TY - JOUR
T1 - Exact solution to Navier-Stokes equation for developed radial flow between parallel disks
AU - Guo, Junke
AU - Shan, Haoyin
AU - Xie, Zhaoding
AU - Li, Chen
AU - Xu, Haijue
AU - Zhang, Jianmin
N1 - Publisher Copyright:
© 2016 American Society of Civil Engineers.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - 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.
AB - 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.
KW - Exact solution
KW - Jacobi elliptic function
KW - Laminar flow
KW - Navier-Stokes equation
KW - Nonlinear inertia
KW - Radial flow
KW - Velocity distribution
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U2 - 10.1061/(ASCE)EM.1943-7889.0001227
DO - 10.1061/(ASCE)EM.1943-7889.0001227
M3 - Article
AN - SCOPUS:85017516348
VL - 143
JO - Journal of Engineering Mechanics - ASCE
JF - Journal of Engineering Mechanics - ASCE
SN - 0733-9399
IS - 6
M1 - 04017026
ER -