Both necessary and sufficient conditions are derived in a systematic, rigorous way for a Reynolds stress model to preserve both frame indifference and positive semi-definiteness of the Reynolds stresses and to satisfy the principle of material frame indifference (PMFI). Also developed are two approximate theories, a linear theory and a quadratic theory, to simplify the work of developing models satisfying such conditions. The results are also valid for the higher-order correlations and the SGS Reynolds stresses. This leads to the results either confirming the previous intuitive arguments or offering new insights into turbulence modelling, and is of significance in clarifying some controversies in the literature, examining how well existing models preserve the physics, and developing new models.

An examination of various existing models with respect to the necessary and sufficient conditions obtained in the present work shows that no one model can simultaneously satisfy the invariance, the realizability and the PMFI. This offers answers to some fundamental questions regarding the existing models, such as the reason why k−ε and k−l models fail to give accurate predictions for the normal Reynolds stresses.

We report an abrupt change in the diffusive transport of inertial objects in wave-driven turbulence as a function of the object size. In these non-equilibrium two-dimensional flows, the turbulent diffusion coefficient $D$ of finite-size objects undergoes a sharp change for values of the object size $r_{p}$ close to the flow forcing scale $L_{f}$. For objects larger than the forcing scale ($r_{p}>L_{f}$), the diffusion coefficient is proportional to the flow energy $U^{2}$ and inversely proportional to the size $r_{p}$. This behaviour, $D\sim U^{2}/r_{p}$ , observed in a chaotic macroscopic system is reminiscent of a fluctuation–dissipation relation. In contrast, the diffusion coefficient of smaller objects ($r_{p}<L_{f}$) follows $D\sim U/r_{p}^{0.35}$. This result does not allow simple analogies to be drawn but instead it reflects strong coupling of the small objects with the fabric and memory of the out-of-equilibrium flow. In these turbulent flows, the flow structure is dominated by transient but long-living bundles of fluid particle trajectories executing random walk. The characteristic widths of the bundles are close to $L_{f}$. We propose a simple phenomenology in which large objects interact with many bundles. This interaction with many degrees of freedom is the source of the fluctuation–dissipation-like relation. In contrast, smaller objects are advected within coherent bundles, resulting in diffusion properties closely related to those of fluid tracers.

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_{B}^{2}$ imposed in the third, vertical direction in a fluid with kinematic viscosity $\unicode[STIX]{x1D708}$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using ‘recurrent flow analysis’. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the ‘mixing efficiency’ of the flow $\mathscr{E}=\unicode[STIX]{x1D712}/(\unicode[STIX]{x1D712}+{\mathcal{D}})$ (where ${\mathcal{D}}$ is the volume-averaged kinetic energy dissipation rate and $\unicode[STIX]{x1D712}$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_{B}=\overline{{\mathcal{D}}}/(\unicode[STIX]{x1D708}N_{B}^{2})$, and is bounded above by $1/6$, consistently with the classical model of Osborn (J. Phys. Oceanogr., vol. 10 (1), 1980, pp. 83–89). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_{B}\sim O(1)$ and those with nearly optimal $\mathscr{E}\lesssim 1/6$ at intermediate $\overline{Re}_{B}\sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or ‘scouring’ motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

The primary goal of the paper is to demonstrate that a circular jet of inviscid liquid, which issues from a nozzle at an angle to the gravitational field, can sustain a self-excited linear global mode. The mode is found for the flow that is locally convectively unstable at every point, i.e. has no region of local absolute instability. A simple experiment shows that the mode can evolve into a wave of finite amplitude, which is localized in space near the nozzle.

Section 2 of the paper summarizes some recent developments in the linear theory of stability of weakly inhomogeneous flows.

Syringomyelia is a medical condition in which one or more fluid-filled cavities (syrinxes) form in the spinal cord. The syrinxes often form near locations where the spinal subarachnoid space (SSS; the fluid-filled annular region surrounding the spinal cord) is partially obstructed. Previous studies showed that nonlinear interactions between the pulsatile fluid flow in the SSS and the elastic deformation of the tissues surrounding it can generate a fluid pressure distribution that would tend to drive fluid from the SSS into the syrinx if the tissue separating the two regions was porous. This provides a potential explanation for why a partial occlusion of the SSS can induce the growth of an already existing nearby syrinx. We study this hypothesis by analysing the mass transfer between the SSS and the syrinx, using a poroelastic fluid–structure interaction model of the spinal cord that includes a representation of the partially obstructed SSS, the syrinx and the poroelastic tissues surrounding these fluid-filled cavities. Our numerical simulations show that poroelastic fluid–structure interaction can indeed cause an increase (albeit relatively small) in syrinx volume. We analyse the seepage flows and show that their structure can be captured by an analytical model which explains why the increase in syrinx volume tends to be relatively small.

