Determination of the transition mass ratio for onset of galloping of a square cylinder at the least permissible Reynolds number of 150

The Den Hartog stability criterion tests for galloping of an oscillator. A square cylinder satisfies this and is susceptible to galloping. This criterion being necessary for occurrence of galloping appears insufficient as certain parameters, i.e., angle of incidence, α; mass ratio, m*; damping ratio, ζ; reduced speed, U*; and Reynolds number, Re; also assume key roles in determining if the oscillator motion is vortex-induced vibrations (VIV) or galloping. At Re ≈ 150 and U* ≈ 10 or smaller, a square cylinder does not gallop despite satisfying the Den Hartog criterion. By coupling U* and Re, K. Sourav and S. Sen [“Transition of VIV-only motion of a square cylinder to combined VIV and galloping at low Reynolds numbers,” Ocean Eng. 187, 106208-1–106208-19 (2019)] two-degrees-of-freedom motion over Re ≤ 250, the minimum m* or mtr* as 3.4 below which galloping cannot develop. For transverse-only motion, X. Li et al. [“Mode competition in galloping of a square cylinder at low Reynolds number,” J. Fluid Mech. 867, 516–555 (2019)] considered U* = 40 only and determined mtr*≈4 at the least permissible Re of 150. For α = 0° and ζ = 0, we determine the mtr* numerically at Re = 150. By analyzing the transverse response and oscillation frequency over an extended U* range of 10–60, a novel “VIV-galloping transition map” is generated in the m*–U* plane. From this map, the value of mtr* converges to 3.4. The mtr* decays as U*−2.36. The conditions leading to “VIV forever” of a square cylinder are also identified.