Fluid transport in human physiology, such as air exchange in the pulmonary system or blood flow through the heart's chambers, depends on the proper function of biological valves that provide passive flow control. During their operation, the valve leaflets may experience self-excited oscillations caused by fluid-induced instability, potentially resulting in impaired fluid transport \cite{johnson2020thinner}. Although extensive research into the realm of fluttering motion, the onset of a limit-cycle oscillation in a valve-like configuration remains not well understood. To the end, this study seeks to bridge this knowledge gap by correlating the onset of self-sustained oscillations with operational parameters in a generalized configuration using high-fidelity simulations. The investigation is carried out with an extensively validated software which employs a partitioned framework to handle Fluid-Structure Interaction (FSI) problems\cite{nitti2020immersed}. The fluid domain is resolved using a finite-difference fractional step scheme on a staggered grid, while the structural domain discretization employs an efficient and accurate NURBS-based isogeometric method suited for capturing large strain gradients with a minimal number of degrees of freedom. The coupling at the interface is managed by an immersed boundary method based on a moving-least-squares (MLS) approach \cite{detullio2016moving}. Our findings confirm the existence of critical reduced velocity that triggers the flapping motion of valve leaflets, influenced by geometric parameters, structural properties, and flow conditions. Indeed, the third-mode natural frequency predominantly governs the flapping dynamics of the valve structure for all the parameters examined. At high Reynolds numbers ($Re$), flow passing through the valve exhibits jet-like behavior, causing asymmetry and spanwise bending in the flapping structures.