Abstract
This paper examines the effect of coupling several equations for vertical velocity profiles in a depth-integrated model to clarify the roles of three-dimensional flow structures minimizing the errors from the two-dimensional calculation model. In addition, this paper develops a numerical discretization method for the dispersion terms in the horizontal momentum equations. It is revealed that the use of the 2DC model underestimates the water surface elevation through the comparisons with the experimental datasets, advanced 2DC and 3DC results. The prediction of water surface elevation is considerably improved by taking into account of secondary flow effect with vorticity equations. It is clarified that the accuracy in predicting the water surface elevation is increased with increasing the degree of the function for vertical velocity profiles in advanced 2DC models or the number of vertical grids in 3DC model, while non-hydrostatic pressure and variation in vertical velocity have a second importance.
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Abbreviations
- d ij, D ij :
-
Numerical diffusion and depth-integrated numerical diffusion in the dispersion term
- d ωiξ, D ωiη :
-
Horizontal vorticity flux due to advection, rotation, dispersion, and turbulence diffusion
- ER σi :
-
Rotation term of vertical vorticity (ERσi = usiωσs− ubiωσb)
- g :
-
Acceleration due to gravity
- h :
-
Water depth
- J :
-
Jacobian (area of computational grid J = xξyη− xηyξ)
- k s :
-
Equivalent sand grain roughness
- P si :
-
Production term due to shear stress acting on the thin water surface layer δZS
- P ωi :
-
production term of vorticity from the bottom vortex layer
- T ξξ, T ηη, T ξη :
-
Horizontal momentum transfer, comprising shear stress terms of turbulent motions and a dispersion term resulting from the vertical velocity distribution
- u i, U i :
-
Horizontal velocity and depth-averaged horizontal velocity
- u si, u bi :
-
Horizontal velocity at the water surface and bottom
- U ξ, U η :
-
Depth-averaged horizontal velocity in the ξ and η directions
- x, y :
-
x: streamwise direction (x1); y: lateral direction (x2)
- δu i, Δu ij :
-
Velocity differences between the water surface and bottom, water surface and depth-averaged (δui = usi − ubi), (Δuij = Usi-− Ui)
- Δξ, ΄η :
-
Calculation grid spacing of covariant in the ξ and η directions
- ε ij :
-
Levi–Civita symbol
- κ :
-
von Karman constant (κ = 0.41)
- v t :
-
Kinematic eddy viscosity coefficient
- v tb :
-
Eddy viscosity coefficient at the bottom
- ξ, η :
-
General curvilinear coordinates
- τ 0ξ, τ 0η :
-
Bed shear stress in the ξ and η directions
- ω bi :
-
Horizontal vorticity at the bottom
- ω bei :
-
Equilibrium bottom vorticity at the bottom
- ω i, Ωi :
-
Vorticity and depth-averaged vorticity
- Ω σs, Ω σb :
-
Rotation of Usi, Ubi
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The authors acknowledge the support provided through a JSPS KAKENHI grant (Number 18H01546).
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Lugina, F.P., Uchida, T. & Kawahara, Y. Numerical Calculations for Curved Open Channel Flows with Advanced Depth-Integrated Models. KSCE J Civ Eng 28, 1026–1040 (2024). https://doi.org/10.1007/s12205-024-1431-7
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DOI: https://doi.org/10.1007/s12205-024-1431-7