Preventing transition to turbulence using streamwise traveling waves: theoretical analysis

Abstract

We assess the efficacy of a zero-net-mass-flux blowing and suction in the form of streamwise traveling waves for controlling the onset of turbulence in a channel flow. For small amplitude actuation along the walls, we utilize perturbation analysis to determine modifications in the base flow and to examine the resulting net power balance. Sensitivity of the velocity fluctuations around this base flow is then employed as a basis for selection of traveling wave parameters. Our simulation-free approach reveals that, relative to the uncontrolled flow, the velocity fluctuations around the upstream traveling waves at best exhibit similar sensitivity to background disturbances. In contrast, the downstream traveling waves with properly designed speed and frequency can significantly reduce sensitivity which makes them wellsuited for preventing transition. These theoretical predictions are facilitated by perturbation analysis (in the wave amplitude) of the linearized Navier-Stokes equations, and they are verified using full-scale simulations of the nonlinear flow dynamics in companion paper, [1].

Preventing transition to turbulence using streamwise traveling waves: direct numerical simulations

“…The traveling waves that are considered theoretically in [1] are tested in DNS of a 3D transitional channel flow. All simulations performed in this work use the code developed in [4].…”

“…The base velocity, u b = (U, V, W = 0), represents the steady-state solution to (1) in the presence of the following boundary conditions…”

“…However, we show that this comes at the expense of poor net power balance in flows driven by fixed pressure gradient. Our numerical simulations elucidate the predictive power of the theoretical framework developed in [1] and suggest that the linearized NS equations with uncertainty represent an effective control-oriented model for maintaining the laminar flow.…”

“…For example, an extensive number of simulations on turbulent drag reduction by means of spanwise wall oscillation was conducted in [3] where 37 cases of different control parameters were considered. Instead of an extensive numerical study, this paper builds directly on the theoretical findings of a companion paper [1] where system-theoretic approach was used to show that downstream traveling waves (DTWs) are capable of weakening the sensitivity to background disturbances of the most energetic structures in a flow with no control. The effectiveness of DTWs and UTWs in preventing or enhancing transition is examined in this work.…”

“…Financial support from the National Science Foundation under CAREER Award CMMI-06-44793 is gratefully acknowledged. We use high-fidelity simulations to demonstrate that the DTWs with parameters selected in [1] can prevent transition and achieve positive net efficiency relative to the uncontrolled flow that becomes turbulent. On the contrary, the UTWs induce turbulence even when the uncontrolled flow stays laminar.…”