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On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Abstract

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: \({Z\propto{\rm Re}^{0.8}}\) and \({P\propto {\rm Re}^{2.25}}\) for 5 × 102 ≤ Re ≤ 2 × 104 and \({Z\propto{\rm Re}^{0.5}}\) and \({P\propto{\rm Re}^{1.5}}\) for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re c (here, \({{\rm Re}_c\approx 2\times 10^4}\)) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: \({Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}}\) , and dP/dt \({\propto {\rm Re}^{11/4}}\) in agreement with the numerically obtained scaling laws. For Re ≥ Re c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: \({Z\propto{\rm Re}^{1/2}}\) and \({P\propto {\rm Re}^{3/2}}\).

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Acknowledgements

The authors gratefully acknowledge financial support from the Dutch Foundation for Fundamental Research on Matter (FOM) and from the Technology Foundation (STW) under Grant No. ESF.6239, which are both financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scientific Research, NWO). Part of this study was sponsored by the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation, NCF) for the use of supercomputer facilities, with financial support from NWO.

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Correspondence to H. J. H. Clercx.

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Communicated by P. R. Spalart

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Keetels, G.H., Kramer, W., Clercx, H.J.H. et al. On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions. Theor. Comput. Fluid Dyn. 25, 293–300 (2011). https://doi.org/10.1007/s00162-010-0205-7

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Keywords

  • Dipole-wall collision
  • Confined 2D turbulence
  • Boundary layers
  • Vorticity production