Wave Drift Forces - Revision history

Adi Kurniawan: /* Mean Drift Force on a Floating 2D Body */

2010-08-06T15:13:16Z

Mean Drift Force on a Floating 2D Body

← Older revision		Revision as of 15:13, 6 August 2010
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	where the momentum conservation principle derived earlier has been applied. Mean values in time are taken leading to a vanishing mean value for the momentum rate of change within the volume <math>V(t)\,</math>, for fixed <math>S^\pm\,</math>.		where the momentum conservation principle derived earlier has been applied. Mean values in time are taken leading to a vanishing mean value for the momentum rate of change within the volume <math>V(t)\,</math>, for fixed <math>S^\pm\,</math>.

	* The mean momentum flux across <math>S_F(t)\,</math> is zero.		The mean momentum flux across <math>S_F(t)\,</math> is zero.

	* The mean horizontal momentum flux across the body section is the drift force <math>D\equiv{\overline{F_x}}^t\,</math>.		The mean horizontal momentum flux across the body section is the drift force <math>D\equiv{\overline{F_x}}^t\,</math>.

	* There remains to evaluate the integrals over <math>S^\pm\,</math> which are surfaces fixed in space and therefore, <math>U_n=0\,</math> on <math>S^\pm\,</math>. Recall that <math>\vec{V}=\nabla\phi\,</math> is the total fluid velocity and <math>P\,</math> is given by the Bernoulli equation.		There remains to evaluate the integrals over <math>S^\pm\,</math> which are surfaces fixed in space and therefore, <math>U_n=0\,</math> on <math>S^\pm\,</math>. Recall that <math>\vec{V}=\nabla\phi\,</math> is the total fluid velocity and <math>P\,</math> is given by the Bernoulli equation.

	* The definition of the plane progressive wave forms defined above will be introduced and an expression will be derived for <math>D\,</math> in terms of <math>A,A^+,A^-\,</math>. The result is "mildly" surprising!		* The definition of the plane progressive wave forms defined above will be introduced and an expression will be derived for <math>D\,</math> in terms of <math>A,A^+,A^-\,</math>. The result is "mildly" surprising!

Adi Kurniawan: /* Mean Drift Force on a Floating 2D Body */

2010-08-06T15:07:39Z

Mean Drift Force on a Floating 2D Body

← Older revision		Revision as of 15:07, 6 August 2010
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	Applying the momentum conservation principle in the x-direction within the volume <math>V(t)\,</math> bounded by <math>S^+ + S^- + S_F(t) + S_B(t) \, </math> we obtain:		Applying the momentum conservation principle in the x-direction within the volume <math>V(t)\,</math> bounded by <math>S^+ + S^- + S_F(t) + S_B(t) \, </math> we obtain:

	<center><math> D \equiv {\overline{F_x}}^t = - {\overline{\iint_{S^++S^-} \left[ P \vec{n}+\rho\vec{V}\left(V_n-U_n\right)\right]dS}}^t </math></center>		<center><math> D \equiv {\overline{F_x}}^t = - {\overline{\iint_{S^++S^-} \left[ P \vec{n}+\rho\vec{V}\left(V_n-U_n\right)\right]\mathrm{d}S}}^t </math></center>

	where the momentum conservation principle derived earlier has been applied. Mean values in time are taken leading to a vanishing mean value for the momentum rate of change within the volume <math>V(t)\,</math>, for fixed <math>S^\pm\,</math>.		where the momentum conservation principle derived earlier has been applied. Mean values in time are taken leading to a vanishing mean value for the momentum rate of change within the volume <math>V(t)\,</math>, for fixed <math>S^\pm\,</math>.
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	Define the velocity potentials for the plane progressive waves described above:		Define the velocity potentials for the plane progressive waves described above:

	<center><math> \phi_I = \~~mathbb~~{~~R}\mathbf{e~~} \left\{ \frac{igA}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>		<center><math> \phi_I = \mathrm{Re} \left\{ \frac{igA}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>
	<center><math> \phi^+ = \~~mathbb{R}\mathbf~~{e} \left\{ \frac{igA^+}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>		<center><math> \phi^+ = \mathrm{Re} \left\{ \frac{igA^+}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>
	<center><math> \phi^- = \~~mathbb{R}\mathbf~~{e} \left\{ \frac{igA^-}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>		<center><math> \phi^- = \mathrm{Re} \left\{ \frac{igA^-}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>

	Upon substitution into the momentum flux expression defined above and after some simple algebra that has been illustrated earlier, it follows that:		Upon substitution into the momentum flux expression defined above and after some simple algebra that has been illustrated earlier, it follows that:

Adi Kurniawan: /* Mean Drift Force on a Floating 2D Body */

2010-08-06T15:02:35Z

Mean Drift Force on a Floating 2D Body

Adi Kurniawan: /* General case */

2010-08-06T14:57:42Z

General case

← Older revision		Revision as of 14:57, 6 August 2010
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	====General case====		====General case====
	In the more general case of surface waves interacting with floating bodies oscillating due to the ambient waves, a more general expression may be derived by directly integrating the pressure over the <u>instantaneous</u> position of the vessel wetted surface and linearizing about its mean position keeping consistently terms of <math> O \left( A^2 \right), </math>. The expression is not given here for simplicity.		In the more general case of surface waves interacting with floating bodies oscillating due to the ambient waves, a more general expression may be derived by directly integrating the pressure over the <u>instantaneous</u> position of the vessel wetted surface and linearizing about its mean position keeping consistently terms of <math> O \left( A^2 \right)</math>. The expression is not given here for simplicity.

	===Drift Forces by Momentum Conservation===		===Drift Forces by Momentum Conservation===

Adi Kurniawan: /* Drift Forces by Momentum Conservation */

2010-08-06T14:55:25Z

Drift Forces by Momentum Conservation

← Older revision		Revision as of 14:55, 6 August 2010
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	===Drift Forces by Momentum Conservation===		===Drift Forces by Momentum Conservation===

	It is possible to derive an expression for the mean drift force by applying the momentum conservation principle derived earlier. This approach provides an expression for the drift force in terms of the wave systems in the far field and can be shown to produce theoretically an identical result to that obtained from the considerably more elaborate task of using near-field pressure integration. The full three-dimensional result will not be presented here, as for the pressure integration, and can be found in Faltinsen 1990 and ~~W&L~~. It is noted that it is numerically superior to pressure integration and is often preferred in practice.		It is possible to derive an expression for the mean drift force by applying the momentum conservation principle derived earlier. This approach provides an expression for the drift force in terms of the wave systems in the far field and can be shown to produce theoretically an identical result to that obtained from the considerably more elaborate task of using near-field pressure integration. The full three-dimensional result will not be presented here, as for the pressure integration, and can be found in [[Faltinsen 1990]] and [[Wehausen and Laitone 1960]]. It is noted that it is numerically superior to pressure integration and is often preferred in practice.

	Below we prove from first principles the result in two dimensions which is itself quite useful in practice and educational.		Below we prove from first principles the result in two dimensions which is itself quite useful in practice and educational.

Adi Kurniawan: /* Nonlinear Effects */

2010-08-06T14:51:17Z

Nonlinear Effects

← Older revision		Revision as of 14:51, 6 August 2010
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	Some of the most important nonlinear effects arising in connection with wave-body interactions are		Some of the most important nonlinear effects arising in connection with wave-body interactions are

	* Drift forces. They are the mean forces exerted on floating or submerged bodies by ambient waves. They may be treated very well by perturbation theory.		* '''Drift forces'''. They are the mean forces exerted on floating or submerged bodies by ambient waves. They may be treated very well by perturbation theory.

