A fourth order accurate Discontinuous Galerkin (DG) method coupled with cellwise relaxation implicit (CRI) scheme for solving the RANS equations is presented. Our efforts are focused on reducing computational time needed for setting Jacobian matrix. It is shown that the computational efficiency of the fourth order accurate DG-CRI scheme is substantially improved by employing several numerical techniques. Among the techniques, a use of the quadrature simplification by orthogonality is highly recommended in terms of the performance in reducing computational cost, numerical instability and required memory size. In the calculation of the turbulent boundary layer flow over a flat plate, it is shown that the quadrature simplification by orthogonality successfully reduces the computational time per step to 1/20. In the calculation of the vortical flowfield over a delta wing, the favorable spatial accuracy of the present fourth order accurate DG-CRI scheme is well demonstrated.