Efficient Numerical Methods for Viscous Incompressible Flows

该项目将针对不可压缩流的一类新的和非常有效的数值方法的开发，该方法基于已被证明是适定的Navier-Stokes方程的重新表述，并提供了涡度循环方面的粘性对压力的贡献的边界条件。 主要的见解是，总的创造边界涡可以精确计算的拉普拉斯和亥姆霍兹投影算子之间的换向器。此外，该项由粘度项支配，因此我们可以显式地处理它以获得效率和稳定性。该方法的主要优点是可以在标准的连续有限元空间中进行计算，并且在速度和压力近似空间中不需要协调条件。精确、高效地模拟复杂区域的三维流动仍然是许多科学和工程问题的主要挑战。物理边界附近流动的分辨率对于准确预测升力、阻力、旋涡脱落和边界层分离等物体力是必不可少的。这个项目的成功将对科学和工程的许多分支产生重要影响。它将提供更准确的体力预测，这将导致更好的能源效率为运输相关的应用。它将为科学研究和工程应用提供快速、可靠的模拟。

This project will address the development of a class of novel and very efficient numerical methods for incompressible flows based on a reformulation of the Navier-Stokes equations that has been proved well-posed and provides boundary conditions for the viscous contribution to pressure in terms of vorticity circulation. The main insight is that the total creation of boundary vorticity can be computed exactly by a commutator between the Laplacian and Helmholtzprojection operators. Moreover, this term is dominated by the viscosity term, hence we can treat it explicitly to gain efficiency and stability. The major advance is that the method can be formulated in standard continuous finite element space, and there is no compatibility condition needed for the velocity and pressure approximation spaces. Hence the standard fast solver can be applied to Navier-Stokes equation directly.Accurate, efficient simulations of 3D flows in complicated domain are still major challeges for many scientific and engineering problems. The resolution of the flows near physical boundaries is essential to the accurate prediction of the body force such as the lift and drag, shedding of vortex and boundary layer separation. The success of this project will have an important impact on many branches of science and engineering. It will give more accurate predictions of body force which will result in better energy efficiency for transportation related applications. It will provide fast and reliable simulations for scientific research and engineering application.