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The tendency of turbulent flows to produce fine-scale motions from large-scale energy injection is often viewed as a scale-wise cascade of kinetic energy driven by vorticity stretching. This has been recently evaluated by an exact, spatially local relationship (Johnson, P.L. Phys. Rev. Lett., vol. 124, 2020, p. 104501), which also highlights the contribution of strain self-amplification. In this paper, the role of these two mechanisms is explored in more detail. Vorticity stretching and strain amplification interactions between velocity gradients filtered at the same scale account for approximately half of the energy cascade rate, directly connecting the restricted Euler dynamics to the energy cascade. Multiscale strain amplification and vorticity stretching are equally important, however, and more closely resemble eddy viscosity physics. Moreover, ensuing evidence of a power-law decay of energy transfer contributions from disparate scales supports the notion of an energy cascade, albeit a ‘leaky’ one. Besides vorticity stretching and strain self-amplification, a third mechanism of energy transfer is introduced and related to the vortex thinning mechanism important for the inverse cascade in two dimensions. Simulation results indicate this mechanism also provides a net source of backscatter in three-dimensional turbulence, in the range of scales associated with the bottleneck effect. Taken together, these results provide a rich set of implications for large-eddy simulation modelling.

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Breaking waves generate a distribution of bubble sizes that evolves over time. Knowledge of how this distribution evolves is of practical importance for maritime and climate studies. The analytical framework developed in Part 1 (Chan, Johnson & Moin, J. Fluid Mech., vol. 912, 2021, A42) examined how this evolution is governed by the bubble-mass flux from large- to small-bubble sizes which depends on the rate of break-up events and the distribution of child bubble sizes. These statistics are measured in Part 2 as ensemble-averaged functions of time by simulating ensembles of breaking waves, and identifying and tracking individual bubbles and their break-up events. The large-scale break-up dynamics is seen to be statistically unsteady, and two intervals with distinct characteristics were identified. In the first interval, the dissipation rate and bubble-mass flux are quasi-steady, and the theoretical analysis of Part 1 is supported by all observed statistics, including the expected $-10/3$ power-law exponent for the super-Hinze-scale size distribution. Strong locality is observed in the corresponding bubble-mass flux, supporting the presence of a super-Hinze-scale break-up cascade. In the second interval, the dissipation rate decays, and the bubble-mass flux increases as small- and intermediate-sized bubbles become more populous. This flux remains strongly local with cascade-like behaviour, but the dominant power-law exponent for the size distribution increases to $-8/3$ as small bubbles are also depleted more quickly. This suggests the emergence of different physical mechanisms during different phases of the breaking-wave evolution, although size-local break-up remains a dominant theme. Parts 1 and 2 present an analytical toolkit for population balance analysis in two-phase flows.

Large-eddy simulations (LES) are widely used for computing high Reynolds number turbulent flows. Spatial filtering theory for LES is not without its shortcomings, including how to define filtering for wall-bounded flows, commutation errors for non-uniform filters and extensibility to flows with additional complexity, such as multiphase flows. In this paper, the theory for LES is reimagined using a coarsening procedure that imitates nature. This physics-inspired coarsening approach is equivalent to Gaussian filtering for single-phase wall-free flows but opens up new insights for both physical understanding and modelling even in that simple case. For example, an alternative to the Germano identity is introduced and used to define a dynamic procedure without the need for a test filter. Non-uniform resolution can be represented in this framework without commutation errors, and the divergence-free condition is retained for incompressible flows. Potential extensions of the theory to more complex physics such as multiphase flows are briefly discussed.

The statistics of the velocity gradient tensor in turbulent flows is of both theoretical and practical importance. The Lagrangian view provides a privileged perspective for studying the dynamics of turbulence in general, and of the velocity gradient tensor in particular. Stochastic models for the Lagrangian evolution of velocity gradients in isotropic turbulence, with closure models for the pressure Hessian and viscous Laplacian, have been shown to reproduce important features such as non-Gaussian probability distributions, skewness and vorticity strain-rate alignments. The recent fluid deformation (RFD) closure introduced the idea of mapping an isotropic Lagrangian pressure Hessian as the upstream initial condition using the fluid deformation tensor. Recent work on a Gaussian fields closure, however, has shown that even Gaussian isotropic velocity fields contain significant anisotropy for the conditional pressure Hessian tensor due to the inherent velocity–pressure couplings, and that assuming an isotropic pressure Hessian as the upstream condition may not be realistic. In this paper, Gaussian isotropic field statistics is used to generate more physical upstream conditions for the recent fluid deformation mapping. In this new framework, known isotropy relations can be satisfied by tuning the free model parameters and the original Gaussian field coefficients can be directly used without direct numerical simulation (DNS)-based re-adjustment. A detailed comparison of results from the new model, referred to as the recent deformation of Gaussian fields (RDGF) closure, with existing models and DNS shows the improvements gained, especially in various single-time statistics of the velocity gradient tensor at moderate Reynolds numbers. Application to arbitrarily high Reynolds numbers remains an open challenge for this type of model, however.

