  We propose a new formula for the nonlinear viscous numerical flux and extend
the direct discontinuous Galerkin method with interface correction (DDGIC) of
Liu and Yan (H. Liu, J. Yan, The direct discontinuous Galerkin (DDG) method for
diffusion with interface corrections, Communications in Computational Physics 8
(3) (2010) 541) to compressible Navier-Stokes equations. The new DDGIC
framework is based on the observation that the nonlinear diffusion can be
represented as a sum of multiple individual diffusion processes corresponding
to each conserved variable. A set of direction vectors corresponding to each
individual diffusion process is defined and approximated by the average value
of the numerical solution at the cell interfaces. The new framework only
requires the computation of conserved variables' gradient, which is linear and
approximated by the original direct DG numerical flux formula. The proposed
method greatly simplifies the implementation, and thus, can be easily extended
to general equations and turbulence models. Numerical experiments with $P_1$,
$P_2$, $P_3$ and $P_4$ polynomial approximations are performed to verify the
optimal $(k+1)^{th}$ high-order accuracy of the method. The new DDGIC method is
shown to be able to accurately calculate physical quantities such as lift,
drag, and friction coefficients as well as separation angle and Strouhal
number.
