Diminishing effect of a pressure gradient on large-scale rolls of plane Couette flow: A singular value analysis

We investigate the transition from pure Couette flow to Poiseuille flow at <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mtext>Re</a:mtext><a:mo>≈</a:mo><a:mn>2250</a:mn></a:mrow></a:math> with a focus on the observed disappearance of single-circle coherent structures occupying the whole channel height. According to the DNS study by Gandía-Barberá [], these are accompanied by a change of sign in the Reynolds stress tensor at sufficiently large pressure gradients, although from a deeper theoretical point of view the actual mechanisms for the disappearance of these structures remained completely uncovered. In order to investigate the behavior of these structures theoretically, we use resolvent and structured singular value analysis, with the aforementioned DNS serving as a data basis for comparison, hence simultaneously exploring the capabilities of resolvent and structured singular value analysis in capturing the characteristics of these structures. Through detailed parameter studies, we examine how the pressure gradient influences the length and width of streamwise elongated rolls, noting a shortening and narrowing effect as the flow becomes increasingly pressure driven, and compare these effects to the ones observed in the DNS work. Additional insights into the development of coherent structures are gained, as their wall-normal positions are successfully linked to the occurrence of critical layers within the flow, particularly through structured singular value analysis. The results show that the modified response modes from structured singular value analysis show the best agreement with the DNS observations, suggesting that these analytical approaches can effectively replicate the phenomena otherwise obtained only through costly DNS, and provide deeper insights into the underlying fluid dynamics.