We propose a two-dimensional model of three-dimensional Rayleigh–Bénard convection in the limit of very high Prandtl number and Rayleigh number, as in the Earth's mantle. The model equation describes the evolution of the first moment of the temperature anomaly in the thermal boundary layer, which is assumed thin with respect to the scale of motion. This two-dimensional field is transported by the velocity that it induces and is amplified by surface divergence. This model explains the emergence of thermal plumes, which arise as finite-time singularities. We determine critical exponents for these singularities. Using a smoothing method we go beyond the singularity and reach a stage of developed convection. We describe a process of plume merging, leaving room for the birth of new instabilities. The heat flow at the surface predicted by our two-dimensional model is found to be in good agreement with available data.

Liquid sheets are a generic occurrence in nature. They provide a key pathway by which liquid matter breaks up and is dispersed into smaller entities, as in the formation of sprays. However, the stability of liquid sheets is surprisingly subtle, and involves a range of physical effects, without which a quantitative understanding is impossible. As a result, even the linear stability of liquid sheets has remained open for over half a decade. Now, by a combination of experiment and thorough theoretical analysis by Tammisola et al. (J. Fluid Mech., this issue, vol. 672, 2011, pp. 5–32), we have finally gained a solid understanding of the key factors that control the initial stages of sheet breakup.

Hydrodynamic measurements were made with a triaxial hot wire in the full-coverage region and the recovery region following an array of injection holes inclined downstream, at 30° to the surface. The data were taken under isothermal conditions at ambient temperature and pressure for two blowing ratios: M = 0·9 and M = 0·4. (The ratio M = ρjetUjet/ρ∞U∞, where U is the mean velocity and ρ is the density. Subscripts jet and ∞ stand for injectant and free stream, respectively.) Profiles of the three mean-velocity components and the six Reynolds stresses were obtained at several spanwise positions at each of five locations down the test plate.

In the full-coverage region, high levels of turbulence kinetic energy (TKE) were found for low blowing and low TKE levels for high blowing. This observation is especially significant when coupled with the fact that the heat transfer coefficient is high for high blowing, and low for low blowing. This apparent paradox can be resolved by the hypothesis that entrainment of the mainstream fluid must be more important than turbulent mixing in determining the heat transfer behaviour at high blowing ratios (close to unity).

In the recovery region, the flow can be described in terms of a two-layer model: an outer boundary layer and a two-dimensional inner boundary layer. The inner layer governs the heat transfer.

Direct numerical simulation was used to study laminar and turbulent flows in circular pipes with smoothly corrugated walls. The corrugation wavelength was kept constant at $0.419D$, where $D$ is the mean diameter of the wavy-wall pipe and the corrugation height was varied from zero to $0.08D$. Flow rates were varied in steps between low values that generate laminar flow and higher values where the flow is in the post-transitional turbulent regime. Simulations in the turbulent regime were also carried out at a constant Reynolds number, $\mathit{Re}_{{\it\tau}}=314$, for all corrugation heights. It was found that even in the laminar regime, larger-amplitude corrugations produce flow separation. This leads to the proportion of pressure drop attributable to pressure drag being approximately 50 %, and rising to approximately 85 % in transitional rough-wall flow. The near-wall structure of turbulent flow is seen to be heavily influenced by the effects of flow separation and reattachment. Farther from the wall, the statistical profiles examined exhibit behaviours characteristic of smooth-wall flows or distributed roughness rough-wall flows. These observations support Townsend’s wall-similarity hypothesis. The organized nature of the present roughness allows the mean pressure drop to be written as a function of the corrugation height. When this is exploited in an analysis of the mean dynamical equation, the scaling problem is explicitly revealed to result from the combined influences of roughness and Reynolds number. The present results support the recent analysis and observations of Mehdi et al. (J. Fluid Mech., vol. 731, 2013, pp. 682–712), indicating that the length scale given by the distance from the wall at which the mean viscous force loses leading order is important to describing these combined influences, as well as providing a dynamically self-consistent connection to the scaling structure of smooth-wall pipe flow.

Direct numerical simulation (DNS), based on high-order numerical schemes, is used to study the effects of distributed pressure gradients on the redistribution of fluctuating kinetic energy in supersonic nozzle and diffuser flow with incoming fully developed turbulent pipe flow. Axisymmetric geometries and flow parameters have been selected such that shock waves are avoided and streamline curvature remains unimportant. Although mean extra rates of strain are quite small, strong changes in Reynolds stresses and their production/redistribution mechanisms are observed, in agreement with findings of Bradshaw (J. Fluid Mech., vol. 63, 1974, pp. 449–464). The central role of pressure–strain correlations in changing the Reynolds stress anisotropy is highlighted. A Green’s function-based analysis of pressure–strain correlations is presented, showing remarkable agreement with DNS data.

