Boundary Conditions

Boundary Conditions

Wall Boundary Condition

At closed boundaries, two boundary conditions are applied. The first is zero flow normal to the boundary, and the second is the tangential shear stress due to flow parallel to the wall boundary. In CMS, two boundary conditions are available for tangential flow to walls. The first is a free-slip boundary condition in which the tangential shear stress set to zero, and the second is partial-slip boundary condition in which a friction term is included by assuming a log-law for a rough wall.

$\overrightarrow{\tau}_{wall} = - \rho c_{wall}\mid \overrightarrow{U_\Box} \mid \overrightarrow{U_\Box}$ (2-34)

where $c_{wall}$ is the wall friction coefficient equal to

$c_{wall} = \left[\frac{\kappa}{ln(y_p / y_0) - 1} \right]^2$ (2-35)

where $y_0$ is the roughness length of the wall and is assumed to be equal to that of the bed (i.e.$y_0 = z_0 ). y_p$ is the distance from the wall to the cell center.

Flux Boundary Condition

The flux boundary condition is typically applied to the upstream end of a river or stream and is specified as either a constant or time-series of total water volume flux Q is defined as

$Q = \sum_i h_i (\overrightarrow{U_i}\cdot \hat{n})\Delta l_i$ (2-36)

where

i = subscript indicating a boundary cell
Q = total volume flux across the boundary [m3/s]
h = total water depth [m]
$\overrightarrow{U}$ = depth-average current velocity [m/s]
$\hat{n}$ = boundary face unit vector (positive outward)
$\Delta l$ = cell width in the transverse direction to flow [m]

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 (i.e. $U \propto h^{\gamma} / n$ ). 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, qi , at each boundary cell i is calculated as

$\overrightarrow{q_i} = h \overrightarrow{U_i} = \frac {f_{Ramp}Q}{\mid \sum_i (\hat{e}\cdot \hat{n}) \frac{h_i^(r+1)}{n_i} \Delta l_i\mid} \frac{h_i^{r+1}}{n_i} \hat{e}$

where

i = subscript indicating a boundary cell
$\overrightarrow{q_i}$ = volume discharge at boundary cell i per unit width [m2/s]
$\hat{e}$ = unit vector for inflow direction = $(sin \varphi, cos \varphi) ::[itex]\varphi$ = inflow direction measured clockwise from North [deg]
$\hat{n}$ = boundary face unit vector (positive outward)
Q = total volume flux across the boundary [m3/s]
n = Manning’s coefficient [s/m1/3]
r =empirical constant equal to approximately 2/3
$\Delta l$ = cell width in the transverse direction to flow [m]
$f_{Ramp} =$ramp function [-]

The total volume flux is positive into the computational domain. Since it is not always possible to orient all flux boundaries to be normal to the inflow direction, the option is given to specify an inflow direction $\varphi$ . The angle is specified in degrees clockwise from true North. If the angle is not specified, then the inflow angle is assumed to be normal to the boundary. The total volume flux is conserved independently of the inflow direction.

Water Level Boundary Condition

Water level time series, both spatially constant and varying can be applied. A small degree of relaxation is obtained by applying the water level forcing as a source term rather than assigning the water level at the boundaries. This technique is common practice in finite volume models and improves stability and convergence. When applying a water level boundary condition to the nearshore, local flow reversals and boundary problems may result if the wave-and wind-induced setup are not included. This problem is avoided by adjusting the local water level to account for the cross-shore wind and wave setup similar that described in Reed and Militello (2005). The general formula for the boundary water surface elevation is given by

$\bar{n}_B = f_{Ramp}(\bar{n}_E + \Delta \bar{n} + \bar{n}_C + \bar{n}_G) + (1 - f_{Ramp})\bar{n}_0$ (2-38)

where

$\bar{n}_B$ = boundary water surface elevation [m]
$\bar{n}_E$ = external boundary water surface elevation [m]
$\Delta \bar{n}$ = water surface elevation offset [m]
$\bar{n}_0$= initial boundary water surface elevation [m]
$\bar{n}_C$ = correction to the boundary water surface elevation which is a function of the wind and wave forcing [m]
$\bar{n}_G$ = water surface elevation component derived from user speci-fied gradients [m]
$f_{Ramp}$ = ramp function [-]

