Three-dimensional numerical simulations of four rows elliptical finned-tube heat exchanger are performed within a Reynolds numbers range usually used in many applications including the laminar-turbulence transition. A highly-resolved Large Eddy Simulation (LES) approach is chosen in order to get reliable transition-to-turbulence data. This paper presents an analysis of the row-by-row evolution of flow topology and unsteadiness of thecomplex vortical structure throughout the heat exchanger for various Reynolds numbers. The simulations show several important unsteady and turbulence transition fluid flow features that impact heat transfer. It is found that the topology of the horseshoe vortex system is highly dependent on the row considered. Moreover the unsteadiness of the flow occurs sooner upstream from the trailing edge of the heat exchanger when the Reynoldsnumber increases. At last an important increase of global Nusselt number value is correlated with the vortical turbulent structures and unsteadiness.