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2021 | OriginalPaper | Chapter

Validation of a Simple Empirical Model for Calculating the Vibration of Flat Plates Excited by Incompressible Homogeneous Turbulent Boundary Layer Flow

The vibration responses of three flat rectangular plates excited by turbulent boundary layer flow are calculated and compared to measured data. The measurements were made in three different facilities by Wilby at ISVR [1] (high speeds typical of aircraft), Han at Purdue University [2] (moderate speeds typical of automobiles), and Robin at University of Sherbrooke [3] (lowest speeds), spanning 50 years of time. The plates are different sizes, made from different materials, and have different boundary conditions. The boundary layers have different heights and flow speeds. The ratios of plate flexural and convective wavenumbers k_{b}/k_{c} over the three cases range from about 0.1 to 2. Plate vibrations are normalized by wall pressure fluctuation autospectra measured by the previous investigators. This wide range of structural and flow conditions and the use of plate vibration spectra normalized by wall pressure autospectra allows for an objective assessment of various TBL wall pressure fluctuation cross-spectral empirical models. Two cross-spectral models are considered: the widely used Corcos model [4] and the less well-known elliptical extension by Mellen [5]. Smolyakov’s empirical models for convection velocity and streamwise and spanwise surface pressure length scales [6] supplement the Corcos and Mellen models. Calculations using the Corcos cross-spectral model overestimate the vibrations by about an order of magnitude at lower speed (and lower k_{b}/k_{c}) conditions. Including Smolyakov’s convection velocity and length scale formulations improves accuracy at low frequencies. The Mellen cross-spectral pressure model, supplemented with Smolyakov’s empirical models for convective wave speed and streamwise and spanwise surface pressure length scales, is therefore well suited for calculating plate vibrations due to TBL flows with flow speed/flexural wave speed ratios ranging from 0.1 to 2.

Appendix 1: Autospectra Comparisons
Appendix 2: Diffuse Acoustic Field Effects

Although this study focuses on the cross-spectral behavior of TBL wall pressure fluctuations, autospectra were measured in the three facilities also. The ISVR pressure spectra are not available in digital form, but the empirical model of Goody [
24] was derived in part on these data. The wall pressure spectra at multiple speeds in the Purdue and Sherbrooke facilities scale well with each other, and single speed data from each are compared here. There are many empirical wall pressure spectrum models [
8], but we compare the measurements to only two: the model of Goody, and a modified form of a model suggested by Howe [
25].
The single-sided cyclic frequency Goody model is:

where
\(\rho\) is the fluid density,
\(U_{\tau }\) is the friction velocity,
\(\delta\) is the boundary layer thickness,
\(U_{e}\) is the boundary layer edge velocity (free-stream velocity for a flat plate),
\(\nu\) is the kinematic viscosity, and the empirical constants are
\(a = 3.0, b = 2, c = 0.75, d = 0.5, e = 3.7, f = 1.1, g = - 0.57, h = 7\). Goody performed a thorough analysis of a large number of datasets to arrive at the empirical constants, but it should be noted that there is considerable spread in the reported data, which limits the precision of any empirical model.
The Howe model [
25] is based on a simplification of the autospectrum developed by Chase [
12], and several authors (following the lead of Goody) have referred to it as the Chase-Howe model. The single sided cyclic frequency Chase-Howe model can be written in the form

where
\(f_{\delta } = 0.12U_{e} /\left( {2\pi \delta^{*} } \right)\) and
\(\delta^{*}\) is the displacement thickness. As Goody and others have noted, the model does not include the steep roll-off at high frequencies associated with viscous dissipation, and therefore cannot be integrated to get the mean-square pressure.
Lysak [
26] developed a model in terms of integration of the turbulence—mean shear interaction sources in the boundary layer, based on the solution of the pressure Poisson equation. This model can be evaluated numerically for a given boundary layer mean velocity profile and assumed turbulence characteristics. The mid-to-high frequency range of the solution depends entirely on the log and viscous regions of the boundary layer, so it can be normalized using inner variable scaling (
\(\rho\),
\(U_{\tau }\),
\(\nu\)). Due to the characteristics of the log region, the mid-frequency range asymptotically approaches an
\(f^{ - 1}\) slope as the Reynolds number tends to infinity, so it can be matched to the Chase-Howe model. At high frequencies, an exponential decay factor was found to closely match the computed viscous region. This leads to following modification to the Chase-Howe model:

where the constant has been changed from 2 to 3 to match the computed
\(f^{ - 1}\) region, and the exponential decay factor has been included to model the viscous region. The form of the low frequency peak has not been changed from the Chase-Howe model. We refer to Eq.
12 as the ‘Chase–Howe–Lysak’ TBL autospectrum model.
The Purdue and Sherbrooke wall pressure measurements are corrected for sensor spatial resolution bias errors, scaled on outer variables and compared to the Goody and Chase–Howe–Lysak empirical models in Fig.
15. The agreement is mixed, with both empirical models matching well over different frequency ranges and datasets. The Sherbrooke low frequency pressures are known to be affected by the non-TBL flow-induced noise source in their facility. This comparison is typical of the results found when comparing pressure spectra reported in the literature, and illustrates the difficulty in choosing appropriate wall pressure autospectra models for general use. The low frequency corruption shown here is, unfortunately, quite typical of most wind and water tunnel facilities. These low frequency sources will induce structural vibration of unknown strength (since their spatial correlations are not readily known).

×

All test facilities have background vibration and sound levels caused by various sources. Acoustic background noise in wind and water tunnels can be caused by the turbulent flow itself (usually quite low at small Mach Numbers), and more importantly from sound generated by the turbulence convecting over surfaces, exiting ducts, and impinging on bodies in the flow path. Sound within tunnels can be particularly strong at low frequencies, where ‘humps’ in the spectra caused by turbulent flow are often centered. Also, acoustic waves can be one dimensional at very low frequencies in tunnels, and therefore of comparable amplitude throughout the facility. These effects are visible in all three facilities considered in this study.
However, at higher frequencies the modal density in a tunnel acoustic volume increases significantly such that the acoustic sound field becomes statistically diffuse. The effects of Diffuse Acoustic Fields (DAF) on panel vibration can be significant, particularly for stiff panels where flexural and acoustic wavenumbers are comparable. The DAF acoustic loading must therefore always be considered when measuring TBL-excited structural response. The relative response of the Sherbrooke panel to slow TBL flow and DAF loading is shown in Fig.
16. Once again, we compare the normalized G
_{aa}/G
_{pp} functions. The DAF spatial pressure distribution is modeled with the usual sin(
k_{o}r)/
k_{o}r empirical model, where
r is the separation distance between points on the panel.

×

The plot shows that the panel is much more responsive to the DAF, by one to two orders of magnitude at low frequencies, and three orders of magnitude at higher frequencies.
This implies that any diffuse acoustic background noise that is three orders of magnitude lower than the TBL wall pressure autospectrum will cause plate vibration comparable to that of the TBL pressures. In most cases, three orders of magnitude of signal to noise is acceptable. This is not the case with TBL-excited plate measurements, however. It is therefore quite common to see ‘tail-ups’ at higher frequencies in TBL-excited plate vibration measurements. This is a sign that acoustic background noise has corrupted the data.

Validation of a Simple Empirical Model for Calculating the Vibration of Flat Plates Excited by Incompressible Homogeneous Turbulent Boundary Layer Flow