A general stability condition for vortices in a two-dimensional incompressible inviscid flow field is presented. This condition is first applied to analyse the stability of symmetric vortices behind elliptic cylinders and circular cylinders with a splitter plate at the rear stagnation point. The effect of the size of the splitter plate on the stability of the vortices is studied. It is also shown that no stable symmetric vortices exist behind two-dimensional bodies based on the stability condition. The two-dimensional stability condition is then extended to analyse the absolute (temporal) stability of a symmetric vortex pair over three-dimensional slender conical bodies. The threedimensional problem is reduced to a vortex stability problem for a pair of vortices in two dimensions by using the conical flow assumption, classical slender-body theory, and postulated separation positions. The bodies considered include circular cones and highly swept flat-plate wings with and without vertical fins, and elliptic cones of various eccentricities. There exists an intermediate cone with a finite thickness ratio between the circular cone and the flat-plate delta wing for which the symmetric vortices change from being unstable to being stable at a given angle of attack. The effects of the fin height and the separation position on the stability of the vortices are studied. Results agree well with known experimental observations.

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The higher-order gas-kinetic scheme for solving the Navier-Stokes equations has been studied in recent years. In addition to the use of higher-order reconstruction techniques, many terms are used in the Taylor expansion of the equilibrium and non-equilibrium gas distribution functions in the higher-order gas kinetic flux function. Therefore, a large number of coefficients need to be determined in the calculation of the time evolution of the gas distribution function at cell interfaces. As a consequence, the higher-order flux function takes much more computational time than that of a second-order gas-kinetic scheme. This paper aims to simplify the evolution model by two steps. Firstly, the coefficients related to the higher-order spatial and temporal derivatives of a distribution function are redefined to reduce the computational cost. Secondly, based on the physical analysis, some terms can be removed without loss of accuracy. As a result, through the simplifications, the computational efficiency of the higher-order scheme is increased significantly. In addition, a self-adaptive numerical viscosity is designed to minimize the necessary numerical dissipation. Several numerical examples are tested to demonstrate the accuracy and robustness of the current scheme.

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