论文标题
没有压力？不可压缩的Navier-Stokes方程与时间相关边界条件的能源一致的ROM
No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions
论文作者
论文摘要
这项工作为不可压缩的Navier-Stokes方程式提供了一种新颖的减速模型（ROM），并具有时间相关的边界条件。该ROM仅速度 - 即速度的仿真不需要压力的计算，并保留了动能演化的结构。新型ROM的关键要素是将速度分解为具有均匀边界条件的磁场，并且具有满足规定的不均匀边界条件的质量方程的提升函数。这种分解的灵感来自于Helmholtz-Hodge分解和两个组件的正交性。这种正交性对于保留动能演化的结构至关重要。为了评估提升功能的有效效率，我们提出了一种新的方法，涉及使用POD模式明确近似边界条件，同时保留了速度分解的正交性，从而保留了动能演化的结构。我们表明，所提出的仅速度的ROM等于速度压力ROM，即模拟速度和压力的ROM。这种等效性可以推广到其他现有的速度压力ROM，并揭示其行为的宝贵见解。对具有流出流量边界条件的测试用例的数值实验证实了新ROM的正确性和效率，以及与速度压力公式的等效性。
This work presents a novel reduced-order model (ROM) for the incompressible Navier-Stokes equations with time-dependent boundary conditions. This ROM is velocity-only, i.e. the simulation of the velocity does not require the computation of the pressure, and preserves the structure of the kinetic energy evolution. The key ingredient of the novel ROM is a decomposition of the velocity into a field with homogeneous boundary conditions and a lifting function that satisfies the mass equation with the prescribed inhomogeneous boundary conditions. This decomposition is inspired by the Helmholtz-Hodge decomposition and exhibits orthogonality of the two components. This orthogonality is crucial to preserve the structure of the kinetic energy evolution. To make the evaluation of the lifting function efficient, we propose a novel method that involves an explicit approximation of the boundary conditions with POD modes, while preserving the orthogonality of the velocity decomposition and thus the structure of the kinetic energy evolution. We show that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. This equivalence can be generalized to other existing velocity-pressure ROMs and reveals valuable insights in their behaviour. Numerical experiments on test cases with inflow-outflow boundary conditions confirm the correctness and efficiency of the new ROM, and the equivalence with the velocity-pressure formulation.
