On using high-order polynomial curve fits in the quasi-steady theory for square-cylinder galloping

Quasi-steady theory shows that the galloping response of a square cylinder exhibits a hysteresis phenomenon. The equation of motion, which was derived based on a seventh-order polynomial curve fit on the side force (Cy) versus angle of attack (α) curve, shows that the number of positive real roots corresponds to the number of stationary oscillation amplitudes. In this investigation, we use polynomials of even higher order (ninth and eleventh) to curve fit the Cy versus α curve, in an attempt to see if additional positive real roots occur, which may reveal even more flow physics. The results show that only extra negative real roots and/or complex roots are obtained when higher than seventh-order polynomial curve fits are used. Hence, the use of a seventh-order polynomial curve fit in the quasi-steady theory is shown to be sufficient in describing the flow physics which includes the prediction of the hysteresis phenomenon.