Matrix-Free Methods for the Stability and Control of Boundary Layers

Abstract

This paper presents matrix-free methods for the stability analysis and control design of high-dimensional systems arising from the discretized linearized Navier-Stokes equations. The methods are applied to the twodimensional spatially developing Blasius boundary-layer. A critical step in the process of systematically investigating stability properties and designing feedback controllers is solving very large eigenvalue problems by storing only velocity fields at different times instead of large matrices. For stability analysis, where the entire dynamics of perturbations in space and time is of interest, iterative and adjoint-based optimization techniques are employed to compute the global eigenmodes and the optimal initial conditions. The latter are the initial conditions yielding the largest possible energy growth over a finite time interval. The leading global eigenmodes take the shape of Tollmien-Schlichting wavepackets located far downstream in streamwise direction, whereas the leading optimal disturbances are tilted structures located far upstream in the boundary layer. For control design on the other hand, the input-output behavior of the system is of interest and the snapshot-method is employed to compute balanced modes that correctly capture this behavior. The inputs are external disturbances and wall actuation and the outputs are sensors that extract wall shear stress. A lowdimensional model that capture the input-output behavior is constructed by projection onto balanced modes. The reduced-order model is then used

Massively Parallel Solution of the BiGlobal Eigenvalue Problem Using Dense Linear Algebra

“…Typically, these problems are addressed by using a time-stepping method for the numerical solution of the BiGlobal EVP [10][11][12]. Time-stepping approaches were devised at a time when storage of large matrices was impractical and work optimally for the recovery of a small number of leading eigenmodes.…”

Transition delay using control theory

“…The reduced order was used to design a small optimal feedback controller, which could damp the energy of the twodimensional TS wavepackets forced upstream by orders of magnitudes. A similar analysis was performed by Bagheri et al [28], where instead of body forces, blowing/suction actuators and wall-shear stress were employed. However, the fact that the inputs and outputs were modelled as a body force does not mean that they were unrealistic.…”

A Domain Decomposition Matrix - Free Method for Global Linear Stability

“…Such a method is well detailed by Saad (2003) and is applied with success in several single domain configurations in both three- (Tezuka and Kojiro (2006), Bagheri et al (2009b) and Feldman and Gelfat (2010)) and two-dimensional geometries (Barkley and Blackburn (2008), Bagheri et al (2009a) and Bres and Colonius (2008)). We will briefly introduce the notations and numerical methods within a single domain framework.…”

“…For demonstration purposes, we consider partition P 1 and the mesh (300 × 300). ∆T is taken to verify the Nyquist criterion (see Bagheri et al (2009a)). In this context, we focus on the highest frequency mode, corresponding to T 2 .…”

“…By contrast, within a DNS framework, the convective velocity of the perturbation reaching the outlet may be updated at each time step allowing disturbances to exit smoothly out of the domain (see for instance Pauley et al (1990) ). On the other hand, although the matrix method has been used successfully in several cases, (see Theofilis (2003Theofilis ( ) orÅkervik et al (2007 for instance) the storage of the discretized global stability equations still poses a great computational challenge due to its very large dimension (see Bagheri et al (2009a)). Recently, Rodriguez and Theofilis (2009) proposed a methodology based on a massively parallel solution.…”