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Chapter 9
Turbulence-Induced Vibration in Cross-Flow

## Excerpt

The response of one-dimensional structures such as beams and tubes to cross-flow turbulence has important applications to heat exchanger design, operation and maintenance. In cross-flows, because the front of the pressure wave reaches all points on the structure at the same time, the forces along the length of the structure are always in phase, contrary to the case of parallel flow-induced vibration. This greatly simplifies the acceptance integral method of turbulence-induced vibration analysis discussed in the previous chapter. By making the additional assumption that the random pressure is fully coherent across the width of the structure, the equation for the mean square response of the structure derived in the last chapter is simplified to:

$=∑nLGF(fn)ϕn2(x)64π3mn2fn3ζnJnn+cross-terms$

Here GF =D2Gp is the random force PSD, expressed in (force/length)2/Hz and Jnn are the joint acceptance integrals in the axial (cross-stream) direction. Because of the absence of the phase angle in the coherence function, Jnn are much easier to compute in cross-flow over 1D structures compared with those for parallel flows discussed in the previous chapter. Using the finite-element technique, the joint and cross-acceptances for cross-flow over beams and tubes with arbitrary boundary conditions are computed as a function of λ/L and are given as design charts in Appendix 9B. In particular, it is shown that as long as the correlation length is small compared with the half flexural wavelength of the structure, the following relationship:

$Jnn=2λ∕Lasnπλ∕L→0$
is true irrespective of the boundary conditions of the beam.

• Summary
• Acronyms
• Nomenclature
• 9.1 Introduction
• 9.2 Exact Solution for the Joint Acceptance Integrals
• 9.3 Finite Element Method of Evaluating the Acceptance Integrals
• Case 1: Spring-Supported Rigid Beam with Uniform Mass Density
• Case 2: Simply-Supported Beam with Uniform Mass Density
• 9.4 Correlation Length and Power Spectral Density for Single-Phase Cross-Flows
• Correlation Length
• The Random Pressure PSD for Single-Phase Cross-Flow
• Vortex-Induced Response
• Example 9.1
• 9.5 The Acceptance Integral for a Structure with Non-uniform Mass Density
• 9.6 Practical Heat Exchanger Tubes with Multiple Supports
• Example 9.2
• Example 9.3
• 9.7 Cross-Modal Contributions to Responses
• Example 9.4
• 9.8 Vibration due to Turbulent Two-Phase Cross-Flow
• Two-Phase Flow Regimes
• Two-Phase Parameters
• Two-Phase Damping
• Random Pressure PSD in Two-Phase Flows
• Example 9.5
• Appendix 9A: Derivation of Equation (9.11)
• Appendix 9B: Charts of Acceptance Integrals
• References
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