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The perturbation of a turbulent boundary layer by a two-dimensional hill

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Abstract

Turbulent boundary layers over flat walls in the presence of a hill are frequently found in nature and industry. Some examples are the airflows over hills and desert dunes, and also water flows over aquatic dunes inside closed conduits. The perturbation of a two-dimensional boundary layer by a hill introduces new scales in the problem, changing the way in which velocities and stresses are distributed along the flow. When in the presence of sediment transport, the stress distribution along the hill is strongly related to bed instabilities. This paper presents an experimental study on the perturbation of a fully developed turbulent boundary layer by a two-dimensional hill. Water flows were imposed over a hill fixed on the bottom wall of a closed conduit and the flow field was measured by particle image velocimetry. From the flow measurements, mean and fluctuation fields were computed. The general behaviors of velocities and stresses are compared to published asymptotic analyses and the surface shear stress is analyzed in terms of instabilities of a granular bed.

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Abbreviations

A :

Constant

B :

Constant

B e :

Constant

f :

Darcy friction factor

g :

Acceleration of gravity (ms−2)

H :

Channel height (m)

h :

Hill’s local height (m)

H eff :

Distance from the PVC bed to the top wall (m)

i :

Imaginary number

k :

Wavenumber (m−1)

L :

Longitudinal distance between the crest and the position where the local height is half of its maximum value (m)

Q :

Water flow rate (m3/h)

Re :

Channel Reynolds number, \(Re=\overline{U}2H_\text{eff}/\nu\)

\(\overline{U}\) :

Cross-sectional mean velocity of the fluid (m/s)

u :

Longitudinal component of the mean fluid velocity (ms−1)

u′:

Longitudinal component of the velocity fluctuation (ms−1)

u * :

Shear velocity (ms−1)

u + :

Dimensionless velocity, u + = u/u *

\(-\overline{u'v'}\) :

xy component of the Reynolds shear stress, (m/s)2

\(\mathbf{V}\) :

Mean fluid velocity (ms−1)

v :

Vertical component of the mean fluid velocity (ms−1)

v′:

Vertical component of the velocity fluctuation (ms−1)

x :

Longitudinal coordinate (m)

y :

vertical coordinate, m

y d :

Displaced vertical coordinate (m)

y 0 :

Roughness length (m)

y + :

Dimensionless vertical coordinate, y + = yu */ν

κ:

von Kármán constant

λ:

Wavelength (m)

ν:

Kinematic viscosity (m2/s)

ρ:

Specific mass of the fluid (kg/m3)

τ:

Shear stress on the bed (N/m2)

ξ:

Integration variable (m)

k :

Relative to the Fourier space

x :

Relative to the real space

0:

Relative to the flat wall (except in y 0)

\(\hat{}\) :

Relative to the perturbation

′:

Fluctuation

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Acknowledgments

The authors are grateful to Petrobras S.A. (contract number 0050.0045763.08.4). Erick M. Franklin is grateful to FAEPEX/UNICAMP (conv. 519.292, project 1435/12) and to FAPESP (contract number 2012/19562-6).

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Author notes

  1. Guilherme Augusto Ayek

    Present address: Benteler Automotive, Campinas, Brazil 

Authors and Affiliations

  1. Faculty of Mechanical Engineering, University of Campinas-UNICAMP, Campinas, Brazil 

    Erick de Moraes Franklin & Guilherme Augusto Ayek

Authors
  1. Erick de Moraes Franklin
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  2. Guilherme Augusto Ayek
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Correspondence to Erick de Moraes Franklin.

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Technical Editor: Francisco Ricardo Cunha.

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Franklin, E.d.M., Ayek, G.A. The perturbation of a turbulent boundary layer by a two-dimensional hill. J Braz. Soc. Mech. Sci. Eng. 35, 337–346 (2013). https://doi.org/10.1007/s40430-013-0024-z

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  • DOI: https://doi.org/10.1007/s40430-013-0024-z

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