The growth of small perturbations in isotropic turbulence is studied using massive ensembles of direct numerical simulations. These ensembles capture the evolution of the ensemble-averaged flow field and the ensemble variance in the fully nonlinear regime of perturbation growth. Evolution equations for these two fields are constructed by applying the ensemble average operator to the Navier–Stokes equations and used to study uncertainty growth in scale and physical space. It is shown that uncertainty growth is described by a flux of energy from the ensemble-averaged flow to the ensemble variance. This flux is formally equivalent to the subgrid scale (SGS) energy fluxes of the turbulence cascade, and can be interpreted as an inverse uncertainty cascade from small to large scales. In the absence of information sources (measurements), the uncertainty cascade is unsteady and leads to the progressive filtering of the small scales in the ensemble-averaged flow, a process that represents the loss of predictability due to chaos. Similar to the kinetic energy cascade, the uncertainty cascade displays an inertial range with a constant average uncertainty flux, which is bounded from below by the average kinetic energy dissipation. Locally in space, uncertainty fluxes differ from the SGS energy fluxes at the same scale, but both have similar statistics and are significantly correlated with each other in space. This suggests that uncertainty propagation is partly connected to the energy cascade and that they share similar mechanisms. These findings open avenues to model uncertainty propagation in turbulence following an approach similar to the SGS models in large-eddy simulations. This is relevant not only to efficiently assess the reliability and accuracy of turbulence forecasts, but also to design uncertainty-robust reconstruction techniques for data assimilation or SGS modelling.

The simulation of turbulent flow requires many degrees of freedom to resolve all the relevant time and length scales. However, due to the dissipative nature of the Navier–Stokes equations, the long-term dynamics is expected to lie on a finite-dimensional invariant manifold with fewer degrees of freedom. In this study, we build low-dimensional data-driven models of pressure-driven flow through a circular pipe. We impose the ‘shift-and-reflect’ symmetry to study the system in a minimal computational cell (e.g. the smallest domain size that sustains turbulence) at a Reynolds number of 2500. We build these models by using autoencoders to parametrise the manifold coordinates and neural ordinary differential equation to describe their time evolution. Direct numerical simulations (DNSs) typically require of the order of $\mathcal{O}(10^5)$ degrees of freedom, while our data-driven framework enables the construction of models with fewer than 20 degrees of freedom. Remarkably, these reduced-order models effectively capture crucial features of the flow, including the streak breakdown. In short-time tracking, these models accurately track the true trajectory for one Lyapunov time, as well as the leading Lyapunov exponent, while at long-times, they successfully capture key aspects of the dynamics such as Reynolds stresses and energy balance. The model can quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. Additionally, we report new exact coherent states found in the DNS with the aid of these low-dimensional models. This approach leads to the discovery of seventeen previously unknown solutions within the turbulent pipe flow system, notably featuring relative periodic orbits characterised by the longest reported periods for such flow conditions.

Gas-phase turbulence in a bubbling gas–solid fluidised bed is modelled using the data from particle-resolved direct numerical simulations. The subgrid particle-induced turbulent kinetic energy (TKE) is modelled as a function of filter width, filtered solid volume fraction, particle Reynolds number and filtered gas-phase strain rate tensor. Within the volume-filtered framework, we demonstrate that the fluid Reynolds stress models originally developed for a homogeneous system remain applicable to the inhomogeneous fluidised bed, provided that the inhomogeneous drag and particle-induced TKE models are used for the dissipation rate interfacial term. An algebraic model for the anisotropy of gas-phase velocity variance is developed by simplifying the proposed Reynolds stress equation model, which incorporates the effects from both filtered slip velocity and filtered fluid strain rate. The new models are shown to agree well with the direct numerical simulation data of clustered particle settling systems, indicating good applicability of our models for various clustered particle-laden flows.

This study compares turbulent channel flows over elastic walls with those over rough walls, to explore the role of the dynamic change of shape of the wall in turbulence. The comparison is made meaningful by generating rough walls from instantaneous configurations of elastic cases. The aim of this comparison is to individually understand the role of fluid–structure interaction effects and the role of wall shape/undulations in determining the overall physics of flow near elastic walls. With an increase in the compliance of the wall, qualitatively similar trends for many of the effects produced by a rough wall are also seen in the elastic wall. However, specific features can be observed for the elastic-wall cases only, arising from the mutual interaction between the solid and fluid, leading to a further increase in drag. To understand them, we look at the turbulent structures, which exhibit clear differences across the various configurations: roughness induces only a slight reduction of streamwise coherency, resulting in a situation qualitatively similar to what is found in classical turbulent channel flows, whereas elasticity causes the emergence of a novel dominant spanwise coherency. Additionally, we explored the effect of vertical disturbances on elastic-wall dynamics by comparing with permeable walls having similar (average) wall-normal velocity fluctuations at the interface. The permeable walls were found to have minimal similarities to elastic walls. Overall, we can state that the wall motion caused by the complex fluid–structure interaction contributes significantly to the flow and must be considered when modelling it. In particular, we highlight the emergence of strong wall-normal fluctuations near the wall, which result in strong ejection events, an attribute not observed for rigid walls.

The locomotion of microorganisms in complex fluids at low Reynolds numbers has been widely studied by ignoring fluid inertia. Here, we combine the asymptotic analysis and numerical simulations to explore the effect of fluid inertia on the dynamic mechanism of microorganisms swimming through viscoelastic fluids using Taylor’s swimming sheet model, undergoing small-amplitude undulations. Surprisingly, fluid inertia can enhance the speed and efficiency of the infinite-length sheet in viscoelastic fluids at finite Reynolds numbers, in stark contrast to the previous results found in Newtonian fluids. Moreover, speed and efficiency slightly exceed those Newtonian values at the small Weissenberg number due to a passive inertial response of the sheet. We associate this with the magnitude of the hydrodynamic force increasing at finite Reynolds numbers. These insights contribute to a deeper understanding of the inertial effect on the locomotion of microorganisms through complex fluids.

Turbulence accounts for most of the energy losses associated with the pumping of fluids in pipes. Pulsatile drivings can reduce the drag and energy consumption required to supply a desired mass flux, when compared with steady driving. However, not all pulsation waveforms yield reductions. Here, we compute drag- and energy-optimal driving waveforms using direct numerical simulations and a gradient-free black-box optimisation framework. Specifically, we show that Bayesian optimisation is vastly superior to ordinary gradient-based methods in terms of computational efficiency and robustness, due to its ability to deal with noisy objective functions, as they naturally arise from the finite-time averaging of turbulent flows. We identify optimal waveforms for three Reynolds numbers and two Womersley numbers. At a Reynolds number of $8600$ and a Womersley number of 10, optimal waveforms reduce total energy consumption by 22 % and drag by 37 %. These reductions are rooted in the suppression of turbulence prior to the acceleration phase, the resulting delay in turbulence onset, and the radial localisation of turbulent kinetic energy and production towards the pipe centre. Our results pinpoint that the predominant, steady operation mode of pumping fluids through pipes is far from optimal.