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Stabilisation and drag reduction of pipe flows by flattening the base profile

Published online by Cambridge University Press:  28 January 2019

Elena Marensi Open the ORCID record for Elena Marensi [Opens in a new window] ,
Ashley P. Willis Open the ORCID record for Ashley P. Willis [Opens in a new window]  and
Rich R. Kerswell
Elena Marensi*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
Ashley P. Willis
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
Rich R. Kerswell
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK
*
†Email address for correspondence: e.marensi@sheffield.ac.uk
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Abstract

Recent experimental observations (Kühnen et al., Nat. Phys., vol. 14, 2018b, pp. 386–390) have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon exists for laminar pipe flow profiles in the sense that the nonlinear stability of the laminar state is enhanced as the profile becomes more flattened. The flattening of the laminar base profile is produced by an artificial localised body force designed to mimic an obstacle used in the experiments of Kühnen et al. (Flow Turbul. Combust., vol. 100, 2018a, pp. 919–943) and the nonlinear stability measured by the size of the energy of the initial perturbations needed to trigger transition. Significant drag reduction is also observed for the turbulent flow when triggered by sufficiently large disturbances. In order to make the nonlinear stability computations more efficient, we examine how indicative the minimal seed – the disturbance of smallest energy for transition – is in measuring transition thresholds. We first show that the minimal seed is relatively robust to base profile changes and spectral filtering. We then compare the (unforced) transition behaviour of the minimal seed with several forms of randomised initial conditions in the range of Reynolds numbers $Re=2400$–$10\,000$ and find that the energy of the minimal seed after the Orr and oblique phases of its evolution is close to that of a critical localised random disturbance. In this sense, the minimal seed at the end of the oblique phase can be regarded as a good proxy for typical disturbances (here taken to be the localised random ones) and is thus used as initial condition in the simulations with the body force. The enhanced nonlinear stability and drag reduction predicted in the present study are an encouraging first step in modelling the experiments of Kühnen et al. and should motivate future developments to fully exploit the benefits of this promising direction for flow control.

JFM classification

Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 850 - 875
Copyright
© 2019 Cambridge University Press 

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