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== Continuity and Momentum Equations ==
= Governing Equations =
On the basis of the definitions [[CMS-Flow_Hydrodnamics:_Variable_Definitions | Variable Definitions]], and assuming depth-uniform currents, the general depth-integrated and wave-averaged continuity and momentum equations may be written as (Phillips 1977; Mei 1989; Svendsen 2006)
The depth-averaged 2-D continuity equation may be written as
{{Equation|
{{Equation|<math> \frac{\partial h}{\partial t}+\nabla \cdot (h\mathbf{U})=S </math>|2=1}}
<math>\frac{\partial h}{\partial t} + \frac {\partial(hV_j)} {\partial x_j} = S^M</math>
|1}}


where <math>h</math> is the total water depth <math>h=\zeta+\eta</math>, <math>\eta</math> is the water surface elevation, <math>\zeta</math> is the still water depth, <math> \mathbf{U}=\left( {{U}_{1}},{{U}_{2}} \right) </math> is the depth-averaged current velocity, <math> S </math> is a source term due to precipitation and evaporation, and <math> \nabla =\left( {{\nabla }_{1}},{{\nabla }_{2}} \right) </math> is the divergence operator.
{{Equation|
<math>\frac{\partial(hV_i)}{\partial t} + \frac {\partial(hV_jV_i)}{\partial x_j} - \varepsilon_{ij}f_chV_j = -gh\frac{\partial \bar{\eta}}{\partial x_i} - \frac{h}{\rho} \frac{\partial p_{a}}{\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 h U_{wi}U_{wj} \right) + \frac{\tau_{si}}{\rho} - m_{b}\frac{\tau_{bi}}{\rho}</math>''   
The momentum equation can be written as
|2}}
{{Equation| <math> \frac{\partial (h{{U}_{i}})}{\partial t}+\nabla \cdot (h\mathbf{U}{{U}_{i}})-\mathbf{BU}=-gh{{\nabla }_{i}}\eta +\nabla \cdot \left( {{\nu }_{t}}h\nabla {{U}_{i}} \right)+\frac{1}{\rho }\left( {{\tau }_{wi}}+{{\tau }_{Si}}-{{\tau }_{bi}} \right) </math>|2=2}} 
 
where <math>g</math> is the gravitational constant, <math> \mathbf{B}=\left( \begin{matrix} 0 & {{f}_{c}}  \\  -{{f}_{c}} & 0  \\ \end{matrix} \right) </math> where <math>f_{c}</math> is the Coriolis parameter, <math>\nu_t</math> is the eddy viscosity, <math> \tau_{wi} </math> is the wind stress, <math> \tau_{Si} </math> is the wave stresses, and <math> \tau_{bi} </math> is the combined wave-current mean bed shear stress.


= Numerical Methods =
where
== Temporal Term ==
The temporal term of the momentum equations is discretized using a first order implicit Euler scheme
{{Equation| <math> \int\limits_{A}{\frac{\partial (h\phi )}{\partial t}}\text{d}A=\frac{\partial }{\partial t}\int\limits_{A}{(h\phi )\text{d}A}=\frac{{{h}^{n+1}}\phi _{{}}^{n+1}-{{h}^{n}}\phi _{{}}^{n}}{\Delta t}\Delta A </math>|2=3}} 
where <math> \Delta A </math> is the cell area, and <math> \Delta t </math> is the hydrodynamic time step.


== Advection Term ==
: t = time[s]
The advection scheme obtained using the divergence theorem as 
where  is the outward unit normal on cell face f,  is the cell face length and  is the total water depth linearly interpolated to the cell face. Here the overbar indicates a cell face interpolation operator described in the following section. For Cartesian grids the cell face unit vector is always aligned with one of the Cartesian coordinates which simplifies the calculation. Defining the cell face normal velocity as  the above equation simplifies to 