	* Slamming. These are highly nonlinear effects arising when a ship section acts upon the water surface or when a steep or breaking wave impinges upon a floating structure. They may be modeled by fully or partially nonlinear potential flow models of analytical or numerical nature		* '''Slamming'''. These are highly nonlinear effects arising when a ship section acts upon the water surface or when a steep or breaking wave impinges upon a floating structure. They may be modeled by fully or partially nonlinear potential flow models of analytical or numerical nature

	* Forces due to viscous flow separation around floating structures and their subsystems, e.g. risers, mooring lines etc. Vortex induced vibrations (VIV) is an important example. Such effects can be treated experimentally and computationally by solving the Navier-Stokes equations.		* Forces due to '''viscous flow separation''' around floating structures and their subsystems, e.g. risers, mooring lines etc. Vortex induced vibrations (VIV) is an important example. Such effects can be treated experimentally and computationally by solving the Navier-Stokes equations.

	* Nonlinear ship motions in steep waves. These effects are mostly of potential flow nature and are being treated by nonlinear Rankine panel methods. The primary nonlinearity is the variable wetness of the ship hull, the nonlinearity of the kinematics of ambient waves and the numerical solution of the equations of motion in the time domain.		* Nonlinear ship motions in '''steep waves'''. These effects are mostly of potential flow nature and are being treated by nonlinear Rankine panel methods. The primary nonlinearity is the variable wetness of the ship hull, the nonlinearity of the kinematics of ambient waves and the numerical solution of the equations of motion in the time domain.

	* Nonlinear responses of deep water offshore platforms in certain flexural modes of their tethers. These effects are known as springing & ringing and are treated by a combination of perturbation and nonlinear methods and experiments.		* Nonlinear responses of deep water offshore platforms in certain flexural modes of their tethers. These effects are known as '''springing & ringing''' and are treated by a combination of perturbation and nonlinear methods and experiments.

	==Drift Forces==		==Drift Forces==

Adi Kurniawan at 14:24, 19 October 2009

2009-10-19T14:24:11Z

← Older revision		Revision as of 14:24, 19 October 2009
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	Drift forces will first be considered in regular waves. The main results will then be extended in random waves. One very important property of drift forces other than their practical significance is that they depend only on the linear solution.		Drift forces will first be considered in regular waves. The main results will then be extended in random waves. One very important property of drift forces other than their practical significance is that they depend only on the linear solution.

	===Mean Drift Force on a Vertical Wall===		===Drift Forces by Pressure Integration===
			====Mean Drift Force on a Vertical Wall====

	The incident and diffracted velocity potentials can be expressed as		The incident and diffracted velocity potentials can be expressed as
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	# It is possible to extend ray theorem with little additional complexity to the case where the vessel advances with a moderate forward speed (see Faltinsen, 1990).		# It is possible to extend ray theorem with little additional complexity to the case where the vessel advances with a moderate forward speed (see Faltinsen, 1990).

	===~~Drift Forces by Pressure Integration~~===		====General case====

	In the more general case of surface waves interacting with floating bodies oscillating due to the ambient waves, a more general expression may be derived by directly integrating the pressure over the <u>instantaneous</u> position of the vessel wetted surface and linearizing about its mean position keeping consistently terms of <math> O \left( A^2 \right), </math>. The expression is not given here for simplicity.		In the more general case of surface waves interacting with floating bodies oscillating due to the ambient waves, a more general expression may be derived by directly integrating the pressure over the <u>instantaneous</u> position of the vessel wetted surface and linearizing about its mean position keeping consistently terms of <math> O \left( A^2 \right), </math>. The expression is not given here for simplicity.

Adi Kurniawan: /* Mean Drift Force on a Floating 2D Body */

2009-10-19T14:22:23Z

Mean Drift Force on a Floating 2D Body

← Older revision		Revision as of 14:22, 19 October 2009
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	Below we prove from first principles the result in two dimensions which is itself quite useful in practice and educational.		Below we prove from first principles the result in two dimensions which is itself quite useful in practice and educational.

