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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]
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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

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