User:Tohline/PGE/RotatingFrame - Revision history

Tohline at 17:02, 11 July 2015

2015-07-11T17:02:14Z

← Older revision		Revision as of 17:02, 11 July 2015
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	</math>		</math>
	</div>		</div>



	=Related Discussions=		=Related Discussions=

Tohline at 17:01, 11 July 2015

2015-07-11T17:01:32Z

← Older revision		Revision as of 17:01, 11 July 2015
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	</b></font>		</b></font>
	I have move the earlier contents of this page to a new Wiki location called		I have move the earlier contents of this page to a new Wiki location called
	[~~http://www.vistrails.org/index.php/~~User:Tohline/Apps/RiemannEllipsoids_Compressible Compressible Riemann Ellipsoids].		[[User:Tohline/Apps/RiemannEllipsoids_Compressible\|Compressible Riemann Ellipsoids]].

	=Rotating Reference Frame=		=Rotating Reference Frame=

Tohline: /* Nonlinear Velocity Cross-Product */

2015-07-11T16:58:04Z

Nonlinear Velocity Cross-Product

← Older revision		Revision as of 16:58, 11 July 2015
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	==Nonlinear Velocity Cross-Product==		==Nonlinear Velocity Cross-Product==

	In some contexts —for example, our discussion of [~~http://www.vistrails.org/index.php/~~User:Tohline/Apps/RiemannEllipsoids_Compressible Riemann ellipsoids] or the analysis by [~~http://www.vistrails.org/index.php/~~User:Tohline/Apps/Korycansky_Papaloizou_1996 Korycansky & Papaloizou (1996)] of nonaxisymmetric disk structures — it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,		In some contexts — for example, our discussion of [[User:Tohline/Apps/RiemannEllipsoids_Compressible\|Riemann ellipsoids]] or the analysis by [[User:Tohline/Apps/Korycansky_Papaloizou_1996\|Korycansky & Papaloizou (1996)]] of nonaxisymmetric disk structures — it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: Move references to wikipedia

2015-07-11T16:55:44Z

Move references to wikipedia

← Older revision		Revision as of 16:55, 11 July 2015
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	</math>		</math>
	</div>		</div>



	==Continuity Equation (rotating frame)==		==Continuity Equation (rotating frame)==
	Applying these transformations to the standard, inertial-frame representations of the continuity equation presented [~~http://www.vistrails.org/index.php/~~User:Tohline/PGE/ConservingMass#Continuity_Equation elsewhere], we obtain the:		Applying these transformations to the standard, inertial-frame representations of the continuity equation presented [[User:Tohline/PGE/ConservingMass#Continuity_Equation\|elsewhere]], we obtain the:

	<div align="center">		<div align="center">
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	<math>\biggl[ \frac{d\rho}{dt} \biggr]_{rot} + \rho \nabla \cdot {\vec{v}}_{rot} = 0</math> ;		<math>\biggl[ \frac{d\rho}{dt} \biggr]_{rot} + \rho \nabla \cdot {\vec{v}}_{rot} = 0</math> ;
	</div>		</div>




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	<math>\biggl[ \frac{\partial\rho}{\partial t} \biggr]_{rot} + \nabla \cdot (\rho {\vec{v}}_{rot}) = 0</math> .		<math>\biggl[ \frac{\partial\rho}{\partial t} \biggr]_{rot} + \nabla \cdot (\rho {\vec{v}}_{rot}) = 0</math> .
	</div>		</div>



	==Euler Equation (rotating frame)==		==Euler Equation (rotating frame)==
	Applying these transformations to the standard, inertial-frame representations of the Euler equation presented [~~http://www.vistrails.org/index.php/~~User:Tohline/PGE/Euler#Euler_Equation elsewhere], we obtain the:		Applying these transformations to the standard, inertial-frame representations of the Euler equation presented [[User:Tohline/PGE/Euler#Euler_Equation\|elsewhere]], we obtain the:

	<div align="center">		<div align="center">
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	<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math> ;		<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x})</math> ;
	</div>		</div>




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	<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}\|{\vec{\Omega}}_f \times \vec{x}\|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math> ;		<math>\biggl[\frac{\partial\vec{v}}{\partial t}\biggr]_{rot} + ({\vec{v}}_{rot}\cdot \nabla) {\vec{v}}_{rot}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi - \frac{1}{2}\|{\vec{\Omega}}_f \times \vec{x}\|^2 \biggr] - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} </math> ;
	</div>		</div>




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	==Centrifugal and Coriolis Accelerations==		==Centrifugal and Coriolis Accelerations==

	Following along the lines of the discussion presented in Appendix 1.D, §3 of [~~http://www.vistrails.org/index.php/~~User:Tohline/Appendix/References BT87], in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,		Following along the lines of the discussion presented in Appendix 1.D, §3 of [[User:Tohline/Appendix/References\|BT87]], in a rotating reference frame the Lagrangian representation of the Euler equation may be written in the form,
	<div align="center">		<div align="center">
	<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi + {\vec{a}}_{fict} </math>,		<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi + {\vec{a}}_{fict} </math>,
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	</div>		</div>

	(see the related [~~http~~://~~en.wikipedia.org/wiki/Coriolis_effect~~#~~Formula~~ Wikipedia discussion]) and the		(see the related [[User:Tohline/PGE/RotatingFrame#WikiCoriolis\|Wikipedia discussion]]) and the
	<div align="center">		<div align="center">
	<font color="#770000">'''Centrifugal Acceleration'''</font>		<font color="#770000">'''Centrifugal Acceleration'''</font>
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	</math>		</math>
	</div>		</div>
	(see the related [~~http~~://~~en.wikipedia.org/wiki/Centrifugal_force~~#~~Derivation_using_vectors~~ Wikipedia discussion]).		(see the related [[User:Tohline/PGE/RotatingFrame#WikiCentrifugal\|Wikipedia discussion]]).

