Sound transmission through double cylindrical shells lined with porous material under turbulent boundary layer excitation
Introduction
Double-walled constructions are widely used in transportation vehicles, e.g. trains, ships, aircraft and space shuttles, due to their superior mechanical, acoustical and thermal properties. In acoustics, it is of great current interest to study sound propagation through such constructions [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In the context of vehicles moving in air, the flow in one side of the double-walled system can affect the acoustics of transmission. Therefore, some studies investigated the effect of mean flow on sound transmission through double-wall system [9], [10], [11], [12], [13], [14], [15]. However, mean flow is accounting for just the convective effect due to flow and ignores the high velocity gradients which present within a boundary layer close to the surface of the structure over which flow takes place. In high speed vehicle, e.g. aircraft, the turbulent boundary layer (TBL) is a major contributor to the interior noise [16]. A reliable prediction of the interior noise generated by the external TBL excitation is important at the design stage of aircraft which requires taking TBL effects into account. The present work is motivated by these considerations.
Many researchers in the past have paid attention to the noise radiated by a single plate [17], [18], [19], [20], [21], [22], [23], [24], [25]. However, these studies are still quite far away from real industrial problems. A study based on the plate model neglects the curvature of the fuselage or the effect of the neighbouring panel and even both. The plate model is only suitable for describing the subsystem instead of the whole fuselage. For more accurate prediction, these effects need to be included. A structure of cylindrical geometry is a good generic benchmark case to study these effects. Only few studies consider such geometry of the structure. Tang et al. [26] used the modal expansion method and the Galerkin approach to develop an analytical model of sound transmission through cylindrical shell structures excited by a TBL. Rocha et al. [27] developed an analytical model with closed-form expressions for the prediction of flow-induced noise in aircraft cylindrical cabins. Two cases, a whole circular cylindrical shell and a set of individual open circular cylindrical shells, were considered in this model. Gardonio [28] gave an introduction of the general procedure using Green׳s function approach for analysing the interior noise, in a cylindrical enclosure with a flexible thin wall, caused by a TBL.
However, in practice, the aircraft fuselage is not a single wall structure. It is a double wall system which consists of a skin panel and a trim panel with an annular space between them. Therefore, Tang et al. [29] studied the sound transmission into two concentric cylindrical sandwich shells subject to turbulent flow on the exterior surface of the outer shell. However, Tang et al. [29] did not consider the effect of the porous lining in their study, which is widely used to control interior noise. Maury et al. [30] studied the case of sound transmission through a flat double-panel, representative of an aircraft sidewall excited by a TBL. The effect of porous material filled in the air gap between the skin and trim panels was included in the study. An empirical model [31] is used by Maury et al. [30] to model the porous material as an equivalent fluid. Several active control approaches for reducing the interior noise were also developed and compared. Although the effect of the porous lining was considered, the plate model was used by Maury et al. [30]. As the frame waves in the fiberglass material are neglected, Maury et al.׳s result was not reasonable for the case when the fiberglass is directly bonded to both the two face plates. Allard and Atalla [32] shown that, when the material is bonded onto a vibrating structure, the frame wave should be considered.
The literature review shows that only one study [30] considered the effect of porous material and TBL, and these are for flat plate geometry. Unfortunately, a flat plate model is not accurate in some cases due to the reasons mentioned above. Since the TBL is a random excitation, the space correlation functions need to be used to describe the pressure fluctuation caused by a TBL. Therefore, the length of the structure should be considered as finite. In the present study, sound transmission through finite double cylindrical shells lined with porous material under external TBL excitation is considered.
This paper is organised as follows. In Section 2, the construction of the system is described first. Three types of configurations are considered and the relevant boundary conditions are presented. The models for the turbulent boundary layer are also presented in Section 2. In Section 3, results and discussions for kinetic energy of inner shell are presented and compared. Finally, concluding remarks are made in Section 4.
Section snippets
Description of model system
A schematic diagram of the configuration of the system under study is shown in Fig. 1. Consider two concentric finite cylindrical shells with hard end caps of length L and radii R1 and R2 for the outer and the inner shell, respectively. The outer shell is assumed to be excited by a fully developed turbulent boundary layer. The fluid medium in the exterior and the interior cavity has the density and the speed of sound as (ρe, ce) and (ρi, ci), respectively. The porous material is lined within
Results and discussions
The two elastic shells are made of aluminium. All the geometry and material properties used in our calculation are listed in Table 1. The dimensions of shell chosen here are the same as those used by Tang et al. [29]. The properties of the polyimide foams are obtained from the work of Silcox et al. [49], which are listed in Table 2. The parameters of the structure and the porous material are chosen from Table 1 and the column of 9.6 kg/m3 in Table 2 unless stated otherwise, respectively. The
Conclusions
In this study, an analytical model is developed to calculate the sound transmission through a double shell lined with poroelastic material under the exterior turbulent boundary layer excitation. The axial length of the shell is kept finite. The equivalent fluid method based on Biot׳s model is used to describe the porous material. The power spectrum of the inner shell kinetic energy, an indicator of the spatially averaged vibration and also of the near field sound radiation, is calculated for
Acknowledgements
The first author should thank for the help from the Prof. Stephen Elliott of ISVR in University of Southampton and Prof. Paolo Gardonio in Università degli Studi di Udine. The authors also thank the anonymous reviewers for their valuable comments to this paper.
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