Efficient Numerical Methods for Viscous Incompressible Flows

研究者建议继续发展高效和精确的非定常不可压缩流动数值方法，特别强调(i)寻找新的等效的Navier-Stokes方程公式，通常受到物理考虑的启发，更适合数值计算；（ii）适当处理边界条件，这是本课题的一个关键难点，因为重要的物理相互作用经常发生在边界附近；（iii）三维流动的快速计算和复杂几何流动方案的推广；（iv）开发复杂流体的有效方案，如磁流体动力学、液晶聚合物、地球发电机、气候模型和大涡流湍流模拟；(v)设计保留数值守恒量的数值格式，如无粘流的能量和螺旋度，并准确地捕获它们在高雷诺数下的耗散率。人们普遍认为，不同的能量耗散率和螺旋度耗散率是导致湍流大型相干结构形成的关键机制。不可压缩流动的数值模拟在当前许多科学和工业应用中起着重要作用，对数值分析人员和计算流体动力学家来说都是一项具有挑战性的任务。所提出的快速、精确的不可压缩流动数值方法有望成为模拟和分析复杂湍流现象的重要工具，包括涡旋破裂、（大质量）流分离、涡旋脱落、横流瞬态射流、尾迹-体相互作用、高旋流等。它也是设计先进流动控制机制的重要工具，例如，减少流动引起的噪音和振动，并在降低能耗的情况下提高升阻性能。例子包括流过地面或水下交通工具等钝体的水流；在引擎;在旋转机械内/周围或在具有旋转和运动部件的数据存储单元内。

The investigator proposes to continue the development of efficient and accurate numerical methods for unsteady incompressible flow, with particular emphasis on (i) finding new equivalent formulations of Navier-Stokes equations, often inspired by physical considerations, better suited to numerical computation; (ii) proper handling of boundary conditions, a key difficulty in the subject since important physical interactions frequently occur near the boundary; (iii) rapid computation of three dimensional flows and generalization of schemes for flows in complex geometries; (iv) development of efficient schemes for complex fluids such as magneto-hydrodynamics, liquid crystal polymers, geodynamo, climate modeling, and large eddy turbulence simulations; (v) design of numerical schemes that preserves numerically conserved quantities such as energy and helicity for inviscid flow, and accurately captures their dissipation rates at high Reynolds number. It is widely believed that different dissipation rates for energy and helicity is the key mechanism leading to the formation of large coherent structures of turbulent flows.The numerical simulation of incompressible flows, which plays an important role in numerous scientific and industrial applications of current interest, is a challenging task for both numerical analysts and computational fluid dynamicists. The proposed fast and accurate numerical methods for incompressible flow is expected to become an important tool for simulation and analysis of complex turbulent flow phenomena including vortex breakdown, (massive) flow separation, vortex shedding, transient jets in cross-stream, wake-body interaction, high-swirl flow, etc. It is also an essential tool for the design of advanced flow control mechanisms used, for example, to reduce flow-induced noise and vibration, and to improve lift/drag performance at reduced energy consumption rates. Examples include flow over bluff bodies such as ground or under-water vehicles; in engines; in/around rotating machinery or in data storage units with rotating and moving parts.