Based on the immersed-boundary method, a numerical simulation for an oscillating cascade is established and the relevant analysis is presented with emphasis on the physical understanding of fluid-structure interaction. To validate the method, two simulation cases, an oscillating circular cylinder at a low Keulegan-Carpenter number and a flapping airfoil, are performed and the results are in good agreement with the previous research. In the oscillating cascade simulation, it is found that the reduced velocity U is a very sensitive factor which affects the critical stable boundary in the present examples. On the other hand, the effects of interblade phase angles on the system stability are also discussed. In particular, it is worth noting that the same process is applied to several test cases without generating any body-fitting grid. Therefore, the method shows a significant time savings in the computational process for such a complicated fluid-structure interaction problem.damping ratio in transverse direction c = structural damping ratio in rotational direction D = diameter of the cylinder Fx; t = external force F h = lift in transverse direction F = moment around the airfoil elastic center f = oscillating frequency fx s ; t = force per unit area applied by the body to the fluid f n = natural frequency of the oscillating blade h = transverse movement displacement _ h = transverse movement velocity h = transverse movement acceleration h x = mesh size in x direction h y = mesh size in y direction I = inertial moment around the airfoil elastic center KC = Keulegan-Carpenter number k h = structural stiffness ratio in transverse direction k = structural stiffness ratio in rotational direction l = reference length m = mass of the airfoil m = nondimensional mass of the body px; t = flow pressure Re = Reynolds number r = length between surface point and elastic center S = static moment around the airfoil elastic center T = overall period of the system Ux; t = flow velocity Ux s ; t = flow velocity on the body surface U max = maximum velocity of the cylinder motion U 1 = uniform flow velocity U = reduced velocity vx s ; t = body surface velocity x = computational mesh coordinates x s = body surface coordinates x = Cartesian components of x xt = translation motion position x 0 t = translation motion velocity y = Cartesian components of x = feedback function negative constants 0= rotational initial angle = feedback function negative constants 0 = rotational amplitude = airfoil surface = two-dimensional delta function = structural damping ratio = rotational angle _ = rotational angular velocity = rotational angular acceleration 1 = density of fluid = difference phase between xt and t r = function to construct

Based on the immersed boundary method, a numerical simulation for an oscillating airfoil is established and a preliminary analysis of the oscillating airfoil is presented with an emphasis on the physical understanding of fluid-structure interaction. In order to validate the method, two simulation cases: oscillating circular cylinder at low K-C number and two degrees of freedom oscillating cylinder are carried out first and the results are in good agreement with the previous researches. In the oscillating airfoil simulation，it is found that the reduced velocity U * is a very sensitive factor and especially U *~2 .8 is the critical stable boundary in the present work. The method shows the predominance of time saving in computational process for such a complicated fluid-structure interaction problem.

Based on the immersed boundary method, a fast simulation for solving unsteady, incompressible, viscous flow associated with the oscillating cascade is established on a quasi-three-dimensional coordinate system. The numerical method is applied to the simulation of the flow passing an oscillating circular cylinder which is forced to move in X direction under prescribed motions in water at rest at low Keulegan-Carpenter numbers. Then vortex-induced vibration of a cylinder with two degrees of freedom which oscillates in in-line direction and transverse direction is simulated using this method. The results are in good agreement with the previous research. Then the method is extended to the oscillating cascade simulation of making various comparisons. It is found that the IBPA (inter blade phase angle) will change as the time goes on, because of the non-uniformity of the flow in the circumferential direction, until the oscillating cascade goes to a stable situation. The reduced velocity and the number of blades are chosen to investigate the effects of them on IBPA. The results indicate that both the reduced velocity and the number of blades are the main factors which influence IBPA. It is worth noting that the coupling process is not necessary to generate any body-fitting grids, which makes it much faster in computational process for such a complicated fluid-structure interaction problem.