We examine theoretically the flow interactions and forward flight dynamics of tandem or in-line flapping wings. Two wings are driven vertically with prescribed heaving motions, and the horizontal propulsion speeds and positions are dynamically selected through aero- or hydro-dynamic interactions. Our simulations employ an improved vortex-sheet method to solve for the locomotion of the pair within the collective flow field, and we identify ‘schooling states’ in which the wings travel together with nearly constant separation. Multiple terminal configurations are achieved by varying the initial conditions, and the emergent separations are approximately integer multiples of the wavelength traced out by each wing. We explain the stability of these states by perturbing the follower and mapping out an effective potential for its position in the leader’s wake. Each equilibrium position is stabilised since smaller separations are associated with in-phase follower-wake motions that constructively reinforce the flow but lead to decreased thrust on the follower; larger separations are associated with antagonistic follower-wake motions, increased thrust and a weakened collective wake. The equilibria and their stability are also corroborated by a linearised theory for the motion of the leader, the wake it produces and its effect on the follower. We also consider a weakly flapping follower driven with lower heaving amplitude than the leader. We identify ‘keep-up’ conditions for which the wings may still ‘school’ together despite their dissimilar kinematics, with the ‘freeloading’ follower passively assuming a favourable position within the wake that permits it to travel significantly faster than it would in isolation.

The behaviour of internal waves propagating in a background shear flow is studied in the case where the direction of shear is orthogonal to gravity. Ray-tracing theory is used to predict properties of the wave state at locations where instability occurs. Local wave energy growth is found to result from two distinct mechanisms: an increase in wave steepness due to refraction by the shear or an increase in streamwise velocity perturbations due to wave advection of the background flow. Based on the initial conditions, a dimensionless perturbation energy ratio $F$ is constructed to predict the relative importance of these two mechanisms in facilitating wave breaking. When $F$ is small and waves become locally steep, perturbation kinetic and potential energy remain approximately equipartitioned and subsequent instabilities are expected to develop due to a combination of shear and convection. On the other hand, as $F$ increases, kinetic energy dominates and wave advection of momentum may instead cause breaking to become increasingly driven by enhanced vertical shear. To test these predictions, fully nonlinear direct numerical simulations are conducted, spanning a range of wave-breaking dynamics. Good qualitative agreement with the theory is found despite substantial departures from the underlying assumptions. Wave breaking leads to significant turbulent dissipation, which in some cases greatly exceeds the initial wave energy. Momentum and energy transfers between the wave, background flow and turbulence are found to be sensitive to the dynamics of breaking, as are the mixing properties.

The addition of polymers to turbulent pipe flows induces significant drag reduction and fundamentally modifies turbulent flow structures. This study presents a fractal dimension analysis of polymeric turbulent pipe flows using velocity fields captured via two-dimensional particle image velocimetry in the streamwise-radial plane. Two experimental datasets were generated: one by varying the polymer concentration at a constant Reynolds number ($\textit{Re}$) and another by varying $\textit{Re}$ at a fixed polymer concentration. Friction factors were measured concurrently to quantify the extent of drag reduction. The two-dimensional fractal dimension was evaluated for isosurfaces of turbulent kinetic energy. While Newtonian turbulence exhibits a nearly constant fractal dimension at length scales exceeding a critical threshold, the introduction of polymers causes the fractal dimension to decrease monotonically with increasing concentration. Conversely, the fractal dimension remains insensitive to changes in the Reynolds number. The ratio of the critical length scale to the Kolmogorov scale varies according to both $\textit{Re}$ and polymer concentration; however, this scale ratio becomes independent of both parameters once the maximum drag reduction asymptote is reached. Spatial analysis of the one-dimensional fractal dimension across radial positions helps to further reveal the evolution of turbulence fractality. The results demonstrate that while flow inertia promotes the formation of space-filling structures, viscoelastic effects smooth these structures and transition them towards sheet-like or linear geometries. Finally, the correlation between the fractal dimension and turbulence intermittency is discussed.

Inertial waves in fluid regions of planets and stars play an important role in their dynamics and evolution, through energy, heat and angular momentum transport and mixing of chemicals. While inertial wave propagation in flows prescribed by solid-body rotation is well understood, natural environments are often characterised by convection or zonal flows. In these more realistic configurations, we do not yet understand the propagation of inertial waves or their transport properties. In this work, we focus on the interaction between inertial waves and geostrophic currents, which has thus far only been investigated using ray theory, where the wavelength is assumed to be small relative to the length scale of the current, or averaging/statistical approaches. We develop a quasi-two-dimensional analytical model to investigate the reflection and transmission of inertial waves in the presence of a localised geostrophic shear layer of arbitrary width and compare our theoretical findings with a set of numerical simulations. We demonstrate that, in contrast to ray theory predictions, partial reflections occur even in subcritical shear layers and tunnelling with almost total transmission is possible in supercritical shear layers, if the layer is thin compared with the wavelength. That is, supercritical shear layers act as low-pass filters for inertial wave beams allowing the low-wavenumber waves to travel through. Thus, our analytical model allows us to predict interactions between inertial waves and geostrophic shear layers not addressed by ray-based or statistical theories and conceptually understand the behaviour of the full wave field around and inside such layers.

Building on classical oblique jump theory, we develop a one-dimensional (1-D) analytical framework that incorporates non-Newtonian rheology to predict the onset of hydraulic jumps, their internal structure and the associated Mach-front geometry. Source terms representing bed slope and wall friction are included, and the resulting formulation is systematically assessed against laboratory experiments, two-dimensional (2-D) shallow-water simulations and fully three-dimensional (3-D) computational fluid dynamics. Experiments with Newtonian, shear-thinning and shear-thickening fluids on converging sidewalls demonstrate a good match with the 1-D formulation. For Newtonian and shear-thinning fluids on mild slopes, the 1-D formulation with source terms closely reproduces the measured shock-front geometry and the 2-D simulation results. The analysis shows that upstream flow deceleration governs the reduction of the Mach angle and the resulting curvature. By contrast, in tests with shear-thickening fluids and steeper slopes, gravitational contributions produce detachment and strong front curvature that are not captured by the 1-D model. Comparisons of the transverse front position confirm that 1-D models lose validity when the upstream Froude number decreases sharply along the front. Fully 3-D simulations reveal concave front deformation driven by shear, strong dominance of tangential over normal velocities and flow features absent in depth-averaged models. The results demonstrate that 2-D shallow-water models capture the key dynamics for mild slopes and shear-thinning conditions, while accurate prediction for shear-thickening fluids requires 3-D approaches, motivating future hybrid strategies.