The discontinuous Galerkin (DG) method has attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. In this work, we present a novel DG discretization for solving the two-dimensional INS equations in which the inviscid term of the INS equations is split into two parts, the Stokes operator and the nonlinear convective term, and treated separately. The Stokes operator is discretized using the artificial compressibility flux which is provided by the (exact) solution of a Riemann problem associated with a local artificial compressibility perturbation of the Stokes system, while the nonlinear term is discretized in divergency form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Unlike the existing artificial compressibility flux for the DG discretization of the INS equations which needs to solve a Riemann problem for a nonlinear system by numerical iteration, the separate treatment of the nonlinear term from the Stokes operator makes the Riemann problem become linear and can be solved explicitly and straightforwardly, therefore, no iterative procedure is further required. A number of test cases with a wide range of Reynolds number are presented to assess the performance of the proposed method, which demonstrates its potential to be an alternative approach for high order numerical simulations of incompressible flows.