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1. Small scale creation in active scalars

   https://doi.org/10.1007/978-3-030-54899  

   Alexander Kiselev (January 2020, Progress in mathematical fluid dynamics, Lecture Notes in Mathematics)

   Berselli, Luigi G; Ruzicka, Michael (Ed.)

   The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions. 

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2. The role of fluid flow in the dynamics of active nematic defects

   https://doi.org/10.1088/1367-2630/abe8a8  

   Angheluta, Luiza; Chen, Zhitao; Marchetti, M Cristina; Bowick, Mark J (March 2021, New Journal of Physics)

   Abstract We adapt the Halperin–Mazenko formalism to analyze two-dimensional active nematics coupled to a generic fluid flow. The governing hydrodynamic equations lead to evolution laws for nematic topological defects and their corresponding density fields. We find that ±1/2 defects are propelled by the local fluid flow and by the nematic orientation coupled with the flow shear rate. In the overdamped and compressible limit, we recover the previously obtained active self-propulsion of the +1/2 defects. Non-local hydrodynamic effects are primarily significant for incompressible flows, for which it is not possible to eliminate the fluid velocity in favor of the local defect polarization alone. For the case of two defects with opposite charge, the non-local hydrodynamic interaction is mediated by non-reciprocal pressure-gradient forces. Finally, we derive continuum equations for a defect gas coupled to an arbitrary (compressible or incompressible) fluid flow. 

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3. Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

   https://doi.org/10.1016/j.jfa.2023.110057  

   Koganemaru, Junichi; Tice, Ian (October 2023, Journal of Functional Analysis)

   This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest. 

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4. Fluid Simulation on Neural Flow Maps

   https://doi.org/10.1145/3618392  

   Deng, Yitong; Yu, Hong-Xing; Zhang, Diyang; Wu, Jiajun; Zhu, Bo (December 2023, ACM Transactions on Graphics)

   We introduce Neural Flow Maps, a novel simulation method bridging the emerging paradigm of implicit neural representations with fluid simulation based on the theory of flow maps, to achieve state-of-the-art simulation of in-viscid fluid phenomena. We devise a novel hybrid neural field representation, Spatially Sparse Neural Fields (SSNF), which fuses small neural networks with a pyramid of overlapping, multi-resolution, and spatially sparse grids, to compactly represent long-term spatiotemporal velocity fields at high accuracy. With this neural velocity buffer in hand, we compute long-term, bidirectional flow maps and their Jacobians in a mechanistically symmetric manner, to facilitate drastic accuracy improvement over existing solutions. These long-range, bidirectional flow maps enable high advection accuracy with low dissipation, which in turn facilitates high-fidelity incompressible flow simulations that manifest intricate vortical structures. We demonstrate the efficacy of our neural fluid simulation in a variety of challenging simulation scenarios, including leapfrogging vortices, colliding vortices, vortex reconnections, as well as vortex generation from moving obstacles and density differences. Our examples show increased performance over existing methods in terms of energy conservation, visual complexity, adherence to experimental observations, and preservation of detailed vortical structures. 

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5. Fluid-structure interaction simulations with a LES filtering approach in solids4Foam

   https://doi.org/10.2478/caim-2021-0002  

   Girfoglio, Michele; Quaini, Annalisa; Rozza, Gianluigi (January 2021, Communications in Applied and Industrial Mathematics)

   Abstract The goal of this paper is to test solids4Foam, the fluid-structure interaction (FSI) toolbox developed for foam-extend (a branch of OpenFOAM), and assess its flexibility in handling more complex flows. For this purpose, we consider the interaction of an incompressible fluid described by a Leray model with a hyperelastic structure modeled as a Saint Venant-Kirchho material. We focus on a strongly coupled, partitioned fluid-structure interaction (FSI) solver in a finite volume environment, combined with an arbitrary Lagrangian-Eulerian approach to deal with the motion of the fluid domain. For the implementation of the Leray model, which features a nonlinear differential low-pass filter, we adopt a three-step algorithm called Evolve-Filter-Relax. We validate our approach against numerical data available in the literature for the 3D cross flow past a cantilever beam at Reynolds number 100 and 400. 

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