Solution of Wave-Body Flows, Green's Theorem - Revision history

Adi Kurniawan: /* Boundary-value problem */

2012-07-26T19:55:16Z

Boundary-value problem

Adi Kurniawan: /* Example */

2010-02-27T10:56:13Z

Example

← Older revision		Revision as of 10:56, 27 February 2010
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	<center><math> V(\vec{X}) = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \, </math></center>		<center><math> V(\vec{X}) = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \, </math></center>

	==Example ==		===Example===
	Uniform flow past <math> S_B\,</math>:		Uniform flow past <math> S_B\,</math>:
	<center><math> \Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>		<center><math> \Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>

Adi Kurniawan: /* Example */

2010-02-27T10:26:52Z

Example

← Older revision		Revision as of 10:26, 27 February 2010
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	So the RHS of the Green equation becomes:		So the RHS of the Green equation becomes:

	<center><math> RHS = \iint_{S_B} G(\vec{X};\vec{\xi}) \left(-~~Un_1X~~\right) \mathrm{d} S_X \, </math></center>.		<center><math> RHS = \iint_{S_B} G(\vec{X};\vec{\xi}) \left(-Un_1\right) \mathrm{d} S_X \, </math></center>.


	[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]		[[Ocean Wave Interaction with Ships and Offshore Energy Systems]]

Adi Kurniawan: /* Example */

2010-02-27T10:25:19Z

Example

← Older revision		Revision as of 10:25, 27 February 2010
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	Uniform flow past <math> S_B\,</math>:		Uniform flow past <math> S_B\,</math>:
	<center><math> \Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>		<center><math> \Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>
	<center><math> \Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(~~U_X~~\right) = - n_1 U \equiv V(\vec{X}) </math></center>		<center><math> \Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(U X\right) = - n_1 U \equiv V(\vec{X}) </math></center>

	So the RHS of the Green equation becomes:		So the RHS of the Green equation becomes:

Adi Kurniawan: /* Example */

2010-02-27T10:24:00Z

Example

← Older revision		Revision as of 10:24, 27 February 2010
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	==Example ==		==Example ==
	Uniform flow past <math> S_B\,</math>:		Uniform flow past <math> S_B\,</math>:
	<center><math> \Phi = U X + \~~phi_1~~, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>		<center><math> \Phi = U X + \phi, \ \frac{\partial\Phi}{\partial n} = 0, \ \mbox{on} \ S_B \, </math></center>
	<center><math> \Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(U_X\right) = - n_1 U \equiv V(\vec{X}) </math></center>		<center><math> \Longrightarrow \frac{\partial\phi}{\partial n} = -\frac{\partial}{\partial n} \left(U_X\right) = - n_1 U \equiv V(\vec{X}) </math></center>

Adi Kurniawan: /* Infinite domain potential flow solutions */

2010-02-27T10:23:39Z

Infinite domain potential flow solutions

← Older revision		Revision as of 10:23, 27 February 2010
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	For closed boundaries <math>S_B\,</math> with no shed wakes responsible for lifting effects the resulting integral equation for <math>\phi(\vec{X})\,</math> over the body boundary becomes:		For closed boundaries <math>S_B\,</math> with no shed wakes responsible for lifting effects the resulting integral equation for <math>\phi(\vec{X})\,</math> over the body boundary becomes:

	<center><math> \frac{1}{2} \phi)(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G(\vec{X};\vec{\xi})}{\partial n_X} \mathrm{d} S_X = \iint_{S_B} G(\vec{X};\vec{\xi}) V(\vec{X}) \mathrm{d} S_X </math></center>		<center><math> \frac{1}{2} \phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G(\vec{X};\vec{\xi})}{\partial n_X} \mathrm{d} S_X = \iint_{S_B} G(\vec{X};\vec{\xi}) V(\vec{X}) \mathrm{d} S_X </math></center>

	with		with

Adi Kurniawan: /* Infinite domain potential flow solutions */

2010-02-27T10:20:07Z

Infinite domain potential flow solutions

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	with		with

	<center><math> B(\vec{X} = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \, </math></center>		<center><math> V(\vec{X}) = \frac{\partial\phi}{\partial n}, \quad \mbox{on} \ S_B \, </math></center>

	==Example ==		==Example ==

Adi Kurniawan: /* Havelock's wave source potential */

2010-02-27T10:18:51Z

Havelock's wave source potential

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	where now both <math>\vec{X}\,</math> and <math>\vec{\xi}\,</math> lie no the body surface. This becomes an integral equation for <math>\phi(\vec{X})\,</math> over a surface <math> S_B\,</math> of boundary extent. Its solution is carried out with panel methods described below.		where now both <math>\vec{X}\,</math> and <math>\vec{\xi}\,</math> lie no the body surface. This becomes an integral equation for <math>\phi(\vec{X})\,</math> over a surface <math> S_B\,</math> of boundary extent. Its solution is carried out with panel methods described below.

