This work details the development and validation of an unsteady unstructured high order ( 3) h/p Discontinuous Galerkin (DG) - Fourier solver for the incompressible Navier-Stokes equations on static and rotating sliding meshes in 3D. This general purpose solver is used to provide insight into cross-flow (wind or tidal) turbine physical phenomena. To account for the relative mesh motion, the system of equations is written in arbitrary Lagrangian-Eulerian form and a non-conformal DG formulation is used for spatial discretisation. The DG method, together with a novel sliding mesh technique,...