用于Navier-Stokes方程的带变分多尺度稳定化的Helmholtz-Leray投影方法
A Helmholtz-Leray projection method with variational multiscale stabilization for the Navier-Stokes equations
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中文总结 AI 辅助
本研究提出带变分多尺度稳定化的增量Helmholtz-Leray投影方法求解Navier-Stokes方程，通过省略压力细尺度降低阻力过预测与耗散，提升求解效率，且数值结果吻合参考数据。
中文摘要 AI 辅助
不可压缩Navier-Stokes方程的Galerkin有限元格式存在两个主要挑战：保持速度-压力耦合的稳定性，以及控制对流主导区域的不稳定性；此外，整体格式会产生耦合的非线性鞍点问题。本研究提出一种基于残差的变分多尺度（VMS）稳定化的增量Helmholtz-Leray投影方法，将该耦合鞍点问题替换为非线性速度预测、压力泊松方程和速度投影。多尺度分解仅应用于预测速度，压力和校正后的弱散度自由速度均不分解为粗、细尺度；建模的速度细尺度一致作用于三个子问题，在动量预测中引入类SUPG稳定化，在压力泊松方程中引入类PSPG残差贡献。研究提供了形式化误差分解，将BDF2时间离散、投影分裂和空间VMS误差分离，在给定稳定性和空间逼近假设下，得到具有二阶时间精度的组合速度误差估计。对 manufactured 解、顶盖驱动空腔流、圆柱绕流和Taylor-Green涡的数值结果与已有的参考数据吻合良好；与整体VMS的对比显示，省略压力细尺度可减小阻力过预测和过度建模耗散，代价是散度误差增加；在相同求解器设置下，Taylor-Green测试中该投影格式每步平均求解时间减少约1.3倍至2.7倍。
英文摘要
The Galerkin finite element formulation of the incompressible Navier-Stokes equations presents two principal challenges: maintaining stable velocity-pressure coupling and controlling instability in advection-dominated regimes. Moreover, the monolithic formulation produces a coupled nonlinear saddle-point problem. In this work, we present a residual-based variational multiscale (VMS) stabilization of an incremental Helmholtz-Leray projection method that replaces this coupled saddle-point problem with a nonlinear velocity predictor, a pressure Poisson equation, and a velocity projection. The multiscale decomposition is applied only to the predicted velocity; neither the pressure nor the corrected, weakly divergence-free velocity is decomposed into coarse and fine scales. The modeled velocity fine scale contributes consistently to all three subproblems, introducing SUPG-like stabilization in the momentum predictor and a PSPG-like residual contribution in the pressure Poisson equation. We provide a formal error decomposition that separates the BDF2 time-discretization, projection-splitting, and spatial-VMS errors and, under stated stability and spatial-approximation assumptions, yields a combined velocity error estimate with second-order temporal accuracy. Numerical results for manufactured solutions, lid-driven cavity flow, flow past a cylinder, and the Taylor-Green vortex agree closely with established reference data. Comparisons with monolithic VMS indicate that omitting the pressure fine scale reduces drag overprediction and excess modeled dissipation, at the cost of increased divergence error. In the Taylor-Green tests, the projection formulation also reduces the average solution time per step by factors ranging from approximately $1.3\times$ to $2.7\times$ under identical solver settings.
