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== Continuity and Momentum Equations ==
== Continuity and Momentum Equations ==
Phillips (1977), Mei (1983), and Svendsen (2006) provide a detailed deri-vation of the depth-integrated and wave-averaged hydrodynamic equa-tions. Here, only variable definitions are provided and derivations may be obtained from the preceding references. The instantaneous current velocity u<sub>i</sub>  is split into:
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)
{{Equation|
{{Equation|
<math>u_i = \bar{u_i} + \tilde{u_i} + u_i^'</math>
<math>\frac{\partial h}{\partial t} + \frac {\partial(hV_j)} {\partial x_j} = S^M</math>
|1}}
|1}}


in which
:<math>\bar{u_i}</math>'' = current (wave-averaged) velocity [m/s]
:<math>\tilde{u_i}</math>'' =  wave (oscillatory) velocity with wave-average <math>\bar{\tilde{u_i}} = 0 [m/s]</math>
:<math>u_i^'</math> = turbulent fluctuation with ensemble average ''<math>\langle u_i^' \rangle</math>'' = 0 and wave average <math>\bar{u_i^'}</math> = 0 [m/s]
The wave-averaged total volume flux is defined as
{{Equation|
{{Equation|
<math>hV_i</math> = <math>\bar{{\int_z^\eta} {u_i dz }}</math>
<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>''   
|2}}
|2}}


where
where


:<math>h</math> = wave-averaged water depth <math>h=\bar{\eta} - z_b </math> [m]
: t = time[s]  
 
:<math>V_i</math> = total mean mass flux velocity or simply total flux velocity for short [m/s]
 
:<math>u_i</math> = instantaneous current velocity [m/s]
 
:<math>\eta</math>  = instantaneous water level with respect to the Still Water Level (SWL) [m]
 
:<math>z_b</math>  = bed elevation with respect to the SWL [m]
 
For simplicity in the notation, the over bar in subsequent wave-averaged variables is omitted. The total flux velocity is also referred to as the mean transport velocity (Phillips 1977) and mass transport velocity (Mei 1983). The current volume flux is defined as
{{Equation|
<math>hU_i = \int^\eta_{z} \bar{u_i}dz</math>
|3}}


where <math>U_i</math> is the depth-averaged current velocity. Similarly, the wave volume flux is defined as by
:<math>x_j</math> = Cartesian coordinate in the <math>j^{th}</math> direction [m], j = 1,2 or x, y
{{Equation|
<math>Q_{wi} = hU_{wi} = \bar{\int \tilde{u_i} dz}</math>
|4}}


where <math>U_{wi}</math> is the depth-averaged wave flux velocity [m/s], and ''<math>\eta_t</math>'' = wave trough elevation [m]. Therefore the total flux velocity may be written as
:<math>S^m = </math> source term due to precipitation, evaporation and structures (e.g. culverts) [m/s]
{{Equation|
<math>V_i = U_i + U_{wi}</math>
|5}}


On the basis of the above 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)
:<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
{{Equation|
<math>\frac{\partial h}{\partial t} + \frac {\partial(hV_j)} {\partial x_j} = S_M</math>
|6}}
 
{{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_{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 h U_{wi}U_{wj} \right) + \frac{\tau_{si}}{\rho} - m_{b}\frac{\tau_{bi}}{\rho}</math>''   
|7}}
 
where
 
:<math> t </math> = time[s]
 
:<math>x_j</math> = Cartesian coordinate in the <math>j^{tj}</math> direction [m]
 
:math>f_c</math> = Coriolis parameter [rad/s] <math> = 2\Omega sin\phi</math> where <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>h</math> = wave-averaged total water depth <math> h = \zeta + \bar{\eta}</math> [m]
 
:<math>\bar{\eta} = </math> wave-averaged water surface elevation with respect to reference datum [m]
 
:<math>S_M = </math> water source/sink term due to precipitation, evaporation and structures (e.g. culverts) [m/s]
 
:<math>V_i = </math> total flux velocity defined as <math> V_i = U_i + U_{wi}</math> [m/s]
 
:<math>U_i = </math> wave- and depth-averaged current velocity [m/s]
 
:<math>U_{wi} = </math> mean wave mass flux velocity or wave flux velocity for short [m/s]


:<math>g = </math> gravitational constant (~9.81 m/s<sup>2</sup>)
:<math>g = </math> gravitational constant (~9.81 m/s<sup>2</sup>)


:<math>p_{atm} = </math> atmospheric pressure [Pa]
:<math>p_a</math> = atmospheric pressure [Pa]


:<math>\rho = </math> water density (~1025 kg/m<sup>3</sup>)
:<math>\rho = </math> water density (~1025 kg/m<sup>3</sup>)
Line 84: Line 29:
:<math>\tau_{si} = </math> wind surface stress [Pa]
:<math>\tau_{si} = </math> wind surface stress [Pa]


:<math>S_{ij} = </math> wave radiation stress [Pa]
:<math>S_{ij}</math> = wave radiation stress [Pa]


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


:<math>m_b = </math> bed slope coefficient [-]
:<math>m_b</math> = bed slope coefficient [-]


:<math>\tau_{bi} = </math> combined wave and current mean bed shear stress [Pa]
:<math>\tau_{bi}</math> = combined wave and current mean bed shear stress [Pa].


The equations above are similar to those derived by Svendsen (2006), ex-cept for the inclusion of the water source/sink term in the continuity equation and the atmospheric pressure and surface roller terms in the momentum equation. It is also noted that the horizontal mixing term is formulated 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.
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.
 
Figure 2-1 shows the vertical conventions used for mean water surface elevation and bed elevations. The arrows indicate the reference elevation for each variable. Note that the total water depth is always positive.
 
 
Figure 2-1 here.
[[File:fig_2_1.png]]
 
Figure 2.1 Vertical conventions used for mean water surface elevation and bed elevations.


= References =
= 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.
* 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.
* Ferziger, J. H., and Peric, M. (1997). “Computational Methods for Fluid Dynamics”, Springer-Verlag, Berlin/New York, 226 p.
* Mei, C. 1989. The applied dynamics of ocean surface waves. New York: John Wiley.
* 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.  
* 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.
* 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.
* 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.
* 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.
 
 
----
 
 
= Variable Index =
{| border=1
! Symbol !! Description !! Units
|-
| <math> t </math> || Time || sec
|-
| <math> h </math> ||  Total water depth <math> h = \zeta + \eta </math> || m
|-
| <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/secsup2/sup
|-
| <math> \rho </math> || Water density || kg/msup3/sup
|-
| <math> p_a  </math> || Atmospheric pressure || Pa
|-
| <math> \nu_t  </math> || Turbulent eddy viscosity || msup2/sup/sec
|}
 


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[[CMS#Documentation_Portal | Documentation Portal]]
[[CMS#Documentation_Portal | Documentation Portal]]

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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