Bottom Friction: Difference between revisions
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'''Bottom Friction''' | '''Bottom Friction''' | ||
The mean (short-wave averaged) bed shear stress, | The mean (short-wave averaged) bed shear stress, <math>\tau_{bi}</math>, is calculated as | ||
<math>\tau_{bi} = \lambda_{wc}\tau_{ci}</math> (2-8) | |||
where | where | ||
<math>\lambda_{wc} = </math> nonlinear bottom friction enhancement factor <math> \left( \lambda_{wc} \geq 1 \right)</math> [-] | |||
<math>\tau_c = </math> current-related bed shear stress vector [Pa] | |||
The current bed shear stress is given by | The current bed shear stress is given by | ||
<math>\tau_{ci} = \rho c_b UU_i</math> (2-9) | |||
where | where | ||
<math>\rho = </math> water density (~1025 kg/m<sup>3</sup>) | |||
c<sub>b</sub> = bed friction coefficient [-] | c<sub>b</sub> = bed friction coefficient [-] | ||
<math>U =</math> current magnitude = <math>\sqrt{U_i^2 + U_j^2}</math> [m/s] | |||
The bottom roughness is specified with either a Manning's roughness coefficient | The bottom roughness is specified with either a Manning's roughness coefficient <math>\eta</math> , Nikuradse roughness height k<sub>s</sub> , or bed friction coefficient c<sub>b</sub> . 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 c<sub>b</sub> is related to the Manning’s roughness coefficient ''<math>\eta</math>'' by (Graf and Altinakar 1998) | The bed friction coefficient c<sub>b</sub> is related to the Manning’s roughness coefficient ''<math>\eta</math>'' by (Graf and Altinakar 1998) | ||
<math>c_b = \left( \frac{\kappa}{ln (z_0/h) + 1 }\right)^2</math> (2-11) | |||
where <math>\kappa =</math> 0.4. The roughness length is related to the Nikuradse roughness (roughness height specified in SMS interface for CMS-Flow) by <math> z_0 = k_z /30.</math> | |||
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), | |||
<math>\lambda_{wc} = \frac{\sqrt{ U^2 + c_w u^2_w}}{U}</math> (2-12) | |||
where c<sub>w</sub> is an empirical coefficient equal to approximately 0.5 and u<sub>w</sub> is the bottom wave orbital velocity amplitude. For random waves, u<sub>w</sub> is calculated based on the significant wave height and peak wave period, from linear wave theory as | |||
<math>u_{wc} = \frac{\pi H_s}{T_p sinh(kh)}</math> (2-13) | |||
The DATA2, DATA13, F84, and HT91 formulations are calculated using general parameterization of Soulsby (1995), | |||
<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) | |||
<math>u_{rms} = \frac{\pi H_{rms}}{T_p \sqrt{2} sinh(kh)}</math> (2-18) | |||
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 </math> < 8.8 s) 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 (e.g. JONSWAP, Pierson-Moskowitz, etc.), 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 in Equation 2-19 below based on the JONSWAP (<math>\gamma</math> = 3.3) spectrum following the work of Soulsby (1987) | |||
<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> | |||
(2-19) | |||
where <math> T_n = \sqrt {h/g}</math>. | |||
The above expression agrees closely with the curves presented by Soulsby (1987; 1997). | |||
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 2008) | |||
<math>m_b = \mid \bigtriangledown z_b \mid = \sqrt {\left(\frac {\partial z_b}{\partial x} \right)^2 + \left(\frac {\partial z_b}{\partial y} \right)^2 + 1 }</math> (2-20) | |||
where <math>z_b</math> is the bed elevation and <math> \bigtriangledown = \left(\frac {\partial}{\partial x}, \frac {\partial}{\partial y}, 1 \right)</math>. 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. For bottom slopes of 1/5 and 1/3, the above expression leads to an increase in bottom friction of 2.0% and 5.4 %, respectively. | |||
[[Eddy Viscosity]] | |||
Latest revision as of 19:58, 18 July 2014
Bottom Friction
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}} (2-8)
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 \left( \lambda_{wc} \geq 1 \right)} [-]
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 = } current-related bed shear stress vector [Pa]
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 UU_i} (2-9)
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 \rho = } water density (~1025 kg/m3)
cb = 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 =} current magnitude = 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 \sqrt{U_i^2 + U_j^2}} [m/s]
The bottom roughness is specified 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 \eta} , 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 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 \eta} by (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} (2-11)
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 \kappa =} 0.4. The roughness length is related to the Nikuradse roughness (roughness height specified in SMS interface for CMS-Flow) 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_z /30.} 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),
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^2_w}}{U}} (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
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_{wc} = \frac{\pi H_s}{T_p sinh(kh)}} (2-13)
The DATA2, DATA13, F84, and HT91 formulations are calculated using general parameterization of Soulsby (1995),
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} (2-14)
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 that depend on the formulation selected. 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} 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 \tau_w} are the current- and wave-related bed shear stress magnitudes. 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 = \rho c_b U^2} (2-15)
and the wave 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_w = \frac{1}{2}\rho f_w u^2_w} (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 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 urms the root-mean-squared bottom orbital velocity amplitude defined here as (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_{rms} = \frac{\pi H_{rms}}{T_p \sqrt{2} sinh(kh)}} (2-18)
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 (e.g. JONSWAP, Pierson-Moskowitz, etc.), 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 in Equation 2-19 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]} (2-19)
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).
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 2008)
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 = \mid \bigtriangledown z_b \mid = \sqrt {\left(\frac {\partial z_b}{\partial x} \right)^2 + \left(\frac {\partial z_b}{\partial y} \right)^2 + 1 }} (2-20)
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)} . 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. For bottom slopes of 1/5 and 1/3, the above expression leads to an increase in bottom friction of 2.0% and 5.4 %, respectively.