Linear stability analysis in fluid–structure interaction with transpiration. Part I: Formulation and mathematical analysis Original Research Article

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.