CAREER: New Algorithms and Models for Turbulence in Incompressible Fluids

湍流在自然界中无处不在，影响我们生活的决定每天都是基于对湍流的预测做出的。在全球变化估计、天气预报、淡水供应、提高发动机的能源效率、控制污染物的扩散和设计生物医学设备方面，获得湍流的准确预测是一个核心挑战。湍流是一个高度不规则的系统，其特征是在非线性相互作用中涉及大范围尺度的混沌性质变化。这些特征产生了很高的计算复杂性，这使得旨在解决最小尺度下的所有特征的湍流的直接数值模拟即使在现代超级计算机上也是不可行的。取而代之的是，湍流模型被用于实际的湍流模拟，以绕过混沌细节，降低计算复杂性。该项目旨在开发一类新的集合平均湍流模型及其求解的新的数值方法，扩展了现有的有效湍流模拟的适用性和计算局限性，考虑到湍流在几乎所有地球物理和工业流动中的显著影响，这可能会对航空、水力学、化工、海洋学、气象学、天体物理学和地球物理中的许多应用产生重大影响。将开发一个全面的教育计划，为学生提供计算流体力学的系统培训，并使他们了解该领域当前的研究课题。湍流模拟仍然是最重要的科学挑战之一。湍流模拟的基本方法是寻求近似流体速度的适当(集合、时间或空间)平均值，而不是逐点速度本身。系综平均是湍流统计理论中最直观的方法，但由于与系综模拟相关的计算成本极高，目前还没有用于工业流动的实际湍流模拟。最近，新开发的系综算法打破了这一僵局，这些算法可以在每个时间步长访问完整的系综，从而为建立系综平均Navier-Stokes方程的湍流模型开辟了新的直接可能性。在这个项目中，研究人员将在新的系综平均框架下开发一系列新的基于系综的变分多尺度方法(VMS)湍流模型和新的数值方法，用于实际的湍流模拟。新的无条件稳定的系综算法将被用来快速求解新的基于系综的VMS湍流模式，并有效地克服开边界条件下湍流的回流不稳定性。这个项目将为湍流建模和模拟提供新的途径，并建立一个新的严格的数值分析，解决如何在牛顿混沌面前进行有效的近似。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Turbulence is ubiquitous in nature, and decisions that affect our life are made daily based on predictions of turbulent flows. Obtaining accurate predictions of turbulent flows is a central challenge in global change estimation, weather forecasting, freshwater supply, improving the energy efficiency of engines, controlling dispersal of contaminants, and designing biomedical devices. A turbulent flow is a highly irregular system, characterized by chaotic property changes involving a wide range of scales in nonlinear interaction with each other. These features yield a high computational complexity, which makes direct numerical simulations of turbulent flows that aim at resolving all features down to the smallest scales infeasible even with modern supercomputers. Instead, turbulence models are used for practical turbulence simulations to bypass the chaotic details and reduce the computational complexity. This project aims to develop a new family of ensemble averaged turbulence models and novel numerical methods for their solution, extending current applicability and computational limitations of effective turbulence simulations, which may have a great impact on numerous applications in aeronautics, hydraulics, chemical engineering, oceanography, meteorology, astrophysics, and geophysics, considering turbulence’s prominent influence in almost all geophysical and industrial flows. A comprehensive educational program will be developed to provide students with systematic training in computational fluid dynamics and bring them up to date on current research topics in this field. Turbulence modeling remains one of the most important scientific challenges. The fundamental approach for turbulence modeling is to seek to approximate suitable (ensemble, time, or spatial) averages of fluid velocity instead of pointwise velocity itself. Ensemble averaging is the most intuitive approach from the statistical theory for turbulence, but it is currently not in use for practical turbulence simulations of industrial flows due to the extremely high computational cost associated with ensemble simulations. This deadlock is recently broken with newly developed ensemble algorithms that give access to the full ensemble at every time step and thus open new and direct possibilities for developing turbulence models for the ensemble averaged Navier-Stokes equations. In this project the investigator will develop a new family of ensemble-based variational multiscale method (VMS) turbulence models and novel numerical methods under the new framework of ensemble averaging for practical turbulent flow simulations. New unconditionally stable ensemble algorithms will be developed for fast solution of the new ensemble-based VMS turbulence models and to effectively overcome the backflow instability for turbulent flows with open boundary conditions. This project will provide new avenues to turbulence modeling and simulations and build a new rigorous numerical analysis addressing how to make effective approximations in the face of Newtonian chaos.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.