The energy spectrum is commonly used to describe the scale dependence of the turbulent fluctuations in homogeneous isotropic turbulence. In contrast, one-point statistical quantities, such as the turbulent kinetic energy, are employed for inhomogeneous turbulence modelling. To obtain a better understanding of inhomogeneous turbulence, some attempts have been made to describe its scale dependence by using the second-order structure function and the two-point velocity correlation. However, previous expressions for the energy density in the scale space do not satisfy the requirement that it should be non-negative. In this work, a new expression for the energy density in the scale space is proposed on the basis of the two-point velocity correlation; the integral with a filter function is introduced to satisfy the non-negativity of the energy density. Direct numerical simulation (DNS) data of homogeneous isotropic turbulence were first used to assess the role of the energy density by comparing it with the energy spectrum. DNS data of a turbulent channel flow were then used to investigate the energy density and its transport equation in inhomogeneous turbulence. It was shown that the new energy density is positive in the scale space of the homogeneous direction. The energy transfer was successfully examined in the scale space both in the homogeneous and inhomogeneous directions. The energy cascade from large to small scales was clearly observed. Moreover, the inverse energy cascade from large to very large scales was observed in the scale space of the spanwise direction.

Time-resolved planar particle image velocimetry was used to analyse the structuring of a turbulent boundary layer into uniform momentum zones (UMZs). The instantaneous peak-detection method employed by Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) and de Silva et al. (J. Fluid Mech., vol. 786, 2016, pp. 309–331) is extended to account for temporal coherence of UMZs. The resulting number of zones detected appears to follow a normal distribution at any given instant. However, the extreme cases in which the number of zones is either very high or very low, are shown to be linked with two distinct flow states. A higher than average number of zones is associated with a large-scale $Q2$ event in the log region which creates increased small-scale activity within that region. Conversely, a low number of zones corresponds to a large-scale $Q4$ event in the log region and decreased turbulent activity away from the wall. The residence times, within the measurement plane, of zones belonging to the latter scenario are shown to be on average four times larger than those of zones present during higher than average zone structuring states. For both cases, greater residence times are observed for zones of higher momentum that are generally closer to the free stream.

This article discusses the description of wall-bounded turbulence as a deterministic high-dimensional dynamical system of interacting coherent structures, defined as eddies with enough internal dynamics to behave relatively autonomously from any remaining incoherent part of the flow. The guiding principle is that randomness is not a property, but a methodological choice of what to ignore in the flow, and that a complete understanding of turbulence, including the possibility of control, requires that it be kept to a minimum. After briefly reviewing the underlying low-order statistics of flows at moderate Reynolds numbers, the article examines what two-point statistics imply for the decomposition of the flow into individual eddies. Intense eddies are examined next, including their temporal evolution, and shown to satisfy many of the properties required for coherence. In particular, it is shown that coherent structures larger than the Corrsin scale are a natural consequence of the shear. In wall-bounded turbulence, they can be classified into coherent dispersive waves and transient bursts. The former are found in the viscous layer near the wall, and as very large structures spanning the entire boundary layer. Although they are shear-driven, these waves have enough internal structure to maintain a uniform advection velocity. Conversely, bursts exist at all scales, are characteristic of the logarithmic layer, and interact almost linearly with the shear. While the waves require a wall to determine their length scale, the bursts are essentially independent from it. The article concludes with a brief review of our present theoretical understanding of turbulent structures, and with a list of open problems and future perspectives.

‘Chance is the name we give to what we choose to ignore (Voltaire)’

We extend the local theory of available potential energy (APE) to a general multicomponent compressible stratified fluid, accounting for the effects of diabatic sinks and sources. As for simple compressible fluids, the total potential energy density of a fluid parcel is the sum of its available elastic energy and APE density. These respectively represent the adiabatic compression/expansion work needed to bring it from its reference pressure to its actual pressure and the work against buoyancy forces required to move it from its reference state position to its actual position. Our expression for the APE density is new and is derived using only elementary manipulations of the equations of motion; it is significantly simpler than existing published expressions, while also being more transparently linked to the relevant form of APE density for the Boussinesq and hydrostatic primitive equations. Our new framework is used to clarify the links between some aspects of the energetics of Boussinesq and real fluids, as well as to shed light on the physical basis underlying the choice of reference state(s) in local APE theory.

Equal coaxial symmetrically located helical vortices translate and rotate steadily while preserving their shape and relative position if they move in an unbounded inviscid incompressible fluid. In this paper, the linear and angular velocities of this set of vortices ($U$ and $\unicode[STIX]{x1D6FA}$ respectively) are computed as the sum of the mutually induced velocities found by Okulov (J. Fluid Mech., vol. 521, 2004, pp. 319–342) and the self-induced velocities found by Velasco Fuentes (J. Fluid Mech., vol. 836 2018). Numerical computations of the velocities using the Helmholtz integral and the Biot–Savart law, as well as numerical simulations of the flow evolution under the Euler equations, are used to verify that the theoretical results are accurate for $N=1,\ldots ,4$ vortices over a broad range of values of the pitch and radius of the vortices. An analysis of the flow topology in a reference system that translates with velocity $U$ and rotates with angular velocity $\unicode[STIX]{x1D6FA}$ serves to determine the capacity of the vortices to transport fluid.