Abstract: This work presents a new Cartesian-based ghost cell method for two-dimensional high Reynolds number compressible viscous flows. Based on the six fundamental assumptions used in the law of the wall, a wall function ghost cell method (WF-GCM) is developed to treat turbulent wall boundary conditions. Reference points are employed to compute primitive variables and turbulent properties at ghost cells. Meanwhile, the turbulent variables at the near wall cells and boundary cells are modified by using the wall function model. The turbulent boundary conditions are incorporated into a Reynolds average Navier-Stokes (RANS) finite volume solver that includes the SST k-ω turbulence model. Finally, the transonic flow past a RAE2822 airfoil and supersonic flow past a circle cylinder are simulated with adaptive Cartesian grid. Good agreement with the experimental datas shows the accuracy and efficiency of the presented WF-GCM.