SD BL - Revision history

Robertson at 12:08, 18 May 2016

2016-05-18T12:08:30Z

← Older revision		Revision as of 12:08, 18 May 2016
Line 1:		Line 1:
	<div class="title">Soulsby & Damgaard Bedload Formulation</div>		<div class="title">Soulsby & Damgaard Bedload Formulation</div>
	~~<wikitex>~~
	The [[Bibliography#SoulsbyR_2005b \| Soulsby and Damgaard (2005)]] formulae account for the combined effects of mean currents and asymmetrical waves on bedload flux. Their formulations are based on numerical integration, over a wave cycle, of the non-dimensional transport equation		The [[Bibliography#SoulsbyR_2005b \| Soulsby and Damgaard (2005)]] formulae account for the combined effects of mean currents and asymmetrical waves on bedload flux. Their formulations are based on numerical integration, over a wave cycle, of the non-dimensional transport equation
	$$
	\overrightarrow \Phi = \max \left[ {A_2 \theta ^{0.5} \left( {\theta _{sf} - \theta _c } \right)\frac{{\overrightarrow {\theta _{sf} } }}		<math display="block">\overrightarrow \Phi = \max \left[ {A_2 \theta ^{0.5} \left( {\theta _{sf} - \theta _c } \right)\frac{{\overrightarrow {\theta _{sf} } }}
	{{\theta _{sf} }},\quad 0} \right]		{{\theta _{sf} }},\quad 0} \right]</math>
	$$
	where $\overrightarrow \Phi$ and $\overrightarrow {\theta _{sf} }$ are vectors with components in the direction of the mean current and in the direction perpendicular to the current, e.g., $ \overrightarrow \Phi = \left( {\Phi _\parallel ,\,\Phi _ \bot } \right)$ ,		where <math>\overrightarrow \Phi</math> and <math>\overrightarrow {\theta _{sf} }</math> are vectors with components in the direction of the mean current and in the direction perpendicular to the current, e.g., <math>\overrightarrow \Phi = \left( {\Phi _\parallel ,\,\Phi _ \bot } \right)</math>,
	${\overrightarrow {\theta _{sf} }}=\left( {\theta _{sf\parallel} ,\,\theta _{sf\bot} } \right)$, $\theta _{sf} = \left\| {\overrightarrow {\theta _{sf} } } \right\| $, $\theta_c$ is the critical Shields parameter, and $A2$ = 12 is a semi-empirical coefficient. The implementation of this method requires computation of transport rates in the directions parallel and perpendicular to the currents as:		<math>{\overrightarrow {\theta _{sf} }}=\left( {\theta _{sf\parallel} ,\,\theta _{sf\bot} } \right)</math>, <math>\theta _{sf} = \left\| {\overrightarrow {\theta _{sf} } } \right\|</math>, <math>\theta_c</math> is the critical Shields parameter, and <math>A2 = 12</math> is a semi-empirical coefficient. The implementation of this method requires computation of transport rates in the directions parallel and perpendicular to the currents as:
	$$
	\Phi _\parallel = \max \left[ {\Phi _{\parallel 1} ,\quad \Phi _{\parallel 2} } \right]		<math display="block">\Phi _\parallel = \max \left[ {\Phi _{\parallel 1} ,\quad \Phi _{\parallel 2} } \right]</math>
	$$
	where		where
	$$
	~~\Phi _{\parallel 1} = A_2 \theta _m^{0.5} \left( {\theta _m - \theta _c } \right)~~
	$$

	$$		<math display="block">\Phi _{\parallel 1} = A_2 \theta _m^{0.5} \left( {\theta _m - \theta _c } \right)</math>
	\Phi _{\parallel 2} = A_2 ~~\left( {0.9534 + 0.1907\cos 2\varphi } \right)~~\theta _w^{0.5} ~~\theta _m + A_2~~ \left( {~~0.229~~\~~gamma _w~~ \theta ~~_w^{1.5} \cos \varphi~~ } \right)
	$$

	$$
	\Phi _ \bot = A_2 \frac{{0.1907\theta _w^2 }}		<math display="block">\Phi _{\parallel 2} = A_2 \left( {0.9534 + 0.1907\cos 2\varphi } \right)\theta _w^{0.5} \theta _m + A_2 \left( {0.229\gamma _w \theta _w^{1.5} \cos \varphi } \right)</math>


