发表机构
Hubei Provincial Key Laboratory of Chemical Equipment Intensification and Intrinsic Safety, School of Mechanical and Electrical Engineering, Wuhan Institute of Technology; AECC Hunan Aviation Powerplant Research Institute; School of Aeronautics and Astronautics, Zhejiang University（武汉理工大学机电工程学院化工装备强化与本质安全湖北省重点实验室; 中国航发湖南航空发动机研究所; 浙江大学航空航天学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究基于压力与引力的类比，建立压力几何化的牛顿-嘉当框架，统一解释涡识别准则，并为二维湍流双级联场景提供新的拓扑解释。
AI 中文摘要
流体力学中的压力与广义相对论中的引力之间存在深刻的数学和概念类比，这为流动提供了一种新颖的几何重新解释。本文提出了一个压力几何化的理论框架，通过建立牛顿-嘉当（Newton-Cartan）几何并对空间度量应用共形变换来实现。压力场被重新解释为一种几何势，它诱导出有效空间曲率，由里奇张量（Ricci tensor）Rij和相关的类爱因斯坦张量（Einstein-like tensor）Gij编码。在这个弯曲的有效空间中，仅受压力影响的流体粒子遵循测地线运动，且在弱场极限下可恢复经典欧拉方程。压力泊松方程作为测地线方程和比安基恒等式的自然结果出现，揭示压力是一种瞬时几何约束而非外力。该框架为广泛使用的涡识别准则提供了统一的几何解释，表明主流的欧拉（Eulerian）和拉格朗日（Lagrangian）方案对应于有效曲率的不同几何不变量或投影。在退化的二维（2D）极限下，有效空间张量完全消失，总曲率的全局积分也为零。这一拓扑约束为二维湍流中的双级联场景提供了新的解释：逆能量级联源于正曲率结构向更大尺度的几何凝聚，而正涡量级联则源于负曲率结构向更小尺度的几何色散。
英文摘要
The profound mathematical and conceptual analogy between pressure in fluid mechanics and gravity in general relativity motivates a novel geometric reinterpretation of flows. This article proposes a theoretical framework for geometrizing pressure by establishing a Newton-Cartan geometry and applying a conformal transformation to the spatial metric. The pressure field is reinterpreted as a geometric potential that induces an effective spatial curvature, encoded in the Ricci tensor Rij and the associated Einstein-like tensor Gij. Within this curved effective space, fluid particles under the exclusive influence of pressure follow geodesics, and the classical Euler equation is recovered in the weak-field limit. The pressure Poisson equation emerges naturally as a consequence of the geodesic equation and the Bianchi identity, revealing pressure as an instantaneous geometric constraint rather than an external force. This framework provides a unified geometric interpretation of widely used vortex identification criteria, demonstrating that the prevailing Eulerian and Lagrangian schemes correspond to different geometric invariants or projections of the effective curvature. In the degenerate two-dimensional (2D) limit, the effective spatial tensor vanishes identically, and the global integral of the total curvature vanishes as well. This topological constraint offers a novel interpretation of the dual-cascade scenario in 2D turbulence: the inverse energy cascade arises from geometric condensation of positive-curvature structures toward larger scales, while the forward enstrophy cascade results from geometric dispersion of negative-curvature structures toward smaller scales.
