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Chapter 8
Turbulence-Induced Vibration in Parallel Flow

## Excerpt

In spite of recent advances in computational fluid dynamics, today the most practical method of turbulence-induced vibration analysis follows a hybrid experimental/analytical approach. The forcing function is determined by model testing, dimensional analysis and scaling, while the response is computed by finite-element probabilistic structural dynamic analysis using the acceptance integral approach formulated by Powell in the 1950s. The following equation is often used to estimate the root-mean-square (rms) response, or to back out the forcing function from the rms response, of structures excited by flow turbulence:

$=ΣαAGp(fα)ψα(x⃗)J⃗αα2(fα)64π3mα2fα3ζα$
where Jαα is the familiar joint acceptance. As it stands Equation (8.50) is general and applicable to one-dimensional as well as two-dimensional structures in either parallel flow or cross-flow; the latter will be covered in the following chapter. It is also independent of mode shape normalization as long as the same normalization convention is used throughout the equation. However, Equation (8.50) is derived under many simplifying assumptions, of which the most important ones are that cross-modal contribution to the response is negligible, and the turbulence is homogeneous, isotropic and stationary. In addition, if one assumes that the coherence function can be factorized into a streamwise component, assumed to be in the x1, or longitudinal direction, and a cross-stream x2, or lateral component, and that each factor can be completely represented by two parameters—the convective velocity and the correlation length—then the acceptance integral in the two directions can be expressed in the form:
$ReJmr=1L1∫0L1dx″ψm(x″)∫0x″ψr(x′)e−(x′−x″)∕λcos2πf(x′−x″)Ucdx′+1L1∫0L1dx″ψm(x″)∫x″L1ψr(x′)e−(x″−x′)∕λcos2πf(x″−x′)Ucdx′$
$ImJmr=1L1∫0L1dx″ψm(x″)∫0x″ψr(x′)e−(x′−x″)∕λcos2πf(x′−x″)Ucdx′+$

• Summary
• Nomenclature
• 8.1 Introduction
• 8.2 Random Processes and the Probability Density Function
• 8.3 Fourier Transform, Power Spectral Density Function and the Parseval Theorem
• 8.4 The Acceptance Integral in Parallel Flow over the Surface of Structures
• 8.5 Factorization of the Coherence Function and the Acceptance Integral
• The One-Dimensional Coherence Function
• The Longitudinal and Lateral Acceptances
• 8.6 The Mean Square Response
• Validity of the Response Equation
• 8.7 The Physical Meaning of the Acceptance Integral
• 8.8 Upper Bound of the Joint Acceptance
• 8.9 The Turbulence PSD, Correlation Length and Convective Velocity
• The Convective Velocity
• The Displacement Boundary Layer Thickness
• Correlation Length
• Turbulence Random Pressure Power Spectral Density
• 8.10 Examples
• Example 8.1
• Example 8.2 and Case Study
• Appendix 8A: Charts of Acceptance Integrals
• References
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