Breaking waves entrain gas beneath the surface. The wave-breaking process energizes turbulent fluctuations that break bubbles in quick succession to generate a wide range of bubble sizes. Understanding this generation mechanism paves the way towards the development of predictive models for large-scale maritime and climate simulations. Garrett et al. (J. Phys. Oceanogr., vol. 30, 2000, pp. 2163–2171) suggested that super-Hinze-scale turbulent break-up transfers entrained gas from large- to small-bubble sizes in the manner of a cascade. We provide a theoretical basis for this bubble-mass cascade by appealing to how energy is transferred from large to small scales in the energy cascade central to single-phase turbulence theories. A bubble break-up cascade requires that break-up events predominantly transfer bubble mass from a certain bubble size to a slightly smaller size on average. This property is called locality. In this paper, we analytically quantify locality by extending the population balance equation in conservative form to derive the bubble-mass-transfer rate from large to small sizes. Using our proposed measures of locality, we show that scalings relevant to turbulent bubbly flows, including those postulated by Garrett et al. (J. Phys. Oceanogr., vol. 30, 2000, pp. 2163–2171) and observed in breaking-wave experiments and simulations, are consistent with a strongly local transfer rate, where the influence of non-local contributions decays in a power-law fashion. These theoretical predictions are confirmed using numerical simulations in Part 2 (Chan et al., J. Fluid. Mech. vol. 912, 2021, A43), revealing key physical aspects of the bubble break-up cascade phenomenology. Locality supports the universality of turbulent small-bubble break-up, which simplifies the development of cascade-based subgrid-scale models to predict oceanic small-bubble statistics of practical importance.

In wall-bounded turbulent flows, the wall-normal gradient in turbulence intensity causes inertial particles to move preferentially toward the wall, leading to elevated concentration levels in the viscous sublayer. At first glance, wall-modelled large-eddy simulations may seem ill suited for accurately simulating this behaviour, given that the sharp gradients and coherent structures in the viscous sublayer and buffer region are unresolved in this approach. In this paper, a detailed inspection of conservation equations describing the influence of turbophoresis and near-wall structures on particle concentration profiles reveals a more nuanced view depending on the friction Stokes number. The dynamics of low and moderate Stokes number particles indeed depends strongly on the complex spatio-temporal details of streaks, ejections, and sweeps in the near-wall region. This significantly impacts the near-wall particle concentration through a biased sampling effect which provides a net force away from the wall on the particle ensemble caused by the tendency of inertial particles to accumulate in low-speed ejection regions. At higher Stokes numbers, however, this biased sampling is of minimal importance, and the particle concentration becomes inversely proportional to the wall-normal particle velocity variance at a given distance from the wall. As a result, wall-modelled large-eddy simulations can predict concentration profiles with more accuracy in the high Stokes number regime than low Stokes numbers simply by modifying the interpolation scheme for particles between the first grid point and the boundary. However, accurate representation of low and moderate Stokes number particles depends critically on information not present in standard wall-modelled large-eddy simulations.

The detailed dynamics of small-scale turbulence are not directly accessible in large-eddy simulations (LES), posing a modelling challenge, because many micro-physical processes such as deformation of aggregates, drops, bubbles and polymers dynamics depend strongly on the velocity gradient tensor, which is dominated by the turbulence structure in the viscous range. In this paper, we introduce a method for coupling existing stochastic models for the Lagrangian evolution of the velocity gradient tensor with coarse-grained fluid simulations to recover small-scale physics without resorting to direct numerical simulations (DNS). The proposed approach is implemented in LES of turbulent channel flow and detailed comparisons with DNS are carried out. An application to modelling the fate of deformable, small (sub-Kolmogorov) droplets at negligible Stokes number and low volume fraction with one-way coupling is carried out and results are again compared to DNS results. Results illustrate the ability of the proposed model to predict the influence of small-scale turbulence on droplet micro-physics in the context of LES.

In homogeneous and isotropic turbulence, the relative contributions of different physical mechanisms to the energy cascade can be quantified by an exact decomposition of the energy flux (Johnson, Phys. Rev. Lett., vol. 124, 2020, 104501; J. Fluid Mech., vol. 922, 2021, A3). We extend the formalism to the transfer of kinetic helicity across scales, important in the presence of large-scale mirror‐breaking mechanisms, to identify physical processes resulting in helicity transfer and quantify their contributions to the mean flux in the inertial range. All subfluxes transfer helicity from large to small scales. Approximately 50 % of the mean flux is due to the scale-local vortex flattening and vortex twisting. We derive a new exact relation between these effects, similar to the Betchov relation for the energy flux, revealing that the mean contribution of the former is three times larger than that of the latter. Multi-scale effects account for the remaining 50 % of the mean flux, with approximate equipartition between multi-scale vortex flattening, twisting and entangling.

The fate of small particles in turbulent flows depends strongly on the velocity gradient properties of the surrounding fluid, such as rotation and strain rates. For non-inertial (fluid) particles, the restricted Euler model provides a simple low-dimensional dynamical system representation of Lagrangian evolution of velocity gradients in fluid turbulence, at least for short times. Here, we derive a new restricted Euler dynamical system for the velocity gradient evolution of inertial particles, such as solid particles in a gas, or droplets and bubbles in turbulent liquid flows. The model is derived in the limit of small (sub-Kolmogorov-scale) particles and low Stokes number. The system exhibits interesting fixed points, stability and invariant properties. Comparisons with data from direct numerical simulations show that the model predicts realistic trends such as the tendency of increased straining over rotation along heavy particle trajectories and, for light particles such as bubbles, the tendency of reduced self-stretching of the strain rate.