This paper presents a higher order, three-dimensional boundary element method for investigating the dynamics and stability of elastic structures containing and/or submerged in flowing fluid. The method developed can be applied to any shape of elastic structures partially or completely in contact with fluid. In the mathematical model, it is assumed that the fluid is ideal (i.e. inviscid, incompressible and its motion is irrotational). The fluid structure interaction forces are calculated using the higher order boundary element method, and the finite element method is employed for the structural analysis. In this study, it is assumed that the elastic structure vibrates in its in vacuo modes when it is in contact with flowing fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. The in vacuo dynamic properties of the dry elastic structure are obtained by using standard finite element software. In the wet part of the analysis, the wetted surface of the elastic structure is idealized by using appropriate boundary elements, referred to as hydrodynamic panels. Over each hydrodynamic panel, higher order distributions (linear and quadratic) are adopted in the present study in order to obtain a better convergence, in contrast to Ugurlu and Ergin (2006) assuming constant distribution over each hydrodynamic panel. The fluid structure interaction forces are calculated in terms of the generalized added mass coefficients, generalized Coriolis fluid force coefficients, and generalized centrifugal fluid force coefficients. To assess the influence of flowing fluid and end support conditions (e.g. simply supported ends, clamped ends, and cantilever cylindrical shell) on the dynamic response behaviour and stability of the cylindrical shells, the non-dimensional wet frequencies and associated vibration modes are presented as a function of the non-dimensional axial flow velocity, and the calculations compare well with the analytical solutions found in the open literature.