The Navier-Stokes equations are solved with the second-order LU-SGS-τTS implicit time-stepping method. The sixth-order symmetric TVD scheme with fifth-order WENO interpolation based on Roe is applied to the convection terms. A boundary layer flow around a flat plate, a low speed flow around the Aerospatial A-profile at high incidence and a transonic flow with shock/boundary interaction around RAE2822 airfoil are computed by Baldwin-Lomax and Spalart-Allmaras models with the transition criterion, respectively. The results show that the approaches capture the transition automatically.

A newly developed gas-kinetic BGK scheme is further simplified. Numerical tests show that the new simplification not only keeps the advantages of the old scheme, such as its robustness and automatic satisfaction of the entropy condition but also makes the scheme simpler and more efficient.