Passivity-based output-feedback control of turbulent channel flow

Abstract

This paper describes a robust linear time-invariant output-feedback control strategy to reduce turbulent fluctuations, and therefore skinfriction drag, in wall-bounded turbulent fluid flows, that nonetheless gives performance guarantees in the nonlinear turbulent regime. The novel strategy is effective in reducing the supply of available energy to feed the turbulent fluctuations, expressed as reducing a bound on the supply rate to a quadratic storage function. The nonlinearity present in the equations that govern the dynamics of the flow is known to be passive and can be considered as a feedback forcing to the linearised dynamics (a Lur'e decomposition). Therefore, one is only required to control the linear dynamics in order to make the system close to passive. The ten most energy-producing spatial modes of a turbulent channel flow were identified. Passivity-based controllers were then generated to control these modes. The controllers require measurements of streamwise and spanwise wall-shear stress, and they actuate via wall transpiration. Nonlinear direct numerical simulations demonstrated that these controllers were capable of significantly reducing the turbulent energy and skin-friction drag of the flow.

A framework for input–output analysis of wall-bounded shear flows

“…This is a well-known dissipation inequality that corresponds to passivity. Passivity has been used to study finite-dimensional linear discretizations of the Navier-Stokes equation with the nonlinearity being modeled as an input (Sharma et al (2011);Heins et al (2016)). The general dissipation inequality framework allows us to consider more general energy inequalities rather than only the passivity inequality.…”

Dynamically correct formulations of the linearised Navier–Stokes equations

“…As in many previous studies (see, e.g. [48,12,13]), we assume actuation in the form of wall transpiration, in this case at the upper wall. The actuator is modelled as a first-order system:…”

“…If the objective of the control system is to suppress perturbations around a mean flow, then designing feedback controllers from linearised approximations can give rise to acceptable closed-loop performance, as has been shown in a number of studies (e.g. [4,[6][7][8][9][10][11][12][13]). However, obtaining low-order approximations that are accurate in…”

“…the sense of retaining the important dynamics of a given flow remains an open problem [12]. A typical approach relies upon spatial discretisation of the linearised Navier-Stokes equations on a computational mesh in order to obtain finite-dimensional state-space models [13][14][15][16].The construction of state-space models of a flow by spatial discretisation is typically restricted to simple geometries such as plane channel flow, whereby the assumption of periodicity in the streamwise (and spanwise for three-dimensional geometries) direction allows the use of Fourier transforms in space, decoupling the flow by spatial wavenumber [6]. For more complex geometries, such assumptions are less readily applicable, and hence, computational fluid dynamics codes often use methods based on local interpolants, such as finite difference, finite volume or finite element discretisation.…”

“…the sense of retaining the important dynamics of a given flow remains an open problem [12]. A typical approach relies upon spatial discretisation of the linearised Navier-Stokes equations on a computational mesh in order to obtain finite-dimensional state-space models [13][14][15][16].…”