CMS-Flow:Bottom Friction: Difference between revisions

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== Bottom Friction ==
== Bed Roughness ==
The bed roughness is specified for the hydrodynamic calculations with either a Manning's roughness coefficient (<math> n </math>), Nikuradse roughness height (<math>k_s</math>), or bed friction coefficient ( <math>c_b</math>). 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.


'''Flow without Waves'''
The bed friction coefficient (<math> c_b</math>) is related to the Manning’s roughness coefficient (<math>n</math> ) by (Soulsby 1997)
{{Equation|
<math> c_b = g n^2 h^{-1/3} </math>|1}}


In the situation without waves, the bottom shear stress is calculated based on the quadratic formula
Commonly, the bed friction coefficient is calculated by assuming a logarithmic velocity profile as (Graf and Altinakar 1998)
{{Equation|
<math> c_b=\left [\frac{\kappa}{\ln(z_0/h)+1} \right ]^2 </math>
|2}}
where <math>\kappa</math> = 0.4 is Von Karman constant, and <math>z_0</math> is the bed roughness length which is related to the Nikuradse roughness (<math>k_s</math>) by <math>z_0 = k_s/30</math> (hydraulically rough flow).


      <math> \tau_m = \tau_c = \rho c_b |u_c| u_c  </math>
== Current-Related Shear Stress ==
The current bed shear stress is given by


where <math> c_b </math> is the bottom friction coefficient, <math>u_c</math> is the depth-averaged current velocity.
{{Equation|<math> \tau_{ci} = \rho c_b U U_i </math>|3}}


'''Flow with Waves'''
where:


There are five models available in CMS for calculating the combined wave and current mean shear stress:
:<math>\rho</math> = water density (~1025 kg/m<sup>3</sup>)


1. Simplified quadratic formula (named W09 in CMS)
:<math>c_b</math> = bed friction coefficient [-]


2. Soulsby (1995) Data2 (named DATA2 in CMS)
:<math>U = \sqrt{U_i U_i}</math> = current velocity magnitude [m/s]


3. Soulsby (1995) Data13 (named DATA13 in CMS)
The magnitude of the current-related bed shear stress is simply


4. Fredsoe (1984) (names F84 in CMS)
{{Equation|<math>  \tau_c = \rho c_b U^2 </math>|2}}


5. Huynh-Thanh and Temperville (1991) (named HT91 in CMS)
== Wave-Related Shear Stress ==
The wave-related bed shear stress amplitude is given by (Jonsson 1966)


In this case the simplified expression for the combined wave and current mean shear stress is given by
{{Equation|<math> \tau_w = \frac{1}{2} \rho f_w u_w^2 </math>|5}}


      <math> \tau_m = \rho c_b u_c \sqrt{ u_c^2 + c_w u_w^2 } </math>
where <math>f_w</math> = wave friction factor, and <math>u_w</math> is an equivalent or representative bottom wave orbital velocity amplitude.  The wave friction factor (<math>f_w</math>) is estimated using one of the following:


where <math> u_w </math> is the wave bottom orbital velocity based on the significant wave height, and <math> c_w </math> is an empirical coefficient approximately equal to 0.5 (default).
{{Equation|<math> f_w = \exp(5.5 r^{-0.2} - 6.3 )  </math> (Nielson 1982)|6}}
For all of the other models, the mean shear stress is calculated as


      <math> \tau_m = \lambda_{wc} \tau_c </math>
{{Equation|<math> f_w = 0.237 r^{-0.52} </math> (Soulsby 1997)|7}}


where <math> \lambda_{wc} </math> is the nonlinear wave enhancement factor which is parameterized in the generalized form (Soulsby, 1995)
{{Equation|<math> f_w = \left\{
\begin{align}
&exp(5.21 r^{-0.19} - 6.0 ) for r>1.57 \\ 
&0.3  \ \ \ \ \ \  for r\leq 1.57  (Swart 1974)
\end{align}\right.</math>|8}}


      <math> \lambda_{wc} = 1 + bX^p(1-X)^q </math>


where <math>b</math>, <math>p</math>, and <math>q</math> are coefficients that depend on the model selected and
where:


      <math> X=\frac{\tau_w}{\tau_c + \tau_w} </math>
: r = relative roughness = <math>A_w/k_s</math> [-]


The default method for calculating the mean shear stress is the simplified quadratic formula, but the user may change this by using the advanced card
: <math>k_s</math> = Nikuradse roughness [m]


    WAVE-CURRENT_MEAN_STRESS      W09  !W09 | DATA2 | DATA13 | F84 | HT91
: <math>A_w =</math> semi-orbital excursion = <math>u_w T /(2\pi)\ </math> [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 (<math>\tau_{bi}</math>) is calculated as
 
{{Equation|<math> \tau_{bi} = \lambda_{wc} \tau_{ci} </math>|9}}
 
where:
 
:<math>\lambda_{wc}</math> = nonlinear bottom friction enhancement factor <math>(\lambda_{wc} \geq 1)</math> [-]
 
:<math>\tau_{ci}</math> = current-related bed shear stress [Pa].
 
