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)

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

(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 $z_{0}$ 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

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

(1)

where

$\rho$ = water density (~1025 kg/m³)

$c_{b}$ = bed friction coefficient [-]

$U={\sqrt {U_{i}U_{i}}}$ = current velocity magnitude [m/s]

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

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

(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)

where $f_{w}$ = wave friction factor, and $u_{w}$ is an equivalent or representative bottom wave orbital velocity amplitude. The wave friction factor ($f_{w}$) is estimated using one of the following:

$f_{w}=\exp(5.5r^{-0.2}-6.3)$ (Nielson 1982)

(3)

$f_{w}=0.237r^{-0.52}$ (Soulsby 1997)

(3)

$f_{w}=\left\{{\begin{aligned}&exp(5.21r^{-0.19}-6.0)&for\ r>1.57\\&0.3&for\ r<1.57\end{aligned}}\right.$

(3)

(Swart 1974)

where:

r = relative roughness = $A_{w}/k_{s}$ [-]

$k_{s}$ = Nikuradse roughness [m]

$A_{w}=$ semi-orbital excursion = $u_{w}T/(2\pi )\$[m]

T = wave period[s]

Mean Bed Shear Stress Due to Waves and Currents

Under combined waves and currents, the mean (wave-averaged) bed shear stress is enhanced compared to the case of currents only. This enhancement of the bed shear stress is due to the nonlinear interaction between waves and currents in the bottom boundary layer. In CMS, the mean (short-wave averaged) bed shear stress ($\tau _{bi}$) is calculated as

$\tau _{bi}=\lambda _{wc}\tau _{ci}$

(4)

where:

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

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

The nonlinear bottom friction enhancement factor ($\lambda _{wc}$) is calculated using one of the following formulations (name abbreviations are given in parenthesis):

1. Wu et al. (2010) 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 QUAD formula, $\lambda _{wc}$ is given by

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

(5)

where $c_{w}$ is an empirical coefficient, and $u_{w}$ is the wave bottom orbital velocity amplitude based on linear wave theory. For random waves, $u_{w}-u_{ws}$ where $u_{ws}$ is the bottom wave orbital velocity amplitude calculated based on the significant wave height and peak wave period (Equation XXX). Wu et al. (2010) originally proposed setting $c_{w}=0.5$. Here, the coefficient $c_{w}$ has been calibrated equal to 1.33 for regular waves and 0.65 for random waves to agree better with DATA2 formula.

A formula similar to Equation (2-17) was independently proposed by Wright and Thompson (1983) and calibrated using field measurements by Feddersen et al. (2000). The main difference in the two formulations is that Wu et al. (2010) uses the bottom wave orbital velocity based on the significant wave height, while the Wright and Thompson (1983) formulation uses the standard deviation of the bottom orbital velocity.

The DATA2, DATA13, F84, HT91, DSK88, and GM79 formulations are calculated using the general parameterization of Soulsby (1993):

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

(6)

where $X=\tau _{c}/(\tau _{c}+\tau _{w})$ and b, P, and q are coefficients given by (Soulsby et al. 1993)

$X=\left(X_{1}+X_{2}|cos|\varphi |^{J}\right)+\left(X_{3}+X_{4}|cos\varphi |^{J}\right)log_{10}\left({\frac {f_{w}}{c_{b}}}\right)$

(7)

where $X=(b,p,q)=f(X_{1},X_{2},X_{3},X_{4})$ are coefficients which have been fitted to each model (Table 2-1).

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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