CMS-Flow:Bottom Friction

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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=\biggl(\frac{\kappa}{\ln(z_0/h)+1} \biggr)^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_{c,i} = \rho c_b U U_i }

(1)

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 } (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 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) (3)
  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) (3)
  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<1.57 \end{align}\right.} (3)
(Swart 1974)

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} } (4)

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} } (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 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 XXX). 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 (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):

  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 } (6)

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)} (7)

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 2-1).


Table 2-1. 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
p1
p2
p3
p4
q1
q2
q3
q4
J

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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