CMS-Flow:Bottom Friction

The bottom roughness is specified in CMS with either a Manning's n coefficient, roughness height (Nikradse bed roughness), or bed friction coefficient. The the roughness value is help constant throughout the simulation and is not changed according to bed composition and bed forms.

Flow without Waves

In the situation without waves, the bottom shear stress is calculated based on the quadratic formula

$\tau _{m}=\tau _{c}=m_{b}\rho c_{b}|u_{c}|u_{c}$

(1)

where $c_{b}$ is the bottom friction coefficient, $u$ is the depth-averaged current velocity and $m_{b}$ is a bed friction slope coefficient equal to

$m_{b}={\sqrt {1+{\biggl (}{\frac {\partial z_{b}}{\partial x}}{\biggr )}^{2}+{\biggl (}{\frac {\partial z_{b}}{\partial y}}{\biggr )}^{2}}}$

(2)

The bed friction coefficient $c_{b}$ is related to the Manning's coefficient $n$ by

$c_{b}={\frac {gn^{2}}{h^{1}/3}}$

(3)

where $g$ is the gravitational constant, and $h$ is the water depth.

Similarly, the bed friction coefficient $c_{b}$ is related to the roughness height $k_{s}$ by

$c_{b}={\biggl (}{\frac {\kappa }{\ln(30k_{s}/h)+1}}{\biggr )}^{2}$

(4)

Flow with Waves

There are five models available in CMS for calculating the combined wave and current mean shear stress:

1. Quadratic formula (named W09 in CMS)
2. Soulsby (1995) two coefficient data fit (named DATA2 in CMS)
3. Soulsby (1995) thirteen coefficient data fit (named DATA13 in CMS)
4. Fredsoe (1984) (named F84 in CMS)
5. Huynh-Thanh and Temperville (1991) (named HT91 in CMS)

In this case the simplified quadratic formula the combined wave and current mean shear stress is given by

$\tau _{m}=m_{b}\rho c_{b}u_{c}{\sqrt {u_{c}^{2}+c_{w}u_{w}^{2}}}$

(5)

where $u_{w}$ is the wave bottom orbital velocity based on the significant wave height, and $c_{w}$ is an empirical coefficient approximately equal to 0.5 (default).

For all of the other models, the mean shear stress is calculated as

$\tau _{m}=\lambda _{wc}\tau _{c}$

(6)

where $\lambda _{wc}$ is the nonlinear wave enhancement factor which is parameterized in the generalized form (Soulsby, 1995)

$\lambda _{wc}=1+bX^{p}(1-X)^{q}$

(7)

where $b$, $p$, and $q$ are coefficients that depend on the model selected and

$X={\frac {\tau _{w}}{\tau _{c}+\tau _{w}}}$

(8)

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