A network-based perspective on coherent structure detection from very-sparse Lagrangian data

Coherent structure detection (CSD) is a long-lasting issue in fluid mechanics research as the presence of spatio-temporal coherent motion enables simpler ways to characterize the flow dynamics. Such reducedorder representation, in fact, has significant implications for the understanding of the dynamics of flows, as well as their modeling and control (Hussain, 1986). While the Eulerian framework has been extensively adopted for CSD, Lagrangian coherent structures have recently received increasing attention, mainly driven by advancements in Lagrangian flow measurement techniques (Haller, 2015; Hadjighasem et al., 2017). Lagrangian particle tracking (LPT), in particular, is widely used nowadays due to its ability to quantity fluid-parcel trajectories in three-dimensional volumes (Schanz et al., 2016).

LaPIV using multigrid warping and proxy regularization

The Lagrangian Particle Image Velocimetry (LAPIV) method was firstly proposed in Yang et al. (2019) as a prototype approach to achieve the goal of accurate and efficient reconstruction of 3D Eulerian velocity field of fluid flow from multi-view particle images. After validating against synthetic datasets, the prototype has already shown significant advantages in revealing more small scale flow structures than other stateof-the-art Eulerian velocity estimation methods, such as TomoPIV (Scarano, 2013) and VIC# (Jeon et al., 2019). However, at this early stage, LAPIV can not be easily applied to other datasets. In the current work, we focus on extending LAPIV to operational search by incorporating several essential and wellestablished paradigms: multi-resolution, warping, and proxy regularization. Recent approaches, Lasinger et al. (2019) and Cornic et al. (2020), function in the same vein as LAPIV, aiming at reconstructs the dense Eulerian volumetric flow directly from multi-view particle-seeded images Another pipeline consists of firstly reconstructing the Lagrangian flow using the Lagrangian Particle Tracking (LPT), then optimally interpolating the Lagrangian flow to Eulerian grids, taking into account the Eulerian dynamics constraints as in Flowfit (Gesemann, 2020) and VIC# (Jeon et al., 2019). If Eulerian flow is required, LAPIV is the preferred approach due to its simplicity and ability to utilize the original rich image features.

Numerical Modelling of Two-Phase Flow in a Gas Separator Using the Eulerian–Lagrangian Flow Model

Gravity-driven separators are broadly used in various engineering applications to remove particulate matters from gaseous fluids to meet legislation demands. This study represents a detailed numerical investigation of a two-phase cyclone separator using the Eulerian–Lagrangian gas flow method. The turbulence is modelled using the Reynolds stress model (RSM). The technique has successfully predicted the typical trends and variations seen in such gas separators with an average error of approximately 5.5%. Also, the computed results show a realistic agreement with the experimental measurements.

Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves

The seminal work of DiPerna and Lions (Invent Math 98(3):511–547, 1989) guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio’s superposition principle, we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Székelyhidi in the recent groundbreaking work (Modena and Székelyhidi in Ann PDE 4(2):38, 2018). On the opposite side, we introduce a new class of asymmetric Lusin–Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna–Lions theory.

Ocean Surface Connectivity in the Arctic: Capabilities and Caveats of Community Detection in Lagrangian Flow Networks

<div><span>To identify barriers to transport in a fluid domain, community detection algorithms from network science have been used to divide the domain into clusters that are sparsely connected with each other. In a previous application to the closed domain of the Mediterranean Sea, communities detected by the <em>Infomap</em> algorithm have barriers that often coincide with well-known oceanographic features. We apply this clustering method to the surface of the Arctic and subarctic oceans and thereby show that it can also be applied to open domains. First, we construct a Lagrangian flow network by simulating the exchange of Lagrangian particles between different bins in an icosahedral-hexagonal grid. Then, <em>Infomap </em>is applied to identify groups of well-connected bins. The resolved transport barriers include naturally occurring structures, such as the major currents. As expected, clusters in the Arctic are affected by seasonal and annual variations in sea-ice concentration. An important caveat of community detection algorithms is that many different divisions into clusters may qualify as good solutions. Moreover, while certain cluster boundaries lie consistently at the same location between different good solutions, other boundary locations vary significantly, making it difficult to assess the physical meaning of a single solution. We therefore consider an ensemble of solutions to find persistent boundaries, trends and correlations with surface velocities and sea-ice cover.</span></div>

Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

We develop Korevaar–Schoen’s theory of directional energies for metric-valued Sobolev maps in the case of $${\textsf {RCD}}$$ RCD source spaces; to do so we crucially rely on Ambrosio’s concept of Regular Lagrangian Flow. Our review of Korevaar–Schoen’s spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of ‘differential of a map along a vector field’ and about the parallelogram identity for $${\textsf {CAT}}(0)$$ CAT ( 0 ) targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.