A time-implicit representation of the lift force for coupled translational–rotational galloping

The lift force acting on a purely translational galloping oscillator can be well approximated by using the quasi-steady theory, which states that the flow around a galloping body in motion is very similar to the known flow around a fixed body provided a minimum free stream velocity threshold and a similarity principle are satisfied. However, for oscillators undergoing coupled translational–rotational galloping, the rotation of the bluff body breaks the similarity principle, and therewith the quasi-steady assumption. Traditionally, the effects of rotation are accounted for by including explicit time-dependent terms in the lift force. However, we argue that the time-explicit representation of the lift force is unnecessary because the phenomenon of galloping is not time-explicit and acts only as a function of the body motion. Thus, we propose a modified time-implicit lift force representation, which has the same form of the quasi-steady theory, but with an additional dependence on the free stream velocity. The modified lift force is obtained by studying the transient growth of the amplitude of the bluff body oscillation. The proposed approach is used to model the lift force for square and trapezoidal prisms undergoing coupled translational–rotational galloping showing excellent prediction capabilities.