Two common definitions of the spatially local rate of kinetic energy cascade at some scale $\ell$ in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’ $R>0$, $Q>0$ quadrant, where $R$ and $Q$ are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when $R$ is large.

An extended turbulent state can coexist with the stable laminar state in pipe flows. We focus here on short pipes with additional discrete symmetries imposed. In this case, the boundary between the coexisting basins of attraction, often called the edge of chaos, is the stable manifold of an edge state, which is a lower-branch travelling wave solution. We show that a low-dimensional submanifold of the edge of chaos can be constructed from velocity data using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of non-resonant spectral subspaces of the linearized system at stationary states. Using very low-dimensional SSM-based reduced-order models, we predict transitions to turbulence or laminarization for velocity fields near the edge of chaos.

We experimentally investigate the rotational dynamics of neutrally buoyant axisymmetric particles in a simple shear flow. A custom-built shearing cell and a multi-view shape-reconstruction method are used to obtain direct measurements of the orientation and period of rotation of particles having oblate and prolate shapes (such as spheroids and cylinders) of varying aspect ratios. By systematically changing the viscosity of the fluid, we examine the effect of inertia (which may be originated from either phase) on the dynamical behaviour of these suspended particles up to a particle Reynolds number of approximately one. While no significant effect on the period of rotation is found in this small-inertia regime, a systematic drift among several rotations toward limiting stable orbits is observed. Prolate particles are seen to drift towards the tumbling orbit in the plane of shear, whereas oblate particles are driven either to the tumbling or to the vorticity-aligned spinning orbits, depending on their initial orientation. These results are compared with recent small-inertia asymptotic theories, assessing their range of validity, as well as to numerical simulations in the small-inertia regime for both prolate and oblate particles.

We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, $\varepsilon$ and $\chi$. Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number $Re_b:=\varepsilon /(\nu N^2)$, where $\nu$ is the kinematic viscosity and $N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for $\varepsilon$ and $\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes $Re_b\ll 1$ and $Re_b\gg 1$, and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of $Re_b$), the PCNN outperforms assumptions of isotropy significantly as $Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.

The recent Lagrangian relaxation towards equilibrium (LaRTE) approach (Fowler et al., J. Fluid Mech., vol. 934, 2022, A44) is a wall model for large-eddy simulations (LES) that isolates quasi-equilibrium wall-stress dynamics from non-equilibrium responses to time-varying LES inputs. Non-equilibrium physics can then be modelled separately, such as the laminar Stokes layers that form in the viscous region and generate rapid wall-stress responses to fast changes in the pressure gradient. To capture additional wall-stress contributions due to near-wall turbulent eddies, a model term motivated by the attached eddy hypothesis is proposed. The total modelled wall stress thus includes contributions from various processes operating at different time scales (i.e. the LaRTE quasi-equilibrium plus laminar and turbulent non-equilibrium wall stresses) and is called the multi-time-scale (MTS) wall model. It is applied in LES of turbulent channel flow subject to a wide range of unsteady conditions from quasi-equilibrium to non-equilibrium. Flows considered include pulsating and linearly accelerating channel flow for several forcing frequencies and acceleration rates, respectively. We also revisit the sudden spanwise pressure gradient flow (considered in Fowler et al., J. Fluid Mech., vol. 934, 2022, A44) to review how the newly introduced model features affect this flow. Results obtained with the MTS wall model show good agreement with direct numerical simulation data over a vast range of conditions in these various non-equilibrium channel flows. To further understand the MTS model, we also describe and test the instantaneous-equilibrium limit of the MTS wall model. In this limit, good wall-stress predictions are obtained with reduced model complexity but providing less complete information about the wall-stress physics.

We carry out direct numerical simulation (DNS) of turbulent flow in rough pipes. Two types of irregular roughness are investigated, namely a grit-blasted and a graphite surface. A wide range of Reynolds numbers is tested, from the laminar up to the fully rough regime, attempting to replicate Nikuradse's pioneering study. Despite the large relative roughness, outer-layer similarity is achieved at high Reynolds number as hypothesised by Townsend, with deviations from the smooth wall case of 4 % for the grit-blasted surface and 13 % for the graphite surface. This makes it possible to define a roughness function and the equivalent sand-grain roughness. The results are compared with those obtained in plane channels, with small differences pointing to the residual influence of the duct cross-sectional shape in the presence of relatively large roughness. The computed friction factors behave similar to those Nikuradse's chart, with differences in terms of the friction factor in the laminar region and of the critical Reynolds number, which are partly absorbed by using the hydraulic radius as reference length scale. The distributions of the velocity fluctuations intensities point to a isotropisation of turbulence in the near-wall region resulting from the roughness, with influence of the roughness geometry. Comparison of the computed equivalent sand-grain roughness height suggest that existing correlations suffer from poor predictive power, at least for surfaces with large relative roughness.

The refraction of surface gravity waves by currents leads to spatial modulations in the wave field and, in particular, in the significant wave height. We examine this phenomenon in the case of waves scattered by a localised current feature, assuming (i) the smallness of the ratio between current velocity and wave group speed, and (ii) a swell-like, highly directional wave spectrum. We apply matched asymptotics to the equation governing the conservation of wave action in the four-dimensional position–wavenumber space. The resulting explicit formulas show that the modulations in wave action and significant wave height past the localised current are controlled by the vorticity of the current integrated along the primary direction of the swell. We assess the asymptotic predictions against numerical simulations using WAVEWATCH III for a Gaussian vortex. We also consider vortex dipoles to demonstrate the possibility of ‘vortex cloaking’ whereby certain currents have (asymptotically) no impact on the significant wave height. We discuss the role of the ratio of the two small parameters characterising assumptions (i) and (ii) above, and show that caustics are significant only for unrealistically large values of this ratio, corresponding to unrealistically narrow directional spectra.

Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number $Re_\tau =1000$. The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.