Reynolds Transport Theorem - Revision history

Admin at 09:15, 11 February 2011

2011-02-11T09:15:11Z

← Older revision		Revision as of 09:15, 11 February 2011
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	\|}		\|}

	where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.~~<ref name="Lilly_1992">{{cite journal~~		where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.
	~~\|author=Lilly~~
	~~\|title=A proposed modification of the Germano subgrid-scale closure method~~
	~~\|journal=Physics of Fluids A~~
	~~\|year=1992~~
	~~\|volume=4~~
	~~\|issue=3~~
	~~\|pages=633-636}}</ref>~~

	= References =		= References =
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	<references />		<references />

			<!--
			<ref name="Lilly_1992">
			-->
	{{cite journal		{{cite journal
			\|author=Lilly
	\|title=A proposed modification of the Germano subgrid-scale closure method		\|title=A proposed modification of the Germano subgrid-scale closure method
	\|journal=Physics of Fluids A		\|journal=Physics of Fluids A
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	\|issue=3		\|issue=3
	\|pages=633-636}}		\|pages=633-636}}
			<!--
			</ref>
			-->


	{{GoverningEquations}}		{{GoverningEquations}}

Admin: /* References */

2010-11-16T07:33:01Z

References

← Older revision		Revision as of 07:33, 16 November 2010
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	<references />		<references />

			{{cite journal
			\|title=A proposed modification of the Germano subgrid-scale closure method
			\|journal=Physics of Fluids A
			\|year=1992
			\|volume=4
			\|issue=3
			\|pages=633-636}}



	{{GoverningEquations}}		{{GoverningEquations}}

Admin: /* Definition */

2010-11-16T07:32:37Z

Definition

← Older revision		Revision as of 07:32, 16 November 2010
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	where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.<ref name="Lilly_1992">{{cite journal		where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.<ref name="Lilly_1992">{{cite journal
			\|author=Lilly
	\|title=A proposed modification of the Germano subgrid-scale closure method		\|title=A proposed modification of the Germano subgrid-scale closure method
	\|journal=Physics of Fluids A		\|journal=Physics of Fluids A
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	\|issue=3		\|issue=3
	\|pages=633-636}}</ref>		\|pages=633-636}}</ref>


	= References =		= References =

Admin at 07:32, 16 November 2010

2010-11-16T07:32:13Z

@@ Line 106: / Line 106: @@
 |}
-where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.
+where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.<ref name="Lilly_1992">{{cite journal
 {{GoverningEquations}}

Admin at 19:42, 7 November 2010

2010-11-07T19:42:57Z

← Older revision		Revision as of 19:42, 7 November 2010
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	where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.		where <math>S_{B}^{M}</math> is the surface of the material volume <math>V_{B}^{M}</math>, <math>\boldsymbol{n}</math> is the vector normal to this surface, and <math>dS</math> is a differential area on the surface.

			{{GoverningEquations}}

Admin at 19:41, 7 November 2010

2010-11-07T19:41:55Z

← Older revision		Revision as of 19:41, 7 November 2010
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	</math>		</math>

	Because the integral is with time, the integral <math>\iiint_{V_{B}^{M}(t)} and the derivative <math>\frac{d}{dt}</math> do not commute.		Because the integral is with time, the integral <math>\iiint_{V_{B}^{M}(t)}</math> and the derivative <math>\frac{d}{dt}</math> do not commute.

	However, if the integration were over a fixed volume that did not change in time (for example, <math>V_{B,0}</math>, the integral and the derivative ''would'' commute.		However, if the integration were over a fixed volume that did not change in time (for example, <math>V_{B,0}</math>, the integral and the derivative ''would'' commute.

Admin: /* Definition */

2010-11-04T00:48:21Z

Definition

← Older revision		Revision as of 00:48, 4 November 2010
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	<math>		<math>
	\frac{dB}{dt} = \iiint_{V_{B,0}} \left[ \frac{ d \rho b }{ dt } + \rho b \left( \nabla \cdot \boldsymbol{v}_{B} \right] J dV		\frac{dB}{dt} = \iiint_{V_{B,0}} \left[ \frac{ d \rho b }{ dt } + \rho b \left( \nabla \cdot \boldsymbol{v}_{B} \right) \right] J dV
	</math>		</math>

Admin: /* Definition */

2010-11-04T00:47:28Z

Definition

← Older revision		Revision as of 00:47, 4 November 2010
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	\|}		\|}

	Now, the definition of the substantial derivative can be used to put this into a different (more familiar) form:		Now, the definition of the substantial derivative, given by:

			<math>
			\frac{d}{dt} = \frac{\partial}{\partial t} + (\boldsymbol{v}_{B} \cdot \nabla)
			</math>


			can be used to put this into a different (more familiar) form:

	{\|style="border:1px black solid;"		{\|style="border:1px black solid;"

Admin at 00:45, 4 November 2010

2010-11-04T00:45:26Z

Show changes

Admin at 19:58, 31 October 2010

2010-10-31T19:58:41Z

← Older revision		Revision as of 19:58, 31 October 2010
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	</math>		</math>

	Reynolds Transport Theorem is a formal way of performing this balance over a fluid control volume, of arbitrary shape, moving at an arbitrary velocity.		Reynolds Transport Theorem is a formal way of performing this balance over a fluid control volume, or fluid material volume, of arbitrary shape, moving at an arbitrary velocity.

	A derivation of Reynolds Transport Theorem is given here: [[Reynolds Transport Theorem Derivation]]		A derivation of Reynolds Transport Theorem is given here: [[Reynolds Transport Theorem Derivation]]
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	</math>		</math>

	where <math>\rho</math> is the fluid density and <math>V_{\mathcal{F}}</math> is the material volume that corresponds to the property <math>F</math> (which moves at a velocity <math>u_{\mathcal{F}}</math>.		where <math>\rho</math> is the fluid density and <math>V_{\mathcal{F}}</math> is the material volume that corresponds to the property <math>F</math> (which moves at a velocity <math>\boldsymbol{v}_{\mathcal{F}}</math>.

	The partial derivative with respect to volume for the intensive property <math>\mathcal{F}</math> is,		The partial derivative with respect to volume for the intensive property <math>\mathcal{F}</math> is,
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	<math>		<math>
	\frac{d}{dt} \iiint_{V_{\mathcal{F}}(t)}		\frac{d}{dt} \iiint_{V_{\mathcal{F}}(t)}
	= \iiint_{V_{\mathcal{F}}(t)} ~~\rho~~ \frac{ \partial \mathcal{F} }{\partial t} dV		= \iiint_{V_{\mathcal{F}}(t)} \frac{ \partial \rho \mathcal{F} }{\partial t} dV
	+ \iint_{S_{\mathcal{F}}(t)} \rho \mathcal{F} \boldsymbol{v}_{\mathcal{F}} \cdot \boldsymbol{n} dS		+ \iint_{S_{\mathcal{F}}(t)} \rho \mathcal{F} \boldsymbol{v}_{\mathcal{F}} \cdot \boldsymbol{n} dS
	</math>		</math>

	where <math>\boldsymbol{v}_{\mathcal{F}}</math> is the velocity vector of the ~~property~~ <math>~~\mathcal{~~F}</math>.		where <math>\boldsymbol{v}_{\mathcal{F}}</math> is the velocity vector of the material volume containing <math>F</math>.