	The interpretation of the derivative under the integral sign as follows:		The interpretation of the derivative under the integral sign is as follows:

	<center><math> \frac{\partial G}{\partial n_X} \equiv \vec{n}_X \cdot \nabla_X G(\vec{X};\vec{\xi}) \equiv \left( n_1 \frac{\partial}{\partial X} + n_2 \frac{\partial}{\partial Y} + n_3 \frac{\partial}{\partial Z} \right) G (\vec{X}; \vec{\xi}) </math></center>		<center><math> \frac{\partial G}{\partial n_X} \equiv \vec{n}_X \cdot \nabla_X G(\vec{X};\vec{\xi}) \equiv \left( n_1 \frac{\partial}{\partial X} + n_2 \frac{\partial}{\partial Y} + n_3 \frac{\partial}{\partial Z} \right) G (\vec{X}; \vec{\xi}) </math></center>

	where derivatives are taken ~~w.r.t.~~ the first argument for a point source centered at point <math> \vec{\xi}\,</math>.		where derivatives are taken with respect to the first argument for a point source centered at point <math> \vec{\xi}\,</math>.

	==Infinite domain potential flow solutions==		==Infinite domain potential flow solutions==

Adi Kurniawan: /* Havelock's wave source potential */

2010-02-27T10:15:52Z

Havelock's wave source potential

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	It follows that a relationship is obtained between the value of <math> \phi(\vec{\xi})\,</math> at some point in the fluid domain and its values <math> \phi_(\vec{X})\,</math> and normal derivatives over the body boundary:		It follows that a relationship is obtained between the value of <math> \phi(\vec{\xi})\,</math> at some point in the fluid domain and its values <math> \phi_(\vec{X})\,</math> and normal derivatives over the body boundary:

	<center><math> \phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G ( \vec{X}; \vec{\xi} )} {\partial n_X} \mathrm{d} S_X = \iint_{S_B} G (\vec{X};\vec{\xi}) B(\vec{X}) \mathrm{d} S_X </math></center>		<center><math> \phi(\vec{\xi}) + \iint_{S_B} \phi(\vec{X}) \frac{\partial G ( \vec{X}; \vec{\xi} )} {\partial n_X} \mathrm{d} S_X = \iint_{S_B} G (\vec{X};\vec{\xi}) V(\vec{X}) \mathrm{d} S_X </math></center>

	Stated differently, knowledge of <math> \phi\,</math> and <math> \frac{\partial\phi}{\partial n}\,</math> over the body boundary allows the determination of <math> \phi\,</math> and upon differentiation of <math>\nabla\phi\,</math> in the fluid domain.		Stated differently, knowledge of <math> \phi\,</math> and <math> \frac{\partial\phi}{\partial n}\,</math> over the body boundary allows the determination of <math> \phi\,</math> and upon differentiation of <math>\nabla\phi\,</math> in the fluid domain.
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	The reduction of Green's Theorem derived above survives almost identically with a factor of <math>\frac{1}{2}\,</math> now multiplying the <math>I_\epsilon\,</math> integral since only <math> \frac{1}{2}\,</math> of the <math>S_\epsilon\,</math> surface lies in the fluid domain in the limit as <math>\epsilon\to 0 \, </math> and for a body surface which is smooth. It follows that:		The reduction of Green's Theorem derived above survives almost identically with a factor of <math>\frac{1}{2}\,</math> now multiplying the <math>I_\epsilon\,</math> integral since only <math> \frac{1}{2}\,</math> of the <math>S_\epsilon\,</math> surface lies in the fluid domain in the limit as <math>\epsilon\to 0 \, </math> and for a body surface which is smooth. It follows that:

	<center><math> \frac{1}{2} \phi(\vec{\xi}) + \iint_{S_B}\phi(\vec{X}) \frac{\partial G (\vec{X}; \vec{\xi})}{\partial n_X} \mathrm{d} S_X = \iint_{S_B} G(\vec{X}; \vec{\xi}) V(\vec{X} \mathrm{d} S_X </math></center>		<center><math> \frac{1}{2} \phi(\vec{\xi}) + \iint_{S_B}\phi(\vec{X}) \frac{\partial G (\vec{X}; \vec{\xi})}{\partial n_X} \mathrm{d} S_X = \iint_{S_B} G(\vec{X}; \vec{\xi}) V(\vec{X}) \mathrm{d} S_X </math></center>

	where now both <math>\vec{X}\,</math> and <math>\vec{\xi}\,</math> lie no the body surface. This becomes an integral equation for <math>\phi(\vec{X})\,</math> over a surface <math> S_B\,</math> of boundary extent. Its solution is carried out with panel methods described below.		where now both <math>\vec{X}\,</math> and <math>\vec{\xi}\,</math> lie no the body surface. This becomes an integral equation for <math>\phi(\vec{X})\,</math> over a surface <math> S_B\,</math> of boundary extent. Its solution is carried out with panel methods described below.

