CMS-Flow:Hydro Eqs

Continuity and Momentum Equations

On the basis of the definitions Variable Definitions, and assuming depth-uniform cur-rents, the general depth-integrated and wave-averaged continuity and momentum equations may be written as (Phillips 1977; Mei 1983; Svendsen 2006)

${\frac {\partial h}{\partial t}}+{\frac {\partial (hV_{j})}{\partial x_{j}}}=S_{M}$

(6)

${\frac {\partial (hV_{i})}{\partial t}}+{\frac {\partial (hV_{j}V_{i})}{\partial x_{j}}}-\varepsilon _{ij}f_{c}hV_{j}=-gh{\frac {\partial {\bar {\eta }}}{\partial x_{i}}}-{\frac {h}{\rho }}{\frac {\partial p_{atm}}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}{\left(v_{t}h{\frac {\partial V_{i}}{\partial x_{j}}}\right)}-{\frac {1}{\rho }}{\frac {\partial }{\partial x_{j}}}\left(S_{ij}+R_{ij}-\rho hU_{wi}U_{wj}\right)+{\frac {\tau _{si}}{\rho }}-m_{b}{\frac {\tau _{bi}}{\rho }}$

(7)

where

$t$ = time[s]

$x_{j}$ = Cartesian coordinate in the $j^{tj}$ direction [m]

math>f_c</math> = Coriolis parameter [rad/s] $=2\Omega sin\phi$ where $\Omega =7.29x10^{-5}$ rad/s is the earth’s angular velocity of rotation and $\phi$ is the latitude in degrees

$h$ = wave-averaged total water depth $h=\zeta +{\bar {\eta }}$ [m]

${\bar {\eta }}=$ wave-averaged water surface elevation with respect to reference datum [m]

$S_{M}=$ water source/sink term due to precipitation, evaporation and structures (e.g. culverts) [m/s]

$V_{i}=$ total flux velocity defined as $V_{i}=U_{i}+U_{wi}$ [m/s]

$U_{i}=$ wave- and depth-averaged current velocity [m/s]

$U_{wi}=$ mean wave mass flux velocity or wave flux velocity for short [m/s]

$g=$ gravitational constant (~9.81 m/s²)

$p_{atm}=$ atmospheric pressure [Pa]

$\rho =$ water density (~1025 kg/m³)

$v_{t}=$ turbulent eddy viscosity [m²/s]

$\tau _{si}=$ wind surface stress [Pa]

$S_{ij}=$ wave radiation stress [Pa]

$R_{ij}=$ surface roller stress [Pa]

$m_{b}=$ bed slope coefficient [-]

$\tau _{bi}=$ combined wave and current mean bed shear stress [Pa]

The above 2DH equations are similar to those derived by Svendsen (2006), except for the inclusion of the water source/sink term in the continuity equation and the atmospheric pressure and surface roller terms and the bed slope coefficient in the momentum equation. It’s also noted that the horizontal mixing term is formulated slightly differently as a function of the total flux velocity, similar to the Generalized Lagrangian Mean (GLM) approach (Andrews and McIntyre 1978; Walstra et al. 2000). This approach is arguably more physically meaningful and also simplifies the discretization in the case where the total flux velocity is used as the model prognostic variable.

References

• Buttolph, A. M., Reed, C. W., Kraus, N. C., Ono, N., Larson, M., Camenen, B., Hanson, H.,Wamsley, T., and Zundel, A. K. (2006). “Two-dimensional depth-averaged circulation model CMS-M2D: Version 3.0, Report 2: Sediment transport and morphology change,” Tech. Rep. ERDC/CHL TR-06-9, U.S. Army Engineer Research and Development Center, Coastal and Hydraulic Engineering, Vicksburg, MS.
• Ferziger, J. H., and Peric, M. (1997). “Computational Methods for Fluid Dynamics”, Springer-Verlag, Berlin/New York, 226 p.
• Huynh-Thanh, S., and Temperville, A. (1991). “A numerical model of the rough turbulent boundary layer in combined wave and current interaction,” in Sand Transport in Rivers, Estuaries and the Sea, eds. R.L. Soulsby and R. Bettess, pp.93-100. Balkema, Rotterdam.
• Phillips, O.M. (1977) Dynamics of the upper ocean, Cambridge University Press.
• Rhie, T.M. and Chow, A. (1983). “Numerical study of the turbulent flow past an isolated airfoil with trailing-edge separation”. AIAA J., 21, 1525–1532.
• Saad, Y., (1993). “A flexible inner-outer preconditioned GMRES algorithm,” SIAM Journal Scientific Computing, 14, 461–469.
• Saad, Y., (1994). “ILUT: a dual threshold incomplete ILU factorization,” Numerical Linear Algebra with Applications, 1, 387-402.
• Saad, Y. and Schultz, M.H., (1986). “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM Journal of Scientific and Statistical, Computing, 7, 856-869.
• Soulsby, R.L. (1995). “Bed shear-stresses due to combined waves and currents,” in Advanced in Coastal Morphodynamics, ed M.J.F Stive, H.J. de Vriend, J. Fredsoe, L. Hamm, R.L. Soulsby, C. Teisson, and J.C. Winterwerp, Delft Hydraulics, Netherlands. 4-20 to 4-23 pp.
• Svendsen, I.A. (2006). Introduction to nearshore hydrodynamics, Advanced Series on Ocean Engineering, 124, World Scientific Publishing, 722 p.
• Wu, W. (2004). “Depth-averaged 2-D numerical modeling of unsteady flow and nonuniform sediment transport in open channels,” Journal of Hydraulic Engineering, ASCE, 135(10) 1013-1024.
• Wu, W., Sánchez, A., and Mingliang, Z. (2011). “An implicit 2-D shallow water flow model on an unstructured quadtree rectangular grid,” Journal of Coastal Research, [In Press]
• Wu, W., Sánchez, A., and Mingliang, Z. (2010). “An implicit 2-D depth-averaged finite-volume model of flow and sediment transport in coastal waters,” Proceeding of the International Conference on Coastal Engineering, [In Press]
• Van Doormal, J.P. and Raithby, G.D., (1984). Enhancements of the SIMPLE method for predicting incompressible fluid flows. Num. Heat Transfer, 7, 147–163.
• Zhu, J. (1991). “A low-diffusive and oscillation-free convection scheme,”Communications in Applied Numerical Methods, 7, 225-232.
• Zwart, P. J., Raithby, G. D., Raw, M. J. (1998). “An integrated space-time finite volume method for moving boundary problems”, Numerical Heat Transfer, B34, 257.

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Variable Index

Symbol	Description	Units
$t$	Time	sec
$h$	Total water depth $h=\zeta +\eta$	m
$\zeta$	Still water depth	m
$\eta$	Water surface elevation with respect to the still water elevation	m
$U_{j}$	Current velocity in the jth direction	m/sec
$S$	Sum of Precipitation and evaporation per unit area	m/sec
$g$	Gravitational constant	m/secsup2/sup
$\rho$	Water density	kg/msup3/sup
$p_{a}$	Atmospheric pressure	Pa
$\nu _{t}$	Turbulent eddy viscosity	msup2/sup/sec

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