Author: Alec Linot, Postdoctoral Scholar and IDRE Fellow – Department of Mechanical and Aerospace Engineering
In industrial processes involving fluids, vast amounts of energy are lost as dissipation due to turbulent drag. The energy production required to sustain these processes accounts for 5% of all manmade carbon dioxide (Jimenez 2013). Consequently, even small improvements toward reducing these losses provide a massive opportunity for reducing energy consumption and carbon emissions. As such, a long-standing goal of fluid dynamicists is to understand the mechanisms through which laminar flows transition to turbulent ones, and how turbulent flows sustain themselves. Due to the fast time scales over which turbulent flows may evolve, and the challenges in measuring the fine details of these flows, it has become common to use direct numerical simulations (DNS), which are numerical solutions of the equations of motion that resolve all time- and length-scales. DNS is a powerful tool for modeling turbulent flows, but due to the high resolution required to resolve all the small-scale turbulent structures typical simulations may call for O(10^8 − 10^9) grid points.
In this work, we aim to find the optimal perturbations — the fastest growing perturbations — in unsteady flows, like the accelerating and decelerating airfoils depicted above. Unsteadiness is the rule, not the exception, yet few studies investigate these flows due to the computational difficulties they introduce. For example, the most naive approach would involve applying random perturbations to a DNS, and running many computationally expensive simulations to find the optimal perturbations. This would be both intractable and suboptimal. Instead, we will approach this problem by finding the fundamental solution operator A(t), which gives the linear mapping from a perturbation at time 0 to time t. Once we find this operator, the optimal perturbation is nothing but the leading left singular vector. Thus, this work involves both efficient ways of computing the fundamental solution operator and the singular value decomposition of this operator. Finding the optimal perturbations will provide insight into control methods for suppressing the transition to turbulence in these flows. Additionally, the tools developed here have broader use cases in problems such as predicting extreme events in geophysics and modeling disease outbreaks in network-based epidemiological models. Both of these problems are ones in which there is a time-varying base state and we are interested in preventing (or knowing) the growth of the optimal perturbation (e.g. extreme weather event or disease outbreak).