Global linear stability of compressible,three-dimensional flow around a sphere is investigated with a quasi-linearization method on Navier-Stokes equations.An implicit restarted Arnoldi approach is adopted to solve the eigenvalue problem.The global stability problem of three-dimensional compressible basic flows around a sphere is investigated at subcritical parameters of Reynolds number Re=200,Mach number M=0.2,and supercritical parameters of Re=300,M=0.6.It shows that the increasing of Mach number(up to 0.6) has no qualitative influence on transition of flow patterns.

In a compressible fluid-mixture model based on volume fraction, a high-order piecewise parabolic method(PPM) is employed to solve multi-fluid flows characterized by the van der Waals equation of state.A double shock approximation is used in the Riemann solver.Simulation on a pure interface problem indicates that the pressure and velocity profiles do not show any spurious oscillation at the contact discontinuity and the numerical diffusion is limited within two or three cell grids.It shows that the method deals with such physical problems as contact discontinuity,shock, multi-dimensional slip line efficiently.

A high order spectral element method with a domain decomposition Stokes solver is presented for hydrodynamic stability analysis.A Jacobian-free Inexact-Newton Krylov algorithm with a Stokes time-stepping preconditioning technique is introduced to the calculation of steady state of incompressible flows.An Anoldi method is used to calculate the leading eigenvalues and corresponding eigenvectors,which are responsible for the hydrodynamic instability.The method deals with steady and unsteady simulations in a similar way without time-splitting divergence error and does not need Jacobian matrix.As a result,it reduces memory allocation and computation cost,and speeds up the convergence.(Numerical) result for Kovasznay flows with an analytic solution shows spectral accuracy with exponentially spacial convergence and superlinear convergence for inexact Newton method.An antisymmetric sinusoidal velocity driven cavity problem is considered at Re=800.The stable and unstable patterns are analyzed with leading eigenvalues of steady states.The symmetric-breaking Hopf bifurcations are considered in the wake of a circular cylinder limitlessly or between two parallel walls.The onset of instabilities agrees well with experimental and numerical results.