			<math display="block">\Phi _ \bot = A_2 \frac{{0.1907\theta _w^2 }}
	{{\theta _w^{3/2} + 1.5\theta _m^{1.5} }}\left( {\theta _m \sin 2\varphi + 1.2\gamma _w \theta _w \sin \varphi } \right)		{{\theta _w^{3/2} + 1.5\theta _m^{1.5} }}\left( {\theta _m \sin 2\varphi + 1.2\gamma _w \theta _w \sin \varphi } \right)
	$$		</math>
	where $\theta _m$ is the mean Shields parameter and $\tau_m$ is
	$$		where <math>\theta _m</math> is the mean Shields parameter and <math>\tau_m</math> is
	\tau _m = \tau _c \left( {1 + 1.2\left( {\frac{{\tau _w }}
	{{\tau _w + \tau _c }}} \right)^{1.5} } \right)		<math display="block">\tau _m = \tau _c \left( {1 + 1.2\left( {\frac{{\tau _w }}
	$$		{{\tau _w + \tau _c }}} \right)^{1.5} } \right)</math>
	and $\tau_c$ is the bottom stress from the currents only, $\tau_w$ is the bottom stress from the waves only calculated in the bottom-boundary layer routines (see below). The asymmetry factor $\gamma _w $ is the ratio between the amplitude of the second harmonic and the amplitude of the first harmonic of the oscillatory wave stress. Following the suggestion of Soulsby and Damgaard (2005), we estimate the asymmetry factor using Stokes second-order theory (e.g., Fredsøe and Deigaard, 1992) and constrain it to be less than 0.2. The non-dimensional fluxes (Eqns. (30) and (33)) are rotated into $x$ and $y$ directions using the directions for mean current and waves and dimensionalized with Eqn. (24) to yield values for $q_{blx}$ and $q_{bly}$ for each sediment class. ~~</wikitex>~~
			and <math>\tau_c</math> is the bottom stress from the currents only, <math>\tau_w</math> is the bottom stress from the waves only calculated in the bottom-boundary layer routines (see below). The asymmetry factor <math>\gamma _w</math> is the ratio between the amplitude of the second harmonic and the amplitude of the first harmonic of the oscillatory wave stress. Following the suggestion of Soulsby and Damgaard (2005), we estimate the asymmetry factor using Stokes second-order theory (e.g., Fredsøe and Deigaard, 1992) and constrain it to be less than 0.2. The non-dimensional fluxes (Eqns. (30) and (33)) are rotated into <math>x</math> and <math>y</math> directions using the directions for mean current and waves and dimensionalized with Eqn. (24) to yield values for <math>q_{blx}</math> and <math>q_{bly}</math> for each sediment class.

Csherwood at 16:30, 3 November 2008

2008-11-03T16:30:09Z

← Older revision		Revision as of 16:30, 3 November 2008
Line 6:		Line 6:
	{{\theta _{sf} }},\quad 0} \right]		{{\theta _{sf} }},\quad 0} \right]
	$$		$$
	where $\overrightarrow \Phi$ and $\overrightarrow {\theta _{sf} }$ are vectors with components in the direction of the mean current and in the direction perpendicular to the current, e.g.,		where $\overrightarrow \Phi$ and $\overrightarrow {\theta _{sf} }$ are vectors with components in the direction of the mean current and in the direction perpendicular to the current, e.g., $ \overrightarrow \Phi = \left( {\Phi _\parallel ,\,\Phi _ \bot } \right)$ ,
	$ \overrightarrow \Phi = \left( {\Phi _\parallel ,\,\Phi _ \bot } \right)$ ,		${\overrightarrow {\theta _{sf} }}=\left( {\theta _{sf\parallel} ,\,\theta _{sf\bot} } \right)$, $\theta _{sf} = \left\| {\overrightarrow {\theta _{sf} } } \right\| $, $\theta_c$ is the critical Shields parameter, and $A2$ = 12 is a semi-empirical coefficient. The implementation of this method requires computation of transport rates in the directions parallel and perpendicular to the currents as:

	$\theta _{sf} = \left\| {\overrightarrow {\theta _{sf} } } \right\| $
	, $\theta_c$ is the critical Shields parameter, and $A2$ = 12 is a semi-empirical coefficient. The implementation of this method requires computation of transport rates in the directions parallel and perpendicular to the currents as:
	$$		$$
	\Phi _\parallel = \max \left[ {\Phi _{\parallel 1} ,\quad \Phi _{\parallel 2} } \right]		\Phi _\parallel = \max \left[ {\Phi _{\parallel 1} ,\quad \Phi _{\parallel 2} } \right]

Csherwood: New page:
Soulsby & Damgaard Bedload Formulation
The Soulsby and Damgaard (2005) formulae account for the combined effects of mea...

