CMS-Flow Numerical Methods:General Transport Equation

General Transport Equation

All of the governing equations may be written in general form

$\underbrace {\frac {\partial (h\phi )}{\partial t}} _{\text{Temporal Term}}+\underbrace {\nabla \cdot (h\ bf{U}\phi )} _{\text{Advection Term}}=\underbrace {\nabla \cdot \left(\Gamma h\nabla \phi \right)} _{\text{Diffusion Term}}+\underbrace {S} _{\text{Source Term}}$

(8)

where $\phi$ is a general scalar, $t$ is time, $h$ is the total water depth, $bf{U}$ 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 Discretization

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

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

(9)

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

Cell-face interpolation operator

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

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

(12)

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.

Advection Schemes

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

$\int \limits _{A}{\nabla \cdot (h\mathbf {U} \phi )}{\text{d}}A=\oint \limits _{L}{h\phi \left(\mathbf {U} \cdot \mathbf {n} \right)}{\text{d}}L=\sum \limits _{f}^{}{{{\bar {h}}_{f}}\Delta {{l}_{f}}{{\left({{\hat {n}}_{i}}{{U}_{i}}\right)}_{f}}{{\tilde {\phi }}_{f}}}$

(10)

where $\mathbf {n} ={{\mathbf {\hat {n}} }_{\mathbf {i} }}=({{\mathbf {\hat {n}} }_{\mathbf {1} }},{{\mathbf {\hat {n}} }_{\mathbf {2} }})$ 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

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

(11)

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.

Diffusion term

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

$\int \limits _{A}{\nabla \cdot \left(\Gamma h\nabla \phi \right)}{\text{d}}A=\oint \limits _{S}{\Gamma h\left(\nabla \phi \cdot \mathbf {n} \right)}{\text{d}}S=\sum \limits _{f}{{\bar {\Gamma }}_{f}{{\bar {h}}_{f}}\Delta {{l}_{f}}{{\left({{\hat {n}}_{i}}{{\nabla }_{i}}\phi \right)}_{f}}}$

(13)

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

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

(14)

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