Data driven reduced order modelling of transonic buffet

Abstract

On aircraft wings operating in the transonic regime, self-sustained shock oscillations brought on by shock wave-boundary layer interactions give rise to the phenomenon known as transonic buffet. The operational flying envelope is limited by the ensuing unstable aerodynamic loads, which, if unchecked, may cause structural fatigue or even catastrophic failure. The non-linear dynamics of Hopf bifurcations and limit cycle oscillations, which control buffet start and progression, are not sufficiently captured by traditional linear reduced-order models (ROMs), despite the fact that they have been used with varying level of success for buffet suppression, with the use of controllers. The current thesis builds a data-driven, non-linear, reduced order modelling framework for describing the transonic buffet dynamics, based on spectral submanifold (SSM) embedding, in order to get over this restriction. The methodology implements the use of unsteady Reynolds-averaged Navier-Stokes (URANS) simulations, which simulate an n-dimensional dynamical system, combined with the use of an invariant manifold in n-dimensions, to construct compact and data-driven ROMs. First, steady and unsteady flow simulations are performed for a supercritical aerofoil across different Mach number-angle of attack combinations to identify the buffet onset boundary. Linear (eigenmode) analysis and Proper Orthogonal Decomposition (POD) are applied to extract eigenmodes and POD modes from the flow field, which are used to define a two-dimensional tangent space, to the n-dimensional manifold. The n-dimensional system dynamics, stored as flowfield data, can be projected onto this tangent space, creating a set of reduced coordinates through an encoder function. The dynamics of the system in the reduced coordinates are then captured with the help of an ordinary differential equation, obtained via polynomial regression. The pair of encoder function and the ordinary differential equations obtained, results in a reduced order model for the system. Both the linear and non-linear behaviour of the original system are captured and combined with a Taylor series expansion of the reduced coordinates to form a decoder function, which reconstructs the n-dimensional system output in the flowfield form. The identified ordinary differential equation is used to predict the evolution of reduced coordinates in the tangent space, while the decoder function enables the evaluation of the full flow field from the predicted reduced coordinates. The validation of the obtained ROM is carried out against independent URANS simulations with alternate initialization condition, for each flow condition evaluated. The results show that the SSM-based ROM is capable of accurately reproducing limit cycle oscillations and reconstructing full flowfields with good accuracy. The reduced order model predictions and the reconstructed flowfields obtained from it show good agreement with actual features of the buffet phenomena like shock location at various points in time. This work establishes the use of spectral submanifold embedding as a promising tool for constructing non-linear ROMs of transonic buffet. The developed framework provides new opportunities for real-time prediction and control of buffet phenomena. Beyond aerospace applications, the methodology has broader potential for analysing other non-linearizable fluid-structure interaction problems.