1.1.1 Jet switch model

(Roberts, 1962, 1966) considered both a single and a double row of cylinders normal to flow. His analysis was limited to in-flow motion (experiments indicated that instability was purely in the in-flow direction). Roberts assumed that the flow downstream of two-adjacent cylinders could be represented by two wake regions, one large and one small, and a jet between them as shown in Figure 1.

Downstream

Considering a downstream cylinder moving upstream; as the two cylinders cross, insufficient fluid flows in to the large wake region to maintain the entrainment, causing the wake to shrink and the jet to switch directions. Roberts has given the flow equation of motion for a cylinder or tube in a row.

where Cpb is the base pressure coefficient, т is non-dimensional time (trnn), D is the tube diameter, mn is the natural frequency, x is the in-flow cylinder displacement, £ is the damping factor or ratio, S is the logarithmic decrement, and m is mass of the tube. Equation 1 was solved using the method of Krylov and Bogoliubov (Minorsky, 1947) giving Vc, the velocity just sufficient to initiate limit cycle motion for any mS / pD2 . Neglecting unsteady terms and fluid damping, the solution reduces to

C = K 2 I (2)

®nsD PD2)

where є is the ratio of fluid-elastic frequency to structural frequency, which is approximately 1.0 and p is the fluid density. This has the same form as the classical Connors equation (Blevins, 1979). Figure 2 presents Robert’s experimental data for pitch-to — dia. ratio (P / D = 1.5 ), showing a good agreement with this theoretical model.

Fig. 2. Theoretical stability boundary for fluid-elastic instability obtained by Roberts for a single flexible cylinder in a row of cylinders (Roberts, 1966).