Based on the close relationship between Hamilton-Jacobi (H-J) equations and hyperbolic conservation laws,a high-order numerical method is developed to solve the H-J equations in the 3rd order and 5th order compact schemes.The upwind compact schemes are tested on a variety of one-dimensional and two-dimensional problems,including a problem related to the Richtmyer-Meshkov instability accelerated by planar shocks.Numerical results show that these schemes yield uniform high-order accuracy in smooth regions and satisfactorily resolve discontinuities in the derivatives.Moreover,since the present methods have less numerical dissipation than WENO scheme with the same order,they could be used to solve the H-J equations more accurately in the simulation of multi-scale complex flows.

Two-dimensional compressible flow field for the interaction of shocks with cylinder interface is directly simulated by using Ghost Fluid method (Ghost) and γ-model method, respectively, with the same discrete order in space and time. Numerical results are compared with the experimental results. They are almost alike in the beginning time and show the right position of interface, right strength and velocity of shocks. With the time developing, the effect of large numerical dissipation of γ-model method becomes greater and greater while the low numerical dissipation of Ghost method makes it efficient to simulate the moving interface well. Comparison with experiments proves that Ghost method is better than γ-model method in simulating the problem of interface instability.

Through examples, the mechanism of the generation of the aliasing error associated with upwind compact difference schemes and spectral method is studied and compared. The comparison shows the upwind compact difference schemes have some advantages in treating the aliasing error.

An approach combining the fifth Upwind Compact Difference Scheme and Group Velocity Control is applied to simulate directly the interaction of shocks and the density stratified interface of light-gas cylinder.Some research about the generation and development of vorticity and the distortion of edge of cylinder is done.The instability of cylinder density-interface is analyzed.Main character of the linear developing phase is that the gas of higher density goes into the one of lower denstiy at a constant velocity and the spike structure is formed and the character of the nonlinear developing phase is that a pair of vortex structures rotating adversely are formed on both sides of the top of the spike.The numerical solution accords with the experiment.

The temporal spatial discrete Fourier transformation is employed to analyze the dissipation and dispersion errors of the numerical algorithm used here. Based on directly solving the two dimensional time dependent compressible Navier Stokes equations, the spatial evolution of primary vortex generated in forced compressible plane free shear layers is simulated numerically by using a new fifth order upwind compact difference scheme. The phenomena, including the primary vortex saturating, the first pairing, and the second pairing, are shown with the passively conserved scalar method and so on. The initial value effect related to subharmonic disturbances in incoming flows is studied. The results show that the spatial evolution type of large scale compressible vortices is associated with the disturbance modes.

An algorithm is proposed to combine Kirchhoff integral and an unsteady flow computation. Axisymmetric jet flows are given by DNS using 5-order accurate upwind compact scheme and 3-order accurate Runge-Kutta scheme. Kirchhoff integral is used to predict the far-field sound. Numerical results show that far-field sound pressure has the directivity and reaches maximal value at 20° diverged from the flow direction. The directivity is attenuating while the propagation distance is increasing.

The linear instabilities in compressible shear layers are investigated by solving the two and three dimensional time dependent compressible Navier Stokes equations.A high accuracy upwind/symmetric compact difference hybrid algorithm is used.A pseudo random number generation technique is employed to simulate the white noise disturbance.Based on numerically solving the three-dimensional viscous disturbance equations by using a new method named as SCD-MNR,some results related to linear stability analysis are first given.Both two-and three-dimensional waves are considered.Compressibility effect can increase the most unstable wave angles besides that damps the growth rates of unstable waves.Numerical results confirm the prediction of linear theory that during the initial evolution of the compressible shear layer unstable waves are linear,and that in this course the wave number and angle of the most unstable disturbance are selective.The algorithm used here appears to be effective.It is a reasonable means for the study on compressible shear layers that linear theory combines with numerical simulation.

The flow fields of 2-D compressible axisymmetric jet are simulated by solving N-S equations. The discretization method has high order accuracy. The information and development of the large vortical structure and it's role in the jet development are studied. The Kelvin-Helmholtz wave appears firstly when jet losts it's stability, and then the single vortex rolls up. After that, with the development of jet, the vortex structures begin to pair and merge because of the increasement of non-linear effects. It is also shown that the pressure distribution is directly relative to the large vortical structures.

Near wake flow is simulated by using numerical method for solving the Navier-Stokes Equations.In order to get fine solutions near the corner coordinate transformation is made in both x and y directions.The differetial equations are approximared with factored implicit scheme developed in⁽¹⁾. The scheme is implicit, and large time step can be used. But the difference solutions can be expressed explicitly. In process of solution there are no numerical inversions of matrices and no matrix operations.The scheme has advantages of both implicit and explicit schemes.Flows with different Mach number M and Peynolds number Pe are computed.

A new implicit factored difference scheme for solving compressible Navier-Stokes equations on the base of the single step scheme developed before is described in this paper. For the inviscid portion of the Navier-stokes equations the flux vectors are split according to the sign of the eigenvalues of the Jacobian matrices using the property that the nonlinear flux vectors are homogeneous functions of degree one. For the viscous portion correction with operator the addition is used for improving convergence rate. Obtained difference scheme is used to solve Couette flow.