A solution algorithm using Hamiltonian paths and strand grids is presented for turbulent flows and unsteady flow calculations around representative geometries that initially consists of a purely unstructured triangular surface mesh. Turbulence models are applied to the present method for both two and three dimensional flows, and the predicted results are validated by comparing against those obtained from established flow solvers. In addition, time accurate methods with dual-time stepping strategies are explored to solve for flows in canonical time-dependent problems. The mesh system is also extended to utilize an overset mesh technique. This overset technique allows for multiple mesh systems, which consists of a near-body Hamiltonian/strand grid and off-body Cartesian background meshes. Finally, the integration framework between the various components of the code suite is performed using Python to allow for ease of integration in the future to other codes. It is observed that up to fifth-order spatial reconstruction schemes are possible starting from an unstructured grid and good temporal accuracy is also observed when compared to traditional solvers.