Wave Forces on a Body - Revision history

Xjwiki: /* Type of Forces */

2009-07-01T09:46:05Z

Type of Forces

← Older revision		Revision as of 09:46, 1 July 2009
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	\| <math> \phi = \, </math> \|\| <math> \underbrace{\phi_I} \, </math> \|\| <math> + \, </math> \|\| <math> \underbrace{\phi_D} \, </math> \|\| <math> + \, </math> \|\| <math> \underbrace{\phi_R} \, </math>		\| <math> \phi = \, </math> \|\| <math> \underbrace{\phi_I} \, </math> \|\| <math> + \, </math> \|\| <math> \underbrace{\phi_D} \, </math> \|\| <math> + \, </math> \|\| <math> \underbrace{\phi_R} \, </math>
	\|- align="center"		\|- align="center"
	\| \|\| Incident wave \|\| \|\| ~~Diffracted~~ wave \|\| \|\| Radiated wave		\| \|\| Incident wave \|\| \|\| Scattered wave \|\| \|\| Radiated wave
	\|- align="center"		\|- align="center"
	\| \|\| potential <math> (a) \, </math> \|\| \|\| potential <math> (b.1) \, </math> \|\| \|\| potential <math> (b.2) \, </math>		\| \|\| potential <math> (a) \, </math> \|\| \|\| potential <math> (b.1) \, </math> \|\| \|\| potential <math> (b.2) \, </math>

Syan077: /* Important parameters */

2007-07-18T09:34:48Z

Important parameters

← Older revision		Revision as of 09:34, 18 July 2007
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	-----		-----

	This article is based on the MIT open course notes and the original article can be found [~~htthttp~~://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-20Spring-2005/E5599747-6768-4532-9601-8E4297E8DAC2/0/lecture22.pdf here].		This article is based on the MIT open course notes and the original article can be found [http://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-20Spring-2005/E5599747-6768-4532-9601-8E4297E8DAC2/0/lecture22.pdf here].

	[[Marine Hydrodynamics]]		[[Marine Hydrodynamics]]

Syan077: /* Important parameters */

2007-07-18T09:34:22Z

Important parameters

← Older revision		Revision as of 09:34, 18 July 2007
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	III. <math> C D \, </math> is not important and <math> F - K \, </math> approximation is not valid.		III. <math> C D \, </math> is not important and <math> F - K \, </math> approximation is not valid.


			-----

			This article is based on the MIT open course notes and the original article can be found [htthttp://ocw.mit.edu/NR/rdonlyres/Mechanical-Engineering/2-20Spring-2005/E5599747-6768-4532-9601-8E4297E8DAC2/0/lecture22.pdf here].

			[[Marine Hydrodynamics]]

Syan077: /* Type of Forces */

2007-07-18T09:30:56Z

Type of Forces

← Older revision		Revision as of 09:30, 18 July 2007
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	\| <math> \hat{n} dS \leftarrow \, </math>		\| <math> \hat{n} dS \leftarrow \, </math>
	\| width="225" \| can calculate knowing (incident) <br> wave kinematics (and body geometry)		\| width="225" \| can calculate knowing (incident) <br> wave kinematics (and body geometry)
	\|- ~~align="center"~~		\|-
	\|		\|
	\|		\|
	~~\| width="25"~~ \| body ~~<br> surface~~		\| body <math> . \qquad \equiv p_I \, </math>
	\| <math> \equiv p_I \, </math>		\|-
			\|
			\|
			\| width="25" \| surface
	\|}		\|}

Syan077: /* Important parameters */

2007-07-18T09:17:50Z

Important parameters

← Older revision		Revision as of 09:17, 18 July 2007
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	* If <math> \frac{\ell}{\lambda} \gg 1/5 \, </math>, must consider wave diffraction, radiation <math> \left( \frac{A}{\ell} \leq \frac{0.07}{\ell / \lambda} \leq 0.035 \right) \, </math>.		* If <math> \frac{\ell}{\lambda} \gg 1/5 \, </math>, must consider wave diffraction, radiation <math> \left( \frac{A}{\ell} \leq \frac{0.07}{\ell / \lambda} \leq 0.035 \right) \, </math>.

