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1. Clebsch Gauge Fluid on Particle Flow Maps

   https://doi.org/10.1145/3731194  

   Li, Zhiqi; Lin, Candong; Chen, Duowen; Zhou, Xinyi; Xiong, Shiying; Zhu, Bo (August 2025, ACM Transactions on Graphics)

   We propose a novel gauge fluid solver that evolves Clebsch wave functions on particle flow maps (PFMs). The key insight underlying our work is that particle flow maps exhibit superior performance in transporting point elements—such as Clebsch components—compared to line and surface elements, which were the focus of previous methods relying on impulse and vortex gauge variables for flow maps. Our Clebsch PFM method incorporates three main contributions: a novel gauge transformation enabling accurate transport of wave functions on particle flow maps, an enhanced velocity reconstruction method for coarse grids, and a PFM-based simulation framework designed to better preserve fine-scale flow structures. We validate the Clebsch PFM method through a wide range of benchmark tests and simulation examples, ranging from leapfrogging vortex rings and vortex reconnections to Kelvin-Helmholtz instabilities, demonstrating that our method outperforms its impulse- or vortex-based counterparts on particle flow maps, particularly in preserving and evolving small-scale features. 

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2. Fluid Simulation on Compressible Flow Maps

   https://doi.org/10.1145/3731192  

   Chen, Duowen; Li, Zhiqi; Zhang, Taiyuan; He, Jinjin; Zhou, Junwei; van-Bloemen-Waanders, Bart G; Zhu, Bo (August 2025, ACM Transactions on Graphics)

   This paper presents a unified compressible flow map framework designed to accommodate diverse compressible flow systems, including high-Mach-number flows (e.g., shock waves and supersonic aircraft), weakly compressible systems (e.g., smoke plumes and ink diffusion), and incompressible systems evolving through compressible acoustic quantities (e.g., free-surface shallow water). At the core of our approach is a theoretical foundation for compressible flow maps based on Lagrangian path integrals, a novel advection scheme for the conservative transport of density and energy, and a unified numerical framework for solving compressible flows with varying pressure treatments. We validate our method across three representative compressible flow systems, characterized by varying fluid morphologies, governing equations, and compressibility levels, demonstrating its ability to preserve and evolve spatiotemporal features such as vortical structures and wave interactions governed by different flow physics. Our results highlight a wide range of novel phenomena, from ink torus breakup to delta wing tail vortices and vortex shedding on free surfaces, significantly expanding the range of fluid systems that flow-map methods can handle. 

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3. An Eulerian Vortex Method on Flow Maps

   https://doi.org/10.1145/3687996  

   Wang, Sinan; Deng, Yitong; Deng, Molin; Yu, Hong-Xing; Zhou, Junwei; Chen, Duowen; Komura, Taku; Wu, Jiajun; Zhu, Bo (November 2024, ACM Transactions on Graphics)

   We present an Eulerian vortex method based on the theory of flow maps to simulate the complex vortical motions of incompressible fluids. Central to our method is the novel incorporation of the flow-map transport equations forline elements, which, in combination with a bi-directional marching scheme for flow maps, enables the high-fidelity Eulerian advection of vorticity variables. The fundamental motivation is that, compared to impulsem, which has been recently bridged with flow maps to encouraging results, vorticityωpromises to be preferable for its numerical stability and physical interpretability. To realize the full potential of this novel formulation, we develop a new Poisson solving scheme for vorticity-to-velocity reconstruction that is both efficient and able to accurately handle the coupling near solid boundaries. We demonstrate the efficacy of our approach with a range of vortex simulation examples, including leapfrog vortices, vortex collisions, cavity flow, and the formation of complex vortical structures due to solid-fluid interactions. 

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4. An Impulse Ghost Fluid Method for Simulating Two-Phase Flows

   https://doi.org/10.1145/3687963  

   Sun, Yuchen; Chen, Linglai; Zeng, Weiyuan; Du, Tao; Xiong, Shiying; Zhu, Bo (November 2024, ACM Transactions on Graphics)

   This paper introduces a two-phase interfacial fluid model based on the impulse variable to capture complex vorticity-interface interactions. Our key idea is to leverage bidirectional flow map theory to enhance the transport accuracy of both vorticity and interfaces simultaneously and address their coupling within a unified Eulerian framework. At the heart of our framework is an impulse ghost fluid method to solve the two-phase incompressible fluid characterized by its interfacial dynamics. To deal with the history-dependent jump of gauge variables across a dynamic interface, we develop a novel path integral formula empowered by spatiotemporal buffers to convert the history-dependent jump condition into a geometry-dependent jump condition when projecting impulse to velocity. We demonstrate the efficacy of our approach in simulating and visualizing several interface-vorticity interaction problems with cross-phase vortical evolution, including interfacial whirlpool, vortex ring reflection, and leapfrogging bubble rings. 

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5. Generalized Clebsch Variables for Compressible Ideal Fluids: Initial Conditions and Approximations of the Hamiltonian

   https://doi.org/10.3390/fluids7040122  

   Rumpf, Benno; Lvov, Yuri V. (April 2022, Fluids)

   Clebsch variables provide a canonical representation of ideal flows that is, in practice, difficult to handle: while the velocity field is a function of the Clebsch variables and their gradients, constructing the Clebsch variables from the velocity field is not trivial. We introduce an extended set of Clebsch variables that circumvents this problem. We apply this method to a compressible, chemically inhomogeneous, and rotating ideal fluid in a gravity field. A second difficulty, the secular growth of canonical variables even for stationary states of stratified fluids, makes expansions of the Hamiltonian in Clebsch variables problematic. We give a canonical transformation that associates a stationary state of the canonical variables with the stationary state of the fluid; the new set of variables permits canonical approximations of the dynamics. We apply this to a compressible stratified ideal fluid with the aim to facilitate forthcoming studies of wave turbulence of internal waves. 

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