Bottom Friction

Bottom Friction

The mean (short-wave averaged) bed shear stress, $\tau _{bi}$, is calculated as

$\tau _{bi}=\lambda _{wc}\tau _{ci}$ (2-8)

where

$\lambda _{wc}=$ nonlinear bottom friction enhancement factor $\left(\lambda _{wc}\geq 1\right)$ [-]

$\tau _{c}=$ current-related bed shear stress vector [Pa]

The current bed shear stress is given by

$\tau _{ci}=\rho c_{b}UU_{i}$ (2-9)

where

$\rho =$ water density (~1025 kg/m³)

c_b = bed friction coefficient [-]

$U=$ current magnitude = ${\sqrt {U_{i}^{2}+U_{j}^{2}}}$ [m/s]

The bottom roughness is specified with either a Manning's roughness coefficient $\eta$ , Nikuradse roughness height k_s , or bed friction coefficient c_b . It is important to note that the roughness value is held constant throughout the simulation and is 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, it option to automatically estimate the bed roughness from the bed composition and bedforms will be added.

The bed friction coefficient c_b is related to the Manning’s roughness coefficient $\eta$ by (Graf and Altinakar 1998)

$c_{b}=\left({\frac {\kappa }{ln(z_{0}/h)+1}}\right)^{2}$ (2-11)

where $\kappa =$ 0.4. The roughness length is related to the Nikuradse roughness (roughness height specified in SMS interface for CMS-Flow) by $z_{0}=k_{z}/30.$ SOMETHING IS MISSING provides mean roughness heights from a large number of field measurements over natural sea beds.

In the presence of waves, the mean (wave-averaged) bed friction is enhanced beyond the value that would result from a linear superposition of the current- and wave-related components. This enhancement of the bed friction is due a nonlinear interaction between waves and currents in the bottom boundary layer. In CMS, the bottom friction enhancement is included through an explicit factor in the quadratic bottom friction equation (Eq. 2-4) that simplifies the numerical discretization. The nonlinear bot-tom friction enhancement factor is calculated using one of the following formulations (name abbreviations are given in parenthesis):

1. Quadratic formula (QUAD)

2. Soulsby (1995) empirical two-coefficient data fit (DATA2)

3. Soulsby (1995) empirical thirteen-coefficient data fit (DATA13)

4. Fredsoe (1984) analytical wave-current boundary layer model (F84)

5. Huynh-Thanh and Temperville (1991) numerical wave-current boundary layer model (HT91)

6. Davies et al. (1988) numerical wave-current boundary layer model (DSK88)

7. Grant and Madsen (1979) analytical wave-current boundary layer model (GM79)

In the case of the quadratic formula (QUAD),

$\lambda _{wc}={\frac {\sqrt {U^{2}+c_{w}u_{w}^{2}}}{U}}$ (2-12)

where c_w is an empirical coefficient equal to approximately 0.5 and u_w is the bottom wave orbital velocity amplitude. For random waves, u_w is calculated based on the significant wave height and peak wave period, from linear wave theory as

$u_{wc}={\frac {\pi H_{s}}{T_{p}sinh(kh)}}$ (2-13)

The DATA2, DATA13, F84, and HT91 formulations are calculated using general parameterization of Soulsby (1995),

$\lambda _{wc}=1+bX^{P}(1-X)^{q}$ (2-14)

where $X=\tau _{c}/(\tau _{c}+\tau _{w})$ and b, P and q are coefficients that depend on the formulation selected. $\tau _{c}$ and $\tau _{w}$ are the current- and wave-related bed shear stress magnitudes. The current bed shear stress is given by

$\tau _{c}=\rho c_{b}U^{2}$ (2-15)

and the wave bed shear stress is given by

$\tau _{w}={\frac {1}{2}}\rho f_{w}u_{w}^{2}$ (2-16)

where f_w is the wave friction factor. For random waves, u_w is set to an equivalent or representative bottom orbital velocity amplitude equal to $u_{w}={\sqrt {2}}u_{rms}$ , where u_rms the root-mean-squared bottom orbital velocity amplitude defined here as (Soulsby 1987; 1997)

$u_{rms}={\frac {\pi H_{rms}}{T_{p}{\sqrt {2}}sinh(kh)}}$ (2-18)

Wiberg and Sherwood (2008) reported that $u_{rms}$ estimates using $H_{rms}$ and $T_{p}$ agree reasonably well with field measurements (except for $T_{p}$ < 8.8 s) and produces better estimates than other combinations with $H_{rms},H_{s},T_{p}$ and the zero-crossing wave period $T_{z}$ . The zero-crossing wave period is calculated as the average period (time lapse) between consecutive upward or downward intersections of the water level time series with the zero water line. A better approach is to assume a spectral shape (e.g. JONSWAP, Pierson-Moskowitz, etc.), and obtain an explicit curve for $u_{rms}$ by summing the contributions from each frequency (Soulsby 1987; Wiberg and Sherwood 2008). A simple explicit expression is provided in Equation 2-19 below based on the JONSWAP ($\gamma$ = 3.3) spectrum following the work of Soulsby (1987)

$u_{rms}=0.134{\frac {H_{s}}{T_{n}}}\left[1+tanh\left(-7.76{\frac {T_{n}}{T_{p}}}+1.34\right)\right]$ (2-19)

where $T_{n}={\sqrt {h/g}}$. The above expression agrees closely with the curves presented by Soulsby (1987; 1997).

It is noted that in the presence of a sloping bed, the bottom friction acts on a larger surface area for the same horizontal area. This increase in bottom friction is included through the coefficient (Mei 1989; Wu 2008)

$m_{b}=\mid \bigtriangledown z_{b}\mid ={\sqrt {\left({\frac {\partial z_{b}}{\partial x}}\right)^{2}+\left({\frac {\partial z_{b}}{\partial y}}\right)^{2}+1}}$ (2-20)

where $z_{b}$ is the bed elevation and $\bigtriangledown =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},1\right)$. In most morphodynamic models, the bottom slope is assumed to be small and the above term is neglected. However, it is included here for completeness. For bottom slopes of 1/5 and 1/3, the above expression leads to an increase in bottom friction of 2.0% and 5.4 %, respectively.

Eddy Viscosity