A summary of some recent efforts on the analysis of fluid-elastic instability in heat exchanger and steam generator tube bundles is given in Table 4.

1.2

Vorticity induced instability

Flow across a tube produces a series of vortices in the downstream wake formed as the flow separates alternatively from the opposite sides of the tube. This shedding of vortices produces alternating forces, which occur more frequently as the velocity of flow increases. For a single cylinder, frequency of vortex shedding fvs is given below by a dimensionless Strouhal number S.

_ SV vs _ D

where V is the flow velocity and D is the tube diameter. For a single cylinder, the vortex shedding Strouhal number is a constant with a value of about 0.2 (Chenoweth, 1993). Vortex shedding occurs for the range of Reynolds number 100 < Re <5×105 and > 2 x 106 whereas it dies out in-between. The gap is due to a shift of the flow separation point in vortices in the intermediate transcritical Reynolds number range. Vortex shedding can excite tube vibration when it matches with the natural frequency of the tubes. For tube banks with vortex shedding, Strouhal number is not constant, but varies with the arrangement and spacing of tubes, typical values for in-line and staggered tube bundle geometries are given in (Karaman, 1912, Lienhard, 1966). Strouhal numbers for in-line tube banks are given in Figure 6.

The vortex shedding frequency can become locked-in to the natural frequency of a vibrating tube even when flow velocity is increased (Blevins, 1977). Earlier on, the mechanism of vortex shedding has been investigated by a number of researchers. These include Sipvack (Sipvack, 1946) and, Thomas and Kraus (Thomas & Kraus, 1964) who investigated the vortex shedding of two cylinders arranged parallel and perpendicular to flow direction respectively. Grotz and Arnold (Groth & Arnold, 1956) measured for the first time systematically the vortex shedding frequencies in in-line tube bank for various tube spacing ratios.

The cause of vorticity excitation has been disputed in literature (Owen, 1965), but recent studies of (Weaver, 1993) and, (Oengoren & Ziada, 1993) have confirmed its cause of existence as periodic vortex formation. Vorticity shedding can cause tube resonance in liquid flow or acoustic resonance of the tube bundles or acoustic resonance of the tube bundles’ containers in gas flows (Oengoren & Ziada, 1995). (Chen, 1990), (Zaida & Oengoren, 1992) and (Weaver, 1993) have summarized the recent research efforts targeted at improvement in Strouhal number charts for vortex shedding and acoustic resonance for in­line tube bundles.

L

D

Fig. 6. Strouha! numbers for in-line tube banks (Karaman, 1912).

(Oengoren and Ziada, 1992) have investigated the coupling between the acoustic mode and vortex shedding, which may occur near the condition of frequency coincidence. They have investigated the system response both in the absence and in the presence of a splitter plate, installed at the mid-height of the bundle to double the acoustic resonance frequencies and therefore double the Reynolds number at which frequency coincidence occurs. They have also investigated the effect of row number on vortex shedding and have carried out flow visualization in Reynolds number range of < 355000. Figure 7 is a typical example of the mechanism of vortex shedding from the tubes of the first two rows displaying a time series of symmetric and anti-symmetric patterns (Oengoren & Ziada, 1993).

(Liang et al., 2009) has addressed numerically the effect of tube spacing on vortex shedding characteristics of laminar flow past an inline tube arrays. The study employs a six row in­line tube bank for eight pitch to diameter (^/^) ratios with Navier-Strokes continuity equation based unstructured code (validated for the case of flow past two tandem cylinders) (Axisa & Izquierdo, 1992) . A critical spacing range between 3.0 and 3.6 is identified at which mean drag as well as rms lift and drag coefficients for last three cylinders attain maximum values. Also at critical spacing, there is 180o phase difference in the shedding cycle between successive cylinders and the vortices travel a distance twice the tube spacing within one period of shedding.

(Williamson & Govardhan, 2008) have reviewed and summarized the fundamental results and discoveries related to vortex induced vibrations with particular emphasis to vortex dynamics and energy transfer which give rise to the mode of vibrations. The importance of mass and damping and the concept of "critical mass", "effective elasticity" and the relationship between force and vorticity. With reference to critical mass, it is concluded that

as the structural mass decreases, so the regime of velocity (non-dimensional) over which there is large amplitude of vibrations increases. The synchronizing regime become infinitely wide not simply when mass become zero but when a mass falls below special critical value when the numerical value depends upon the vibrating body shape.

(Williamson & Govardhan, 2000) present a large data set for the low branch frequency flower plotted versus m* (mass ratio) yielding a good collapse of data on to single curve base equation 7.

(7)

This equation provides a practical and simple means to calculate the frequency attained by vortex induced vibrations. The critical mass ratio is given by

m*crit = 0.54 ±0.002 (8)

Below which the lower branch of response can never be attained. With respect to combine mass-damping parameter’s capability to reasonably collapse peak amplitude data in Griffins plot, a number of parameters like stability parameter, Scrutom number and combined response parameter termed as Skop-Griffins parameter given by (SG):

Where S stands for single vortices and Sc is the Scruton number.

(Hamakawa & Matsue, 2008) focused on relation between vortex shedding and acoustic resonance in a model (boiler plant) for tube banks to clarify the interactive characteristics of vortex shedding and acoustic resonance. Periodic velocity fluctuation due to vortex shedding was noticed inside the tube banks at the Reynolds number (1100-10000) without acoustic resonance and natural vortex shedding frequency of low gap velocities. Kumar et al., 2008 in their review stated that controlling or suppressing vortex induced vibrations is of importance in practical applications where active or passive control could be applied.

(Paidoussis, 2006) specially addressed real life experiences in vortex induced vibrations and concludes with this mechanism in addition of other already clarified mechanisms of flow induced vibrations. Vortex induced vibrations of ICI nozzles and guide tubes in PWR for ICI thimble guiding into the core of the reactor to monitor reactivity may witness breakage of ICI nozzles resulting in strange noises experience in the reactor. Analysis of shedding frequencies confirmed the vortex induced vibrations to be the culprit partially due to the large values of varying lift coefficients and partially due to lock-in.

(Hamakawa & Fukano, 2006) also focused vortex shedding in relation with the acoustic resonance in staggered tube banks and observe three Strouhal number (0.29, 0.22 and 0.19). In cases with no resonance inside tube banks, the last rows of tube banks and in both regimes respectively. The vortices of 0.29 and 0.22 components alternatively irregularly originated.

(Pettigrew & Taylor, 2003) discussed and overviewed procedures and recommended design guidelines for periodic wave shedding in addition to other flow induced vibration considerations for shell and tube heat exchangers. It concludes that the fluctuating forces due to periodic wave shedding depends on the number of considerations like geometric configuration of tube bundles, its location, Reynolds number, turbulence, density of fluid and pitch to diameter ratio.