ denoting spectral bandwidth and angular spreading. Our refinement leads to a new analytical solution for the dynamic kurtosis of narrowband directional waves described with a Gaussian-type spectrum. For multidirectional or short-crested seas initially homogeneous and Gaussian, in a focusing (defocusing) regime dynamic kurtosis grows initially, attaining a positive maximum (negative minimum) at the intrinsic time scale ${\it\tau}_{c}={\it\nu}^{2}{\it\omega}_{0}t_{c}=1/\sqrt{3R}$, or $t_{c}/T_{0}\approx 0.13/{\it\nu}{\it\sigma}_{{\it\theta}}$, where ${\it\omega}_{0}=2{\rm\pi}/T_{0}$ denotes the dominant angular frequency. Eventually the dynamic excess kurtosis tends monotonically to zero as the wave field reaches a quasi-equilibrium state characterized by nonlinearities mainly due to bound harmonics. Quasi-resonant interactions are dominant only in unidirectional or long-crested seas where the longer-time dynamic kurtosis can be larger than that induced by bound harmonics, especially as the Benjamin–Feir index increases. Finally, we discuss the implication of these results for the prediction of rogue waves.

The purpose of the present paper is to derive expressions for the pressure fields of various high frequency convected singularities immersed in a unidirectional sheared flow. These expressions include the simultaneous effects of fluid and source convection and refraction. These results are then combined to predict the far-field directivity of cold round jets. It is found that the agreement between experiment and the present theory is quite good at a source Strouhal number of unity but that this agreement deteriorates as the source frequency is increased. Our theoretical results show the explicit form of the ‘refraction integral’ and that convective amplification for the pressure of a quadrupole is increased by a factor of (1 — MJ cos Θ)−1 over the classical results, where MJ is the jet Mach number and Θ is the angle from the jet axis. Thus acoustic/mean-flow interaction not only implies refraction but also additional convective amplification due not to source convection but to fluid motion.