基于极小作用方法和高精度算法的湍流生成机理研究

The onset of turbulence is a classical challenging problem in fluid mechanics. The study of transition mechanism and a clear picture of the transition process have great theoretical significance and practical application values. Traditional research methods include linear stability analysis, nonlinear stability theory, direct numerical simulation, etc. Lots of investigations indicate that the natural transitions usually undergo several stages, including external disturbance receptivity, major modes linear growth, nonlinear interaction, secondary instability, local turbulence spots, fully developed turbulence, etc. But the exact transition process also depends on other factors such as the character of disturbance, inflow conditions, domain shape and boundary curvature and roughness. However, there is no unified theory or method for the laminar-turbulence transition. Recently, the minimum action method developed in the framework of large deviation theory has achieved great successes in finding the optimal transition paths of non-gradient systems such as Lorenz system, interface growing KPZ equation, flame front KS equation and 2-dimensional incompressible Navier-Stokes equations, etc. It opens a new direction to the study of turbulence onset. In this project, we combine the minimum action method with high-order algorithms and parallel implementation, to study the lose of stability and the onset of turbulence in several typical incompressible shear flows such as Couette flow and Poiseuille flow, etc. We expect the research will provide a new analysis and computing tool to study the mechanism of laminar-turbulence transitions.

层流解到湍流的转捩是流体力学的一个经典难题.对转捩机理的研究和转捩过程的精确刻画有着重大的理论意义和实际应用价值.传统的研究方法包括线性稳定性分析,非线性稳定性理论,直接数值模拟等.大量研究表明,自然转捩大致经过扰动感知,线性发展,非线性相互作用,二次失稳,局部涡团,完全发展湍流等几个阶段.但具体转捩过程还依赖于扰动形式,入流条件,边界曲率,光滑度等众多因素,尚没有一个统一的理论或方法.最近在大偏差理论框架下发展起来的极小作用方法在寻找诸如洛伦兹系统,界面增长KPZ方程,火焰燃烧KS方程,以及二维不可压缩NS方程等非梯度系统的最优转变路径问题上取得了巨大的成功,为湍流的生成机理研究提供了新的方向.本项目将极小作用方法同高精度算法和并行实现相结合,研究平面剪切流,压力驱动流等几种典型不可压流场的层流失稳和湍流生成机理,期望能给流体转捩机理研究提供一种新的分析计算工具选择.

层流失稳和湍流生成是流体力学中的一个经典难题。 对流体稳定性机理的研究和精确刻画有着重大的理论意义和实际应用价值。 本项目针对典型不可压壁限剪切流场的层流失稳机理问题，设计了高效的 Laguerre 极小作用谱方法。针对偏微分方程应用极小作用方法计算量大的问题，构建了将一种推广的 Onsager 原理同深度整流幂次单元神经网络相结合的模型降维方法，能够为复杂动力学系统构建可高保真重现动力学定性结构的低维模型，为进一步分析和计算提供快速替代模型。本项目中还构建了多个高阶低耗散湍流直接数值模拟格式。项目中创建的数值方法可为可压缩和不可压缩 Navier-Stokes 方程组的高精度直接数值模拟以及相关的稳定性分析提供有力工具。