First instability of the flow past two tandem cylinders with different diameters

Two-dimensional direct stability analysis and numerical simulation are used to identify and characterize the primary instability of the flow around two tandem cylinders with different diameters. The ratio of the diameter d of the upstream cylinder to the diameter D of the downstream cylinder is varied from 1.0 to 0.4, while the distance L from the center of the upstream cylinder to the front stagnation point of the downstream cylinder is varied from 1.0D to 8.0D. Three typical spacings are chosen to describe the different scenarios observed, and the results for each configuration are described in detail. Three different unstable modes are identified, and the structural sensitivity and nonlinear characteristics of each mode are investigated. The results show that when d/D = 0.6 and L/D ≥ 3.5, the migration of the eigenvalues corresponding to the most unstable modes can be observed for a range of Reynolds numbers. This is ultimately shown to be caused by the transition of the unstable mode. At the same time, this mode transition changes the core region of the instability mechanism. Moreover, two different mechanisms that lead to vortex shedding in the gap between the two cylinders and a secondary bifurcation behavior of the flow around the downstream cylinder under a specific configuration are observed. To obtain a more comprehensive view of the first instability, a transient growth analysis of the flow is also performed, and the critical Reynolds number in each case is discussed.