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)

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

(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

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

(3)

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)

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

(5)

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)

(6)

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

(7)

$f_{w}=\left\{{\begin{aligned}&exp(5.21r^{-0.19}-6.0)forr>1.57\\&0.3\ \ \ \ \ \ forr\leq 1.57(Swart1974)\end{aligned}}\right.$

(8)

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

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

(9)

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

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

(10)

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 15). 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 (10) 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):

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

(11)

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

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

(12)

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

Table 1. Fitting coefficients for combined wave-current mean bottom friction.

Coefficient	DATA2	DATA13	F84	HT91	DSK88	GM79
b1	1.2	0.47	0.29	0.27	0.22	0.73
b2	0.0	0.69	0.55	0.51	0.73	0.40
b3	0.0	-0.09	-0.10	-0.10	-0.05	-0.23
b4	0.0	-0.08	-0.14	-0.24	-0.35	-0.24
p1	0.0	-0.53	-0.77	-0.75	-0.0.86	-0.68
p2	0.0	0.47	0.10	0.13	0.26	0.13
p3	0.0	0.07	0.27	0.12	0.34	0.24
p4	0.0	-0.2	0.14	0.02	-0.07	-0.07
q1	3.2	2.34	0.91	0.89	-0.89	1.04
q2	0.0	-2.41	0.25	0.40	2.33	-0.56
q3	0.0	0.45	0.50	0.50	2.60	0.34
q4	0.0	-0.61	0.45	-0.28	-2.50	-0.27
J	0.0	8.8	3.00	2.70	2.70	0.50

The GM79, DATA2, and DATA13 models use the logarithmic relationship for the bed friction coefficient given by Equation (2). In the case of the F84, HT91, and DSK88 models, the bed friction coefficient is linearly interpolated in log-space using the tabulated values presented in Soulsby (1997).

In the case of the F84, HT91, DSK88, and GM79 models, the wave friction factors are linearly interpolated in log-space using the tabulated values found in Soulsby (1997). In the case of the DATA2 and DATA13 formulas, the wave friction factor is estimated using Equation (6).

Bottom Wave Orbital Velocity

The bottom wave orbital velocity amplitude for regular waves $(u_{w})$ is calculated based on linear wave theory as

$u_{w}={\frac {\pi H}{T\ sinh(kh)}}$

(13)

where:

H = wave height [m]

T = wave period [s]

k = wave number [rad/m]

Unless specified otherwise, 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 wave orbital velocity amplitude is defined here following Soulsby (1987; 1997):

$u_{rms}^{2}=var({\tilde {u}}_{b})=\int _{0}^{\infty }S_{u}(f)df$

(14)

where:

var() = variance function,

${\tilde {u}}_{b}$ = instantaneous bottom orbital velocity [m/s]

$S_{u}$ = wave orbital velocity spectrum density [s m²/s²]

f = wave frequency [1/s] .

It is noted that the definition of $u_{rms}$ is slightly different from others such as Madsen (1994), Myrhaug et al. (2001), and Wiberg and Sherwood (2008) which include factor of 2 in their definition. A simple approximation for $u_{rms}$ from linear wave theory and the root-mean-squared wave height $H_{rms}=H_{s}/{\sqrt {2}}$ (for a Rayleigh distribution) is given by

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

(15)

Wiberg and Sherwood (2008) reported that $u_{rms}$ estimates using $H_{rms}$ and $T_{p}$ agree reasonably well with field measurements (except for $T_{p}<8.8s)$ 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 such as the Joint North Sea Wave Project (JONSWAP) (Hasselman et al. 1973) 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 below based on the JONSWAP ($\gamma$ = 3.3) spectrum following the work of Soulsby (1987):

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

(16)

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

In some cases the bottom wave orbital velocity amplitude is calculated based on the significant wave height and peak wave period $(u_{ws})$ as

$u_{ws}={\frac {\pi \ H_{s}}{T_{p}\ sinh(kh)}}$

(17)

Bed slope Friction Coefficient

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 2007)

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

(18)

where $z_{b}$ is the bed elevation, and $\bigtriangledown =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},1\right)$ For bottom slopes of 1/5 and 1/3, the above expression leads to an increase in bottom friction of 2.0 percent and 5.4 percent, respectively. 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.

References

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