Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows

This paper presents a new gas-kinetic scheme for simulation of compressible inviscid flows. It starts to simplify the integral domain of Maxwellian distribution function over the phase velocity ξ and phase energy ζ to the integral domain of modified Maxwellian function over the phase velocity ξ only. The influence of integral over phase energy ζ is embodied as the particle internal energy ep. The modified Maxwellian function is further simplified to a circular function with the assumption that all the particles are concentrated on a circle. Then two circular function-based gas-kinetic schemes are presented for simulation of compressible inviscid flows. In the new schemes, no error and exponential functions, which are often appeared in the Maxwellian function-based gas-kinetic schemes, are involved. As a result, the new schemes can be implemented in a more efficient way. To validate the proposed new gas-kinetic schemes, test examples in the transonic flow, supersonic flow and hypersonic flow regimes are solved. Numerical results showed that the solution accuracy of the circular function-based gas-kinetic schemes is comparable to that of corresponding Maxwellian function-based gas-kinetic schemes. However, the circular function-based gas-kinetic schemes need less computational effort.