	* If <math> K_C \gg 1 \, </math>: separation important, viscous forces can not be neglected. Further on if <math> \frac{\ell}{lambda} \leq \frac{0.07}{A/\ell} \, </math> so <math>		* If <math> K_C \gg 1 \, </math>: separation important, viscous forces can not be neglected. Further on if <math> \frac{\ell}{\lambda} \leq \frac{0.07}{A/\ell} \, </math> so <math>
	\frac{\ell}{\lambda} \ll 1 \, </math> ignore diffraction, i.e., the Froude-Krylov approximation is valid.		\frac{\ell}{\lambda} \ll 1 \, </math> ignore diffraction, i.e., the Froude-Krylov approximation is valid.

Syan077: /* Type of Forces */

2007-07-18T09:13:46Z

Type of Forces

← Older revision		Revision as of 09:13, 18 July 2007
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	\| width="225" \| can calculate knowing (incident) <br> wave kinematics (and body geometry)		\| width="225" \| can calculate knowing (incident) <br> wave kinematics (and body geometry)
	\|- align="center"		\|- align="center"
	\|		\|
	\| width="25" \| body surface		\|
			\| width="25" \| body <br> surface
	\| <math> \equiv p_I \, </math>		\| <math> \equiv p_I \, </math>
	\|}		\|}

Syan077: /* Type of Forces */

2007-07-18T09:00:33Z

Type of Forces

← Older revision		Revision as of 09:00, 18 July 2007
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	{\| border="0"		{\| border="0"
	\| Associated with skin friction <math> \tau_w, \ i.e., \ ~~\vec{F}~~ \, </math>		\| Associated with skin friction <math> \tau_w, \ i.e., \ \, </math>
	\| <math> \sim \iint \tau_w \, </math> \| <math> dS \, </math>.		\| <math> \vec{F} \sim \iint \tau_w \, </math>
			\| <math> dS \, </math>.
	\|- align="center"		\|- align="center"
			\|
	\| body		\| body
	\|- align="center"		\|- align="center"
			\|
	\| (wetted surface)		\| (wetted surface)
	\|}		\|}
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	{\| border="0" align="center"		{\| border="0" align="center"
	\| <math> \vec{F} ~~\, </math>~~		\| <math> \vec{F} = \iint p \hat{n} \, </math>
	~~\| <math>~~ = \~~iint_{body (wetted surface)}~~ p \hat{n} \, </math>
	\| <math> dS \, </math>,		\| <math> dS \, </math>,
	\| where <math> \ p = - \rho \left( \frac{\partial\phi}{\partial t} + g y \right. \, </math>		\| where <math> \ p = - \rho \left( \frac{\partial\phi}{\partial t} + g y \right. \, </math>
	\| <math> + \left. \underbrace{ \frac{1}{2} \left\| \nabla \phi \right\|^2} \right) \, </math>		\| <math> + \left. \underbrace{ \frac{1}{2} \left\| \nabla \phi \right\|^2} \right) \, </math>
	\|- align="center"		\|- align="center"
	~~\| \|~~\| body \|\| \|\| \|\| <math> =0 \, </math> , for linear theory,		\| body \|\| \|\| \|\| <math> =0 \, </math> , for linear theory,
	\|- align="center"		\|- align="center"
	~~\| \|~~\| (wetted surface) \|\| \|\| small amplitude waves		\| (wetted surface) \|\| \|\| \|\| small amplitude waves
	\|}		\|}

