On assessing the control of transition to turbulence: the example of plane Couette flow

  • Beaume, Cedric (University of Leeds)
  • Pershin, Anton (Huawei Technologies Co. Ltd.)
  • Eaves, Tom (University of Dundee)
  • Li, Kuan (University of Leeds)
  • Tobias, Steve (University of Leeds)

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Transition to turbulence is a basic fluid instability that can be triggered by the presence of internal shear within the flow. Depending on the application, either the laminar flow or turbulence is preferred: fluid transport is more efficient when the flow is laminar due to lower wall friction while turbulence can be advantageous in mixing processes. In either case, being able to enhance or suppress the effects of turbulence is crucial to improve the efficiency of many fluid flows. In plane Couette flow, transition to turbulence takes place subcritically: the laminar flow is linearly stable but transition can be initiated by a suitably large perturbation. Our approach to tackle the flow control problem is not to try to directly control the turbulent flow properties but rather to control the robustness of the laminar flow. Where turbulence is undesired, we aim to increase the latter, thereby reducing the risks of transition and statistically alleviating its effects. Unfortunately, assessing the robustness of the laminar flow is not straightforward: whether a perturba- tion will decay or trigger transition depends on its amplitude but also on its shape. The finite-amplitude nature of the instability implies that we need an integral understanding of the basin of attraction of the laminar flow. To achieve it, we partition perturbations from the laminar flow according to the value of their kinetic energy. For each energy partition, we calculate numerically the probability that randomly shaped perturbations decay or laminarize. We represent the results via the laminarization probability as a function of the perturbation energy, which gives a measure of the relative size of the basin of attraction of the laminar flow. We apply this protocol to several values of the Reynolds number to confirm the fact that the sensitivity of the laminar flow to perturbation increases with the Reynolds number and to establish a benchmark against which the efficiency of flow control strategies can be tested (Pershin et al., 2020). We then control the flow by adding in-phase, spanwise oscillations to the walls, a well-established strategy used to control developed turbulence (Quadrio & Ricco, 2004) and to stabilize the laminar flow by increasing the minimal energy required to trigger turbulence (Rabin et al., 2014). Reproducing the computation of the laminarization probability for a range of control parameter values (oscillation amplitude and frequency), we determine optimal control