To improve finite difference scheme, Fourier analysis is used to optimize dispersion and dissipation of WENO scheme. And optimal linear weights are given. A class of hybrid schemes is designed by combing optimized WENO schemes with monotonicity-preserving scheme. A weighted hybrid WENO scheme(H-WENO) is obtained. The scheme is tested with one-dimensional shock tube problem, Shu-Osher problem, two-dimensional Mach reflection problem and R-T instability problem. It shows that the scheme has strong ability to capture shock wave and high resolution for small scale wave structure, which is improved obviously compared with original WENO scheme.

Numerical simulation on superhydrophobic surface of turbulent flow is shown. Velocity or shear stress of turbulent flow are provided and basic law between microscopic structure and drag reduction rate is analyzed. In simulation, we use an unsteady Reynolds averaged simulation model and VOF model for gas-liquid two phase flow. It shows that slip flow and cyclical shear stress exist on superhydrophobic surface. It can reduce resistance and increased resistance as well. Microstructure size has significant effect on drag reduction rate. In order to increase drag reduction rate, the aspect ratio of rectangular mierostructure should be greater than 3:2 and the groove spacing should be small, while the groove width should be less than 200 micron.

With WTC algorithm developed by Weiss, Tabor and Carnevale, we concluded that a nonlinear vibrating string equation has Painlevé property. Bäcklund transformation is obtained. Furthermore, soliton fusion is analyzed by means of Hirota's direct method and Bäcklund transformation.

A stochastic model of heavy particle dispersion is proposed.It is based on the coupling of the stochastic equation of turbulent fluid motion with the heavy particle crossing trajectory equation in which Saffman force,particle-wall collision and particle-particle collision are included.The model is also tested in a turbulent boundary layer with heavy particles.The numerical results are in good agreement with previous experimental data.The effect of Saffman force and particle collision on the heavy particle dispersion is investigated numerically as well.