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	\| <math> \left. \begin{matrix} & \phi \approx \phi_I \\ & p \approx - \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right) \end{matrix} \right\} </math>		\| <math> \left. \begin{matrix} & \phi \approx \phi_I \\ & p \approx - \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right) \end{matrix} \right\} </math>
	\| <math> \Rightarrow \vec{F}_{FK} = \, </math>		\| <math> \Rightarrow \vec{F}_{FK} = \, </math>
	\| <math> \iint ~~\,</math>~~		\| <math> \iint \underbrace{- \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right)} </math>
	~~\| <math>~~ \underbrace{- \rho \left( \frac{\partial\phi_I}{\partial t} + g y \right)} </math>
	\| <math> \hat{n} dS \leftarrow \, </math>		\| <math> \hat{n} dS \leftarrow \, </math>
	\| width="225" \| can calculate knowing (incident) <br> wave kinematics (and body geometry)		\| width="225" \| can calculate knowing (incident) <br> wave kinematics (and body geometry)
	\|- align="center"		\|- align="center"
	~~\| \|~~\|		\|
	\| width="25" \| body surface		\| width="25" \| body surface
	\| <math> \equiv p_I \, </math>		\| <math> \equiv p_I \, </math>
	\|}		\|}

	* Mathematical approximation After applying the divergence theorem, the <math> \vec{F}_{FK} \, </math> can be rewritten as <math> \vec{F}_{FK} = - \~~iint_{\mbox{body surface}}~~ p_I \hat{n} dS = - \~~iiint_{\mbox{body volume}}~~ \nabla p_I d\forall </math>.		* Mathematical approximation After applying the divergence theorem, the <math> \vec{F}_{FK} \, </math> can be rewritten as
			{\| border'"0" align="center"
			\| <math> \vec{F}_{FK} \, </math>
			\| <math> = - \iint p_I \hat{n} \, </math>
			\| <math> dS = - \iiint \nabla p_I d\forall \, </math>
			\|- align="center"
			\|
			\| body <br> surface
			\| body <br> volume
			\|}

	If the body dimensions are very small comparable to the wave length, we can assume that <math> \nabla_{p_I} \, </math> is approximately constant through the body volume <math> \forall \, </math> and 'pull' the <math> \nabla_{p_I} \, </math> out of the integral. Thus, the <math> \vec{F}_{FK} \, </math> can be approximated as		If the body dimensions are very small comparable to the wave length, we can assume that <math> \nabla_{p_I} \, </math> is approximately constant through the body volume <math> \forall \, </math> and 'pull' the <math> \nabla_{p_I} \, </math> out of the integral. Thus, the <math> \vec{F}_{FK} \, </math> can be approximated as

	<center><math> \vec{F}_{FK} \cong \left( - \nabla_{p_I} \right) \left\| {}_{\~~mbox{~~at body ~~center}}~~ \~~iiint_{\mbox{body volume}}~~ d\forall = \underbrace{\forall}_{\~~mbox{body volume}}~~ \left( - \nabla_{p_I} \right) \~~right\|~~ {}_{\~~mbox{~~at body center}} <~~/math~~></center>		{\| border="0" align="center"
			\| <math> \vec{F}_{FK} \cong \left( - \nabla_{p_I} \right) \left. \frac{}{} \right\| \, </math>
			\| valign="bottom" align="center" \| at body
			\| <math> \iiint d\forall = \, </math>
			\| <math> \underbrace{\forall} \, </math>
			\| <math> \left( - \nabla_{p_I} \right) \left. \frac{}{} \right\| \, </math>
			\| valign="bottom" \| at body
			\|-
			\|
			\| align="right" valign="top" \| center
			\| body <br> volume
			\| body <br> volume
			\|
			\| align="right" valign="top" \| center
			\|}

	The last relation is particularly useful for small bodies of non-trivial geometry for 13.021, that is all bodies that do not have a rectangular cross section.		The last relation is particularly useful for small bodies of non-trivial geometry for 13.021, that is all bodies that do not have a rectangular cross section.