The external water surface elevation may be spatially and temporally con-stant or variable. When a time series is specified, the values are interpolated using piecewise Lagrangian polynomials. By default, second order interpolation is used, but can be changed by the user. The water surface elevation offset $\Delta \bar{n}$ is assumed spatially and temporally constant and may be used to correct the boundary water surface elevation for vertical datums, surge, and sea level rise. The correction $\bar{n}_C$ is only applicable when $\bar{n}_E$ is spatially constant as in the case of a single water surface elevation time-series. The component $\bar{n}_G$ is intended to represent regional gradients in the water surface elevation, is assumed to be constant in time, and is only applicable when $\bar{n}_E$ is spatially constant.

Tidal/Harmonic Boundary Condition

Tidal or astronomic water level predictions are based on the prediction formula

$\bar{n}_E (t) = \sum f_i A_i cos(\omega t + V_i^0 + \hat{u}_i - \kappa_i)$ (2-39)

where

i = subscript indicating a tidal constituent
$A_i$= mean amplitude [m]
$f_t$ = node (nodal) factor [-]
$\omega_i$ = frequency [deg/hr]
t = elapsed time from midnight of the starting year [hrs]
$V_i^0 + \hat{u}_i$ = equilibrium phase [deg]
$\kappa_i$= phase lag or epoch [deg]

The nodal factor is a time-varying correction to the mean amplitude The equilibrium phase has a uniform component $V_i^0$ and a relatively smaller periodic component. The zero-superscript of $V_i^0$ indicates that the constituent phase is at time zero. Table 2 1 below provides a list of tidal constituents currently supported in CMS. More information on U.S. tidal constituent values can be obtained from U.S. National Oceanographic and Atmospheric Administration (http://tidesonline.nos.noaa.gov) and National Ocean Service (http://co-ops.nos.noaa.gov).

Table 2-1 Tidal Constituents names and speeds in solar hours implemented in CMS

</tr></tr>

 Constituent Speed Constituent Speed Constituent Speed Constituent Speed SA* 0.041067 SSA* 0.082137 MM* 0.54438 MSF* 1.0159 MF* 1.098 2Q1* 12.8534 Q1* 13.3987 RHO1 13.4715 O1* 13.943 M1* 14.4967 P1* 14.9589 S1* 15.0 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* 60.0 M6 86.9523 S6* 90.0 M8* 115.9364

If a harmonic boundary condition is applied, then the node factors are set to 1.0 and the and equilibrium arguments are set to zero.

Similarly to the water level boundary condition, the local water level at the boundary is adjusted to account for the wind and wave setup in order to avoid local flow reversals or instabilities (Reed and Militello 2005).

Cross-shore Boundary Condition

In the implicit flow solver, a cross-shore boundary condition is applied by solving the 1-D cross-shore momentum equation including wave and wind forcing (Wu et al. 2011a, 2011b). Along a cross-shore boundary, it is assumed that a well-developed longshore current exists. Thus, the along-shore (y-direction) momentum equation can be reduced to

$\frac{\partial}{\partial x}\left(v_t h \frac{\partial V_y}{\partial x} \right) = \frac{1}{\rho} (\tau_{sy} + \tau_{wy} - \tau_{by})$ (2-40)

where $\tau_{sy},\tau_{wy}, \text{and} \tau_{by}$ are the surface, wave, and bottom stresses in the long-shore direction, respectively. The equation above is solved iteratively for the longshore current velocity. The cross-shore (x) component of the velocity is assigned a zero-gradient boundary condition.

The water level due to waves and wind at the cross-shore boundary can be determined by assuming a zero alongshore gradient of flow velocity and negligible cross-shore current velocity. For this case, the cross-shore momentum equation reduces to

$\rho g h \frac{\partial \bar{n}}{\partial x} = \tau_{sx} + \tau_{wx}$ (2-41)

where $\tau_{sx}$ and $\tau_{wx}$ are the wind and wave stresses in the cross-shore direction.