CMS-Flow:Hydro Eqs: Difference between revisions

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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)

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

(1)

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

(2)

where

t = time[s]

x_{j}

= Cartesian coordinate in the

j^{th}

direction [m], j = 1,2 or x, y

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 = } source 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 f_c = 2\Omega sin \phi = } Coriolis parameter [rad/s] in which

\Omega =7.29\ x\ 10^{-5}

rad/s is the Earth’s angular velocity of rotation and

\phi

is the latitude in degrees

g=

gravitational constant (~9.81 m/s²)

p_{a}

= 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

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.

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@@ Line 1: / Line 1: @@
-<big>
+== Continuity and Momentum Equations ==
-== Governing Equation ==
+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, and <math> \mathbf{U}=\left( {{U}_{1}},{{U}_{2}} \right) </math> is the depth-averaged current velocity, 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,  is the eddy viscosity,  is the wind stress,  is the wave stresses, and  is the combined wave-current mean bed shear stress.
-for <math> i=1,2 </math> and <math> j=1,2 </math>
+where
-{| border="1"
+: t = time[s]
-! Symbol !! Description !! Units
-|-
+:<math>x_j</math> = Cartesian coordinate in the <math>j^{th}</math> direction [m], j = 1,2 or x, y
-| <math> t </math> || Time || sec
-|-
+:<math>S^m = </math> source term due to precipitation, evaporation and structures (e.g. culverts) [m/s]
-| <math> h </math> ||  Total water depth <math> h = \zeta + \eta </math> || m
-|-
+:<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
-| <math> \zeta </math> ||  Still water depth || m
-|-
+:<math>g = </math> gravitational constant (~9.81 m/s<sup>2</sup>)
-| <math> \eta </math> ||  Water surface elevation with respect to the still water elevation || m
-|-
+:<math>p_a</math> = atmospheric pressure [Pa]
-| <math> U_j </math> || Current velocity in the jth direction || m/sec
-|-
+:<math>\rho = </math> water density (~1025 kg/m<sup>3</sup>)
-| <math> S </math> || Sum of Precipitation and evaporation per unit area || m/sec
-|-
+:<math>v_t = </math> turbulent eddy viscosity [m<sup>2</sup>/s]
-| <math> g </math> || Gravitational constant || m/sec<sup>2</sup>
-|-
+:<math>\tau_{si} = </math> wind surface stress [Pa]
-| <math> \rho </math> || Water density || kg/m<sup>3</sup>
-|-
+:<math>S_{ij}</math> = wave radiation stress [Pa]
-| <math> p_a  </math> || Atmospheric pressure || Pa
-|-
+:<math>R_{ij}</math> = surface roller stress [Pa]
-| <math> \nu_t  </math> || Turbulent eddy viscosity || m<sup>2</sup>/sec
-|}
+:<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.
+= 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.
 ----
-</big>
 [[CMS#Documentation_Portal | Documentation Portal]]

CMS-Flow:Hydro Eqs: Difference between revisions

Latest revision as of 15:39, 18 February 2015

Continuity and Momentum Equations

References

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