Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes

Abstract

Two-dimensional global eigenmodes are used as a projection basis both for analysing the dynamics and building a reduced model for control in a prototype separated boundary-layer flow. In the present configuration, a high-aspect-ratio smooth cavity-like geometry confines the separation bubble. Optimal growth analysis using the reduced basis shows that the sum of the highly non-normal global eigenmodes is able to describe a localized disturbance. Subject to this worst-case initial condition, a large transient growth associated with the development of a wavepacket along the shear layer followed by a global cycle related to the two unstable global eigenmodes is found. The flow simulation procedure is coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced model for the control optimization is obtained by a projection on the least stable global eigenmodes, and the resulting linear-quadratic-Gaussian controller is applied to the Navier–Stokes time integration. It is shown that the controller is able to damp out the global oscillations.

Convective instability and transient growth in steady and pulsatile stenotic flows

“…Other methods, e.g. based on the full eigenspectrum of the linearized Navier-Stokes equations, may also be used to study transient growth in non-parallel open flows (Ehrenstein & Gallaire 2005;Akervik et al 2007).…”

Control of a separated boundary layer: reduced-order modeling using global modes revisited

“…The possibility of model reduction, using global modes, in view of feedback flow control has been explored recently, for incompressible cavity flows considering a smooth shallow geometry in [1] sharp-edge geometry in [4]. Given the degrees of freedom of real full flow states, only a partial-state control approach is feasible in general and hence the flow state has to be estimated.…”

“…For time-horizon T → ∞, the control gain K can be obtained as K = −l −2 B X c , the matrix X c being solution of an algebraic Riccati equation. Details are not provided here, the procedure being outlined in the review [13] and it has been used in [4], [3], [1], among others. The reduced estimation error x = x − x e is solution of…”

“…Some attempts have also been made to exploit in the model reduction procedure the underlying physical mechanisms of the flow instability and in particular to use the global modes for projection. This approach has been applied to a shallow cavity [1] with some success. Very recently, the reliability of model reduction using global modes has been questioned in [5], considering an open square cavity.…”

Structural changes of laminar separation bubbles induced by global linear instability

“…In the last decade instability analyses based on solutions of partial-derivative eigenvalue problems (often referred to as BiGlobal analyses, Theofilis 2003) and/or DNS have elucidated global instability mechanisms in a variety of LSB flows, including the aforementioned adverse-pressure-gradient flat-plate flow , backward-facing step flow , as well as geometry-induced separation (Marquillie & Ehrenstein 2003;Gallaire, Marquillie & Ehrenstein 2007), NACA 0012 airfoil in incompressible (Theofilis, Barkley & Sherwin 2002) and compressible flow (Crouch, Garbaruk & Magidov 2007), low-pressure-turbine flow (Abdessemed, Sherwin & Theofilis 2004, 2009), shock-induced separation (Boin et al 2006;Robinet 2007), open cavity (Akervik et al 2007) and S-shaped duct flows (Marquet et al 2008(Marquet et al , 2009. In all configurations examined consensus has been built that two different primary instability mechanisms coexist.…”