Bottom Friction: Difference between revisions

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<math>\lambda_{wc} = 1 + bX^P (1 - X)^q</math>  (2-14)
<math>\lambda_{wc} = 1 + bX^P (1 - X)^q</math>  (2-14)
where <math>X = \tau_c / (\tau_c + \tau_w)</math> and b, P and q
are coefficients that depend on the formulation selected. <math>\tau_c</math> and <math>\tau_w</math> are the current- and wave-related bed shear stress magnitudes. The current bed shear stress is given by
<math>\tau_c = \rho c_b U^2</math> (2-15)
and the wave bed shear stress is given by
<math>\tau_w = \frac{1}{2}\rho f_w u^2_w</math>  (2-16)
where f<sub>w</sub>  is the wave friction factor. For random waves, u<sub>w</sub>  is set to an equivalent or representative bottom orbital velocity amplitude equal to <math>u_w = \sqrt{2}u_{rms}</math> , where u<sub>rms</sub>  the root-mean-squared bottom orbital velocity amplitude defined here as (Soulsby 1987; 1997)

Revision as of 20:16, 16 July 2014

Bottom Friction

The mean (short-wave averaged) bed shear stress, , is calculated as

(2-8)

where

nonlinear bottom friction enhancement factor [-]

current-related bed shear stress vector [Pa]

The current bed shear stress is given by

(2-9)

where

water density (~1025 kg/m3)

cb = bed friction coefficient [-]

current magnitude = [m/s]

The bottom roughness is specified with either a Manning's roughness coefficient , Nikuradse roughness height ks , or bed friction coefficient cb . 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 cb is related to the Manning’s roughness coefficient by (Graf and Altinakar 1998)

(2-11)

where 0.4. The roughness length is related to the Nikuradse roughness (roughness height specified in SMS interface for CMS-Flow) by 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),

(2-12)

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

(2-13)

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

(2-14)

where and b, P and q are coefficients that depend on the formulation selected. and are the current- and wave-related bed shear stress magnitudes. The current bed shear stress is given by

(2-15)

and the wave bed shear stress is given by

(2-16)

where fw is the wave friction factor. For random waves, uw is set to an equivalent or representative bottom orbital velocity amplitude equal to , where urms the root-mean-squared bottom orbital velocity amplitude defined here as (Soulsby 1987; 1997)