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 \begin{equation} \tag{1}

\tau_b = \lambda_{wc} m_b \rho c_b |U| U

\end{equation}

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 \begin{equation} \tag{2}

m_b = \sqrt{1+\biggl(\frac{\partial  z_b}{\partial x}\biggr)^2 + \biggl(\frac{\partial z_b}{\partial  y}\biggr)^2 }  

\end{equation}

The bed friction coefficient $c_{b}$ is related to the Manning's coefficient $n$ by \begin{equation} \tag{3} c_b = \frac{g n^2}{h^{1/3}} \end{equation}

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 \begin{equation} \tag{4} c_b=\biggl(\frac{\kappa}{\ln(30k_s/h)+1} \biggr)^2 \end{equation}

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

\begin{equation} \tag{5} \lambda_{wc} = \frac{\sqrt{ U^2 + c_w U_w^2 }}{U} \end{equation}

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)

\begin{equation} \tag{6} \lambda_{wc} = 1 + bX^p(1-X)^q \end{equation}

where $b$, $p$, and $q$ are coefficients that depend on the model selected and \begin{equation} \tag{7} X=\frac{\tau_w}{\tau_c + \tau_w} \end{equation}

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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