Adi Kurniawan: /* Frequency-domain radiation-diffraction. U=0 */

2009-12-11T11:53:30Z

Frequency-domain radiation-diffraction. U=0

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	A consensus has been reached over the past two decades that the most efficient and robust solution methods are based on Green's Theorem using either a wave-source potential or the Rankine source as the Green function. The numerical solution of the resulting integral equations in practice is in almost all cases carried out by panel methods.		A consensus has been reached over the past two decades that the most efficient and robust solution methods are based on Green's Theorem using either a wave-source potential or the Rankine source as the Green function. The numerical solution of the resulting integral equations in practice is in almost all cases carried out by panel methods.

	==Frequency-domain radiation-diffraction. U=0==		==Frequency-domain radiation-diffraction. U = 0==

	===Boundary-value problem===		===Boundary-value problem===

← Older revision		Revision as of 19:55, 26 July 2012
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	<math> S_B \, </math>: mean position of body surface <br>		<math> S_B \, </math>: mean position of body surface <br>
	<math> S_F \, </math>: mean position of the free surface <br>		<math> S_F \, </math>: mean position of the free surface <br>
	<math> S_\infty \, </math>: Bounding cylindrical surface with radius <math> R = \left( X^2 +y^2 \right)^{1/2} \, </math>. Will be allowed to expand after the statement of Green's Theorem		<math> S_\infty \, </math>: Bounding cylindrical surface with radius <math> R = \left( x^2 +y^2 \right)^{1/2} \, </math>. Will be allowed to expand after the statement of Green's Theorem <br>
	<math> S_H \, </math>: Seafloor (assumed flat) of a surface which will be allowed to approach <math> Z=-\infty\,</math> <br>		<math> S_H \, </math>: Seafloor (assumed flat) of a surface which will be allowed to approach <math> Z=-\infty\,</math> <br>
	<math> S_E \, </math>: Spherical surface with radius <math> V = \epsilon \ , </math> centered at point <math> \vec\xi \, </math> in the fluid domain <br>		<math> S_E \, </math>: Spherical surface with radius <math> V = \epsilon \ , </math> centered at point <math> \vec\xi \, </math> in the fluid domain <br>
	<math> \vec{n}\,</math>: Unit normal vector on <math>S\,</math>, at point <math>\vec{X}\,</math> on <math>S\,</math>		<math> \vec{n}\,</math>: Unit normal vector on <math>S\,</math>, at point <math>\vec{x}\,</math> on <math>S\,</math>

	Define two velocity potentials <math>\phi_i(\vec{X})\,</math>:		Define two velocity potentials <math>\phi_i(\vec{x})\,</math>:

	<math> \phi_1(\vec{X}) = \phi(\vec{X}) \equiv \,</math> Unknown complex radiation or diffraction potential <br>		<math> \phi_1(\vec{x}) = \phi(\vec{x}) \equiv \,</math> Unknown complex radiation or diffraction potential <br>
	<math> \phi_2(\vec{X}) = G(\vec{X};\vec{\xi}) \equiv \,</math> Green function value at point <math>\vec{X}\,</math> due to a singularity centered at point <math>\vec{\xi}\,</math>.		<math> \phi_2(\vec{x}) = G(\vec{x};\vec{\xi}) \equiv \,</math> Green function value at point <math>\vec{x}\,</math> due to a singularity centered at point <math>\vec{\xi}\,</math>.

	Two types of Green functions will be used:		Two types of Green functions will be used:

	<u>Rankine source</u>: <math> \~~nabla_X~~^2 G = 0 \, </math>		<u>Rankine source</u>: <math> \nabla_x^2 G = 0 \, </math>

	<center><math> G(\vec{X};\vec{\xi}) = - \frac{1}{4\pi} \left\|\vec{X}-\vec{\xi}\right\|^{-1} = - \frac{1}{4\pi r} \, </math></center>		<center><math> G(\vec{x};\vec{\xi}) = - \frac{1}{4\pi} \left\|\vec{x}-\vec{\xi}\right\|^{-1} = - \frac{1}{4\pi r} \, </math></center>
	<center><math> = - \frac{1}{4\pi} \left\{ (X-\xi)^2 + (Y-n)^2 + (Z-\zeta)^2 \right\}^{-1/2} \, </math></center>		<center><math> = - \frac{1}{4\pi} \left\{ (x-\xi)^2 + (Y-n)^2 + (Z-\zeta)^2 \right\}^{-1/2} \, </math></center>

	Note that the flux of fluid emitted from <math>\vec{\xi}\,</math> is equal to <math>1\,</math>.		Note that the flux of fluid emitted from <math>\vec{\xi}\,</math> is equal to <math>1\,</math>.