{{Equation| <math> \int\limits_{A}{\nabla \cdot (h\mathbf{U}\phi )}\text{d}A=\oint\limits_{L}{h\phi \left( \mathbf{U}\cdot \mathbf{n} \right)}\text{d}L=\sum\limits_{f}^{{}}{{{{\bar{h}}}_{f}}\Delta {{l}_{f}}{{\left( {{{\hat{n}}}_{i}}{{U}_{i}} \right)}_{f}}{{{\tilde{\phi }}}_{f}}} </math>|2=4}} 
:<math>x_j</math> = Cartesian coordinate in the <math>j^{th}</math> direction [m], j = 1,2 or x, y
 
where <math> \mathbf{n}={{\hat{n}}_{i}}=({{\hat{n}}_{1}},{{\hat{n}}_{2}}) </math> is the outward unit normal on cell face f, <math> \Delta {{l}_{f}} </math> is the cell face length and <math> {{\bar{h}}_{f}} </math> is the total water depth linearly interpolated to the cell face. Here the overbar indicates a cell face interpolation operator described in the following section. For Cartesian grids the cell face unit vector is always aligned with one of the Cartesian coordinates which simplifies the calculation. Defining the cell face normal velocity as <math> {{U}_{f}}={{U}_{i}}\in f\bot i </math> the above equation simplifies to
{{Equation| <math> \sum\limits_{f}^{{}}{{{{\bar{h}}}_{f}}\Delta {{l}_{f}}{{\left( {{{\hat{n}}}_{i}}{{U}_{i}} \right)}_{f}}{{{\tilde{\phi }}}_{f}}}=\sum\limits_{f}^{{}}{{{n}_{f}}{{F}_{f}}{{{\tilde{\phi }}}_{f}}} </math>|2=5}}


where <math> {{F}_{f}}={{\bar{h}}_{f}}\Delta {{l}_{f}}{{U}_{f}} </math>, <math> {{n}_{f}}={{n}_{\bot }}={{\left( {{{\hat{e}}}_{i}}{{{\hat{n}}}_{i}} \right)}_{f}} </math>, with <math> {{\hat{e}}^{i}}=({{\hat{e}}_{1}},{{\hat{e}}_{2}}) </math> being the basis vector. <math> n_f </math> is equal to 1 for West and South faces and equal to -1 for North and East cell faces. Lastly, <math> \tilde{\phi }_{f}^{{}} </math> is the advective value of <math> \phi </math> on cell face f, and is calculated using either the Hybrid, Exponential, HLPA (Zhu 1991) schemes. The cell face velocities <math> U_f </math> are calculated using the momentum interpolation method of Rhie and Chow (1983) described in the subsequent section.
:<math>S^m = </math> source term due to precipitation, evaporation and structures (e.g. culverts) [m/s]


== Cell-face interpolation operator ==
:<math>f_c = 2\Omega sin \phi = </math>Coriolis parameter [rad/s] in which <math>\Omega = 7.29 \ x \ 10^{-5} </math> rad/s is the Earth’s angular velocity of rotation and <math>\phi</math> is the latitude in degrees
The general formula for estimating the cell-face value of <math> \tilde{\phi }_{f}^{{}} </math> is given by
{{Equation| <math> {{\bar{\phi }}_{f}}={{L}_{\bot }}{{\phi }_{N}}+(1-{{L}_{\bot }}){{\phi }_{P}}+{{\left( {{\nabla }_{\parallel }}\phi  \right)}_{N}}{{L}_{\bot }}({{x}_{\parallel ,O}}-{{x}_{\parallel ,N}})+{{\left( {{\nabla }_{\parallel }}\phi  \right)}_{P}}(1-{{L}_{\bot }})({{x}_{\parallel ,O}}-{{x}_{\parallel ,P}}) </math>|2=6}}
where <math> {{L}_{\bot }} </math> is a linear interpolation factor given by <math> {{L}_{\bot }}=\Delta {{x}_{\bot ,P}}/(\Delta {{x}_{\bot ,P}}+\Delta {{x}_{\bot ,N}}) </math> and <math> {{\nabla }_{\parallel }} </math> is the gradient operator in the direction parallel to face f. By definition <math> \parallel \,=2\left| {{{\hat{n}}}_{1}} \right|+1\left| {{{\hat{n}}}_{2}} \right| </math>. Note that for neighboring cells without any refinement <math< {{x}_{\parallel ,O}}-{{x}_{\parallel ,P}} </math> and <math>{{x}_{\parallel ,O}}-{{x}_{\parallel ,N}} </math> are zero and thus the above equation is consistent with non-refined cell faces.


