Koopman Analysis and Dynamic Mode Decomposition of a Flexible Flag behind a Semi-Circular Cylinder

Abstract

Flexible flag behind a semi-circular cylinder exhibiting complex dynamics, is studied numerically using immersed boundary method and data driven methods like Koopman Analysis and Dynamic Mode Decomposition. The effect of vortices shed by the D-shaped cylinder were studied by varying the distance between semi-circular cylinder and flag (Gx) in range 0-2.8 by an increment of 0.1 while keeping Re=300, ρ = 1 and bending rigidity ϒ=0.007 as constants. Five regions were identified having different flapping behaviors like symmetric/periodic flapping, chaotic flapping etc. The dynamics of these regions, transition between regions and drag observed by the flag is studied corresponding to these regions. Further more Dynamic Mode decomposition is used for modal analysis of flexible flag behind a semi-circular cylinder for three cases Re 100, 300 and 500 while keeping the fag at a stream-wise distance (Gx) 1.8. Other parameters, ρ = 1 and bending rigidity ϒ=0.007 are kept as constants. The DMD decomposes the system as a quasiperiodic system having two fundamental frequencies, one is frequency of the whole system which is also the frequency of flapping of flag, while the other is a secondary frequency that account for the local interactions of the flag with the fluid that results damping. DMD and kernel variant of DMD which numerically approximates Koopman Operator are used for the reconstruction of posttransient and full transient Re100 and Re300 case.