Model-based design of transverse wall oscillations for turbulent drag reduction

Abstract

Over the last two decades, both experiments and simulations have demonstrated that transverse wall oscillations with properly selected amplitude and frequency can reduce turbulent drag by as much as 40%. In this paper, we develop a model-based approach for designing oscillations that suppress turbulence in a channel flow. We utilize eddyviscosity-enhanced linearization of the turbulent flow with control in conjunction with turbulence modeling to determine skin-friction drag in a simulation-free manner. The Boussinesq eddy viscosity hypothesis is used to quantify the effect of fluctuations on the mean velocity in the flow subject to control. In contrast to the traditional approach that relies on numerical simulations, we determine the turbulent viscosity from the second order statistics of the linearized model driven by white-in-time stochastic forcing. The spatial power spectrum of the forcing is selected to ensure that the linearized model for the uncontrolled flow reproduces the turbulent energy spectrum. The resulting correction to the turbulent mean velocity induced by small amplitude wall movements is then used to identify the optimal frequency of drag reducing oscillations. In addition, the control net efficiency and the turbulent flow structures that we obtain agree well with the results of numerical simulations and experiments. This demonstrates the predictive power of our model-based approach to controlling turbulent flows and is expected to pave the way for successful flow control at higher Reynolds numbers than currently possible. arXiv:1206.0101v1 [physics.flu-dyn]

The use of the r* heuristic in covariance completion problems

“…Figures 3a and 3b respectively show the r-dependence and γ-dependence of the relative error of solutions to (8) and (9). For problem (8), minimum relative error is achieved with r = 2, as expected for a system with two inputs.…”

“…For problem (8), minimum relative error is achieved with r = 2, as expected for a system with two inputs. On the other hand, γ = 8.46 gives the best completion in problem (9).…”

“…Figure 4 shows the recovered covariance matrix of the discretized heat equation resulting from problems (8) and (9) and for various values of r and γ. Figures 4a and 4b correspond to the case of nuclear norm minimization (r = 1) and optimal covariance completion (r = 2) for problem (8). On the other hand, Figs.…”

“…On the other hand, nuclear norm minimization, which corresponds to solving (8) with r = 1 and (9) with γ = ∞, results in the worst completion (46%). Figure 7 shows the recovered covariance matrices of position X pp and velocity X vv resulting from optimization problems (8) and (9) with various values of r and γ. While nuclear norm minimization yields poor recovery of the diagonally dominant covariance matrix of velocities Σ vv (cf.…”

Opposition control within the resolvent analysis framework

“…Bechert et al 1997;Xu, Rempfer & Lumley 2003;Fukagata et al 2008;Choi et al 2012), open-loop active control (transverse wall oscillations, upstream-travelling waves of blowing and suction, streamwise waves of spanwise velocity at the wall (see e.g. Quadrio & Ricco 2004;Min et al 2006;Quadrio, Ricco & Viotti 2009;Moarref & Jovanovic 2012)), as well as feedback flow control.…”