2008-11-03T16:21:38Z

New page: <div class="title">Soulsby & Damgaard Bedload Formulation</div> <wikitex> The Soulsby and Damgaard (2005) formulae account for the combined effects of mea...

New page

<div class="title">Soulsby & Damgaard Bedload Formulation</div>
<wikitex>
The [[Bibliography#SoulsbyR_2005b | Soulsby and Damgaard (2005)]] formulae account for the combined effects of mean currents and asymmetrical waves on bedload flux. Their formulations are based on numerical integration, over a wave cycle, of the non-dimensional transport equation
$$
\overrightarrow \Phi = \max \left[ {A_2 \theta ^{0.5} \left( {\theta _{sf} - \theta _c } \right)\frac{{\overrightarrow {\theta _{sf} } }}
{{\theta _{sf} }},\quad 0} \right]
$$
where $\overrightarrow \Phi$ and $\overrightarrow {\theta _{sf} }$ are vectors with components in the direction of the mean current and in the direction perpendicular to the current, e.g.,
$ \overrightarrow \Phi = \left( {\Phi _\parallel ,\,\Phi _ \bot } \right)$ ,

$\theta _{sf} = \left| {\overrightarrow {\theta _{sf} } } \right| $
, $\theta_c$ is the critical Shields parameter, and $A2$ = 12 is a semi-empirical coefficient. The implementation of this method requires computation of transport rates in the directions parallel and perpendicular to the currents as:
$$
\Phi _\parallel = \max \left[ {\Phi _{\parallel 1} ,\quad \Phi _{\parallel 2} } \right]
$$
where
$$
\Phi _{\parallel 1} = A_2 \theta _m^{0.5} \left( {\theta _m - \theta _c } \right)
$$

$$
\Phi _{\parallel 2} = A_2 \left( {0.9534 + 0.1907\cos 2\varphi } \right)\theta _w^{0.5} \theta _m + A_2 \left( {0.229\gamma _w \theta _w^{1.5} \cos \varphi } \right)
$$

$$
\Phi _ \bot = A_2 \frac{{0.1907\theta _w^2 }}
{{\theta _w^{3/2} + 1.5\theta _m^{1.5} }}\left( {\theta _m \sin 2\varphi + 1.2\gamma _w \theta _w \sin \varphi } \right)
$$
where $\theta _m$ is the mean Shields parameter and $\tau_m$ is
$$
\tau _m = \tau _c \left( {1 + 1.2\left( {\frac{{\tau _w }}
{{\tau _w + \tau _c }}} \right)^{1.5} } \right)
$$
and $\tau_c$ is the bottom stress from the currents only, $\tau_w$ is the bottom stress from the waves only calculated in the bottom-boundary layer routines (see below). The asymmetry factor $\gamma _w $ is the ratio between the amplitude of the second harmonic and the amplitude of the first harmonic of the oscillatory wave stress. Following the suggestion of Soulsby and Damgaard (2005), we estimate the asymmetry factor using Stokes second-order theory (e.g., Fredsøe and Deigaard, 1992) and constrain it to be less than 0.2. The non-dimensional fluxes (Eqns. (30) and (33)) are rotated into $x$ and $y$ directions using the directions for mean current and waves and dimensionalized with Eqn. (24) to yield values for $q_{blx}$ and $q_{bly}$ for each sediment class. </wikitex>

SD BL - Revision history

Robertson at 12:08, 18 May 2016

Csherwood at 16:30, 3 November 2008

Csherwood: New page: Soulsby & Damgaard Bedload Formulation The Soulsby and Damgaard (2005) formulae account for the combined effects of mea...

Csherwood: New page:
Soulsby & Damgaard Bedload Formulation
The Soulsby and Damgaard (2005) formulae account for the combined effects of mea...