Linear control of coherent structures in wall-bounded turbulence at
Introduction
A growing number of studies have successfully utilized linear models for estimation (e.g. Chevalier et al., 2006, Jones et al., 2011, Illingworth et al., 2018, Oehler et al., 2018b, Sasaki et al., 2019) and control (e.g Cortelezzi et al., 1998, Moarref and Jovanović, 2012, Luhar et al., 2014) of wall-bounded turbulent flows. The work of Luhar et al. (2014), in particular, suggests that linear models can qualitatively predict the effect of control on individual scales and also determine at which location they can best be measured. Linear model-based designs are an appealing alternative to direct numerical simulation (DNS) based designs since the cost is several orders of magnitude smaller. One reason for the success of linear models is that linear mechanisms play an important role in the sustenance of turbulence (Schoppa and Hussain, 2002, Kim, 2011). In the linearized Navier–Stokes (LNS) equations, where the flow is linearized around the turbulent mean, these linear mechanisms result in large transient growth that is due to the non-normality of the LNS operator (Trefethen et al., 1993). In particular, it was shown that the LNS operator could predict the typical widths of near-wall streaks and large-scale structures in the outer layer (del Alamo and Jiménez, 2006, Pujals et al., 2009, Hwang and Cossu, 2010b).
Linear mechanisms play a major role in the formation and maintenance of large-scale structures in turbulent wall-bounded flows. These large-scale structures contribute significantly to the turbulent kinetic energy and Reynolds stresses (in the outer region), and there is evidence that they affect the small scales near the wall (Hutchins and Marusic, 2007, Mathis et al., 2009, Marusic et al., 2010a, Marusic et al., 2010b, Duvvuri and McKeon, 2015). Hence, the control of these structures is crucial for any efforts to control wall-bounded flows (see Kim and Bewley (2007) for a review). It was shown that linear estimation, which is closely related to linear control, performs best for those structures that have the greatest potential for transient growth (del Alamo and Jiménez, 2006, Pujals et al., 2009), are the most amplified in stochastically and harmonically forced settings (Hwang and Cossu, 2010b) and are coherent over large wall-normal distances (Madhusudanan et al., 2019). These observations presumably also apply to linear control, which would simplify the controller design process.
This work studies linear feedback control of the largest structures in a turbulent channel flow at a relatively high Reynolds number of Re. It is in part motivated by experimental work that has achieved a reduction in skin-friction drag through real-time control of large-scale structures (Abbassi et al., 2017). The focus of this study is on the sensors and actuators for linear feedback-control. Specifically, we compare control performance when measuring or actuating the full channel (i.e. an ideal set-up) to control performance when measuring or actuating at only one specific wall height (which is a more realistic set-up in a practical application, e.g. hot-wire sensors and synthetic jet actuators). Consequently, it is possible to compare the ideal setting to what is achievable in a laboratory environment.
When considering these control set-ups, we will focus on finding the best control performance possible. Therefore, we (i) assume that sensor noise is insignificant, (ii) remove almost all energy limitations imposed on the actuators, and (iii) ignore the effect of transients. In this way, it is possible to show whether a setup is worth considering in the first place as even the best results might not be sufficient.
Rather than testing various control configurations through the use of DNS, the flow is modeled using the LNS operator for perturbations about the mean flow (Section 2). An eddy viscosity is included in the operator to model the effect of the incoherent scales. This uncontrolled linear model (LM) of the flow is validated by comparing it to DNS in Section 3 before we introduce three specific control set-ups in Section 4 and analyze their performance in Section 5. Finally, we conclude the study in Section 6.
Section snippets
The linear model
A statistically steady incompressible turbulent channel flow at a friction Reynolds number Re is considered, where is the kinematic viscosity, the channel half-height, the friction velocity, the wall shear stress, and the density. Streamwise, spanwise, and wall-normal spatial coordinates are denoted by and the corresponding velocities by . We assume zero initial conditions and apply no-slip boundary conditions. Spatial variables are normalized by ,
Validation of the linear model with DNS
To validate the linear model, we employ a direct numerical simulation (DNS) dataset provided by the Polytechnic University of Madrid (Hoyas and Jiménez, 2006, Encinar et al., 2018). We will look at (i) the DNS model itself, (ii) the flow’s energy as a function of wavenumber ( and ), (iii) the flow’s energy as a function of wall height and (iv) a snapshot of streamwise velocity perturbations.
The control set-up
So far, we have introduced the eddy-viscosity-enhanced Orr–Sommerfeld and Squire operators that are linearized about the mean velocity profile of a turbulent channel flow. We stochastically force the linear operator to generate velocity perturbations that we now want to control. To do so, we include three new signals (, and ) into the state-space model ((4)):
The first new signal represents time-resolved velocity measurements from sensors (e.g. hotwires) that are contaminated by
Control performance
This section is in four parts: Section 5.1 examines the control performance at individual wavenumber pairs; Section 5.2 looks at the overall performance; Section 5.3 at the performance across individual wall heights; and Section 5.4 considers the energy consumed by actuation.
Conclusions
We have considered linear feedback control of a turbulent channel at Re using the linearized Navier–Stokes equations (LNS) which are formed about the turbulent mean. The linear operator is augmented with an eddy viscosity (following many previous studies) and is assumed to be stochastically forced. Applying any type of control will alter the mean velocity profile and with it the linear model itself. As a consequence, any controlled states cannot be fully described with the present
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank Sean Symon and Anagha Madhusudanan for their extensive feedback and support during the creation of this paper and are grateful for the financial support of the Australian Research Council .
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