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Acta Mechanica

, Volume 229, Issue 11, pp 4701–4725 | Cite as

Near-critical turbulent open-channel flows over bumps and ramps

  • Wilhelm Schneider
  • Markus Müllner
  • Youichi Yasuda
Open Access
Original Paper
  • 79 Downloads

Abstract

Steady two-dimensional turbulent free-surface flow in a channel with a slightly uneven bottom is considered. The shape of the unevenness of the bottom can be in the form of a bump or a ramp of very small height. The slope of the channel bottom is assumed to be small, and the bottom roughness is assumed to be constant. Asymptotic expansions for very large Reynolds numbers and Froude numbers close to the critical value \({Fr} = 1\), respectively, are performed. The relative order of magnitude of two small parameters, i.e. the bottom slope and \(({Fr}-1)\), is defined such that no turbulence modelling is required. The result is a steady-state version of an extended Korteweg–de Vries equation for the surface elevation. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. An exact solution describing stationary solitary waves of the classical shape is obtained for a bottom of a particular shape. For more general shapes of ramps and bumps, stationary solitary waves of the classical shape are also obtained as a first approximation in the limit of small, but nonzero, dissipation. With the exception of an eigensolution for a ramp, an outer region has to be introduced. The outer solution describes a ’tail’ that is attached to the stationary solitary wave. In addition to the solutions of the solitary-wave type, solutions of smaller amplitudes are obtained both numerically and analytically. Experiments in a water channel confirm the existence of both types of stationary single waves.

Notes

Acknowledgements

Open access funding provided by TU Wien (TUW). The authors are indebted to Prof. Oscar Castro-Orgaz for his encouragement to include variations in bottom shape into the asymptotic analysis of turbulent free-surface flow. Dr. Richard Jurisits’ yet unpublished numerical solutions of the extended KdV equation helped the authors to cope with numerical problems and find the solutions of the second kind. Dr. Christoph Buchner, Dr. Richard Jurisits, Dr. Bernhard Scheichl and a reviewer provided useful references. The reviewers’ comments led to various improvements of the presentation, including the supply of additional information in three appendices. Mr. Dominik Murschenhofer prepared the Open image in new window file. Finally, financial support by Androsch International Management Consulting GmbH is gratefully acknowledged.

Supplementary material

Supplementary material 1 (mp4 14048 KB)

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Authors and Affiliations

  1. 1.Institute of Fluid Mechanics and Heat TransferTechnische Universität WienViennaAustria
  2. 2.Institute of AerodynamicsRWTH Aachen UniversityAachenGermany
  3. 3.Department of Civil Engineering, College of Science and TechnologyNihon UniversityTokyoJapan

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