Bottom Friction

The bottom shear stress is given by

1)	$\begin{array}{l}\displaystyle \displaystyle \boldsymbol{\tau}_b = \rho C_D \| \boldsymbol{V} \| \boldsymbol{V}\end{array}$

where

$\begin{array}{l}\rho\end{array}$ : water density [M/L³]
$\begin{array}{l}C_D\end{array}$ : non-dimensional drag coefficient [-]

The drag coefficient is computed using the Manning’s roughness coefficient as

2)	$\begin{array}{l}\displaystyle \displaystyle C_D = \frac{n^2 g}{R^{1/3}}\end{array}$

in which

$\begin{array}{l}n\end{array}$ : Manning's roughness coefficient [T/L^1/3]
$\begin{array}{l}R\end{array}$ : hydraulic radius [L]
$\begin{array}{l}g\end{array}$ : gravitational acceleration [L/T²]

Dividing by $\begin{array}{l}\rho R\end{array}$ gives the friction term of the momentum equation,

$\begin{array}{l}\displaystyle \dfrac{ \boldsymbol{\tau}_b }{ \rho R } = c_b \left\| \boldsymbol{V} \right\| \boldsymbol{V} , \qquad c_b = \dfrac{ C_D }{ R } = \dfrac{ g n^2 }{ R^{4/3} }\end{array}$

where $\begin{array}{l}c_b\end{array}$ is the nonlinear friction coefficient [1/L].

For the semi-implicit treatment the term is collected as a linear-plus-quadratic drag acting on the velocity,

$\begin{array}{l}\displaystyle \dfrac{ \boldsymbol{\tau}_b }{ \rho R } = c_f \, \boldsymbol{V}, \qquad c_f = c_L + c_b \left\| \boldsymbol{V} \right\|\end{array}$

where

$\begin{array}{l}c_f\end{array}$ : total friction coefficient [1/T]
$\begin{array}{l}c_L\end{array}$ : linear friction coefficient [1/T]

The linear coefficient is zero by default; it is available for cases where a linear damping is physically appropriate (e.g. very shallow laminar sheet flow) or where mild additional damping is desired. A constant nonlinear drag coefficient may also be specified in place of Manning's law for idealized studies. The quadratic part of the friction term reproduces the familiar Manning friction slope,

$\begin{array}{l}\displaystyle c_b \left\| \boldsymbol{V} \right\| \boldsymbol{V} = g \, \boldsymbol{S}_f , \qquad \boldsymbol{S}_f = \dfrac{ n^2 \left\| \boldsymbol{V} \right\| \boldsymbol{V} }{ R^{4/3} }\end{array}$

where $\begin{array}{l}\boldsymbol{S}_f\end{array}$ is the friction slope vector [-]

The scalar friction slope $\begin{array}{l}S_f\end{array}$ specified for normal-depth boundary conditions (see Flow and Stage Boundary Conditions) is the magnitude of this vector.

The shear velocity is given by:

3)	$\begin{array}{l}\displaystyle \displaystyle u_* = \sqrt{\tau_b / \rho }\end{array}$