 
The nonlinear bottom friction enhancement factor (<math>\lambda_{wc}</math>) is calculated using one of the following formulations (name abbreviations are given in parenthesis):
# Wu et al. (2010) quadratic formula (QUAD)
# Soulsby (1995) empirical two coefficient data fit (DATA2)
# Soulsby (1995) empirical thirteen-coefficient data fit (DATA13)
# Fredsoe (1984) analytical wave-current boundary layer model(F84)
# Huynh-Thanh and Temperville (1991) numerical wave-current boundary layer model ((HT91)
# Davies et al. (1988) numerical wave-current boundary layer model (DSK88)
# Grant and Madsen (1979) analytical wave-current boundary layer model (GM79)
 
In the case of the QUAD formula, <math>\lambda_{wc}</math> is given by
 
{{Equation|<math> \lambda_{wc} = \frac{\sqrt{ U^2 + c_w u_w^2 }}{U} </math>|10}}
 
where <math>c_w</math> is an empirical coefficient, and <math> u_w </math> is the wave bottom orbital velocity amplitude based on linear wave theory.  For random waves, <math>u_w = u_{ws}</math> where <math>u_{ws}</math> 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 <math>c_w = 0.5</math>. Here, the coefficient <math>c_w</math> 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):
 
{{Equation| <math> \lambda_{wc} = 1 + bX^p(1-X)^q </math>|11}}
 
where <math>X = \tau_c /(\tau_c + \tau_w)</math> and ''b, P, and q'' are coefficients given by (Soulsby et al. 1993)
 
 
{{Equation|<math> 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)</math>|12}}
 
where <math>X = (b,p,q) = f(X_1 ,X_2 ,X_3 ,X_4)</math> are coefficients which have been fitted to each model (Table 1).
 
 
'''Table 1. Fitting coefficients for combined wave-current mean bottom friction.'''
 
<table border cellpadding="3">
<tr>
<td><b>Coefficient</b></td>
<td><b>DATA2</b></td>
<td><b>DATA13</b></td>
<td><b>F84</b></td>
<td><b>HT91</b></td>
<td><b>DSK88</b></td>
<td><b>GM79</b></td>
</tr>
 
<tr>
<td>b<sub>1</sub></td>
<td>1.2</td>
<td>0.47</td>
<td>0.29</td>
<td>0.27</td>
<td>0.22</td>
<td>0.73</td>
</tr>
 
 
<tr>
<td>b<sub>2</sub></td>
<td>0.0</td>
<td>0.69</td>
<td>0.55</td>
<td>0.51</td>
<td>0.73</td>
<td>0.40</td>
</tr>
 
<tr>
<td>b<sub>3</sub></td>
<td>0.0</td>
<td>-0.09</td>
<td>-0.10</td>
<td>-0.10</td>
<td>-0.05</td>
<td>-0.23</td>
</tr>
 
<tr>
<td>b<sub>4</sub></td>
<td>0.0</td>
<td>-0.08</td>
<td>-0.14</td>
<td>-0.24</td>
<td>-0.35</td>
<td>-0.24</td>
</tr>
 
<tr>
<td>p<sub>1</sub></td>
<td>0.0</td>
<td>-0.53</td>
<td>-0.77</td>
<td>-0.75</td>
<td>-0.0.86</td>
<td>-0.68</td>
</tr>
 
<tr>
<td>p<sub>2</sub></td>
<td>0.0</td>
<td>0.47</td>
<td>0.10</td>
<td>0.13</td>
<td>0.26</td>
<td>0.13</td>
</tr>
 
<tr>
<td>p<sub>3</sub></td>
<td>0.0</td>
<td>0.07</td>
<td>0.27</td>
<td>0.12</td>
<td>0.34</td>
<td>0.24</td>
</tr>
 
<tr>
<td>p<sub>4</sub></td>
<td>0.0</td>
<td>-0.2</td>
<td>0.14</td>
<td>0.02</td>
<td>-0.07</td>
<td>-0.07</td>
</tr>
 
