Efficient Numerical Methods for Viscous Incompressible Flows

这一项目将致力于发展一类新的、非常有效的不可压缩流动数值方法，该方法基于已被证明是适定的N-S方程的重新表述，并提供了根据涡度环流对压力的粘性贡献的边界条件。主要的见解是，通过拉普拉斯和亥姆霍兹投影算子之间的换位器，可以精确地计算边界涡度的总的产生。此外，这一项是由粘性项主导的，因此我们可以显式处理它，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实现，并且不需要速度和压力近似空间的相容条件。因此，标准的快速求解器可以直接应用于Navier-Stokes方程。准确、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体力，如升力和阻力、涡的脱落和边界层分离是至关重要的。该项目的成功将对科学和工程的许多分支产生重要影响。它将给出更准确的体力预测，从而为与交通相关的应用带来更好的能源效率。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.