A boundary element method is presented to investigate the dynamic behavior of elastic structures partially or completely in contact with uniform axial flow. In the analysis of the linear fluid-structure interaction problem, it is assumed that the fluid is ideal and its motion is irrotational. Furthermore, the elastic structure is assumed to vibrate in relatively high-frequencies, so the infinite frequency limit condition is imposed for fluid free surface, which is satisfied implicitly by using method of images. When in contact with the flowing fluid, the structure is assumed to vibrate in its in vacuo eigen-modes that are obtained by using a finite element software. The wetted surface of the structure is idealized by using appropriate hydrodynamic panels and a boundary element method is formulated for velocity potential function, which is taken as linearly varying over the panels. Using the Bernoulli's equation, the dynamic fluid pressure on the elastic structure is expressed in terms of potential function, and the fluid-structure interaction forces are calculated as generalized added mass, hydrodynamic damping and hydrodynamic stiffness coefficients, due to the inertia, Coriolis and centrifugal effects of fluid, respectively. Solution of the eigenvalue problem associated with the generalized equation of motion gives the dynamic characteristics of the structure in contact with fluid. As an application of the method, the dynamics of a simply supported cylindrical shell Subjected to internal flow is studied. The predictions compare quite well with the previous results in the literature.