<tr>
<td>q<sub>1</sub></td>
<td>3.2</td>
<td>2.34</td>
<td>0.91</td>
<td>0.89</td>
<td>-0.89</td>
<td>1.04</td>
</tr>
 
<tr>
<td>q<sub>2</sub></td>
<td>0.0</td>
<td>-2.41</td>
<td>0.25</td>
<td>0.40</td>
<td>2.33</td>
<td>-0.56</td>
</tr>
 
<tr>
<td>q<sub>3</sub></td>
<td>0.0</td>
<td>0.45</td>
<td>0.50</td>
<td>0.50</td>
<td>2.60</td>
<td>0.34</td>
</tr>
 
<tr>
<td>q<sub>4</sub></td>
<td>0.0</td>
<td>-0.61</td>
<td>0.45</td>
<td>-0.28</td>
<td>-2.50</td>
<td>-0.27</td>
</tr>
 
<tr>
<td>J</td>
<td>0.0</td>
<td>8.8</td>
<td>3.00</td>
<td>2.70</td>
<td>2.70</td>
<td>0.50</td>
</tr>
</table>
 
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 <math>(u_w)</math> is calculated based on linear wave theory as
 
{{Equation|<math>u_w = \frac{\pi H}{T\ sinh(kh)}</math>|13}}
 
where:
 
:H = wave height [m]
 
:T = wave period [s]
 
:k = wave number [rad/m]
 
Unless specified otherwise, for random waves, <math>u_w</math> is set to an equivalent or representative bottom orbital velocity amplitude equal to <math>u_w = \sqrt{2}u_{rms}</math> where <math>u_{rms}</math> the root-mean-squared bottom wave orbital velocity amplitude is defined here following Soulsby (1987; 1997):
 
{{Equation|<math>u^2_{rms} = var(\tilde{u}_b) = \int_0^\infty S_u (f)df</math>|14}}
 
where:
 
:var() = variance function,
 
:<math>\tilde{u}_b</math> = instantaneous bottom orbital velocity [m/s]
 
:<math>S_u</math> = wave orbital velocity spectrum density [s m<sup>2</sup>/s<sup>2</sup>]
 
:f = wave frequency [1/s] .
 
It is noted that the definition of <math>u_{rms}</math> 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 <math>u_{rms}</math> from linear wave theory and the root-mean-squared wave height <math>H_{rms} = H_s /\sqrt{2}</math> (for a Rayleigh distribution) is given by
 
{{Equation|<math>u_{rms} = \frac{\pi\ H_{rms}} {T_p \sqrt{2}\ sinh(kh)}</math>|15}}
 
Wiberg and Sherwood (2008) reported that <math>u_{rms}</math> estimates using <math>H_{rms}</math> and <math>T_p</math> agree reasonably well with field measurements (except for <math>T_p < 8.8 s)</math> and produces better estimates than other combinations with <math>H_{rms}, H_s, T_P</math>, and the zero-crossing wave period <math>(T_z )</math>. 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 <u>J</u>oint <u>N</u>orth <u>S</u>ea <u>W</u>ave <u>P</u>roject (JONSWAP) (Hasselman et al. 1973) and obtain an explicit curve for <math>u_{rms}</math> by summing the contributions from each frequency (Soulsby 1987; Wiberg and Sherwood 2008). A simple explicit expression is provided below based on the JONSWAP (<math>\gamma</math> = 3.3) spectrum following the work of Soulsby (1987):
 
 
{{Equation|<math>u_{rms} = 0.134 \frac{H_s}{T_n} \left[1 + tanh\left(-7.76 \frac{T_n}{T_P} + 1.34  \right)  \right]</math>|16}}
 
where <math>T_n = \sqrt{h/g}</math>. 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 <math>(u_{ws} )</math> as
 
{{Equation|<math>u_{ws} = \frac{\pi\ H_s} {T_p\ sinh(kh)}</math>|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)
 