== Diffusion term ==
:<math>g = </math> gravitational constant (~9.81 m/s<sup>2</sup>)
The diffusion term is discretized in general form using the divergence theorem


{{Equation| <math> \int\limits_{A}{\nabla \cdot \left( \Gamma h\nabla \phi  \right)}\text{d}A=\oint\limits_{S}{\Gamma h\left( \nabla \phi \cdot \mathbf{n} \right)}\text{d}S=\sum\limits_{f}^{{}}{\bar{\Gamma }_{f}^{{}}{{{\bar{h}}}_{f}}\Delta {{l}_{f}}{{\left( {{{\hat{n}}}_{i}}{{\nabla }_{i}}\phi  \right)}_{f}}} </math> |2=7}}
:<math>p_a</math> = atmospheric pressure [Pa]


The discritization of the cell-face gradient is described in the next section. On a Cartesian grid the above expression may be further simplified as
:<math>\rho = </math> water density (~1025 kg/m<sup>3</sup>)


{{Equation| <math> \sum\limits_{f}^{{}}{{{n}_{f}}\bar{\Gamma }_{f}^{{}}{{{\bar{h}}}_{f}}\Delta {{l}_{f}}{{\left( {{\nabla }_{\bot }}\phi  \right)}_{f}}}=\sum\limits_{f}^{{}}{{{D}_{f}}\left[ {{\phi }_{N}}-{{\phi }_{P}}+{{\left( {{\nabla }_{\parallel }}\phi  \right)}_{N}}\left( {{x}_{\parallel ,O}}-{{x}_{\parallel ,N}} \right)-{{\left( {{\nabla }_{\parallel }}\phi  \right)}_{P}}\left( {{x}_{\parallel ,O}}-{{x}_{\parallel ,P}} \right) \right]} </math> |2=8}}
:<math>v_t = </math> turbulent eddy viscosity [m<sup>2</sup>/s]
where <math> {{\nabla }_{\bot }}\phi </math> is gradient in the direction perpendicular to the cell face and
<math> {{D}_{f}}=\frac{\bar{\Gamma }_{f}^{{}}{{{\bar{h}}}_{f}}\Delta {{l}_{f}}}{\left| \delta {{x}_{\bot }} \right|} </math>.
== Cell-centered node-based gradient operator ==


== Cell-centered face-based gradient operator ==
:<math>\tau_{si} = </math> wind surface stress [Pa]


== Cell-face gradient operator ==
:<math>S_{ij}</math> = wave radiation stress [Pa]


== Source terms ==
:<math>R_{ij}</math> = surface roller stress [Pa]


----
:<math>m_b</math> = bed slope coefficient [-]
 
:<math>\tau_{bi}</math> = 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.


{| border="1"
= References =
! Symbol !! Description !! Units
* 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.
|-
* Mei, C. 1989. The applied dynamics of ocean surface waves. New York: John Wiley.
| <math> t </math> || Time || sec
* 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.
| <math> h </math> ||  Total water depth <math> h = \zeta + \eta </math> || m
* 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.
|-
| <math> \zeta </math> ||  Still water depth || m
|-
| <math> \eta </math> ||  Water surface elevation with respect to the still water elevation || m
|-
| <math> U_j </math> || Current velocity in the jth direction || m/sec
|-
| <math> S </math> || Sum of Precipitation and evaporation per unit area || m/sec
|-
| <math> g </math> || Gravitational constant || m/sec<sup>2</sup>
|-
| <math> \rho </math> || Water density || kg/m<sup>3</sup>
|-
| <math> p_a  </math> || Atmospheric pressure || Pa
|-
| <math> \nu_t  </math> || Turbulent eddy viscosity || m<sup>2</sup>/sec
|}


----
----
</big>
[[CMS#Documentation_Portal | Documentation Portal]]
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Latest revision as of 15:39, 18 February 2015

Continuity and Momentum Equations

On the basis of the definitions Variable Definitions, and assuming depth-uniform currents, the general depth-integrated and wave-averaged continuity and momentum equations may be written as (Phillips 1977; Mei 1989; Svendsen 2006)

 

(1)
 

(2)

where

t = time[s]
= Cartesian coordinate in the direction [m], j = 1,2 or x, y
source term due to precipitation, evaporation and structures (e.g. culverts) [m/s]
Coriolis parameter [rad/s] in which rad/s is the Earth’s angular velocity of rotation and is the latitude in degrees
gravitational constant (~9.81 m/s2)
= atmospheric pressure [Pa]
water density (~1025 kg/m3)
turbulent eddy viscosity [m2/s]
wind surface stress [Pa]
= wave radiation stress [Pa]
= surface roller stress [Pa]
= bed slope coefficient [-]
= 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.
  • Mei, C. 1989. The applied dynamics of ocean surface waves. New York: John Wiley.
  • 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.

Documentation Portal

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