In this work the immersed boundary method is applied to simulate incompressible turbulent flows around stationary and moving objects. The goal is to demonstrate that the immersed boundary technique along with a large eddy simulation approach is capable of simulating the effect of the so-called leading edge vortex (LEV), which can be found in flapping wing aerodynamics. A Lagrangian method is used to approximate the solutions in the freshly cleared cells that lay within solid objects at one time step and emerge into fluid domain at the next time step. Flow around a stationary cylinder at ReD = 20, 40, and 3900 (based on cylinder diameter D) is first studied to validate the immersed boundary solver based on the finite volume scheme using a staggered grid. Then, a harmonically oscillating cylinder at ReD = 10 000 is considered to test the solver after the Lagrangian method is implemented to interpolate the solution in the freshly cleared cells. Finally, this approach is used to study flows around a stationary flat-plate at several angles of attack and fast pitching flat-plate. The rapidly pitching plate creates a dynamic LEV that can be used to improve the efficiency of flapping wings of micro air vehicle and to determine the optimum flapping frequency.