	===Mean Drift Force on a Floating 2D Body===		====Mean Drift Force on a Floating 2D Body====

	* Ambient regular waves with amplitude <math>A\,</math> are incident on a floating body in 2D which is allowed to oscillate freely in heave, sway and roll.		* Ambient regular waves with amplitude <math>A\,</math> are incident on a floating body in 2D which is allowed to oscillate freely in heave, sway and roll.

Adi Kurniawan: /* Drift Forces by Momentum Conservation */

2009-10-19T14:22:05Z

Drift Forces by Momentum Conservation

← Older revision		Revision as of 14:22, 19 October 2009
Line 113:		Line 113:
	===Drift Forces by Momentum Conservation===		===Drift Forces by Momentum Conservation===

	* It is possible to derive an expression for the mean drift force by applying the momentum conservation principle derived earlier. This approach provides an expression for the drift force in terms of the wave systems in the far field and can be shown to produce theoretically an identical result to that obtained from the considerably more elaborate task of using near-field pressure integration		It is possible to derive an expression for the mean drift force by applying the momentum conservation principle derived earlier. This approach provides an expression for the drift force in terms of the wave systems in the far field and can be shown to produce theoretically an identical result to that obtained from the considerably more elaborate task of using near-field pressure integration. The full three-dimensional result will not be presented here, as for the pressure integration, and can be found in Faltinsen 1990 and W&L. It is noted that it is numerically superior to pressure integration and is often preferred in practice.

	* The full three-dimensional result will not be presented here, as for the pressure integration, and can be found in the references cited in OMF and W&L. It is noted that it is numerically superior to pressure integration and is often preferred in practice		Below we prove from first principles the result in two dimensions which is itself quite useful in practice and educational.

	* Below we prove from first principles the result in two dimensions which is itself quite useful in practice and educational.

	===Mean Drift Force on a Floating 2D Body===		===Mean Drift Force on a Floating 2D Body===

Adi Kurniawan: /* Drift Forces by Pressure Integration */

2009-10-19T14:19:49Z

Drift Forces by Pressure Integration

← Older revision		Revision as of 14:19, 19 October 2009
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	===Drift Forces by Pressure Integration===		===Drift Forces by Pressure Integration===

	* In the more general case of surface waves interacting with floating bodies oscillating due to the ambient waves, a more general expression may be derived by directly integrating the pressure over the <u>instantaneous</u> position of the vessel wetted surface and linearizing about its mean position keeping consistently terms of <math> O \left( A^2 \right), </math>. The expression is not given here for simplicity.		In the more general case of surface waves interacting with floating bodies oscillating due to the ambient waves, a more general expression may be derived by directly integrating the pressure over the <u>instantaneous</u> position of the vessel wetted surface and linearizing about its mean position keeping consistently terms of <math> O \left( A^2 \right), </math>. The expression is not given here for simplicity.

	===Drift Forces by Momentum Conservation===		===Drift Forces by Momentum Conservation===

← Older revision		Revision as of 15:02, 6 August 2010
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	====Mean Drift Force on a Floating 2D Body====		====Mean Drift Force on a Floating 2D Body====

	* Ambient regular waves with amplitude <math>A\,</math> are incident on a floating body in 2D which is allowed to oscillate freely in heave, sway and roll.		Ambient regular waves with amplitude <math>A\,</math> are incident on a floating body in 2D which is allowed to oscillate freely in heave, sway, and roll.

	* As a result of the diffraction and radiation wave disturbances, a reflected plane progressive wave with complex amplitude <math>A^-\,</math> appears at <math>X=-\infty\,</math> over a control surface <math> S^-\,</math> and a radiated/diffracted wave with amplitude <math>A^+\,</math> appears at <math> X=+\infty\,</math> over <math>S^+\,</math>.		As a result of the diffraction and radiation wave disturbances, a reflected plane progressive wave with complex amplitude <math>A^-\,</math> appears at <math>x=-\infty\,</math> over a control surface <math> S^-\,</math> and a radiated/diffracted wave with amplitude <math>A^+\,</math> appears at <math> x=+\infty\,</math> over <math>S^+\,</math>.