Syan077: /* Wave Forces on a Body */

2007-07-18T08:09:03Z

Wave Forces on a Body

← Older revision		Revision as of 08:09, 18 July 2007
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	== Wave Forces on a Body ==		== Wave Forces on a Body ==

	{\| border="0" align="~~right~~"		{\| border="0" align="center"
	\| <math> U = \omega A \, </math>		\| <math> U = \omega A \, </math>
			\|-
	\| <math> R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \, </math>		\| <math> R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \, </math>
			\|-
	\| <math> K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \, </math>		\| <math> K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \, </math>
	\|}		\|}
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	{\| border="0" align="center"		{\| border="0" align="center"
	\|- align="center"		\|- align="center"
	\| <math> ~~D_F~~ = \frac{F}{\rho g A \ell^2} = f \left( \frac{}{} \right. </math>		\| <math> C_F = \frac{F}{\rho g A \ell^2} = f \left( \frac{}{} \right. </math>
	\| <math> \underbrace{\frac{A}{\lambda}} \,</math>,		\| <math> \underbrace{\frac{A}{\lambda}} \,</math>,
	\| <math> \underbrace{\frac{\ell}{\lambda}} \, </math>,		\| <math> \underbrace{\frac{\ell}{\lambda}} \, </math>,

Syan077: /* Wave Forces on a Body */

2007-07-18T08:06:03Z

Wave Forces on a Body

← Older revision		Revision as of 08:06, 18 July 2007
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	== Wave Forces on a Body ==		== Wave Forces on a Body ==

	~~<center>~~<math> U = \omega A \, </math>~~</center>~~		{\| border="0" align="right"
			\| <math> U = \omega A \, </math>
			\| <math> R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \, </math>
	~~<center>~~<math> R_e = \frac{U\ell}{\nu} = \frac{\omega A \ell}{\nu} \, </math>~~</center>~~		\| <math> K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \, </math>
			\|}

	~~<center>~~<math> K_C = \frac{UT}{\ell} = \frac{A\omega T}{\ell} = 2 \pi \frac{A}{\ell} \, </math~~></center~~>


	{\| border="0" align="center"		{\| border="0" align="center"
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	* ''Friction drag'' <math> ( C_F ) \, </math>		* ''Friction drag'' <math> ( C_F ) \, </math>

	Associated with skin friction <math> \tau_w, \ i.e., \ \vec{F} \sim \~~iint_{~~\~~mbox{body (wetted surface)}}~~ \~~tau_w~~ dS \, </math>.		{\| border="0"
			\| Associated with skin friction <math> \tau_w, \ i.e., \ \vec{F} \, </math>
			\| <math> \sim \iint \tau_w \, </math> \| <math> dS \, </math>.
			\|- align="center"
			\| body
			\|- align="center"
			\| (wetted surface)
			\|}

	2. '''Inertial forces''' Froude-Krylov forces, diffraction forces, radiation forces.		2. '''Inertial forces''' Froude-Krylov forces, diffraction forces, radiation forces.
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	Forces arising from potential flow wave theory,		Forces arising from potential flow wave theory,

	<center><math> \vec{F} = \iint_{body (wetted surface)} p \hat{n} dS~~, \~~ \, </math> where <math> \ p = - \rho \left( \frac{\partial\phi}{\partial t} + g y + \frac{1}{2} \left\| \nabla \phi \right\|^2 \right) </math></center>		{\| border="0" align="center"
			\| <math> \vec{F} \, </math>
			\| <math> = \iint_{body (wetted surface)} p \hat{n} \, </math>
			\| <math> dS \, </math>,
			\| where <math> \ p = - \rho \left( \frac{\partial\phi}{\partial t} + g y \right. \, </math>
			\| <math> + \left. \underbrace{ \frac{1}{2} \left\| \nabla \phi \right\|^2} \right) \, </math>
			\|- align="center"
			\| \|\| body \|\| \|\| \|\| <math> =0 \, </math> , for linear theory,
			\|- align="center"
			\| \|\| (wetted surface) \|\| \|\| small amplitude waves
			\|}

	For linear theory, the velocity potential <math> \phi \, </math> and the pressure <math> p \, </math> can be decomposed to		For linear theory, the velocity potential <math> \phi \, </math> and the pressure <math> p \, </math> can be decomposed to
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	\|}		\|}

