For several applications there are advantages in writing turbulent flow equations in a coordinate frame aligned with the streamlines and several two-dimensional examples of this approach have appeared in the literature. In this paper, we extend this approach to general three-dimensional flows. We find that, in any flow that has a component of its vorticity aligned in the streamline direction, congruences of its streamlines do not form integrable manifolds. This limits the development of a streamline coordinate description of such flows, although some useful results can still be obtained. However, in the case of general three-dimensional complex-lamellar flows, where the mean velocity and mean vorticity are everywhere orthogonal, a complete streamline coordinate description can be derived. Furthermore, we show that general complex-lamellar flows are a good approximation to boundary layers and thin free shear layers. We derive the underlying true coordinate system for such flows, where the orthogonal coordinate surfaces are two stream surfaces and a modified potential surface. From this we obtain physical equations, where flow variables have the same dimensions they would have in a Cartesian coordinate frame. Finally, we show that rational approximations to these equations, which describe small-perturbation flows, contain some terms that have been ignored in previous applications and we detail some practical applications of the theory in modelling and analysis.

The featured article ‘Break-up of a falling drop containing dispersed particles’ (Nitsche and Batchelor, J. Fluid Mech., 1997, vol. 340, pp. 161–175) is G. K. Batchelor's last published paper with his former postdoctoral associate J. M. Nitsche. The objective of the study was to investigate the randomness of the velocities of interacting rigid particles falling under gravity through a viscous fluid at a small Reynolds number and its consequence for the breakup of a falling cloud of particles. The study focused on a quintessential problem of the collective dynamics of interacting particles and has been an inspiration for subsequent work.

Losses due to wake interactions between wind turbines can significantly reduce the power output of wind farms. The possibility of active flow control by wake deflection downstream of yawed horizontal-axis wind turbines has motivated research on the fluid mechanics involved. We summarize the findings of a wind tunnel study (Bastankhah & Porté-Agel, J. Fluid Mech., vol. 806, 2016, pp. 506–541) of the flow associated with a yawed model wind turbine, and the insights and modelling developments that have followed this important study.

Since its publication in 2010, the paper by Schmid (J. Fluid Mech., vol. 656, 2010, pp. 5–28) has wielded considerable influence, an impact we examine here. That seminal work introduced dynamic mode decomposition, a method for performing flow-field spectral analysis of snapshot sequences of data. As a data-driven approach aimed at uncovering spatial and temporal patterns or modes within datasets, its applicability has extended far beyond fluid mechanics, reaching into a wide array of fields.

Boundary layers are present in many natural and industrial fluid flows. The concept of boundary layers can be traced back to Leonardo da Vinci's paintings of pipe flow, where he was aware of a higher velocity away from the walls. During the 19th century, the physics of boundary conditions had been extensively debated, and the well-known Maxwell–Navier slip length was proposed in 1823. In most cases, the no-slip boundary condition is valid at a fluid–solid interface. However, with the advancement of measurement techniques, slip lengths ranging from nanometre to micrometre scales were experimentally measured, raising questions regarding the applicability of the no-slip condition. In 2003, Lauga & Stone (J. Fluid Mech., vol. 489, 2003, pp. 55–77) proposed a simple model to elucidate the effect of surface heterogeneities on the slip length, elegantly bridging the microscopic structure of the wall-boundary conditions to the macroscopic effective slip length.

The entrainment hypothesis states that the mean inflow velocity across the boundary of a turbulent flow is proportional to a characteristic velocity of the flow. Proposed by G. I. Taylor approximately 80 years ago, it is still a common model of turbulence closure widely used in environmental engineering and geophysical fluid mechanics. Although it is a very simple concept and mathematical model, it has proven to be able to predict the entrainment in a variety of geophysical flows, e.g. convective clouds and plumes from erupting volcanoes in the atmosphere; dense water overflows and turbidity currents in the ocean; magma injection in a magma chamber in the interior of the Earth, to name just a few. In a seminal paper, Turner (J. Fluid Mech., vol. 173, 1986, pp. 431–471) presents a variety of laboratory and geophysical flows to illustrate the success of the entrainment hypothesis and discusses why such a simple hypothesis works so well even when the original assumptions are no longer valid.

Resolvent analysis provides a framework to predict coherent spatio-temporal structures of the largest linear energy amplification, through a singular value decomposition (SVD) of the resolvent operator, obtained by linearising the Navier–Stokes equations about a known turbulent mean velocity profile. Resolvent analysis utilizes a Fourier decomposition in time, which has thus far limited its application to statistically stationary or time-periodic flows. This work develops a variant of resolvent analysis applicable to time-evolving flows, and proposes a variant that identifies spatio-temporally sparse structures, applicable to either stationary or time-varying mean velocity profiles. Spatio-temporal resolvent analysis is formulated through the incorporation of the temporal dimension to the numerical domain via a discrete time-differentiation operator. Sparsity (which manifests in localisation) is achieved through the addition of an $l_1$-norm penalisation term to the optimisation associated with the SVD. This modified optimisation problem can be formulated as a nonlinear eigenproblem and solved via an inverse power method. We first showcase the implementation of the sparse analysis on a statistically stationary turbulent channel flow, and demonstrate that the sparse variant can identify aspects of the physics not directly evident from standard resolvent analysis. This is followed by applying the sparse space–time formulation on systems that are time varying: a time-periodic turbulent Stokes boundary layer and then a turbulent channel flow with a sudden imposition of a lateral pressure gradient, with the original streamwise pressure gradient unchanged. We present results demonstrating how the sparsity-promoting variant can either change the quantitative structure of the leading space–time modes to increase their sparsity, or identify entirely different linear amplification mechanisms compared with non-sparse resolvent analysis.