Extremely turbulent flow of the shell-side fluid contains a wide spectrum of frequencies distributed around a central dominant frequency, which increases as the cross-flow velocity increases. This turbulence buffets the tubes, which extract energy from the turbulence at their natural frequency from the spectrum of frequencies present. When the dominant frequency for the turbulent buffeting matches the natural frequency, a considerable transfer of energy is possible leading to significant vibration amplitudes (Chenoweth, 1993). Turbulent flow is characterized by random fluctuation in the fluid velocity and by intense mixing of the fluid. Nuclear fuel bundles and pressurized water reactor (PWR) steam generators are existing examples (Hassan & Ibrahim, 1997).

Turbulence is by nature three-dimensional (Au-Yang, 2000). Large-Eddy Simulation, (LES) incorporated in three-dimensional computer codes has become one of the promising techniques to estimate flow turbulence. (Hassan & Ibrahim, 1997) & (Davis & Hassan, 1993) have carried out Large Eddy Simulation for turbulence prediction in two-and three­dimensional flows. The primary concern in turbulence measurements is how the energy spectrum or the power spectral density (PSD) of the eddies are distributed. The PSD of the

velocity profile E(n) is numerically equal to the square of the Fourier Transform of U"(t), and is defined to be (Hassan & Ibrahim, 1997).

_____ +да

U2 = J E(n)dn

-да

where E(n) is the sum of power at positive and negative frequency n.

where T is the time period over which integration is performed, and a(n) is the Fourier Transform coefficient.

An important parameter of flow turbulence is the correlation function. The Lagrangian (temporal) auto-correlation over a time T gives the length of time (past history) that is related to a given event (Hassan & Ibrahim, 1997).

(Non-dimensional) R(r) = U (t )U ((+ r)

U'(()U'(()

1 (=t

(Dimensional) R(r) = Um r___________ — J U’ (t)U’ (t + z)dt (12b)

T t=0

Physically R(t) represents the average of the product of fluctuating velocity U" values at a given time and at a time r later. R(t) gives information about whether and for how long the instantaneous value of U depends on its previous values. Cross-correlation curves can also be obtained as a function of the time delay to give the correlation between the velocities at consecutive separated location points (Owen, 1965).

1 t=t

Ru(t) = — J U/(t)U2(t + r)dt (13)

T t=0

where R12 gives the cross-correlation of the U-velocity component at 1- and 2- point locations.

Recently (Au-Yang, 2000) has reviewed the acceptance integral method to estimate the random vibration, Root Mean Square (RMS) of structures subjected to turbulent flow (random forcing function). The acceptance integral is given by:

Jap(m) = ~ J0 J0 ^a( x ) [Sp (X > x",m)l Sp (x’ ,®)фр( x )dx’dx” (14)

When a = P, J aa is known as joint acceptance where

Yang obtained closed form solutions for the joint acceptances for two special cases of spring — supported and simply-supported beams. A review of turbulence in two-phase is presented by (Khushnood et al., 2003).

(Endres & Moller, 2009) present the experimental analysis of disturbance propagation with a fixed frequency against cross flow and its effect on velocity fluctuations inside the bank. It is concluded that continuous wavelet transforms of the signals. Figure 8 indicates the disturbance frequency to be showing steady behavior. Generally designing for enhanced heat exchange ratios in thermal equipments ignores the structural effects caused by turbulent flow.

(Pascal-Ribot and Blanchet, 2007) proposed a formulation to collapse the dimensionless spectra of buffeting forces in a single characteristic curve and gives edge to the formulation over previously normalized models in terms of collapse of data.

Figure 9 shows the dimensionless spectra calculated with equations 15 & 16 respectively.

Where a is the void fraction.

(Wang et al., 2006) concludes the physically realistic solutions for turbulent flow in a staggered tube banks can be realized by FLUENT (with 2-D Reynolds stress model).

Figure 10 shows the consistency of turbulence intensity contours obtained through standard wall function approach and non-equilibrium wall function approach whereas near-wall treatment model and near — wall turbulence model predicts much higher results (Wang, et al., 2006).