The purpose of this paper is to consider the effect of one-dimensional random depth variation on the propagation of planetary waves in a homogeneous layer of fluid having a free upper surface. We begin by determining the dispersion relation for the coherent part of the wave using the vorticity equation for the transport stream function and a previously described perturbation method. Then, from the resulting first-order expressions for the wavenumber, we obtain the phase speeds for the two possible planetary-wave solutions. These are compared with the corresponding phase speeds of planetary waves over a smoothly varying topography; the validity limits of the approximations are discussed. For the most physically realizable situation, of random depth correlation lengths much shorter than a typical wavelength, we find that the phase speed of the shorter (longer) wave component is less (greater) over a randomly varying topography than over a smoothly varying topography. In the case of the shorter waves, greatest relative changes in phase speed occur when the associated fluid motions are at right angles to the ‘strike’ of the roughness elements, while for both long and short waves there is no relative change in phase speed if fluid motions are parallel to the roughness contours. Moreover, both types of waves are shown to lose energy in the direction of energy propagation as a result of scattering. Numerical values are then obtained using hydrographic charts of the western North Pacific, and show that the randomness may significantly decrease the phase speed of the shorter planetary-wave component. Finally, we give a brief descriptive explanation of the results based on the effect of the topography on the wave restoring mechanism.

Fibre-level computer simulation is carried out to study the rotational diffusion and structural evolution of semidilute suspensions of non-Brownian, rigid-rod-like fibres under shear flow in a Newtonian fluid. The analyses use a hybrid approach where the lattice-Boltzmann method is coupled with the external boundary force method. The probability distribution of the orbit constant, , in the semidilute regime is predicted with this method. The paper emphasizes assessment of the characteristics of a rotary diffusion model – anisotropic in nature (Koch, Phys. Fluids, vol. 7, 1995, pp. 2086–2088) – when used in suspensions with fibres of different aspect ratios (ranging from to ) and with different volume concentrations (ranging from to ). A measure of the scalar Folgar–Tucker constant, , is extracted from the anisotropic diffusivity tensor, . The scalar is mostly in the semidilute regime and compares very well with the experimental observations of Stover (PhD thesis, School of Chemical Engineering, Cornell University, 1991) and Stover, Koch & Cohen (J. Fluid Mech., vol. 238, 1992, pp. 277–296). The values provide substantial numerical evidence that the range of (0.0038–0.0165) obtained by Folgar & Tucker (J. Rein. Plast. Compos., vol. 3, 1984, pp. 98–119) in the semidilute regime is actually overly diffusive. The paper also branches out to incorporate anisotropic diffusion (through the use of the Koch model) in the second-order evolution equation for (a second-order orientation tensor). The solution of the evolution equation with the Koch model demonstrates unphysical behaviour at low concentrations. The most plausible explanation for this behaviour is error in the closure approximation; and the use of the Koch model in a spherical harmonics-based method (Montgomery-Smith, Jack & Smith, Compos. A: Appl. Sci. Manuf., vol. 41, 2010, pp. 827–835) to solve for the orientation moments corroborates this claim.

Fluid–structure interaction is fundamental to the characteristics of the induced flows due to the motion of structures in fluids and also is crucial to the performance of submerged structures. This paper presents a three-dimensional analytical study of the intrinsic free vibration of an elastic multilayered hollow sphere interacting with an exterior non-Newtonian fluid medium. The fluid is assumed to be characterized by a compressible linear viscoelastic model accounting for both the shear and compressional relaxation processes. For small-amplitude vibrations, the equations governing the viscoelastic fluid can be linearized, which are then solved by introducing appropriate potential functions. The solid is assumed to exhibit a particular material anisotropy, i.e. spherical isotropy, which includes material isotropy as a special case. The equations governing the anisotropic solid are solved in spherical coordinates using the state-space formalism, which finally establishes two separate transfer relations correlating the state vectors at the innermost surface with those at the outermost surface of the multilayered hollow sphere. By imposing the continuity conditions at the fluid–solid interface, two separate analytical characteristic equations are derived, which characterize two independent classes of vibration. Numerical examples are finally conducted to validate the theoretical derivation as well as to investigate the effects of various factors, including fluid viscosity and compressibility, fluid viscoelasticity, solid anisotropy and surface effect, as well as solid intrinsic damping, on the vibration characteristics of the submerged hollow sphere. Particularly, our theoretically predicted vibration frequencies and quality factors of gold nanospheres with intrinsic damping immersed in water agree exceptionally well with the available experimentally measured results. The reported analytical solution is truly and fully three-dimensional, covering from the purely radial breathing mode to the torsional mode to any general spheroidal mode as well as being applicable to various simpler situations, and hence can be a broad-spectrum benchmark in the study of fluid–structure interaction.