General Mean Velocity Distribution Law for Smooth-Wall Plane Couette Flow
Publication: Journal of Engineering Mechanics
Volume 144, Issue 1
Abstract
Plane Couette flow between two parallel smooth walls is one of the classic wall-bounded shear flows. Analytical description of this flow is still limited to the linear law for laminar flow, the classic law of the wall, and the velocity defect law for fully turbulent flow, although extensive direct numerical simulations (DNS) and laboratory experiments are available. This paper integrates the existing knowledge of mean velocity distribution from theory, experiments, and DNS into a single velocity distribution law by introducing a rational eddy viscosity model. Specifically, the eddy viscosity distribution is approximated by an even rational function which is cubic near the wall, linear in the log-law overlap, and symmetrical about the channel centerline. The rational eddy viscosity model leads to a general velocity distribution law in terms of four inverse hyperbolic tangent functions. This law reduces to the linear law for laminar flow, agrees with the classic van Driest law in the inner region, and is antisymmetrical about the channel centerline. Particularly, it well reproduces DNS and laboratory data for transitional and turbulent flows. Furthermore, this general velocity distribution law results in a general friction law. Finally, the rational eddy viscosity model has clear implications for other wall-bounded flows in future studies.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This research was supported by the United States Federal Highway Administration Hydraulics R&D Program (Contract No. DTFH61-11-D-00010) through the Genex System to the University of Nebraska-Lincoln; the open fund research program (Contract No. HESS-1604) at the State Key Lab of Hydraulic Engineering Simulation and Safety, Tianjin University, China; and the open fund research program at the State Key Lab of Hydraulics and Mountain River Engineering (Contract No. SKHL1511), Sichuan University, China. The author also thank the anonymous reviewers for constructive comments which improved the manuscript significantly.
References
Avsarkisov, V., Hoyas, S., Oberlack, M., and Garcia-Galache, J. P. (2014). “Turbulent plane Couette flow at moderately high Reynolds number.” J. Fluid Mech., 751(R1), 1–10.
Aydin, E. M., and Leutheusser, H. J. (1987). “Experimental investigation of turbulent plane-Couette flow.” ASME Forum on Turbulence Flow, FED, ASME, New York, 51–54.
Aydin, E. M., and Leutheusser, H. J. (1991). “Plane-Couette flow between smooth and rough walls.” Exp. Fluids, 11(5), 302–312.
Barkley, D., and Tuckerman, L. S. (2007). “Mean flow of turbulent-laminar patterns in plane Couette flow.” J. Fluid Mech., 576, 109–137.
Bech, K. H., Tillmark, N., Alfredsson, P. H., and Andersson, H. I. (1995). “An investigation of turbulent plane Couette flow at low Reynolds numbers.” J. Fluid Mech., 286(-1), 291–325.
Chien, N., and Wan, Z. (1999). Mechanics of sediment transport, ASCE, Reston, VA.
Chue, S. H., and McDonald, A. T. (1970). “Application of a new effective viscosity model to turbulent plane Couette flow.” AIAA J., 8(11), 2076–2078.
El Telbany, M. M. M., and Reynolds, A. J. (1980). “Velocity distribution in plane turbulent channel flows.” J. Fluid Mech., 100(1), 1–29.
El Telbany, M. M. M., and Reynolds, A. J. (1982). “The structure of turbulent plane Couette flow.” J. Fluids Eng., 104(3), 367–372.
Gavrilakis, S., Tsai, H. M., Voke, P. R., and Leslie, D. C. (1986). “Large-eddy simulation of low Reynolds number channel flow by spectral and finite difference methods.” Direct and large eddy simulation of turbulence: Notes on numerical fluid mechanics, U. Schumann and R. Friedrich, eds., Vol. 15, Springer, Berlin, 105–118.
Gayme, D. F., McKeon, B. J., Papachristodoulou, A., Bamieh, B., and Doyle, J. C. (2010). “A streamwise constant model of turbulence in plane Couette flow.” J. Fluid Mech., 665, 99–119.
Gibson, J. F., Halcrow, J., and Cvitanović, P. (2008). “Visualizing the geometry of state space in plane Couette flow.” J. Fluid Mech., 611, 107–130.