{{Equation|<math>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}</math>|18}}
 
where <math>z_b</math> is the bed elevation, and <math>\bigtriangledown = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y},1  \right)</math> 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 ==
* Davies, A. G., R. L. Soulsby, and H. L. King. 1988. A numerical model of the combined wave and current bottom boundary layer. Journal of Geophysical Research 93(C1):491–508.
* Fredsoe, J. (1984). “Turbulent boundary layer in wave-current motion,” Journal of Hydraulic Engineering, ASCE, 110, 1103-1120.
* Graf, W. H., and M. Altinakar. 1998. Fluvial hydraulics. Hoboken, NJ: Wiley & Sons, Ltd.
* Grant, W. D., and O. S. Madsen. 1979. Combined wave and current interaction with a rough bottom. Journal of Geophysical Research 86(C4):1797–1808.
* Hasselmann, K., T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerbrug, P. Muller, D. J. Olbers, K. Richter, W. Sell, and H. Walden. 1973. Measurements of windwave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Deutsche Hydrographische Zeitschrift A80(12):95.
* 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. 
* Jonsson, I. G. 1966. Wave boundary layers and friction factors. In Proceedings of the 10th Coastal Engineering Conference, ASCE, 127–148.
* Madsen, O. S. 1994. Spectral wave–current bottom boundary layer flows. In Proceedings of the 24th Conference on Coastal Engineering, ASCE, 384–398. Kobe, Japan.
* Myrhaug, D., L. E. Holmedal, R. R. Simons, and R .D. MacIver. 2001. Bottom friction in random waves plus current flow. Coastal Engineering (43):75–92.
* Nielsen, P. 1992. Coastal bottom boundary layers and sediment transport. Singapore: World Scientific.
* 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. 
* Soulsby, R. L. 1997. Dynamics of marine sands. London, England: Thomas Telford Publications.
*  Swart, D. H. 1974. Offshore sediment transport and equilibrium. Beach profiles. Delft, The Netherlands: Delft Hydraulics Laboratory Publications.
* Wiberg, P. L., and C. R. Sherwood. 2008. Calculating wave-generated bottom orbital velocities from surface-wave parameters. Computers and Geosciences 34(10):1243–1262.
* Wu, W., A. Sánchez, and M. Zhang. 2010. An implicit 2-D depth-averaged finite-volume model of flow and sediment transport in coastal waters. In Proceedings of the International Conference on Coastal Engineering, No. 32. Paper Number: Sediment 23. Shanghai, China.
* Wright, D. G., and K. R. Thompson. 1983. Time-averaged forms of the nonlinear stress law. Journal of Physical Oceanography (13):341–346.
 
----
[[CMS#Documentation_Portal | Documentation Portal]]

Latest revision as of 16:20, 18 February 2015

Bed Roughness

The bed roughness is specified for the hydrodynamic calculations with either a Manning's roughness coefficient (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n } ), Nikuradse roughness height (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_s} ), or bed friction coefficient ( Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_b} ) is related to the Manning’s roughness coefficient (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} ) by (Soulsby 1997)

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_b = g n^2 h^{-1/3} }

(1)

Commonly, the bed friction coefficient is calculated by assuming a logarithmic velocity profile as (Graf and Altinakar 1998)

 

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_b=\left [\frac{\kappa}{\ln(z_0/h)+1} \right ]^2 }

(2)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} = 0.4 is Von Karman constant, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_0} is the bed roughness length which is related to the Nikuradse roughness (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_s} ) by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_0 = k_s/30} (hydraulically rough flow).

Current-Related Shear Stress

The current bed shear stress is given by

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{ci} = \rho c_b U U_i } (3)

where:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \rho} = water density (~1025 kg/m3)
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_b} = bed friction coefficient [-]
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U = \sqrt{U_i U_i}} = current velocity magnitude [m/s]

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

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_c = \rho c_b U^2 } (2)

Wave-Related Shear Stress

The wave-related bed shear stress amplitude is given by (Jonsson 1966)

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_w = \frac{1}{2} \rho f_w u_w^2 } (5)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w} = wave friction factor, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w} is an equivalent or representative bottom wave orbital velocity amplitude. The wave friction factor (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w} ) is estimated using one of the following:

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w = \exp(5.5 r^{-0.2} - 6.3 ) } (Nielson 1982) (6)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w = 0.237 r^{-0.52} } (Soulsby 1997) (7)
  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w = \left\{ \begin{align} &exp(5.21 r^{-0.19} - 6.0 ) for r>1.57 \\ &0.3 \ \ \ \ \ \ for r\leq 1.57 (Swart 1974) \end{align}\right.} (8)


where:

r = relative roughness = Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_w/k_s} [-]
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_s} = Nikuradse roughness [m]
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_w =} semi-orbital excursion = Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{bi}} ) is calculated as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{bi} = \lambda_{wc} \tau_{ci} } (9)

where:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda_{wc}} = nonlinear bottom friction enhancement factor Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\lambda_{wc} \geq 1)} [-]
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{ci}} = current-related bed shear stress [Pa].


