An analytic solution is obtained for the plate-velocity statistics of a turbulent-flow excited, simply supported, rectangular flat plate. The radiated acoustic pressure is neglected as contributing to the plate excitation, leaving only the turbulent-boundary-layer pressure fluctuations as the exciting force. The mathematical model for the turbulent-boundary-layer pressure statistics is based on that of Corcos, which agrees well with experiment. Plate-velocity statistics are expressed in the forms of dimensionless cross-power spectral and power spectral densities. Plate-velocity-spectral and cross-spectral densities were obtained with a digital computer for selected flow and plate parameters. From these computed dimensionless spectra, effects of major parameters on the plate-velocity statistics were determined. A “peak spectrum,” constructed by connecting major spectral peaks of the plate-velocity-spectral density, proved to be a useful engineering concept, inasmuch as knowledge of the “peak spectrum” is equivalent to knowledge of the maximum plate velocity spectral density for a given set of input parameters. The computed dimensionless plate-velocity “peak spectrum” compares well with “peak spectra” constructed from available experimental data.
The effects of a fluid environment on the flow-induced vibration of finite and infinite plates are theoretically examined. Qualitative effects of fluid loading are deduced from the mathematical models. Examples of computed flow-induced vibration statistics for identical plates, with and without water loading, illustrate the quantitative effects of fluid loading. The fluid environment exerts forces on the plate in temporal phase with the plate inertial and resistive forces. For heavy fluids (i.e., water), the inertial component of the fluid loading is several orders of magnitude greater than the resistive component. Comparison of fluid-loaded finite and infinite plate statistics reveals that the infinite plate velocity spectral density provides a good approximation to the level and general shape of the finite plate velocity spectral density. However, the cross-spectral characteristics of the infinite and finite plate vibrations are substantially different.
Turbulent-flow-induced plate-vibration statistics are computed for mathematical models of finite and infinite thin plates. The computed statistics for the infinite- and finite-plate models are compared to establish similarities and differences between the vibration statistics for the two plate models. The statistics are also compared with available experimental data to determine which model better approximates reality. These comparisons show little similarity between computed turbulent-flow vibration statistics for the finite- and infinite-plate models. The plate-vibration statistics computed from the finite-plate model are in substantially better agreement with experimental results than those computed from the infinite-plate model.
The results of some experimental measurements, made to determine the hydrophone-size correction factor, are presented and compared with the theoretical results of Corcos. The measurements agree with Corcos' theoretical hydrophone-size correction factors over the range of Strouhal numbers, from 0.01 to 4.0, covered by the experiment. In addition, a practical method of determining effective hydrophone size is presented.
The vibration of an infinite, thin plate, immersed in a fluid and excited by a time-harmonic line-force, is a classical problem in acoustics. The classical solution to the fluid-loaded plate argues that one pair of real roots, equal in magnitude and opposite in sign, exists to the equation defining the free wavenumber. However, recent papers by Stuart [J. Acoust. Soc. Am. 59, 1160–1169; 1170–1174 (1976)] and by Pierucci and Graham [J. Acoust. Soc. Am. 62, S84(A) (1977)] argue that, for certain plate-fluid combinations, three such pairs of real roots exist: the classical pair plus two additional pairs. This paper reexamines this problem to clarify the existence and physical significance of these multiple pairs of real roots. It is shown that only the classical pair of real roots (1) can exist when the radiation condition is satisfied, (2) characterizes the waves in a free plate, and (3) characterizes significant wavenumber contributions to the displacement of the line-force excited plate at large distances from the line excitation. The other two paris of real roots result only if the radiation condition is ignored, and therefore cannot characterize free waves in the plate. Further, these wavenumbers do not characterize any nondecaying contributions to the displacement of the line-force excited plate.
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