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, q_i , 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 = <math>(sin \varphi, cos \varphi)