Analytical study of the water surface fluctuations induced by grid-stirred turbulence

The free surface dynamics excited by grid stirred turbulence is analysed with a Lagrangian approach, with the analytical model calibrated by using experimental results. The Lagrangian model for free surface oscillations has been developed by assuming that coherent structures in the water layer close to the free interface have the shape of blobs and are mobilized by a periodic force. The blobs are assumed to behave like nonlinear oscillators with different spring stiffness in upwelling and downwelling and neglecting viscosity as first approximation. In order to calibrate the model, a set of experiments in a grid stirred tank was used where free surface elevation time series were measured and all the necessary variables were computed. Through calibration and with some working assumptions, the radius of the blobs is shown to be of the order of the integral vertical length scale of turbulence. The forced oscillations are analysed by the method of Rauscher (1938) to predict the amplitude of the oscillations, possible resonances and discontinuities in the amplitude-frequency space. If the forcing term has multiple harmonics, in some cases a blow up of the oscillator is forecast. Some tests including a small damping, in order to evaluate the effects of viscosity, show that a jump is expected in the amplitude-frequency diagram and a range of unstable frequency. The presence of a small damping should also prevent the blow up for multiple harmonics in the forcing term.