Guo, J. (2006). “Self-similarity of mean flow in pipe turbulence.” Proc., 36th AIAA Fluid Dynamics Conf. and Exhibit, American Society of Physics, College Park, MD.
Guo, J. (2017a). “Eddy viscosity and complete log-law for turbulent pipe flow at high Reynolds numbers.” J. Hydraul. Res., 55(1), 27–39.
Guo, J. (2017b). “Exact procedure for Einstein-Johnson’s sidewall correction in open channel flow.” J. Hydraul. Eng., 06016027.
Guo, J., Shan, H., Xu, H., Bai, Y., and Zhang, J. (2017). “Exact solution for asymmetric turbulent channel flow with applications in ice-covered rivers.” J. Hydraul. Eng., 04017041.
Hultmark, M., Vallikivi, M., Bailey, S. C. C., and Smits, A. J. (2013). “Logarithmic scaling of turbulence in smooth and rough-wall pipe flow.” J. Fluid Mech., 728, 376–395.
Kim, J., Moin, P., and Moser, R. (1987). “Turbulence statistics in fully developed channel flow at low Reynolds number.” J. Fluid Mech., 177(-1), 133–166.
Krug, D., Luthi, B., Seybold, H., Holzner, M., and Tsinober, A. (2012). “3D-PTV measurements in a plane Couette flow.” Exp. Fluids, 52(5), 1349–1360.
Kundu, P. (1990). Fluid mechanics, Academic Press, New York.
Laudau, L. D., and Lifshitz, E. M. (1959). Fluid mechanics, Pergamon Press, Oxford, U.K.
Lee, M. J., and Kim, J. (1991). “The structure of turbulence in a simulated plane Couette flow.” Proc., 8th Turbulent Shear Flows, ASME, New York.
Lewtheusser, H. J., and Chu, V. H. (1971). “Experiments on plane Couette flow.” J. Hydraul. Div., 97(9), 1269–1284.
Maple [Computer software]. Maplesoft, McKinney, TX.
Marusic, I., Monty, J. P., Hultmark, M., and Smits, A. J. (2013). “On the logarithmic region in wall turbulence.” J. Fluid Mech., 716(R3), 1–11.
Mathematica [Computer software]. Walfram, Champaign, IL.
MATLAB [Computer software]. MathWorks, Natick, MA.
Monty, J. P. (2005). “Developments in smooth wall turbulent duct flows.” Ph.D. thesis, Univ. of Melbourne, Melbourne, Australia.
Perry, A. E., Hadez, S., and Chong, M. S. (2001). “A possible reinterpretation of the Princeton superpipe data.” J. Fluid Mech., 439, 395–401.
Reichardt, H. (1956). “Über die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couetteströmung.” Zeitschrift für Angewandte Mathematik und Mechanik, 36(S1), S26–S29.
Reichardt, H. (1959). “Gesetzmässigkeiten der geradlinigen turbulenten Couetteströmung.”, Max Planck Institute for Dynamics and Self-Organization, Gottingen, Germany.
Robertson, J. M. (1959). “On turbulent plane Couette flow.” Proc., 6th Midwestern Conf. on Fluid Mechanics, Univ. of Texas, Austin, TX, 169–182.
Robertson, J. M., and Johnson, H. F. (1970). “Turbulence structure in plane Couette flow.” J. Eng. Mech. Div., 96(6), 1171–1182.
Schlichting, H. (1979). Boundary-layer theory, 7th Ed., McGraw-Hill, New York.
Tillmark, N., and Alfredsson, P. H. (1992). “Experiments on transition in plane Couette flow.” J. Fluid Mech., 235, 89–102.
Tsukahara, T., Kawamura, H., and Yamazaki, K. (2006). “DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region.” J. Turbulence, 7(19), 1–15.
Tuckerman, L. S., and Barkley, D. (2011). “Patterns and dynamics in transitional plane Couette flow.” Phys. Fluids, 23(4), 041301.
van Driest, E. R. (1956). “On turbulent flow near a wall.” J. Aeronautical Sci., 23(11), 1007–1011.
Information & Authors
Information
Published In
Copyright
©2017 American Society of Civil Engineers.
History
Received: Dec 19, 2016
Accepted: Jun 8, 2017
Published online: Oct 31, 2017
Published in print: Jan 1, 2018
Discussion open until: Mar 31, 2018
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.