Previous experimental (Kühnen et al., Flow Turb. Combust., vol. 100, 2018, pp. 919–943) and numerical (Marensi et al., J. Fluid Mech., vol. 863, 2019, pp. 850–875) studies have demonstrated that a streamwise-localised baffle can fully relaminarise pipe flow turbulence at Reynolds numbers of $O(10^4)$. Optimising the design of the baffle involves tackling a complicated variational problem built around time stepping turbulent solutions of the Navier–Stokes equations which is difficult to solve. Here instead, we investigate a much simpler ‘spectral’ approach based upon maximising the energy stability of the baffle-modified laminar flow. The ensuing optimal problem has much in common with the variational procedure to derive an upper bound on the energy dissipation rate in turbulent flows (e.g. Plasting & Kerswell, J. Fluid Mech., vol. 477, 2003, pp. 363–379) so well-honed techniques developed there can be used to solve the problem here. The baffle is modelled by a linear drag force $-F(\boldsymbol {x}) \boldsymbol {u}$ (with $F(\boldsymbol {x}) \ge 0 \ \forall \boldsymbol {x}$) where the extent of the baffle is constrained by an $L_{\alpha }$ norm with various choices explored in the range $1 \leq \alpha \leq 2$. An asymptotic analysis demonstrates that the optimal baffle is always axisymmetric and streamwise independent, retaining just radial dependence. The optimal baffle which emerges in all cases has a similar structure to that found to work in experiments: the baffle retards the flow in the pipe centre causing the flow to become faster near the wall thereby reducing the turbulent shear there. Numerical simulations demonstrate that the designed baffle can relaminarise turbulence efficiently at moderate Reynolds numbers ($Re \le 3500$), and an energy saving regime has been identified. Direct numerical simulation at $Re=2400$ also demonstrates that the drag reduction can be realised by truncating the energy-stability-designed baffle to finite length.

We consider the dynamics of a gravity-driven flow coating a vertical fibre rotating about its axis. This flow exhibits rich dynamics including the formation of bead-like structures and different types of steady or oscillatory travelling waves driven by a Rayleigh–Plateau mechanism modified by the presence of gravity and rotation. Linear stability shows that the axisymmetric mode dominates the instability when the rotation is slow, which allows us to derive a two-dimensional model equation under the long-wave assumption. The spatio-temporal dynamics and nonlinear wave solutions are then investigated by the model equation. The spatio-temporal stability analysis showed that the absolute instability is enhanced by the rotation. Steady travelling-wave states and relative periodic states are observed in the numerical simulations of the model equation, which show that the rotation tends to suppress the formation of relative periodic states. To examine this, a linear stability analysis of steady travelling waves is performed, indicating that the rotation has a stabilizing effect on the steady travelling waves. This result is adverse to the destabilizing effect of rotation on the linear stability of initially uniform films. A bifurcation analysis shows that the relative periodic state is born from the instability of steady travelling wave, which represents the coalescence and breakup process between a large droplet and a serial of much smaller droplets.

We revisit the dynamics of a thick liquid film flowing down a vertical fibre. Instead of deriving a long-wave model, we directly solve the Navier–Stokes equations using a domain mapping technique and the exact steady travelling wave solutions are explored using a dynamical systems theory approach. Three distinct flow regimes, labelled ‘a’, ‘b’ and ‘c’, observed in previous experiments (Kliakhandler et al., J. Fluid Mech., vol. 429, 2001, pp. 381–390) are investigated. Flow regime ‘a’ refers to a steady flow state in which large droplets are separated by a long thin film. Flow regime ‘b’ is a necklace-like flow. In flow regime ‘c’, a cyclic process of droplet coalescence and breakup was observed. By matching the mean flow rates of the travelling wave solutions and experimental data, our travelling wave solutions show an excellent agreement with flow regimes ‘a’ and ‘b’. The time-periodic flow regime ‘c’ is compared with direct numerical simulation of the Navier–Stokes equations. A snapshot of the simulation shows a remarkable similarity to an experimental image and the discrepancy of mean wave speed and maximal wave height between our numerical simulation and experimental data is negligible.

Motivated by the results of recent experiments (Kühnen et al., Flow Turbul. Combust., vol. 100, 2018, pp. 919–943), we consider the problem of designing a baffle (an obstacle to the flow) to relaminarise turbulence in pipe flows. Modelling the baffle as a spatial distribution of linear drag ${\boldsymbol {F}}(\boldsymbol {x},t)=-\chi (\boldsymbol {x})\boldsymbol {u}_{tot}(\boldsymbol {x},t)$ within the flow ($\boldsymbol {u}_{tot}$ is the total velocity field and $\chi \ge 0$ a scalar field), two different optimisation problems are considered to design $\chi$ at a Reynolds number $Re=3000$. In the first, the smallest baffle defined in terms of a $L_1$ norm of $\chi$ is sought which minimises the viscous dissipation rate of the flow. In the second, a baffle which minimises the total energy consumption of the flow is treated. Both problems indicate that the baffle should be axisymmetric and radially localised near the pipe wall, but struggle to predict the optimal streamwise extent. A manual search finds an optimal baffle one radius long which is then used to study how the amplitude for relaminarisation varies with $Re$ up to $15\,000$. Large stress reduction is found at the pipe wall, but at the expense of an increased pressure drop across the baffle. Estimates are then made of the break-even point downstream of the baffle where the stress reduction at the wall due to the relaminarised flow compensates for the extra drag produced by the baffle.

We revisit the optimal heat transport problem for Rayleigh–Bénard convection in which a rigorous upper bound on the Nusselt number, $Nu$, is sought as a function of the Rayleigh number, $Ra$. Concentrating on the two-dimensional problem with stress-free boundary conditions, we impose the time-averaged heat equation as a constraint for the bound using a novel two-dimensional background approach thereby complementing the ‘wall-to-wall’ approach of Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662). Imposing the same symmetry on the problem, we find correspondence with their maximal result for $Ra\leqslant Ra_{c}:=4468.8$ but, beyond that, the results from the two approaches diverge. The bound produced by the two-dimensional background field approaches that produced by the one-dimensional background field from below as the length of computational domain $L\rightarrow \infty$. On lifting the imposed symmetry, the optimal two-dimensional temperature background field reverts to being one-dimensional, giving the best bound $Nu\leqslant 0.055Ra^{1/2}$ compared to $Nu\leqslant 0.026Ra^{1/2}$ in the non-slip case. We then show via an inductive bifurcation analysis that introducing two-dimensional temperature and velocity background fields (in an attempt to impose the time-averaged Boussinesq equations) is also unable to lower the bound. This then exhausts the background approach for the two-dimensional (and by extension three-dimensional) Rayleigh–Bénard problem with the bound remaining stubbornly $Ra^{1/2}$ while data seem more to scale like $Ra^{1/3}$ for large $Ra$. Finally, we show that adding a velocity background field to the formulation of Wen et al. (Phys. Rev. E., vol. 92, 2015, 043012), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to $Nu\leqslant O(Ra^{5/12})$, also fails to further improve the bound.