CMS-Flow:Hydro Eqs

Governing Equations

The depth-averaged 2-D continuity equation may be written as

${\frac {\partial h}{\partial t}}+{\frac {\partial (h{{U}_{j}})}{\partial {{x}_{j}}}}=0$

(1)

where $h$ is the total water depth $h=\zeta +\eta$, $\eta$ is the water surface elevation, $\zeta$ is the still water depth, $U_{i}$ is the depth-averaged Langrangian current velocity defined as $U_{i}=U_{i}^{E}+U_{i}^{S}$, where $U_{i}^{E}$ is the phase- and depth-averaged current velocity (i.e. Eulerian velocity), and $U_{i}^{S}$ is the depth-averaged Stokes velocity (Phillips 1977)

$U_{i}^{S}={\frac {{E}_{w}}{\rho hc}}{{w}_{i}}$

(2)

where $E_{w}$ is the wave energy, $\rho$ is the water density, and $c$ is the wave celerity, and ${{w}_{i}}=(\cos \theta ,\sin \theta )$ is the wave unit vector where $\theta$ is the wave direction.

The momentum equation can be written as

${\frac {\partial (h{{U}_{i}})}{\partial t}}+{\frac {\partial (h{{U}_{i}}{{U}_{j}})}{\partial {{x}_{j}}}}-{{\varepsilon }_{ij}}{{f}_{c}}h{{U}_{j}}=-gh{\frac {\partial \eta }{\partial {{x}_{i}}}}-{\frac {h}{{\rho }_{0}}}{\frac {\partial {{p}_{a}}}{\partial {{x}_{i}}}}+{\frac {\partial }{\partial {{x}_{j}}}}\left({{\nu }_{t}}h{\frac {\partial {{U}_{i}}}{\partial {{x}_{j}}}}\right)+{\frac {1}{\rho }}\left(\tau _{i}^{w}+\tau _{i}^{s}-\tau _{i}^{b}\right)$

(3)

where $g$ is the gravitational constant, $f_{c}$ is the Coriolis parameter, $p_{a}$ is the atmospheric pressure, $\rho _{0}$ is a reference water density, $\nu _{t}$ is the turbulent eddy viscosity, $\tau _{w}$ is the wind stress, $\tau _{S}$ is the wave stress, and $\tau _{b}$ is the combined wave-current mean bed shear stress. $\varepsilon _{ij}$ is the permutation parameter equal to 1 for $i,j$ = 1,2, -1 for $i,j$ = 2,1; and 0 for $i=j$.

Bottom Stress

The mean (wave averaged) bed shear stress is calculated as

$\tau _{i}^{b}=m_{b}\lambda _{wc}\rho c_{b}U_{E}U_{i}^{E}$

(4)

where $\lambda _{wc}$ is the nonlinear wave enhancement factor, $m_{b}$ is a bed slope friction coefficient, $c_{b}$ is the bottom friction coefficient, and $U_{E}={\sqrt {U_{i}^{E}U_{i}^{E}}}$ is the Eulerian current magnitude. For additional information on the bottom friction wee Bottom and Wall Friction

Wave Stresses

The wave stresses are calculated as

$\tau _{i}^{w}=-{\frac {\partial }{\partial x_{j}}}\left(S_{ij}^{w}+S_{ij}^{r}-\rho hU_{i}^{S}U_{j}^{S}\right)$

(5)

where $S_{ij}^{w}$ is the wave radiation stresses and $S_{ij}^{r}$ is the roller stresses. $S_{ij}^{w}$ is calculated using linear wave theory

$S_{ij}^{w}=\iint \limits _{}{E_{w}\left[n{{w}_{i}}{{w}_{j}}+{{\delta }_{ij}}\left(n-{\frac {1}{2}}\right)\right]df}d\theta$

(6)

where $f$ is frequency, $\theta$ is the direction, $\delta _{ij}$=1 for $i=j$, $\delta _{ij}$=0 for $i\neq j$, and $n={\frac {1}{2}}\left(1+{\frac {2kh}{\sinh 2kh}}\right)$. For more information on the CMS-Wave model see CMS-Wave.

