Tangential introduction of liquid results in a swirling flow within a cylindro-conical hydrocyclone. Upon continuous feeding with water, the central axial region experiences local low pressure across the height yielding the formation of an air-core, which executes meandering motion similar to the oscillation of an elastic string. We investigated the vortical flow and the induced oscillating behavior of an air column submerged in a water flow field inside a hydrocyclone. Through a series of experiments in a transparent hydrocyclone and subsequent full scale multiphase flow simulations with the Reynolds stress model, we analyzed the morphological characteristics of the air-core (mean and fluctuating properties). Air-core oscillations are characterized in terms of spatial wavelength and frequency. We show that hydrodynamics driven oscillating behavior of the air-core shares an analogy with the vibration of an elastic beam. Following this analogy, we obtain a scaling relationship between the wavelength and air-core radius, which is in good agreement with our experimental data and numerical results.