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This paper presents a high-order discontinuous Galerkin (DG) method for three-dimensional turbulent flows. As an extension of our previous work, the paper further investigates the incorporation of a modified Spalart-and-Allmaras (SA) turbulence model with the Reynolds Averaged Navier-Stokes (RANS) equations that are both discretized using a modal discontinuous Galerkin approach. The resulting system of equations, describing the conservative flow fields as well as the turbulence variable, is solved implicitly by an approximate Newton approach with a local time-stepping method to alleviate the initial transient effects. In the context of high-order methods, curved surface mesh is generated through the use of a CAPRI mesh parameterization tool, followed by a linear elasticity solver to determine the interior mesh deformations. The requirements for the wall coordinate and viscous stretching factor used for viscous mesh generation are studied on a twodimensional turbulent flow case. It has been concluded that, for attached turbulent flows, the conventional parameters often used in low-order methods can be somewhat less stringent when a higher-order method is considered. Several other numerical examples including a direct numerical simulation of the Taylor-Green vortex and turbulent flow over an ONERA M6 wing are considered to assess the solution accuracy and to show the performance of high-order DG methods in capturing transitional and turbulent flow phenomena.