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

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 varying-phase 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 Reynolds-number 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.

Figure 1

Uncontrolled Reynolds stress profiles from DNS and the resolvent model: (a) , DNS data of Lee and Moser [28] (uncontrolled flow); , present DNS (uncontrolled flow) and (b) , resolvent model baseline (uncontrolled flow); , subsampled model (uncontrolled flow); , linear fit to model Reynolds stress profile in the outer flow.

Figure 2

Effect of wave speed and streamwise wave number on the mean Reynolds stress contribution of individual resolvent modes: (a) effect of wave speed $c$ (for $k_x=5.5, k_z=2$, and $c^{+}=\{8,10,12,14\}$), with darker colors indicating faster wave speeds, and (b) effect of streamwise wave number $k_x$ (for $k_x=\{5,5.5,6,6.5\}, k_z=2$, and $c^{+}=10$), with darker colors indicating smaller streamwise wave numbers.

Figure 3

Contour map showing the normalized drag reduction $\xi$ as a function of sensor location $y_d^{+}$ and phase shift $\angle {\widehat{A}}_d$ for (a) the prediction of the resolvent model and (b) the DNS data. Note that the color scale is nonlinear in order to highlight the region of drag reduction. The vertical dashed black line denotes $\angle {\widehat{A}}_d=0$, which is closely related to opposition control, and the contour lines of (a) are replotted as dotted blue lines in (b). Nondimensionalization to inner units is based on ${\left(u_{\tau }\right)}_u$ of the uncontrolled flow and the maximum DR used to normalize the contour maps is (a) resolvent model $max\mathrm{DR}=5\%$ and (b) DNS $max\mathrm{DR}=21\%$.

Figure 4

Reynolds stress profiles of the uncontrolled flow and an example controlled flow ($y_d^{+}=10$ and $\angle {\widehat{A}}_d=-\pi /4$) from DNS and the resolvent model: (a) , present DNS (uncontrolled flow); , present DNS (controlled flow) and (b) , subsampled model (no-slip boundary conditions (BCs) with uncontrolled mean); , subsampled model (control BCs with uncontrolled mean); , subsampled model (no-slip BCs with controlled mean); , subsampled model (control BCs with controlled mean). The vertical line indicates the sensor location $y_d^{+}$, where ${\left(u_{\tau }\right)}_u$ of the uncontrolled flow is used to nondimensionalize to inner units.

Figure 5

Mean Reynolds stress contribution of an example resolvent mode characterized by ${\lambda }_x^{+}=1000, {\lambda }_z^{+}=100$, and $c^{+}=10$ at (a) ${\mathrm{Re}}_{\tau }=180$ (), (b) ${\mathrm{Re}}_{\tau }=6000$ (), and (c) ${\mathrm{Re}}_{\tau }=30000$ (). In all figures, the solid line shows the profile of the uncontrolled flow, while the dotted line displays the profile of an example controlled flow ($y_d^{+}=10$ and $\angle {\widehat{A}}_d=-\pi /4$). (d) Same profiles rescaled according to the Reynolds number scaling law introduced by Moarref et al. [8].