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.

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

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 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 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 y_0} is the roughness length of the wall and is assumed to be equal to that of the bed (i.e.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 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

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 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]

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 \overrightarrow{U} } = depth-average current velocity [m/s]

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 \hat{n} } = boundary face unit vector (positive outward)

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 \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. 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 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, q_i , at each boundary cell i is calculated as

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 \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]

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 \hat{e}} = unit vector for inflow direction = 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 (sin \varphi, cos \varphi) ::<math>\varphi} = inflow direction measured clockwise from North [deg]

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 \hat{n}} = boundary face unit vector (positive outward)

Q = total volume flux across the boundary [m³/s]

n = Manning’s coefficient [s/m^1/3]

r =empirical constant equal to approximately 2/3

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 \Delta l} = cell width in the transverse direction to flow [m]

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 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 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 \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

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 \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

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 \bar{n}_B} = boundary water surface elevation [m]

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 \bar{n}_E} = external boundary water surface elevation [m]

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 \Delta \bar{n}} = water surface elevation offset [m]

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 \bar{n}_0 } = initial boundary water surface elevation [m]

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 \bar{n}_C} = correction to the boundary water surface elevation which is a function of the wind and wave forcing [m]

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 \bar{n}_G} = water surface elevation component derived from user speci-fied gradients [m]

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 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 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 \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 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 \bar{n}_C} is only applicable when 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 \bar{n}_E} is spatially constant as in the case of a single water surface elevation time-series. The component 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 \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 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 \bar{n}_E} is spatially constant.

Tidal/Harmonic Boundary Condition

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

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 \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

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 A_i} = mean amplitude [m]

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 f_t} = node (nodal) factor [-]

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 \omega_i } = frequency [deg/hr]

t = elapsed time from midnight of the starting year [hrs]

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 V_i^0 + \hat{u}_i } = equilibrium phase [deg]

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 \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 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 V_i^0} and a relatively smaller periodic component. The zero-superscript of 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 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

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

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 \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 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 \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

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

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

Sediment Transport 1