Modeling Vortex-induced Fluid-structure Interaction Using an Extension of Jourdain's Principle

Book excerpt: A first-principles variational approach is proposed for reduced-order order modeling of fluid-structure interaction systems, specifically vortex-induced vibration. Fluid-structure interaction has to be taken into account in design and analysis of a large portion of engineering applications, yet a comprehensive theoretical development where analytical equations are derived from first principles is nonexistent. Not only does there exist much ambiguity concerning the general behavior of such systems, but the nature of the Lagrangian-Eulerian transformation is yet to be fully understood. Also, a general variational principle that is purely defined in a Eulerian description is nonexistent. Consequently, the use of variational methods for fluid-structure interaction problems has been relatively successful only for simple problems. Moreover, a review of the literature suggests that the Navier-Stokes (N-S) equations could not be obtained using a variational principle. This can be avoided by using Jourdain's principle (JP). Therefore, we have modified Jourdain's principle and obtained the first purely Eulerian variational formulation. Subsequently, by extending the JP for systems of changing mass, we have shown that the N-S equations can be obtained via a variational approach. Moreover, having shown that conservative terms of the N-S equations do not commute with the Eulerian variational operator, a correction term is obtained that must be added to the classical energy equation in integral form for Newtonian incompressible viscous fluids. Regarding vortex-induced vibration, an elastically supported, inverted pendulum that is immersed in a flow is considered as a study system. The pendulum is allowed to move transversely to the flow direction. This problem has generally been used as a test bed of vortex-induced vibration models, as it provides a simple geometry, yet possesses the nonlinearity of these phenomena. It is shown that the reduced-order modeling can be done without any ad hoc assumptions regarding the fluid forcing function. There exists no reduced-order model in the literature that does not make such assumptions. Based on the theoretical results as well as the reduced-order model, we conclude that the first principles development herein is a viable framework for the modeling of complex fluid-structure interaction problems such as vortex-induced oscillations.