The roller stresses are $S_{ij}^{r}$ are calculated as

$S_{ij}^{r}=2E_{r}w_{i}w_{j}$

(7)

where $E_{r}$ is the roller energy. For more information on the surface roller see Surface Roller.

Numerical Methods

General Transport Equation: Discretization

All of the governing equations may be written in general form

Failed to parse (syntax error): {\displaystyle \underbrace{\frac{\partial (h\phi )}{\partial t}}_{\text{Temporal Term}}+\underbrace{\nabla \cdot (h\<math>bf{U}\phi )}_{\text{Advection Term}}=\underbrace{\nabla \cdot \left( \Gamma h\nabla \phi \right)}_{\text{Diffusion Term}}+\underbrace{S}_{\text{Source Term}} }

(3)

where $\phi$ is a general scalar, $t$ is time, $h$ is the total water depth, Failed to parse (syntax error): {\displaystyle \<math>bf{U} } is the depth averaged current velocity, $\Gamma$ is the diffusion coefficient for $\phi$, $\nabla =({{\nabla }_{1}},{{\nabla }_{2}})$ is the gradient operator, and $S$ includes all other terms. Note that in the case of the continuity and momentum equations $\phi$ is equal to 1 and $U_{i}$ respectively.

Temporal Term

The temporal term of the momentum equations is discretized using a first order implicit Euler scheme

$\int \limits _{A}{\frac {\partial (h\phi )}{\partial t}}{\text{d}}A={\frac {\partial }{\partial t}}\int \limits _{A}{h\phi {\text{d}}A}={\frac {{{h}^{n+1}}\phi _{}^{n+1}-{{h}^{n}}\phi _{}^{n}}{\Delta t}}\Delta A$

(3)

where $\Delta A$ is the cell area, and $\Delta t$ is the hydrodynamic time step.

Advection Term

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

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 \int\limits_{A}{\nabla \cdot (h\<math>bf{U}\phi )}\text{d}A=\oint\limits_{L}{h\phi \left( \<math>bf{U}\cdot \<math>bf{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}}} }

(4)

where 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 \<math>bf{n}={{\hat{n}}_{i}}=({{\hat{n}}_{1}},{{\hat{n}}_{2}}) } is the outward unit normal on cell face f, $\Delta {{l}_{f}}$ is the cell face length and ${{\bar {h}}_{f}}$ 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 ${{U}_{f}}={{U}_{i}}\in f\bot i$ the above equation simplifies to

$\sum \limits _{f}^{}{{{\bar {h}}_{f}}\Delta {{l}_{f}}{{\left({{\hat {n}}_{i}}{{U}_{i}}\right)}_{f}}{{\tilde {\phi }}_{f}}}=\sum \limits _{f}^{}{{{n}_{f}}{{F}_{f}}{{\tilde {\phi }}_{f}}}$

(5)

where ${{F}_{f}}={{\bar {h}}_{f}}\Delta {{l}_{f}}{{U}_{f}}$, ${{n}_{f}}={{n}_{\bot }}={{\left({{\hat {e}}_{i}}{{\hat {n}}_{i}}\right)}_{f}}$, with ${{\hat {e}}^{i}}=({{\hat {e}}_{1}},{{\hat {e}}_{2}})$ being the basis vector. $n_{f}$ is equal to 1 for West and South faces and equal to -1 for North and East cell faces. Lastly, ${\tilde {\phi }}_{f}^{}$ is the advective value of $\phi$ on cell face f, and is calculated using either the Hybrid, Exponential, HLPA (Zhu 1991) schemes. The cell face velocities $U_{f}$ are calculated using the momentum interpolation method of Rhie and Chow (1983) described in the subsequent section. The advection value is calculated as ${{\tilde {\phi }}_{f}}={\tilde {\phi }}_{f}^{L(\exp )}+{\tilde {\phi }}_{f}^{H({\text{imp}})}-{\tilde {\phi }}_{f}^{L({\text{imp}})}$, where the superscripts $L$ and $H$ indicate low and high order approximations and the superscripts $(exp)$ and $(imp)$ 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.

