CMS-Flow:Boundary Conditions: Difference between revisions
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\frac{h_i^{r+1}} {n_i} \Delta l_i\bigg |} | \frac{h_i^{r+1}} {n_i} \Delta l_i\bigg |} | ||
\frac{h_i^{r+1}}{n_i} \hat{e}</math>|3}} | \frac{h_i^{r+1}}{n_i} \hat{e}</math>|3}} | ||
where: | |||
:''i'' = subscript indicating a boundary cell | |||
:<math>\overrightarrow{q}_i</math> = volume discharge at boundary cell i per unit width [m<sup>2</sup>/s] | |||
:<math>\hat{e}</math> = unit vector for inflow direction = <math>(sin\ \varphi, cos\ \varphi)</math> | |||
:<math>\varphi</math> = inflow direction measured clockwise from North [deg] | |||
:<math>\hat{n}</math> = boundary face unit vector (positive outward) | |||
:''Q'' = specified total volume flux across the boundary [m<sup>3</sup>/s] | |||
:''n'' = Manning’s coefficient [s/m<sup>1/3</sup>] | |||
:''r'' = empirical constant equal to approximately 2/3 | |||
:<math>\Delta l</math> = cell width in the transverse direction normal to flow [m] | |||
:<math>f_{ramp}</math>= ramp function [-] (described in Chapter 3). | |||
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 <math>(\varphi)</math>. 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== | |||
The general formula for the boundary water surface elevation is given by | |||
{{Equation|<math> | |||
\bar{n}_B = f_{ramp}(\bar{\eta}_E + \Delta \bar{\eta} + \bar{\eta}_G + \bar{\eta}_C) + (1 - f_{ramp})\bar{\eta}_0 | |||
</math>|4}} | |||
where: | |||
:<math>\bar{\eta}_B</math> = boundary water surface elevation [m] | |||
:<math>\bar{\eta}_E</math> = specified external boundary water surface elevation [m] | |||
:<math>\Delta \bar{\eta}</math> = water surface elevation offset [m] | |||
:<math>\bar{\eta}_0</math> = initial boundary water surface elevation [m] | |||
:<math>\bar{\eta}_C</math> = correction to the boundary water surface elevation based on the wind and wave forcing [m] | |||
:<math>\bar{\eta}_G</math> = water surface elevation component derived from user specified gradients [m] | |||
:<math>f_{ramp}</math> = ramp function [-] (described in Chapter 3). | |||
The external water surface elevation <math>(\bar{\eta}_z )</math> may be specified as a time series, both spatially constant and varying or calculated from tidal/harmonic constituents. 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. If tidal constituents are specified, then <math>\bar{\eta}_E</math> is calculated as | |||
{{Equation|<math> | |||
\bar{\eta}_E (t) = \sum f_i A_i \cos (\omega_i t + V_{i}^0 + \hat{u}_i - \kappa_i) | |||
</math>|5}} | |||
where: | |||
i = subscript indicating a tidal constituent | |||
iA = mean amplitude [m] | |||
if = node (nodal) factor [-] | |||
iω = frequency [deg/hr] | |||
t = elapsed time from midnight of the starting year [hrs] | |||
0ˆiiVu+ = equilibrium phase [deg] | |||
iκ = phase lag [deg]. | |||
Revision as of 19:07, 12 August 2014
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:
- Free-slip: no tangential shear stress (wall friction)
- 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
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_{wall} = \rho c_{wall}U^2_\parallel} | (1) |
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 U_\parallel} is the magnitude of the wall parallel current velocity, 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 c_{wall}} is the wall friction coefficient equal 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 c_{wall} = \left(\frac{\kappa}{ln(y_P / y_0)} \right)^2 } | (2) |
Here, 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 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 (i.e, y_0 = z_0)} . The distance from the wall to the cell center is 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_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 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^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 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_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} {\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) |
where:
- i = subscript indicating a boundary cell
- 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} = 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)}
- 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} = 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 = specified total volume flux across the boundary [m3/s]
- n = Manning’s coefficient [s/m1/3]
- 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 normal 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 [-] (described in Chapter 3).
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
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{\eta}_E + \Delta \bar{\eta} + \bar{\eta}_G + \bar{\eta}_C) + (1 - f_{ramp})\bar{\eta}_0 } | (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 \bar{\eta}_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{\eta}_E} = specified 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{\eta}} = 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{\eta}_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{\eta}_C} = correction to the boundary water surface elevation based on 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{\eta}_G} = water surface elevation component derived from user specified 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 [-] (described in Chapter 3).
The external water surface elevation 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{\eta}_z )} may be specified as a time series, both spatially constant and varying or calculated from tidal/harmonic constituents. 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. If tidal constituents are specified, then 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{\eta}_E} 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 \bar{\eta}_E (t) = \sum f_i A_i \cos (\omega_i t + V_{i}^0 + \hat{u}_i - \kappa_i) } | (5) |
where: i = subscript indicating a tidal constituent
iA = mean amplitude [m]
if = node (nodal) factor [-]
iω = frequency [deg/hr]
t = elapsed time from midnight of the starting year [hrs]
0ˆiiVu+ = equilibrium phase [deg]
iκ = phase lag [deg].
Water Level
Water Level and Current Velocity
Tidal Constituent
The water level predictions are based on a harmonic equation with several arguments
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 \eta(t) = \sum_{i=1}^N f_i A_i \cos (\omega_i t + V_{i}^0 + u_i - \kappa_i) } | (1) |
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 A} is the constituent mean amplitude, 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} is a factor that reduces the mean amplitude and varies in time, 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_0+u} are the constituents equilibrium phase 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 \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
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_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
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 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
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 0 = \rho g \frac{\partial \eta}{\partial x} + \tau_{sx} + \tau_{wx} } | (4) |
Symbol | Description |
---|---|
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 t} | Time |
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} | Constituent mean amplitude |
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} | Constituent 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 V^0} | Constituent equilibrium argument |
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} | Constituent equilibrium argument |
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} | Constituent phase or epoch |