Finite difference method for the acoustic radiation of an elastic plate excited by a turbulent boundary layer: A spectral domain solution

A finite difference method is developed to study, on a two-dimensional mode l, the acoustic pressure radiated when a thin elastic plate, clamped at its boundaries, is excited by a turbulent boundary layer. Consider a homogeneous thin elastic plate clamped at its boundaries and ext ended to infinity by a plane, perfectly rigid, baffle. This plate closes a rectangular cavity. Both the cavity and the outside domain contain a perfec t fluid. The fluid in the cavity is at rest. The fluid in the outside domai n moves in the direction parallel to the system plate/baffle with a constan t speed. A turbulent boundary layer develops at the interface baffle/plate. The wall pressure fluctuations in this boundary layer generates a vibratio n of the plate and an acoustic radiation in the two fluid domains. Modeling the wall pressure fluctuations spectrum in a turbulent boundary la yer developed over a vibrating surface is a very complex and unresolved tas k. Ducan and Sirkis [1] proposed a model for the two-way interactions betwe en a membrane and a turbulent flow of fluid. The excitation of the membrane is modeled by a potential flow randomly perturbed. This potential flow is modified by the displacement of the membrane. Howe [2] proposed a model for the turbulent wall pressure fluctuations power spectrum over an elastomeri c material. The model presented in this article is based on a hypothesis of one-way interaction between the flow and the structure: the flow generates wall pressure fluctuations which are at the origin of the vibration of the plate, but the vibration of the plate does not modify the characteristics of the flow. A finite difference scheme that incorporates the vibration of the plate and the acoustic pressure inside the fluid cavity has been developed and coupl ed with a boundary element method that ensures the outside domain coupling. In this paper, we focus on the resolution of the coupled vibration/interio r acoustic problem. We compare the results obtained with three numerical me thods: (a) a finite difference representation for both the plate displaceme nt and the acoustic pressure inside the cavity; (b) a coupled method involv ing a finite difference representation for the displacement of the plate an d a boundary element method for the interior acoustic pressure; (c) a bound ary element method for both the vibration of the plate and the interior aco ustic pressure. A comparison of the numerical results obtained with two models of turbulent wall pressure fluctuations spectrums - the Corcos model [3] and the Chase model [4] - is proposed. A difference of 20 dB is found in the vibro-acoust ic response of the structure. In [3], this difference is explained by calcu lating a wavenumber transfer function of the plate. In [6], coupled beam-ca vity modes for similar geometry are calculated by the finite difference met hod.