I want to use explicit multistage scheme with upwind sheme(Roe+MUSCL) to solve the NS equation with turbulent model. But I found the implicit with upwind scheme is usually adopted in the literature. I want to know whether my approach is compatible. Could any one recommend the work in this field?

Both implicit and explicit can work theoreticaly. Since turbulence model is present, Linearization of flux Jacobi in implicit scheme presents some cumbersome derivation. Also, the inexact linerization for upwind scheme will slow down the convergence. I will prefer to use multistage RK method, but implement implicit treatment of pure diffusive terms. This enables code robust even at high Re.