Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T19:30:40.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent boundary-layer flow beneath a vortex. Part 1. Turbulent Bödewadt flow

Published online by Cambridge University Press:  03 April 2020

David E. Loper Open the ORCID record for David E. Loper [Opens in a new window]
David E. Loper*
Affiliation:
Professor Emeritus, Florida State University, Tallahassee, FL 32306, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The equations governing the mean fluid motions within a turbulent boundary layer adjoining a stationary plane beneath an axisymmetric circumferential flow $v_{\infty }(r)$, where $r$ is cylindrical radius, are solved by assuming the eddy diffusivity is proportional to $v_{\infty }$ times a diffusivity function $L(r,z)$, where $z$ is axial distance from the plane. The boundary-layer shape and structure depend on the dimensionless vorticity $\unicode[STIX]{x1D703}=\text{d}(rv_{\infty })/2v_{\infty }\,\text{d}r$, but are independent of the strength of the circumferential flow. This problem has been solved using a spectral method in the case of rigid-body motion ($\unicode[STIX]{x1D703}=1$ and $v_{\infty }\sim r$) for two models of $L$: $L$ constant (model A) and $L$ constant within a rough layer of thickness $z_{0}$ adjoining the boundary and increasing linearly with $z$ outside that layer (model B). The influence of the rough layer is quantified by the dimensionless radial coordinate $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D716}r/z_{0}$, where $\unicode[STIX]{x1D716}\ll 1$. The boundary-layer thickness varies parabolically with $r$ for model A and nearly linearly with $r$ for model B. Inertial stability of the outer flow causes the velocity components to decay with axial distance as exponentially damped oscillations, with the radial flow consisting of a sequence of jets. Axial flow is positive (flowing out of the boundary layer). Outflow from the layer, velocity gradients at the bounding plane, meridional-plane circulation and oscillations all increase as radius decreases.

JFM classification

Boundary Layers: Boundary layer structure Geophysical and Geological Flows: Atmospheric flows Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 892 , 10 June 2020 , A16
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bak, P. 1996 How Nature Works, p. 212. Springer.CrossRefGoogle Scholar
Bĕlík, P., Dokken, D. P., Schloz, K. & Shvartsman, M. M. 2014 Fractal powers in Serrin’s swirling vortex solutions. Asymptotic Solutions 90, 53–82.CrossRefGoogle Scholar
Bödewadt, U. T. 1940 Die drehströmung über festum grund. Z. Angew. Math. Mech. 20, 241–253.CrossRefGoogle Scholar
Eliassen, A. 1971 On the Ekman layer in a circular vortex. J. Met. Soc. Japan 49, 784–789.Google Scholar
Fiedler, B. H. & Garfield, G. S. 2010 Axisymmetric vortex simulations with various turbulence models. GFD Lett. 2, 112–122.Google Scholar
Foster, R. C. 2009 Boundary-layer similarity under an axisymmetric, gradient wind vortex. Boundary-Layer Meteorol. 131, 321–344.CrossRefGoogle Scholar
Goldshtik, M. A. 1990 Viscous flow paradoxes. Annu. Rev. Fluid Mech. 22, 441–472.CrossRefGoogle Scholar
Holton, J. R. 2004 An Introduction to Dynamic Meterology, p. 529. Elsevier Academic Press.Google Scholar
von Kármán, T. 1921 Über laminare und turbulente reibung. Z. Angew. Math. Mech. 1, 233–252.CrossRefGoogle Scholar
Kepert, J. D. 2010a Slab- and height-resolving models of the tropical cyclone boundary layer. Part I. Comparing the simulations. Q. J. R. Meteorol. Soc. 136, 1686–1699.CrossRefGoogle Scholar
Kepert, J. D. 2010b Slab- and height-resolving models of the tropical cyclone boundary layer. Part II. Why the simulations differ. Q. J. R. Meteorol. Soc. 136, 1700–1711.CrossRefGoogle Scholar
Kuo, H. L. 1971 Axisymmetric flows in the boundary layer of a maintained vortex. J. Atmos. Sci. 28, 20–41.2.0.CO;2>CrossRefGoogle Scholar
Long, R. R. 1958 Vortex motion in a viscous fluid. J. Meteorol. 15, 108–112.2.0.CO;2>CrossRefGoogle Scholar
Loper, D. E. 2017 Geophysical Waves and Flows: Theory and Application in the Atmosphere, Hydrosphere and Geosphere, p. 505. Cambridge University Press.CrossRefGoogle Scholar
Loper, D. E. 2020 Turbulent boundary-layer flow beneath a vortex. Part 2. Power-law swirl. J. Fluid Mech. 892, A17.Google Scholar
Nolan, D. S. 2013 On the use of doppler radar-derived wind fields to diagnose the secondary circulations of tornadoes. J. Atmos. Sci. 70, 1160–1171.CrossRefGoogle Scholar
Nolan, D. S., Dahl, N. A., Bryan, G. H. & Rotunno, R. 2017 Tornado vortex structure, intensity, and surface wind gusts in large-eddy simulations with fully developed turbulence. J. Atmos. Sci. 74, 1573–1597.CrossRefGoogle Scholar
Oruba, L., Davidson, P. A. & Dormy, E. 2017 Eye formation in rotating convection. J. Fluid Mech. 812, 890–904.CrossRefGoogle Scholar
Oruba, L., Davidson, P. A. & Dormy, E. 2018 Formation of eyes in large-scale cyclonic vortices. Phys. Rev. Fluids 3, 013502.CrossRefGoogle Scholar
Rotunno, R. 2013 The fluid dynamics of tornadoes. Annu. Rev. Fluid Dyn. 45, 59–84.CrossRefGoogle Scholar
Schlichting, H. 1968 Boundary Layer Theory, p. 747. McGraw Hill.Google Scholar
Smith, R. K. & Montgomery, M. T. 2010 Hurricane boundary-layer theory. Q. J. R. Meteorol. Soc. 136, 1665–1670.CrossRefGoogle Scholar
Smith, R. K. & Montgomery, M. T. 2014 On the existence of the logarithmic surface layer in the inner core of hurricanes. Q. J. R. Meteorol. Soc. 140, 72–81.CrossRefGoogle Scholar
Smith, R. K., Montgomery, M. T. & Vogl, S. 2008 A critique of Emanuel’s hurricane model and potential intensity theory. Q. J. R. Meteorol. Soc. 134, 551–561.CrossRefGoogle Scholar
Smith, R. K. & Vogl, S. 2008 A simple model of the hurricane boundary layer revisited. Q. J. R. Meteorol. Soc. 134, 337–351.CrossRefGoogle Scholar
Stewartson, K. 1953 On the flow between two rotating coaxial disks. Proc. Camb. Phil. Soc. 49, 333–341.CrossRefGoogle Scholar
Zandbergen, P. H. & Dijkstra, D. 1987 von Kármán swirling flows. Annu. Rev. Fluid Mech. 19, 465–491.CrossRefGoogle Scholar