Cell-face interpolation operator

The general formula for estimating the cell-face value of ${\tilde {\phi }}_{f}^{}$ is given by

${{\bar {\phi }}_{f}}={{L}_{\bot }}{{\phi }_{N}}+(1-{{L}_{\bot }}){{\phi }_{P}}+{{\left({{\nabla }_{\parallel }}\phi \right)}_{N}}{{L}_{\bot }}({{x}_{\parallel ,O}}-{{x}_{\parallel ,N}})+{{\left({{\nabla }_{\parallel }}\phi \right)}_{P}}(1-{{L}_{\bot }})({{x}_{\parallel ,O}}-{{x}_{\parallel ,P}})$

(6)

where ${{L}_{\bot }}$ is a linear interpolation factor given by ${{L}_{\bot }}=\Delta {{x}_{\bot ,P}}/(\Delta {{x}_{\bot ,P}}+\Delta {{x}_{\bot ,N}})$ and ${{\nabla }_{\parallel }}$ is the gradient operator in the direction parallel to face f. By definition $\parallel \,=2\left|{{\hat {n}}_{1}}\right|+1\left|{{\hat {n}}_{2}}\right|$. Note that for neighboring cells without any refinement ${{x}_{\parallel ,O}}-{{x}_{\parallel ,P}}$ and ${{x}_{\parallel ,O}}-{{x}_{\parallel ,N}}$ are zero and thus the above equation is consistent with non-refined cell faces.

Diffusion term

The diffusion term is discretized in general form using the divergence theorem

Failed to parse (syntax error): {\displaystyle \int\limits_{A}{\nabla \cdot \left( \Gamma h\nabla \phi \right)}\text{d}A=\oint\limits_{S}{\Gamma h\left( \nabla \phi \cdot \<math>bf{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}}} }

(7)

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

$\sum \limits _{f}^{}{{{n}_{f}}{\bar {\Gamma }}_{f}^{}{{\bar {h}}_{f}}\Delta {{l}_{f}}{{\left({{\nabla }_{\bot }}\phi \right)}_{f}}}=\sum \limits _{f}^{}{{{D}_{f}}\left[{{\phi }_{N}}-{{\phi }_{P}}+{{\left({{\nabla }_{\parallel }}\phi \right)}_{N}}\left({{x}_{\parallel ,O}}-{{x}_{\parallel ,N}}\right)-{{\left({{\nabla }_{\parallel }}\phi \right)}_{P}}\left({{x}_{\parallel ,O}}-{{x}_{\parallel ,P}}\right)\right]}$

(8)

where ${{\nabla }_{\bot }}\phi$ is gradient in the direction perpendicular to the cell face and ${{D}_{f}}={\frac {{\bar {\Gamma }}_{f}^{}{{\bar {h}}_{f}}\Delta {{l}_{f}}}{\left|\delta {{x}_{\bot }}\right|}}$.

Wetting and drying

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.

References

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Variable Index

Symbol	Description	Units
$t$	Time	sec
$h$	Total water depth $h=\zeta +\eta$	m
$\zeta$	Still water depth	m
$\eta$	Water surface elevation with respect to the still water elevation	m
$U_{j}$	Current velocity in the jth direction	m/sec
$S$	Sum of Precipitation and evaporation per unit area	m/sec
$g$	Gravitational constant	m/secsup2/sup
$\rho$	Water density	kg/msup3/sup
$p_{a}$	Atmospheric pressure	Pa
$\nu _{t}$	Turbulent eddy viscosity	msup2/sup/sec

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