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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}+\frac{\partial (h{{U}_{j}})}{\partial {{x}_{j}}}=0 /math|2=1}} | | <math>\frac{\partial h}{\partial t} + \frac {\partial(hV_j)} {\partial x_j} = S^M</math> |
| | |1}} |
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| where mathh/math is the total water depth mathh=\zeta+\eta/math, math\eta/math is the water surface elevation, math\zeta/math is the still water depth, mathU_i/math is the depth-averaged Langrangian current velocity defined as math U_i=U_i^E+U_i^S/math, where mathU_i^E/math is the phase- and depth-averaged current velocity (i.e. Eulerian velocity), and mathU_i^S/math is the depth-averaged Stokes velocity (Phillips 1977)
| | {{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>'' |
| | |2}} |
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| {{Equation|mathU_{i}^{S}=\frac{{{E}_{w}}}{\rho hc}{{w}_{i}} /math|2=2}}
| | where |
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| where mathE_w/math is the wave energy, math\rho/math is the water density, and mathc/math is the wave celerity, and math {{w}_{i}}=(\cos \theta ,\sin \theta ) /math is the wave unit vector where math \theta /math is the wave direction.
| | : t = time[s] |
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| The momentum equation can be written as
| | :<math>x_j</math> = Cartesian coordinate in the <math>j^{th}</math> direction [m], j = 1,2 or x, y |
| {{Equation| math \frac{\partial (h{{U}_{i}})}{\partial t}+\frac{\partial (h{{U}_{i}}{{U}_{j}})}{\partial {{x}_{j}}}-{{\varepsilon }_{ij}}{{f}_{c}}h{{U}_{j}}=-gh\frac{\partial \eta }{\partial {{x}_{i}}}-\frac{h}{{{\rho }_{0}}}\frac{\partial {{p}_{a}}}{\partial {{x}_{i}}}+\frac{\partial }{\partial {{x}_{j}}}\left( {{\nu }_{t}}h\frac{\partial {{U}_{i}}}{\partial {{x}_{j}}} \right)+\frac{1}{\rho }\left( \tau _{i}^{w}+\tau _{i}^{s}-\tau _{i}^{b} \right) /math|2=3}}
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| where mathg/math is the gravitational constant, mathf_c/math is the Coriolis parameter, mathp_a/math is the atmospheric pressure, math\rho_0/math is a reference water density, math\nu_t/math is the turbulent eddy viscosity, math \tau_{w} /math is the wind stress, math \tau_{S} /math is the wave stress, and math\tau_{b}/math is the combined wave-current mean bed shear stress. math\varepsilon_{ij}/math is the permutation parameter equal to 1 for mathi,j/math = 1,2, -1 for mathi,j/math = 2,1; and 0 for mathi=j/math.
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| == Bottom Stress ==
| | :<math>S^m = </math> source term due to precipitation, evaporation and structures (e.g. culverts) [m/s] |
| The mean (wave averaged) bed shear stress is calculated as
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| {{Equation|math \tau _i^b= m_b \lambda_{wc} \rho c_b U_E U_i^E /math|2=4}}
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| where math \lambda_{wc} /math is the nonlinear wave enhancement factor, mathm_b/math is a bed slope friction coefficient, math c_b /math is the bottom friction coefficient, and mathU_E= \sqrt{U_i^E U_i^E}/math is the Eulerian current magnitude. For additional information on the bottom friction wee [[CMS-Flow:Bottom_Friction | Bottom and Wall Friction]]
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| == Wave Stresses== | | :<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 wave stresses are calculated as
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| {{Equation|math \tau_i^w= -\frac{\partial }{\partial x_j} \left( S_{ij}^w +S_{ij}^r -\rho h U_i^S U_j^S \right) /math|2=5}}
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| where mathS_{ij}^w/math is the wave radiation stresses and mathS_{ij}^r/math is the roller stresses. mathS_{ij}^w/math is calculated using linear wave theory
| | :<math>g = </math> gravitational constant (~9.81 m/s<sup>2</sup>) |
| {{Equation|math S_{ij}^{w}=\iint\limits_{{}}{E_w \left[ n{{w}_{i}}{{w}_{j}}+{{\delta }_{ij}} \left( n-\frac{1}{2} \right) \right]df}d\theta /math|2=6}}
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| where mathf/math is frequency, math\theta/math is the direction, math\delta_{ij}/math=1 for mathi=j/math, math\delta_{ij}/math=0 for mathi \ne j/math, and mathn=\frac{1}{2}\left( 1+\frac{2kh}{\sinh 2kh} \right) /math. For more information on the CMS-Wave model see [[CMS-Wave]].
| | :<math>p_a</math> = atmospheric pressure [Pa] |
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| The roller stresses are mathS_{ij}^r/math are calculated as
| | :<math>\rho = </math> water density (~1025 kg/m<sup>3</sup>) |
| {{Equation|math S_{ij}^r = 2 E_r w_i w_j /math|2=7}}
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| where math E_r /math is the roller energy. For more information on the surface roller see [[CMS-Flow:Roller | Surface Roller]].