The nonlinear bottom friction enhancement factor (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda_{wc}} is given by

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda_{wc} = \frac{\sqrt{ U^2 + c_w u_w^2 }}{U} } (10)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_w} is an empirical coefficient, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w } is the wave bottom orbital velocity amplitude based on linear wave theory. For random waves, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w = u_{ws}} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_w = 0.5} . Here, the coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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):

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda_{wc} = 1 + bX^p(1-X)^q } (11)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle X = \tau_c /(\tau_c + \tau_w)} and b, P, and q are coefficients given by (Soulsby et al. 1993)


  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (u_w)} is calculated based on linear wave theory as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w} is set to an equivalent or representative bottom orbital velocity amplitude equal to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w = \sqrt{2}u_{rms}} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{rms}} the root-mean-squared bottom wave orbital velocity amplitude is defined here following Soulsby (1987; 1997):

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u^2_{rms} = var(\tilde{u}_b) = \int_0^\infty S_u (f)df} (14)

where:

var() = variance function,
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tilde{u}_b}  = instantaneous bottom orbital velocity [m/s]
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_u} = wave orbital velocity spectrum density [s m2/s2]
f = wave frequency [1/s] .

It is noted that the definition of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{rms}} from linear wave theory and the root-mean-squared wave height Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_{rms} = H_s /\sqrt{2}} (for a Rayleigh distribution) is given by

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{rms} = \frac{\pi\ H_{rms}} {T_p \sqrt{2}\ sinh(kh)}} (15)

Wiberg and Sherwood (2008) reported that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{rms}} estimates using Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_{rms}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_p} agree reasonably well with field measurements (except for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_p < 8.8 s)} and produces better estimates than other combinations with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_{rms}, H_s, T_P} , and the zero-crossing wave period Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} = 3.3) spectrum following the work of Soulsby (1987):


  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (u_{ws} )} as

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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)

  Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_b} is the bed elevation, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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

  • Davies, A. G., R. L. Soulsby, and H. L. King. 1988. A numerical model of the combined wave and current bottom boundary layer. Journal of Geophysical Research 93(C1):491–508.
  • Fredsoe, J. (1984). “Turbulent boundary layer in wave-current motion,” Journal of Hydraulic Engineering, ASCE, 110, 1103-1120.
  • Graf, W. H., and M. Altinakar. 1998. Fluvial hydraulics. Hoboken, NJ: Wiley & Sons, Ltd.
  • Grant, W. D., and O. S. Madsen. 1979. Combined wave and current interaction with a rough bottom. Journal of Geophysical Research 86(C4):1797–1808.
  • Hasselmann, K., T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Gienapp, D. E. Hasselmann, P. Kruseman, A. Meerbrug, P. Muller, D. J. Olbers, K. Richter, W. Sell, and H. Walden. 1973. Measurements of windwave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Deutsche Hydrographische Zeitschrift A80(12):95.
  • 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.
  • Jonsson, I. G. 1966. Wave boundary layers and friction factors. In Proceedings of the 10th Coastal Engineering Conference, ASCE, 127–148.
  • Madsen, O. S. 1994. Spectral wave–current bottom boundary layer flows. In Proceedings of the 24th Conference on Coastal Engineering, ASCE, 384–398. Kobe, Japan.
  • Myrhaug, D., L. E. Holmedal, R. R. Simons, and R .D. MacIver. 2001. Bottom friction in random waves plus current flow. Coastal Engineering (43):75–92.
  • Nielsen, P. 1992. Coastal bottom boundary layers and sediment transport. Singapore: World Scientific.
  • 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.
  • Soulsby, R. L. 1997. Dynamics of marine sands. London, England: Thomas Telford Publications.
  • Swart, D. H. 1974. Offshore sediment transport and equilibrium. Beach profiles. Delft, The Netherlands: Delft Hydraulics Laboratory Publications.
  • Wiberg, P. L., and C. R. Sherwood. 2008. Calculating wave-generated bottom orbital velocities from surface-wave parameters. Computers and Geosciences 34(10):1243–1262.
  • Wu, W., A. Sánchez, and M. Zhang. 2010. An implicit 2-D depth-averaged finite-volume model of flow and sediment transport in coastal waters. In Proceedings of the International Conference on Coastal Engineering, No. 32. Paper Number: Sediment 23. Shanghai, China.
  • Wright, D. G., and K. R. Thompson. 1983. Time-averaged forms of the nonlinear stress law. Journal of Physical Oceanography (13):341–346.

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