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Exact invariant solutions for grooved Couette and channel flows

Exact invariant solutions for grooved Couette and channel flows
Exact invariant solutions for grooved Couette and channel flows
The dynamical systems approach to turbulence has gained a lot of traction since the turn of the century. A large set of exact invariant solutions for canonical wall-bounded flows such as Couette, channel, and pipe flows has been found by researchers. These solutions, and their connections, are thought to form a skeleton for trajectories of turbulent flow. However, this vision of turbulence has not been extended to rough-walled flows despite the practical signficance of such flows in engineering applications. This thesis describes continuation, by numerical homotopy, of known equilibria from smooth-walled plane shear flows to grooved plane shear flows using a domain transformation method, with the hope that this exploratory work would inform later efforts to extend such solutions to rough-walled flows. As a precursor to computing non-laminar equilibria, laminar solutions are computed for grooved channel flows for transverse, longitudinal, and oblique grooves. In addition to the numerical solutions, analytical solutions are also derived for asymptotically long groove-wavelengths, employing the Stokes-flow approximation for transverse and oblique grooves.

Exact invariant solutions can indeed be continued from plane Couette flow (PCoF) with smooth walls to grooved PCoF with longitudinal grooves using a simple domain transformation method. However, smooth PCoF equilibria exist as continuous families of solutions that are identical up to a translational shift; the loss of spanwise homogeneity due to the grooves restricts such continuous families to discrete families due to symmetry-breaking. This phase-based restriction can also be expected to be reflected in turbulent statistics. Continuation of equilibria in grooves of different wavelengths shows a drag increasing tendency for grooves of the same wavelength as the vortex-streak structure, and a drag reducing tendency for grooves of significantly smaller wavelengths. This can relate the optimal spacing of riblets for maximal drag reduction to the spanwise spacing of the vortex-streak structures observed in the self-sustaining near-wall cycle.
University of Southampton
Vadarevu, Sabarish Bharadwaz
2ef2cfd4-0a70-4bfd-8eab-c5f36eb4183e
Vadarevu, Sabarish Bharadwaz
2ef2cfd4-0a70-4bfd-8eab-c5f36eb4183e
Sharma, Atul
cdd9deae-6f3a-40d9-864c-76baf85d8718

Vadarevu, Sabarish Bharadwaz (2017) Exact invariant solutions for grooved Couette and channel flows. University of Southampton, Doctoral Thesis, 207pp.

Record type: Thesis (Doctoral)

Abstract

The dynamical systems approach to turbulence has gained a lot of traction since the turn of the century. A large set of exact invariant solutions for canonical wall-bounded flows such as Couette, channel, and pipe flows has been found by researchers. These solutions, and their connections, are thought to form a skeleton for trajectories of turbulent flow. However, this vision of turbulence has not been extended to rough-walled flows despite the practical signficance of such flows in engineering applications. This thesis describes continuation, by numerical homotopy, of known equilibria from smooth-walled plane shear flows to grooved plane shear flows using a domain transformation method, with the hope that this exploratory work would inform later efforts to extend such solutions to rough-walled flows. As a precursor to computing non-laminar equilibria, laminar solutions are computed for grooved channel flows for transverse, longitudinal, and oblique grooves. In addition to the numerical solutions, analytical solutions are also derived for asymptotically long groove-wavelengths, employing the Stokes-flow approximation for transverse and oblique grooves.

Exact invariant solutions can indeed be continued from plane Couette flow (PCoF) with smooth walls to grooved PCoF with longitudinal grooves using a simple domain transformation method. However, smooth PCoF equilibria exist as continuous families of solutions that are identical up to a translational shift; the loss of spanwise homogeneity due to the grooves restricts such continuous families to discrete families due to symmetry-breaking. This phase-based restriction can also be expected to be reflected in turbulent statistics. Continuation of equilibria in grooves of different wavelengths shows a drag increasing tendency for grooves of the same wavelength as the vortex-streak structure, and a drag reducing tendency for grooves of significantly smaller wavelengths. This can relate the optimal spacing of riblets for maximal drag reduction to the spanwise spacing of the vortex-streak structures observed in the self-sustaining near-wall cycle.

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Published date: December 2017

Identifiers

Local EPrints ID: 417860
URI: http://eprints.soton.ac.uk/id/eprint/417860
PURE UUID: 73363f5a-b770-4da9-8c60-e12f2ae22085
ORCID for Atul Sharma: ORCID iD orcid.org/0000-0002-7170-1627

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Date deposited: 15 Feb 2018 17:31
Last modified: 14 Mar 2019 01:34

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