Modeling of wall pressure fluctuations for finite element structural analysis

Abstract

This paper investigates a modeling technique of wall pressure fluctuations (WPF) due to turbulent boundary layer flows on a surface for finite element structural analysis. This study is motivated by critical issues of structural vibrations due to turbulent WPF over the surface of a body. The WPFs are characterized with random behavior in time and space. The temporal and spatial random behavior of the WPFs is mathematically expressed in the form of the auto- and cross-power spectral density functions (PSDF) in the frequency domain (e.g., Corcos Model). For finite element modeling of the random distributed fluctuations, the cross-PSDF is properly modeled over the finite element structural mesh. The quality of modeling of the cross-PSDF is directly affected by the structural mesh size. We first examine the maximum mesh size required for reliable finite element analysis. The reliability of the FEA results is confirmed by the theoretical results. It is found that the maximum mesh size should be determined under consideration of the spatial distribution of the cross-PSDF in addition to the representation of dynamic behavior of the structure in the frequency range of interest. It is also recognized that the requirement of the maximum mesh size is unrealistic in many practical cases. We then investigate practical modeling schemes under a realistic mesh size condition. We found that the WPF can be modeled without the exact consideration of cross-correlation if the power due to the wall pressure fluctuation can be properly compensated. This is owing to the feature of decreasing the cross-correlation of WPFs at high frequencies and the fact that the WPF does weakly couple into the structural modes at high wavenumbers such that $2\pi f/U_c<k_{\max }$. The wall pressure fluctuations can therefore be modeled as uncorrelated loadings with power compensations.

Introduction

Due to the viscosity, fluid flow over a surface forms a boundary layer [1]. The movement of fluid particles then becomes complicated, and this complicated flow produces wall pressure fluctuations (WPF). The wall pressure fluctuations under a fully developed turbulent boundary layer (TBL) flow were first measured on a flat plate with no pressure gradient [2]. Since then, wall pressure fluctuations have been thoroughly researched theoretically [3] and experimentally [4], [5]. Since the 1960s, the characteristics of WPF under turbulent boundary layer have also been mathematically identified and modeled. Mathematical models of WPF under TBL on an infinite plane surface having no pressure gradient are presented [6], [7], [8]. Wall pressure fluctuations act as excitation forces on flexible structures that lead to structural vibrations.

The flow-induced vibration due to the wall pressure fluctuations is sometimes a critical issue. For aircrafts and space vehicles, for instance, the fuselage suffers from strong vibrations due to wall pressure fluctuations at high speeds. The strong vibrations increases the noise level in the cabin and this increases passengers’ discomfort. The strong vibration at high speed flight is due to a physical phenomenon called wavenumber-matching when the wavenumber of the wall pressure fluctuations coincides with that of the structural response [4], [5]. For underwater vehicles, the WPF on the wet surface can vibrate the ship's hull, and then generate radiated noise. For the SONARs of underwater vehicles including ships, submarines and torpedoes, the WPF on the acoustic window of the array of sensors generates pseudo-sound and increases the self-noise of the SONAR. This self-noise directly affects the SONAR detection performance.

For these reasons, the structural response due to the WPF should be considered in controlling noise and vibrations. In order to reduce the unwanted noise and vibration due to WPF, turbulent boundary layer flow and turbulent intensity should be alleviated by optimizing the lines of a hull. However, noise control by flow optimization is limited due to other requirements. Hence, typical noise and vibration control techniques can be applied to structures; these techniques include the use of damping materials, absorbing materials, decouplers, etc. Prior to applying these noise control techniques to structures the structural dynamics of the vehicles should be characterized.

Structural vibration and radiated noise due to WPF under turbulent boundary layer flows have been investigated experimentally and theoretically. The response of infinite plates to WPF was studied [9], [10]. Furthermore, the response of finite plates to WPF was investigated [11], [12], [13], [14], [15]. The response of water-loaded simply supported plates to WPF for homogeneous boundary layer flow was also investigated [16]. Theoretical solutions, for either infinite plates or simply supported plates, are obtained by the time–space Fourier transform of impulse responses and wall pressure fluctuations. Since the response and the WPF over the surface can be transformed into wavenumber-frequency functions, the response to WPF can be obtained theoretically. For complicated structures that are no longer assumed to be plate structures, however, the theoretical approach becomes no longer applicable and so, a numerical approach, e.g., finite element method (FEM), is employed.

In FEM, the structures are discretized with meshes composed of nodes and elements. The size of the mesh is determined by the two requirements that are used to properly represent the spatial distribution of the input and the response within the frequency range of interest. Firstly, for the representation of the response, the size of the mesh to model the structure should be small enough to depict the spatial behavior of structural response within the frequencies of interest. This mesh size is determined iteratively by modal analysis with reducing mesh size until the highest order natural mode to be superposed is well represented. The corresponding highest natural frequency is greater than double the maximum frequency of interest. Secondly, for spatially distributed input forces, the size of the mesh should be also small enough to represent the spatial variation of the forces. Structural excitation forces due to WPF are spatially distributed over the surface of the structure. The excitation is characterized as random in time as well as in space. The feature of this excitation can be modeled by the cross-correlation functions. The wavenumber spectrum of the cross-correlation of WPF has the highest power at the convective wavenumber. The cross-correlation functions, at least, at the double of the convective wavenumber should be well represented over the discretized mesh.

The mesh required for the spatial variation of WPF is too small in most cases to practically model the structure for finite element analysis. This paper focuses on the modeling schemes for WPF over a discretized surface for finite element analysis. In Section 2, for a simple beam structure, the mathematical formula of the exact solution and a finite element analysis are presented. Finite element modeling of the wall pressure fluctuations due to turbulent boundary layer flows is demonstrated for three different methods in Section 3. Conclusions from this study are summarized in Section 4.