Linear stability analysis in fluid-structure interaction with transpiration. Part I: Formulation and mathematical analysis

The aim of this work is to provide a new Linearization Principle approach particularly suited for problems in fluid-structure stability. The complexity here, and the main difference with respect to the classical approach, comes from the fact that the full non-linear fluid equations are written in a moving (i.e. time dependent) domain. The underlying idea of our approach uses transpiration techniques [J. Fluid Mech. 4 (1958) 383; G. Mortchéléwicz, Application of linearized Euler equations to flutter, in: 85th AGARD SMP Meeting, Aalborg, Denmark, 1997; P. Raj, B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis, in: AIAA 24th Applied Aerodynamics Conference, number 93-3506, Monterey, Canada, 1993; AIAA 27(6) (1989) 777], with the formalization and linearization recently developed in [Rév. Européenne Élém. Finis, 9(6-7) (2000) 681, A. Dervieux (Ed.), Fluid-Structure Interaction, Kogan Page Science, London, 2003 (Chapter 3)]. This allows us to obtain a new grid independent coupled spectral problem involving the linearized Navier-Stokes equations and those of a reduced linear structure. The coupling is realized through specific transpiration conditions acting on a fixed interface, while keeping a fixed fluid domain. We provide a rigorous mathematical treatment of this eigenproblem. We prove that the corresponding eigenmodes, characterizing the free evolution of the system, can be obtained from the characteristic values of a compact operator acting on a Hilbert space. Moreover, we localize the eigenfrequencies of the system in a parabolic region of the complex plan centered along the positive real axis.