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Statistical Stability and Fast Transient Growth in Wall-Bounded Turbulence



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Markeviciute, Vilda 


Shear-driven turbulence can be generated in wall-bounded flows due to the non-slip boundary. The simple geometry of a wall and thus the shear-driven turbulence are ubiquitous in nature and engineering applications. Hence, prediction and modelling of the turbulence appearance and maintenance in wall-bounded flows have major implications to flow control. However, the fully developed turbulent dynamics of wall-bounded turbulence are rich and not yet fully understood. Prevalent methods to explain the driving mechanisms of turbulence structures and their interactions rely on the knowledge of the mean velocity profile and include linear models such as modal stability analysis and transient growth calculation. With a purpose to better the understanding of the relationship between the mean flow and the background turbulence structures, this thesis revisits the linear stability of the mean flow in statistical framework and explores the importance of transient growth in wall-bounded shear flows. The thesis starts with a discussion of the background and motivation of the work in chapter 1. Chapter 2 seeks to generate statistically stable turbulent states by revisiting a two- dimensional channel flow with fixed volume-flux via direct numerical simulations. As a consequence, a bi-stability of turbulent states is discovered for a range of Reynolds numbers providing two types of statistically steady states for the subsequent statistical stability analysis. The new states are asymmetric based on the mean shear at the channel walls and show different properties from their symmetric counterparts. The asymmetric states transition to turbulence by showing heightened turbulent dynamics near one of the channel walls, only resulting in a lower pressure gradient across the channel. Despite the differences, both symmetric and asymmetric states exhibit a strong travelling wave structure which is also found to be the ‘edge’ state separating the two turbulent attractors. The chapter describes how the asymmetric states evolve with increasing Reynolds number. Chapter 3 aims to extend the statistical stability ideas proposed by Malkus (1956) by considering the linear stability analysis of turbulent mean flow in statistical framework. A fully developed turbulent attractor is a stable fixed point from the perspective of the flow statistics. Posing the modal stability problem in the statistical framework should thus reflect this stability. The approach is extended to include not only the mean velocity profile as proposed by Malkus, but also the second order flow statistics. The extended Orr-Sommerfeld stability analysis is developed by mapping the statistical stability problem back to the physical space. The model is then tested on statistically stable turbulent states in two-dimensional channel generated in chapter 2, showing at least partial improvement in statistical stability classification. Possible model limitations and extensions are then discussed. Chapter 4 shifts the focus of the thesis from the statistical stability to transient growth properties of the mean flow following the recent work by Lozano-Duran et al. (2021), where it was shown that the transient growth around the two-dimensional velocity profile (mean velocity profile with a spanwise-dependent streak) is essential to sustain turbulence in three-dimensional channel flow. Therefore, this chapter proposes to model the streak dynamics through a simple two-stage transient growth calculation accurately capturing transient growth levels and timescales observed in simulations. Moreover, the importance of the spanwise dependence of the mean velocity profile (the ‘push-over’ mechanism) to reach substantial levels of transient growth is confirmed. Finally, base streak amplitude needed to achieve energy gains required to sustain turbulence can be identified and predicted for higher Reynolds numbers. Chapter 5 concludes the thesis by discussing the implications and future perspectives of the results.





Kerswell, Rich R


Fluid Dynamics, Turbulence


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge
EPSRC (2089776)