Predicting the response of turbulent channel flow to varying-phase opposition control: Resolvent analysis as a tool for flow control design

Abstract

The present study evaluates the capabilities of a low-order flow model based on the resolvent analysis of McKeon and Sharma [B. J. McKeon and A. S. Sharma, J. Fluid Mech. 658, 336 (2010)] for the purpose of controller design for drag reduction in wall-bounded turbulent flows. To this end, we first show that the model is able to approximate the change in mean wall shear stress, which is commonly used as measure for drag reduction. We also derive an analytical expression that decomposes the drag reduction in internal flows into terms that can be predicted directly by the model and terms that allow for quantification of model error if high-fidelity data are available. We then show by example of varyingphase opposition control in a low-Reynolds-number turbulent channel flow that the drag reduction predicted by the resolvent model captures the trend observed in direct numerical simulation (DNS) over a wide range of controller parameters. The DNS results confirm the resolvent model prediction that the attainable drag reduction strongly depends on the relative phase between sensor measurement and actuator response, which raises interesting flow physics questions for future studies. The good agreement between the resolvent model and DNS further reveals that resolvent analysis, which at its heart is a linear technique, is able to approximate the response of the full nonlinear system to control. We also show that in order to make accurate predictions the model only needs to resolve a small subset of the DNS wave numbers and that the controlled resolvent modes obey the Reynoldsnumber scaling laws of the uncontrolled resolvent operator derived by Moarref et al. [R. Moarref et al., J. Fluid Mech. 734, 275 (2013)]. As a consequence, our results suggest that resolvent analysis can provide a suitable flow model to design feedback flow control schemes for the purpose of drag reduction in incompressible wall-bounded turbulent flows even at technologically relevant Reynolds numbers.

Interaction of forced Orr-Sommerfeld and Squire modes in a low-order representation of turbulent channel flow

“…10(b), respectively, using the scalings derived in Eqs. (29)- (31). For comparison, the leading response and forcing mode for the standard resolvent operator are also shown; as discussed in Section III A, ψ η and φ v are indistinguishable from the OS modes.…”

“…For the results presented in Section III A, which focus on Re τ = 2003, the DNS spectra are interpolated onto a grid of N kx = 30 by N kz = 31 logarithmically spaced wavenumbers, which is sufficient to reproduce the general shape of the spectra. Furthermore, it has recently been shown that statistics such as the uv Reynolds stress can be accurately reproduced even when retaining only about 2% of the wavenumbers from DNS [31]. Both the spectra and resolvent modes are interpolated onto a common grid of N y = 60 logarithmically spaced points in the wall-normal direction, and the wavespeed range c ∈ [0, U cl ] is discretized into N c = 100 linearly spaced wavespeeds.…”

Resolvent-based predictions for turbulent flow over anisotropic permeable substrates

“…2017; Toedtli et al. 2019; Chavarin & Luhar 2020) – is extended to account for the presence of permeable substrates as follows. For an extended discussion of resolvent analysis and its applications, the reader is referred to McKeon (2017).…”

“…The forcing-response gain (i.e. singular value) for this resolvent mode has also proven to be a useful predictor of control performance for both active (Luhar et al 2014;Nakashima et al 2017;Toedtli et al 2019) and passive (Luhar et al 2015;Luhar, Sharma & McKeon 2016;Chavarin & Luhar 2020) techniques. Here, we evaluate whether it can serve as a useful reduced-complexity tool for the evaluation of anisotropic permeable substrates for passive drag reduction.…”

Model-based design of riblets for turbulent drag reduction

“…In these studies, the effect of disturbances was modelled as an additive source of deterministic or stochastic excitation in the NS equations. Such an input-output approach (Jovanovic 2021) has also been used for the model-based design of sensor-free control strategies for suppressing turbulence via streamwise travelling waves (Lieu, Moarref & Jovanovic 2010; Moarref & Jovanovic 2010) and transverse wall oscillations (Jovanovic 2008; Moarref & Jovanovic 2012), as well as feedback control strategies (Kim & Bewley 2007) including opposition control (Luhar, Sharma & McKeon 2014; Toedtli, Luhar & McKeon 2019).…”