CMS-Flow:Bottom Friction

Bed Roughness

The bed roughness is specified for the hydrodynamic calculations with either a Manning's roughness coefficient ($n$), Nikuradse roughness height ($k_{s}$), or bed friction coefficient ( $c_{b}$). It is important to note that the bed roughness is assumed constant in time and not changed according to bed composition and bedforms. This is a common engineering approach which can be justified by the lack of data to initialize the bed composition and the large error in estimating the bed composition evolution and bedforms. In addition using a constant bottom roughness simplifies the model calibration. In future versions of CMS, the option to automatically estimate the bed roughness from the bed composition and bedforms will be added. In addition, the bed roughness used for hydrodynamics may not be the same as that which is used for the sediment transport calculations because each sediment transport formula was developed and calibrated using specific methods for estimating bed shear stresses or velocities, and these cannot be easily changed.

The bed friction coefficient ($c_{b}$) is related to the Manning’s roughness coefficient ($n$ ) by (Soulsby 1997)

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

(1)

Commonly, the bed friction coefficient is calculated by assuming a logarithmic velocity profile as (Graf and Altinakar 1998)

$c_{b}={\biggl (}{\frac {\kappa }{\ln(z_{0}/h)+1}}{\biggr )}^{2}$

(2)

where $\kappa$=0.4 is Von Karman constant, 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 z_} is the bed roughness length which is related to the Nikuradse roughness ($k_{s}$) by $z_{0}=k_{s}/30$ (hydraulically rough flow).

Current-Related Shear Stress

The current bed shear stress is given by

$\tau _{c},i=\rho c_{d}UU_{i}$

(1)

where $\rho$ = water density (~1025 kg/m3) $c_{b}$ = bed friction coefficient [-] $U_{i}$ = current velocity [m/s] $U$ = current velocity magnitude [m/s]

The magnitude of the current-related bed shear stress is simply

$\tau _{c}=\rho c_{d}U^{2}$

(2)

Wave-Related Shear Stress

The wave-related bed shear stress amplitude is given by (Jonsson 1966)

$\tau _{w}={\frac {1}{2}}\rho f_{w}u_{w}^{2}$

(3)

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

$\tau _{b}=\lambda _{wc}m_{b}\rho c_{b}|U|U$

(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)

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 calculated based on one of five models:

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)

For the quadratic formula, the wave enhancement factor is simply

$\lambda _{wc}={\frac {\sqrt {U^{2}+c_{w}U_{w}^{2}}}{U}}$

(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). Therefore, the quadratic formula reduces to $\tau _{b}=m_{b}\rho c_{b}u_{c}{\sqrt {u_{c}^{2}+c_{w}u_{w}^{2}}}$. For all other models, the nonlinear wave enhancement factor $\lambda _{wc}$ is parameterized using the the generalized form proposed by Soulsby (1995)

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

(6)

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

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

(7)

References

• Fredsoe, J. (1984). “Turbulent boundary layer in wave-current motion,” Journal of Hydraulic Engineering, ASCE, 110, 1103-1120.
• Huynh-Thanh, S., and Temperville, A. (1991). “A numerical model of the rough turbulent boundary layer in combined wave and current interaction,” in Sand Transport in Rivers, Estuaries and the Sea, eds. R.L. Soulsby and R. Bettess, pp.93-100. Balkema, Rotterdam.

• Soulsby, R.L. (1995). “Bed shear-stresses due to combined waves and currents,” in Advanced in Coastal Morphodynamics, ed M.J.F Stive, H.J. de Vriend, J. Fredsoe, L. Hamm, R.L. Soulsby, C. Teisson, and J.C. Winterwerp, Delft Hydraulics, Netherlands. 4-20 to 4-23 pp.

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