	* Applying the momentum conservation principle in the x-direction within the volume <math>V(t)\,</math> bounded by <math>S^+ + S^- + S_F(t) + S_B(t) \, </math> we obtain:		Applying the momentum conservation principle in the x-direction within the volume <math>V(t)\,</math> bounded by <math>S^+ + S^- + S_F(t) + S_B(t) \, </math> we obtain:

	<center><math> D \equiv {\overline{~~F_X~~}}^t = - {\overline{\iint_{S^++S^-} \left[ P \vec{n}+\rho\vec{V}\left(V_n-U_n\right)\right]dS}}^t </math></center>		<center><math> D \equiv {\overline{F_x}}^t = - {\overline{\iint_{S^++S^-} \left[ P \vec{n}+\rho\vec{V}\left(V_n-U_n\right)\right]dS}}^t </math></center>

	where the momentum conservation principle derived earlier has been applied. Mean values in time are taken leading to a vanishing mean value for the momentum rate of change within the volume <math>V(t)\,</math>, for fixed <math>S^\pm\,</math>.		where the momentum conservation principle derived earlier has been applied. Mean values in time are taken leading to a vanishing mean value for the momentum rate of change within the volume <math>V(t)\,</math>, for fixed <math>S^\pm\,</math>.
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	* The mean momentum flux across <math>S_F(t)\,</math> is zero.		* The mean momentum flux across <math>S_F(t)\,</math> is zero.

	* The mean horizontal momentum flux across the body section is the drift force <math>D\equiv{\overline{~~F_X~~}}^t\,</math>.		* The mean horizontal momentum flux across the body section is the drift force <math>D\equiv{\overline{F_x}}^t\,</math>.

	* There remains to evaluate the integrals over <math>S^\pm\,</math> which are surfaces fixed in space and therefore, <math>U_n=0\,</math> on <math>S^\pm\,</math>. Recall that <math>\vec{V}=\nabla\phi\,</math> is the total fluid velocity and <math>P\,</math> is given by the Bernoulli equation.		* There remains to evaluate the integrals over <math>S^\pm\,</math> which are surfaces fixed in space and therefore, <math>U_n=0\,</math> on <math>S^\pm\,</math>. Recall that <math>\vec{V}=\nabla\phi\,</math> is the total fluid velocity and <math>P\,</math> is given by the Bernoulli equation.
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	Define the velocity potentials for the plane progressive waves described above:		Define the velocity potentials for the plane progressive waves described above:

	<center><math> \phi_I = \mathbb{R}\mathbf{e} \left\{ \frac{igA}{\omega} e^{KZ-~~iKX~~+i\omega t} \right\} \, </math></center>		<center><math> \phi_I = \mathbb{R}\mathbf{e} \left\{ \frac{igA}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>
	<center><math> \phi^+ = \mathbb{R}\mathbf{e} \left\{ \frac{igA^+}{\omega} e^{KZ-~~iKX~~+i\omega t} \right\} \, </math></center>		<center><math> \phi^+ = \mathbb{R}\mathbf{e} \left\{ \frac{igA^+}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>
	<center><math> \phi^- = \mathbb{R}\mathbf{e} \left\{ \frac{igA^-}{\omega} e^{~~KZ+iKX~~+i\omega t} \right\} \, </math></center>		<center><math> \phi^- = \mathbb{R}\mathbf{e} \left\{ \frac{igA^-}{\omega} e^{kz-ikx+i\omega t} \right\} \, </math></center>

	Upon substitution into the momentum flux expression defined above and after some simple algebra that has been illustrated earlier, it follows that:		Upon substitution into the momentum flux expression defined above and after some simple algebra that has been illustrated earlier, it follows that: