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Turbulence spectra in smooth- and rough-wall pipe flow at extreme Reynolds numbers

Published online by Cambridge University Press:  14 August 2013

B. J. Rosenberg ,
M. Hultmark ,
M. Vallikivi ,
S. C. C. Bailey  and
A. J. Smits
B. J. Rosenberg*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
M. Hultmark
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
M. Vallikivi
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
S. C. C. Bailey
Affiliation:
Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA
A. J. Smits
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Monash University, VIC 3800, Australia
*
†Email address for correspondence: rosenber@princeton.edu
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Abstract

Well-resolved streamwise velocity spectra are reported for smooth- and rough-wall turbulent pipe flow over a large range of Reynolds numbers. The turbulence structure far from the wall is seen to be unaffected by the roughness, in accordance with Townsend’s Reynolds number similarity hypothesis. Moreover, the energy spectra within the turbulent wall region follow the classical inner and outer scaling behaviour. While an overlap region between the two scalings and the associated ${ k}_{x}^{- 1} $ law are observed near ${R}^{+ } \approx 3000$, the ${ k}_{x}^{- 1} $ behaviour is obfuscated at higher Reynolds numbers due to the evolving energy content of the large scales (the very-large-scale motions, or VLSMs). We apply a semi-empirical correction (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26) to the experimental data to estimate how Taylor’s frozen field hypothesis distorts the pseudo-spatial spectra inferred from time-resolved measurements. While the correction tends to suppress the long wavelength peak in the logarithmic layer spectrum, the peak nonetheless appears to be a robust feature of pipe flow at high Reynolds number. The inertial subrange develops around ${R}^{+ } \gt 2000$ where the characteristic ${ k}_{x}^{- 5/ 3} $ region is evident, which, for high Reynolds numbers, persists in the wake and logarithmic regions. In the logarithmic region, the streamwise wavelength of the VLSM peak scales with distance from the wall, which is in contrast to boundary layers, where the superstructures have been shown to scale with boundary layer thickness throughout the entire shear layer. Moreover, the similarity in the streamwise wavelength scaling of the large- and very-large-scale motions supports the notion that the two are physically interdependent.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 731 , 25 September 2013 , pp. 46 - 63
Copyright
©2013 Cambridge University Press 

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