Javier de Vicente Buendía, Eusebio Valero Sánchez
Affiliated Research Center
Universidad Politécnica de Madrid
In this work, several approaches for the stability analysis of complex aerodynamic flows are investigated in order to estimate their applicability to industrial flow problems and their computational limitations. The results obtained in this work significantly contribute to understand of how to choose a numerical method for a stability problem, which was unsufficently investigated until now. The numerical solution of large eigenvalue problems is becoming increasingly demanding by the scientific community since it allows a better understanding of the intrinsic nature of a phenomenon: vibrations in structural mechanics, oscillations in plasma physics, acoustic resonances, magneto-hydrodynamics are different examples of physically relevant eigenvalue problems. In particular, in the field of Fluid Mechanics, the asymptotic behavior of stationary solutions of the Navier Stokes equations is defined by the spectrum of a non-self-adjoint eigenvalue problem. In a temporal framework, the linearization of the aforementioned Navier Stokes equations around a equilibrium state, or base flow, permits the linear stability analysis of aerodynamic flows. Depending on the eigenvalues in the spectrum of the discretized linear differential operator, small disturbances, superimposed on the steady solution, may decay or grow in time, thus characterizing the base flow as linearly stable or unstable. The corresponding eigenvectors provide relevant information about the transition mechanism, which allows to characterize and control the studied aerodynamic flow. Nowadays, the advent of high performance computing facilities, with increasing resources in terms of memory and computational capability, is permitting the study of more demanding problems on complex geometries aligned with industrial real needs. However, as a consequence of this expanded scenario for the stability analysis applications, the leading dimension of the involved matrices has risen from hundreds of thousands to millions. Several two-dimensional analyses evolved into three-dimensional cases, even if not affordable yet but for the simplest configurations. In the last decades, the number of available algorithms has increased significantly and different available variants can give insights into certain properties of the eigenvalue problem. However, the effective applicability of these numerical methods in the stability analysis framework, has been still not clarified. In this context, the aim of this thesis is to evaluate new numerical approaches to overcome the computational cost limitations, opening the stability analysis scenario to complex aerodynamic configurations for industrial applications. Furthermore, a new approach to reduce the computational costs, based on a geometrical concept instead of numerical proprieties, is introduced. This novel method, here presented for the first time, represents in an innovative approach with exceptional results.