	* Mathematical approximation After applying the divergence theorem, the <math> \vec{F}_{FK} \, </math> can be rewritten as <math> \vec{F}_{FK} = - \iint_{\mbox{body surface}} p_I \hat{n} dS = -		* Mathematical approximation After applying the divergence theorem, the <math> \vec{F}_{FK} \, </math> can be rewritten as <math> \vec{F}_{FK} = - \iint_{\mbox{body surface}} p_I \hat{n} dS = - \iiint_{\mbox{body volume}} \nabla p_I d\forall </math>.
	\iiint_{\mbox{body volume}} \nabla p_I d\forall </math>.

	If the body dimensions are very small comparable to the wave length, we can assume that <math> \nabla_{p_I} \, </math> is approximately constant through the body volume <math> \forall \, </math> and 'pull' the <math> \nabla_{p_I} \, </math> out of the integral. Thus, the <math> \vec{F}_{FK} \, </math> can be approximated as		If the body dimensions are very small comparable to the wave length, we can assume that <math> \nabla_{p_I} \, </math> is approximately constant through the body volume <math> \forall \, </math> and 'pull' the <math> \nabla_{p_I} \, </math> out of the integral. Thus, the <math> \vec{F}_{FK} \, </math> can be approximated as

Syan077: /* Important parameters */

2007-07-18T07:52:18Z

Important parameters

← Older revision		Revision as of 07:52, 18 July 2007
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	* If <math> \frac{\ell}{\lambda} \ll 1 \, </math>, ignore diffraction , wave effects in radiation problem (i.e., <math> d_{ij} \approx 0, \ m_{ij} \approx m_{ij} \, </math> infinite fluid added mass). F-K approximation might be used, calculate <math> \vec{F}_{FK} \, </math>.		* If <math> \frac{\ell}{\lambda} \ll 1 \, </math>, ignore diffraction , wave effects in radiation problem (i.e., <math> d_{ij} \approx 0, \ m_{ij} \approx m_{ij} \, </math> infinite fluid added mass). F-K approximation might be used, calculate <math> \vec{F}_{FK} \, </math>.
	* If <math> \frac{\ell}{\lambda} \gg 1/5 \, </math>, must consider wave diffraction, radiation <math> \left( \frac{A}{\ell} \leq \frac{0.07}{\ell / \lambda} \leq 0.035 \right) \, </math>.		* If <math> \frac{\ell}{\lambda} \gg 1/5 \, </math>, must consider wave diffraction, radiation <math> \left( \frac{A}{\ell} \leq \frac{0.07}{\ell / \lambda} \leq 0.035 \right) \, </math>.

			* If <math> K_C \gg 1 \, </math>: separation important, viscous forces can not be neglected. Further on if <math> \frac{\ell}{lambda} \leq \frac{0.07}{A/\ell} \, </math> so <math>
			\frac{\ell}{\lambda} \ll 1 \, </math> ignore diffraction, i.e., the Froude-Krylov approximation is valid.

			{\| border="0" align="center"
			\| <math> F = \frac{1}{2} \rho \ell^2 \, </math>
			\| <math> \underbrace{U(t)} \, </math>
			\| <math> \left\| U(t) \right\| C_D \left( R_e \right) \, </math>
			\|-
			\| \|\| width="50" \| relative velocity
			\|}

			* Intermediate <math> K_c - \, </math> both viscous and inertial effects important, use Morrison's formula.

			<center><math> F= \frac{1}{2} \rho \ell^2 U(t) \left\| U(t) \right\| C_D \left( R_e \right) + \rho \ell^3 \dot{U} C_m \left( R_e, K_C \right) </math></center>

			* Summary

			I. Use: <math> C D \, </math> and <math> F - K \, </math> approximation.

			II. Use: <math> C F \, </math> and <math> F - K \, </math> approximation.

			III. <math> C D \, </math> is not important and <math> F - K \, </math> approximation is not valid.