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. It is important to note that the roughness value is held constant throughout the simulation and is not changed according to bed composition and bed forms. It is also independent of the bed roughness calculation used for various sediment transport formula, since different formulas use different methods for computing the bed shear stresses.

In the CMS, the mean (shot-wave averaged) bottom shear stress is calculated based on the general quadratic formula

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

(1)

where $\lambda _{wc}$ is the nonlinear wave enhancement factor, $m_{b}$ is a bed slope friction coefficient, $c_{b}$ is the bottom friction coefficient, and $u$ is the depth-averaged current velocity.

The bed slope friction coefficient $m_{b}$ is 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)

In the case of currents only the he nonlinear wave enhancement factor equal and $\tau _{m}=\tau _{c}=m_{b}\rho c_{b}|u_{c}|u_{c}$. In the presence of waves, \lambda_{wc} is parametrized using either a simplified quadratic formula 
     

$\lambda _{wc}={\frac {\sqrt {u_{c}^{2}+c_{w}u_{w}^{2}}}{u_{c}^{2}}}$

(7)

The second option is general parametrization of 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)

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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