Abstract
An approximate analytic solution of the Navier–Stokes equations for laminar flow in a circular open channel is presented. The solution is valid for the case of incompressible steady uniform flow in a low-filled channel. (The depth ratio, which is flow depth divided by channel radius, is smaller than one.) The solution is compared with a recently reported exact theoretical solution and found to be in reasonably good agreement. Expressions for variation of the correction factors for momentum and kinetic energy fluxes with flow depth are presented. A correlation between the momentum and kinetic energy flux correction factors is proposed and compared with data reported in the literature.
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Abbreviations
- \({C}_{1}\), \({C}_{2}\) :
-
Integration constants
- Fr:
-
Froude number
- g :
-
Gravitational acceleration
- h :
-
Flow depth
- P :
-
Pressure
- r :
-
Radial coordinate
- \({r}_{\mathrm{s}}\) :
-
Distance from the pipe center to a point on the free surface
- R :
-
Pipe radius
- Re:
-
Reynolds number
- \({T}_{0}\) :
-
Normalized wall shear stress
- \(U\) :
-
Normalized streamwise velocity
- \({U}_{\mathrm{b}}\) :
-
Normalized bulk velocity
- \({U}_{\mathrm{o}}\) :
-
Dimensionless centerline surface velocity
- \({V}_{\mathrm{b}}\) :
-
Bulk velocity
- \({V}_{x}\), \({V}_{r}\), \({V}_{\theta }\) :
-
Velocity components in cylindrical coordinate system
- x, y, z :
-
Spatial coordinates in the Cartesian system
- \({y}_{\mathrm{s}}\) :
-
Vertical distance from a point on the pipe wall to the free surface
- \(\alpha\) :
-
Kinetic energy flux correction factor
- \(\beta\) :
-
Momentum flux correction factor
- \(\mu\) :
-
Dynamic viscosity
- \(\varphi\) :
-
Maximum value of the azimuthal coordinate corresponding to a particular flow depth
- \(\varnothing\) :
-
Pipe inclination angle
- \(\rho\) :
-
Density
- \(\theta\) :
-
Azimuthal coordinate
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Kaya, K., Özcan, O. An approximate analytic solution of uniform laminar flow in a circular open channel. J Braz. Soc. Mech. Sci. Eng. 43, 328 (2021). https://doi.org/10.1007/s40430-021-03037-x
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DOI: https://doi.org/10.1007/s40430-021-03037-x