Exact Solution to Navier-Stokes Equation for Developed Radial Flow between Parallel Disks
Publication: Journal of Engineering Mechanics
Volume 143, Issue 6
Abstract
Laminar radial flow between two parallel disks is a fundamental nonlinear fluid mechanics problem described by the Navier-Stokes (NS) equation, but is unsolved because (1) an exact solution is not found even with extensive references, and (2) it is unclear why radial flow remains laminar at high Reynolds numbers. This paper first presents exact velocity distribution solutions for developed radial inflows and outflows, proving that both flows are described by brief Jacobi elliptic sine-squared functions but with different characteristics. For inflow, a stable velocity distribution forms; for outflow, the velocity distribution may have an inflection point inducing flow instability or separation. Both velocity distributions become the classic parabolic law at low Reynolds numbers, but uniform (similar to turbulent velocity distributions) at high Reynolds numbers. Furthermore, both pressure and boundary shear stress follow an inverse-square law, but the friction factor is invariant. These results are instructive for studying nonuniform open-channel flow for which nonlinear inertia is of importance.
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Acknowledgments
This research was supported by the U.S. Federal Highway Administration Hydraulics Research and Development 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.
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©2017 American Society of Civil Engineers.
History
Received: Jul 17, 2016
Accepted: Nov 4, 2016
Published online: Feb 16, 2017
Published in print: Jun 1, 2017
Discussion open until: Jul 16, 2017
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