对不可压边界层内流体流动的非定常不稳定现象及其转捩过程的研究一直是流体力学领域一个非常重要的研究课题，有很多问题亟待解决，特别是边界层转捩的机理至今仍然并不十分清楚。数值模拟是解决这一类问题的重要手段和有效途径，因此发展精确稳定高效的数值模拟算法具有重要的理论意义和实际应用价值。本申请项目将基于非定常不可压Navier-Sokes方程组的涡量-速度法，充分利用指数型紧致差分格式的高精度和高分辨率的特点、边界值方法无条件稳定的特性以及多重网格方法的加速收敛作用，建立和发展不可压边界层流动问题的新型模拟算法，并对几种典型的不可压边界层流动问题的非定常不稳定现象和平板边界层的转捩过程开展直接数值模拟研究，探讨和分析不可压边界层流动的不稳定性及转捩的机理。通过本项目的研究，可以丰富和完善目前对不可压边界层问题模拟的研究结果，并且所拟建立的新算法可望被推广到流体力学其他更复杂的边界层问题的求解中去。

The research on the unsteady and unstable fluid flow phenomena and laminar-to-turbulence transition of an incompressible boundary layer is a very important subject in the fluid dynamics field.Although much significant progress has been made in the past 100 years,many problems remain unresolved. Especially the mechanism of the transition is still not well understood.One approach to this problem is to use numerical simulation,which not only helps us understand the mechanism of the flow transition,but also helps us carry out engineering applications by boundary layers' controlling. So, it has great theoretical value and practical significance in the research on accurate,stable and efficient simulation algorithms. This project is aiming at proposing and developing accurate and efficient numerical simulation algorithms for solving the incompressible flow problems with boundary layers based on the vorticity-velocity method of the unsteady incompressible Navier-Stokes equations, the exponential high order compact difference method on nonuniform grids,the boundary value method,and the multigrid method. Then these new methods will be employed to conduct direct numerical simulation of the unsteady and unstable dynamics behaviours of the several classical incompressible flow problems with boundary layers and even transition of the flate plate boundary layers and to explore and analyse the mechanism of the flow transition.The main feature of this research is to fully exploit the high order accuracy and high resolution characters of the exponential compact difference schemes and the boundary value method,and the accelerating function of the multigrid method.The research can enrich and replenish the existing study on the incompressible boundary layer problems and the simulation algorithms developed in this application are expected to be applied into numerical simulation of more complicated incompressible flow problems with boundary layers.