| | :<math>v_t = </math> turbulent eddy viscosity [m<sup>2</sup>/s] |
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| = Numerical Methods =
| | :<math>\tau_{si} = </math> wind surface stress [Pa] |
| == General Transport Equation: Discretization ==
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| All of the governing equations may be written in general form
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| {{Equation| math\underbrace{\frac{\partial (h\phi )}{\partial t}}_{\text{Temporal Term}}+\underbrace{\nabla \cdot (h\mathbf{U}\phi )}_{\text{Advection Term}}=\underbrace{\nabla \cdot \left( \Gamma h\nabla \phi \right)}_{\text{Diffusion Term}}+\underbrace{S}_{\text{Source Term}} /math|2=3}}
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| where math \phi /math is a general scalar, math t /math is time, math h /math is the total water depth, math \mathbf{U} /math is the depth averaged current velocity, math \Gamma /math is the diffusion coefficient for math \phi /math, math \nabla =({{\nabla }_{1}},{{\nabla }_{2}}) /math is the gradient operator, and math S /math includes all other terms. Note that in the case of the continuity and momentum equations math \phi /math is equal to 1 and math U_i /math respectively.
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| === Temporal Term ===
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| The temporal term of the momentum equations is discretized using a first order implicit Euler scheme
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| {{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}}
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| where math \Delta A /math is the cell area, and math \Delta t /math is the hydrodynamic time step.
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| === Advection Term === | | :<math>S_{ij}</math> = wave radiation stress [Pa] |
| The advection scheme obtained using the divergence theorem as
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| 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
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| {{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>R_{ij}</math> = surface roller stress [Pa] |
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| 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
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| {{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}}
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| 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. The advection value is calculated as math {{\tilde{\phi }}_{f}}=\tilde{\phi }_{f}^{L(\exp )}+ \tilde{\phi }_{f}^{H(\text{imp})}-\tilde{\phi }_{f}^{L(\text{imp})} /math, where the superscripts mathL/math and mathH/math indicate low and high order approximations and the superscripts math(exp)/math and math(imp)/math indicate either explicit and implicit treatment. The explicit term is solved directly while the implicit term is implemented through a deferred correction in which the terms are approximated using the values from the previous iteration step.
| | :<math>m_b</math> = bed slope coefficient [-] |
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| === Cell-face interpolation operator ===
| | :<math>\tau_{bi}</math> = combined wave and current mean bed shear stress [Pa]. |
| The general formula for estimating the cell-face value of math \tilde{\phi }_{f}^{{}} /math is given by
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| {{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}}
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| 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.
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| === Diffusion term ===
| | 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. |
| 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}}
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| 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
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| {{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}}
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| where math {{\nabla }_{\bot }}\phi /math is gradient in the direction perpendicular to the cell face and
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| math {{D}_{f}}=\frac{\bar{\Gamma }_{f}^{{}}{{{\bar{h}}}_{f}}\Delta {{l}_{f}}}{\left| \delta {{x}_{\bot }} \right|} /math.
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| === Cell-centered node-based gradient operator ===
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| === Cell-centered face-based gradient operator ===
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| === Cell-face gradient operator ===
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| === Source terms ===
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| == Hydrodynamic Solver ==
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| == Wetting and drying ==
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| In the numerical simulation of the surface water flows with sloped beaches, sand bars and islands, the water edges change with time, with part of the nodes being possibly wet or dry. In the present model, a threshold flow depth (a small value such as 0.02 m in field cases) is used to judge drying and wetting. If the flow depth on a node is larger than the threshold value, this node is considered to be wet, and if the flow depth is lower than the threshold value, this node is dry. Because a fully implicit solver is used in the present model, all the wet and dry nodes participate in the solution. Dry nodes are assigned a zero velocity. On the water edges between the dry and wet nodes, the wall-function approach is applied.
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| = 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.
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| * Saad, Y., (1994). “ILUT: a dual threshold incomplete ILU factorization,” Numerical Linear Algebra with Applications, 1, 387-402.
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| * 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.
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| * 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.
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| * 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]
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| * 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]
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| * 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.
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| * 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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| ----
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| = Variable Index =
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| {| border=1
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| ! Symbol !! Description !! Units
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| |-
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| | math t /math || Time || sec
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| |-
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| | math h /math || Total water depth math h = \zeta + \eta /math || m
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| |-
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| | math \zeta /math || Still water depth || m
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| |-
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| | math \eta /math || Water surface elevation with respect to the still water elevation || m
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| |-
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| | math U_j /math || Current velocity in the jth direction || m/sec
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| |-
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| | math S /math || Sum of Precipitation and evaporation per unit area || m/sec
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| |-
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| | math g /math || Gravitational constant || m/secsup2/sup
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| |-
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| | math \rho /math || Water density || kg/msup3/sup
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| |-
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| | math p_a /math || Atmospheric pressure || Pa
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| |-
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| | math \nu_t /math || Turbulent eddy viscosity || msup2/sup/sec
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| |}
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| ---- | | ---- |
| /big
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