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On the skin friction due to turbulence in ducts of various shapes

Published online by Cambridge University Press:  15 January 2018

P. R. Spalart*
Affiliation:
Boeing Commercial Airplanes, Seattle, WA 98124, USA
A. Garbaruk
Affiliation:
Saint Petersburg Polytechnic University, Saint Petersburg 195251, Russia
A. Stabnikov
Affiliation:
Saint Petersburg Polytechnic University, Saint Petersburg 195251, Russia
*
Email address for correspondence: philippe.r.spalart@boeing.com

Abstract

We consider fully developed turbulence in straight ducts of non-circular cross-sectional shape, for instance a square. A global friction velocity $\overline{u}_{\unicode[STIX]{x1D70F}}$ is defined from the streamwise pressure gradient $|\text{d}p/\text{d}x|$ and a single characteristic length $h$, half the hydraulic diameter (shapes with disparate length scales, due to high aspect ratio, are excluded). We reason that as the Reynolds number $Re$ reaches high values, outside the viscous region the streamwise velocity differences and the secondary motion scale with $\overline{u}_{\unicode[STIX]{x1D70F}}$ and the Reynolds stresses with $\overline{u}_{\unicode[STIX]{x1D70F}}^{2}$. This extends the classical defect-law argument, associated with Townsend and many others, and is successful in channel and pipe flows. We then posit matched asymptotic expansions with overlap of the law of the wall and the behaviour we assumed in the core region. The wall may be smooth, or have a Nikuradse roughness $k_{S}$ (such that it is fully rough, with $k_{S}^{+}\gg 1$). The consequences include the familiar logarithmic behaviour of the velocity profile, but also the surprising prediction that the skin friction tends to uniformity all around the duct, except near possible corners, asymptotically as $Re\rightarrow \infty$ or $k_{S}/h\rightarrow 0$. This is confirmed by numerical solutions for a square and two ellipses, using a conventional turbulence model, albeit the trend with Reynolds number is slow. The magnitude of the secondary motion also scales as expected, and the skin-friction coefficient follows the logarithm of the appropriate Reynolds number. This is a validation of the mathematical reasoning, but is by no means independent physical evidence, because the turbulence models embody the same assumptions as the theory. The uniformity of the skin friction appears to be a new and falsifiable deduction from turbulence theory, and a candidate for high-Reynolds-number experiments.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

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References

Bradshaw, P. 1987 Turbulent secondary flows. Annu. Rev. Fluid Mech. 19, 5374.Google Scholar
Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2013 Logarithmic scaling of turbulence in smooth- and rough-wall pipe flow. J. Fluid Mech. 718, 376395.Google Scholar
Millikan, C. 1938 A critical discussion of turbulent flows in channels and circular tubes. In Proceedings of the 5th International Congress for Applied Mechanics, Cambridge, pp. 386392. Wiley.Google Scholar
Nikitin, N. & Yakhot, A. 2005 Direct numerical simulation of turbulent flow in elliptical ducts. J. Fluid Mech. 532, 141164.Google Scholar
Pirozzoli, S., Modesti, D., Orlandi, P. & Grasso, F. 2017 Turbulence and secondary motions in square duct flow. J. Fluid Mech. (submitted).Google Scholar
Pullin, D. I., Inoue, M. & Saito, N. 2013 On the asymptotic state of high Reynolds number, smooth-wall turbulent flows. Phys. Fluids 25, 015116.Google Scholar
Raiesi, H., Piomelli, U. & Pollard, A. 2011 Evaluation of turbulence models using direct numerical and large-eddy simulation data. J. Fluids Engng 133 (2), 021203.Google Scholar
Spalart, P. R.2000 Trends in turbulence treatments. AIAA Paper 2000-2306.Google Scholar
Spalart, P. R. & Allmaras, S. R. 1994 A one-equation turbulence model for aerodynamic flows. La Recherche Aérospatiale 1, 521.Google Scholar
Speziale, C. G. 1982 On turbulent secondary flows in pipes of noncircular cross-section. Intl J. Engg Sci. 20 (7), 863872.Google Scholar
Townsend, A. A. 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97120.Google Scholar