CMS-Flow:Boundary Conditions

Wall Boundary

The wall boundary condition is a closed boundary and is applied at any cell face between wet and dry cells. Any unassigned boundary cell at the edge of the model domain is assumed to be closed and is assigned a wall boundary. A zero normal flux to the boundary is applied at closed boundaries. Two boundary conditions are available for the tangential flow:

1. Free-slip: no tangential shear stress (wall friction)
2. Partial-slip: tangential shear stress (wall friction) calculated based on the log law.

Assuming a log law for a rough wall, the partial-slip tangential shear stress is given by

$\tau _{wall}=\rho c_{wall}U_{\parallel }^{2}$

(1)

where $U_{\parallel }$ is the magnitude of the wall parallel current velocity, and $c_{wall}$ is the wall friction coefficient equal to

$c_{wall}=\left({\frac {\kappa }{ln(y_{P}/y_{0})}}\right)^{2}$

(2)

Here, $y_{0}$ is the roughness length of the wall and is assumed to be equal to that of the bed $(i.e,y_{0}=z_{0})$. The distance from the wall to the cell center is $y_{P}$.

Flux Boundary

The flux boundary condition is typically applied to the upstream end of a river or stream and specified as either a constant or time series of total water volume flux (Q). In a 2DH model, the total volume flux needs to be distributed across the boundary in order to estimate the depth-averaged velocities. This is done using a conveyance approach in which the current velocity is assumed to be related to the local flow depth (h) and Manning’s (n) as $U\propto h^{r}/n$. Here, r is an empirical conveyance coefficient equal to approximately 2/3 for uniform flow. The smaller the r value the more uniform the current velocities are across the flux boundary. The water volume flux $(q_{i})$ at each boundary cell (i) is calculated as

${\overrightarrow {q}}_{i}=h{\overrightarrow {U}}_{i}={\frac {f_{ramp}Q}{{\bigg |}\sum _{i}({\hat {e}}\cdot {\hat {n}}){\frac {h_{i}^{r+1}}{n_{i}}}\Delta l_{i}{\bigg |}}}{\frac {h_{i}^{r+1}}{n_{i}}}{\hat {e}}$

(3)

Tidal Constituent

The water level predictions are based on a harmonic equation with several arguments

$\eta (t)=\sum _{i=1}^{N}f_{i}A_{i}\cos(\omega _{i}t+V_{i}^{0}+u_{i}-\kappa _{i})$

(1)

where $A$ is the constituent mean amplitude, $f$ is a factor that reduces the mean amplitude and varies in time, $V_{0}+u$ are the constituents equilibrium phase and $\kappa$ is the constituent phase lag or epoch. Table 1 shows a list of the currently supported tidal constituents in CMS.

Constituent	Speed	Constituent	Speed	Constituent	Speed	Constituent	Speed
SA	0.041067	SSA	0.082137	MM	0.54438	MSF	1.0159
MF	1.098	2Q1	12.8543	Q1	13.3987	RHO1	13.4715
O1	13.943	M1	14.4967	P1	14.9589	S1	15
K1	15.0411	J1	15.5854	OO1	16.1391	2N2	27.8954
MU2	27.9682	N2	28.4397	NU2	28.5126	M2	28.9841
LDA2	29.4556	L2	29.5285	T2	29.9589	S2	30
R2	30.0411	K2	30.0821	2SM2	31.0159	2MK3	42.9271
M3	43.4762	MK3	44.0252	MN4	57.4238	M4	57.9682
MS4	58.9841	S4	60	M6	86.9523	S6	90
M8	115.9364

Flux

The water flux is specified as m^3/sec per cell along the cell string. This value is multiplied by the number of cells in the cell string to obtain the total flux. The total flux is then redistributed along the boundary according to

$q_{i}={\frac {Q_{b}}{\sum \Delta s_{i}h_{i}^{1+r}/n_{i}}}{\frac {h_{i}^{1+r}}{n_{i}}}$

(2)

Cross-shore

Along a cross-shore boundary, it is assumed that a well-developed longshore current exists. Thus, the y (alongshore) momentum equation can be reduced as follows

$0={\frac {\partial }{\partial x}}{\biggl (}\nu _{t}h{\frac {\partial V}{\partial x}}{\biggr )}+\tau _{sy}+\tau _{wy}-\tau _{by}$

(3)

The water level setup due to waves and winds at the cross-shore boundary can be determined by assuming a zero alongshore gradient of water level, or using the following equation reduced from the x (cross-shore) momentum equation

$0=\rho g{\frac {\partial \eta }{\partial x}}+\tau _{sx}+\tau _{wx}$

(4)

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Symbol	Description
$t$	Time
$A$	Constituent mean amplitude
$f$	Constituent nodal factor
$V^{0}$	Constituent equilibrium argument
$u$	Constituent equilibrium argument
$\kappa$	Constituent phase or epoch

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