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)

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

(1)

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

(2)

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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi} is the latitude in degrees

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h} = wave-averaged total water depth Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h = \zeta + \bar{\eta}} [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar{\eta} = } wave-averaged water surface elevation with respect to reference datum [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_M = } water source/sink term due to precipitation, evaporation and structures (e.g. culverts) [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_i = } total flux velocity defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_i = U_i + U_{wi}} [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_i = } wave- and depth-averaged current velocity [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{wi} = } mean wave mass flux velocity or wave flux velocity for short [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g = } gravitational constant (~9.81 m/s²)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_{atm} = } atmospheric pressure [Pa]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho = } water density (~1025 kg/m³)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_t = } turbulent eddy viscosity [m²/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{si} = } wind surface stress [Pa]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_{ij} = } wave radiation stress [Pa]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{ij} = } surface roller stress [Pa]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m_b = } bed slope coefficient [-]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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

• Andrews, D. G., and M. E. McIntyre. 1978. An exact theory of nonlinear waves on a Lagrangian mean flow. Journal of Fluid Mechanics (89):609–646.
• Phillips, O. M. 1977. Dynamics of the upper ocean, Cambridge University Press.
• Svendsen, I. A. 2006. Introduction to nearshore hydrodynamics, Advanced Series on Ocean Engineering, 124, World Scientific Publishing, 722 p.
• Walstra, D. J. R., J. A. Roelvink, and J. Groeneweg. 2000. Calculation of wave-driven currents in a 3D mean flow model. In Proceedings, 27th International Conference on Coastal Engineering, 1050-1063. Sydney, Australia.

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