A solution algorithm using Hamiltonian paths and strand grids is presented for turbulent flows and unsteady flow calculations around representative geometries initially consisting of a mixed-element unstructured surface mesh. Line-based reconstruction schemes and approximate factorization techniques are robustly implemented on unstructured grids. The Baldwin-Lomax and Spalart-Allmaras turbulence models are integrated to the present method for both two- A nd three-dimensional flows, and the predicted aerodynamic flows are validated by comparing against those obtained from established solvers and/or experiments. In addition, time-accurate methods with dualtime-stepping strategies are explored to predict flows over canonical time-dependent problems. It is observed that a various high-order-type reconstruction scheme can be used on "lines" constructed from purely unstructured grids in the current framework, along with good spatial and temporal accuracy. © 2017 by the American Institute of Aeronautics and Astronautics, Inc.