Behaviour of an immersed boundary method in unsteady flows over sharp-edged bodies

The behaviour of the immersed boundary method proposed by Goldstein et al. [Goldstein D, Handler R, Sirovich L. Modelling a no-slip boundary condition with an external force field. J Comput Phys 1993;105:354-66] as a second-order damped control system is investigated. The natural frequency and the damping coefficient are introduced as driving parameters of the method. The comparison between the velocity response at forced points in the startup flow over a square cylinder with the theoretical response of a second-order damped oscillator is performed. The role of each parameter appears clearly. At the beginning of the startup flow, the response time depends directly on the natural frequency, and this parameter determines the level of residual velocities achieved in an unsteady flow. The damping coefficient drives the oscillation of the velocity response at the beginning of the startup flow, but has negligible influence during the establishment and in the unsteady flow. At forced points facing no unsteady perturbation from the flow, the zero-velocity set point is reached asymptotically, as usual in second-order damped-systems. Through the simulation of the flow over a blunt flat plat at Re=1000, it is observed that the initial thickness of the mixing layer due to the separation at the edge may vary during the simulation because the sharpness of the edge increases as the residual velocities decrease. This insight gained on the behaviour of the response allows a time-step optimisation, which, completed with comparisons to reference literature results, confirms the feedback forcing method a competitive tool for accessing near-wall unsteady flow over sharp-edged bodies.