	==Nonlinear Velocity Cross-Product==		==Nonlinear Velocity Cross-Product==
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	</math>		</math>
	</div>		</div>

			=Related Discussions=
			* <span id="WikiVorticity">Wikipedia discussion of [http://en.wikipedia.org/wiki/Vorticity vorticity].</span>
			* <span id="WikiCoriolis">Wikipedia discussion of [http://en.wikipedia.org/wiki/Coriolis_effect#Formula Coriolis Effect].</span>
			* <span id="WikiCentrifugal">Wikipedia discussion of [http://en.wikipedia.org/wiki/Centrifugal_force#Derivation_using_vectors Centrifugal acceleration].</span>



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Tohline: /* Rotating Reference Frame */ Clean up page before creating PDF-formatted archive

2015-07-11T16:43:38Z

Rotating Reference Frame: Clean up page before creating PDF-formatted archive

← Older revision		Revision as of 16:43, 11 July 2015
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	=Rotating Reference Frame=		=Rotating Reference Frame=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>\Omega_f</math>. In order to transform any one of the [~~http://www.vistrails.org/index.php/~~User:Tohline/PGE#Principal_Governing_Equations principal governing equations] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>d/dt</math> operator, which denotes Lagrangian time-differentiation in the ~~interial~~ frame, must everywhere be replaced as follows:		At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>~\Omega_f</math>. In order to transform any one of the [[User:Tohline/PGE#Principal_Governing_Equations\|principal governing equations]] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>~d/dt</math> operator, which denotes Lagrangian time-differentiation in the inertial frame, must everywhere be replaced as follows:
	<div align="center">		<div align="center">
	<math>		<math>
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	</math>		</math>
	</div>		</div>
	(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in [~~http://www.vistrails.org/index.php/~~User:Tohline/Appendix/References BT87].) Note as well that the relationship between the fluid [~~http://www.vistrails.org/index.php/~~User:Tohline/PGE/~~Euler~~#~~in_terms_of_the_vorticity:~~ vorticity] in the two frames is,		(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in [[User:Tohline/Appendix/References\|BT87]].) Note as well that the relationship between the fluid [[User:Tohline/PGE/RotatingFrame#WikiVorticity\|vorticity]] in the two frames is,
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: Move title image

2012-11-06T17:44:21Z

Move title image

← Older revision		Revision as of 17:44, 6 November 2012
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	__FORCETOC__		__FORCETOC__
	<!-- __NOTOC__ -->		<!-- __NOTOC__ -->
	~~{{LSU_HBook_header}}~~

	<font color="red"><b>		<font color="red"><b>
	NOTE to Eric Hirschmann & David Neilsen...		NOTE to Eric Hirschmann & David Neilsen...
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	I have move the earlier contents of this page to a new Wiki location called		I have move the earlier contents of this page to a new Wiki location called
	[http://www.vistrails.org/index.php/User:Tohline/Apps/RiemannEllipsoids_Compressible Compressible Riemann Ellipsoids].		[http://www.vistrails.org/index.php/User:Tohline/Apps/RiemannEllipsoids_Compressible Compressible Riemann Ellipsoids].





	=Rotating Reference Frame=		=Rotating Reference Frame=
			{{LSU_HBook_header}}
	At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>\Omega_f</math>. In order to transform any one of the [http://www.vistrails.org/index.php/User:Tohline/PGE#Principal_Governing_Equations principal governing equations] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>d/dt</math> operator, which denotes Lagrangian time-differentiation in the interial frame, must everywhere be replaced as follows:		At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>\Omega_f</math>. In order to transform any one of the [http://www.vistrails.org/index.php/User:Tohline/PGE#Principal_Governing_Equations principal governing equations] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>d/dt</math> operator, which denotes Lagrangian time-differentiation in the interial frame, must everywhere be replaced as follows:
	<div align="center">		<div align="center">
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	</math>		</math>
	</div>		</div>


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Tohline: /* Nonlinear Velocity Cross-Product */ Clean up discussion of vector "A"

2010-03-21T19:17:48Z

Nonlinear Velocity Cross-Product: Clean up discussion of vector "A"

← Older revision		Revision as of 19:17, 21 March 2010
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	<math>		<math>
	\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial z}\biggl[- (\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,		\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = - \frac{\partial}{\partial z}\biggl[(\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
	</math><br />		</math><br />

	<math>		<math>
	\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] .		\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] ,
			</math>
			</div>
			where, in this case, the three Cartesian components of the vorticity vector are,
			<div align="center">
			<math>
			\zeta_x = - \frac{\partial v_y}{\partial z} ,
			~~~~~~ \zeta_y = \frac{\partial v_x}{\partial z} ,
			~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .
	</math>		</math>
	</div>		</div>

Tohline: /* Nonlinear Velocity Cross-Product */ Curl of vector "A"

2010-03-21T18:40:31Z

Nonlinear Velocity Cross-Product: Curl of vector "A"

← Older revision		Revision as of 18:40, 21 March 2010
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	<div align="left">		<div align="left">
	<math>		<math>
	\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial ~~xxx~~}\biggl[ \biggr] - \frac{\partial}{\partial ~~xxx~~}\biggl[ \biggr],		\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr],
	</math><br />		</math><br />

	<math>		<math>
	\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial ~~xxx~~}\biggl[ \biggr] - \frac{\partial}{\partial ~~xxx~~}\biggl[ \biggr] ,		\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial z}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
	</math><br />		</math><br />

	<math>		<math>
	\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial ~~xxx~~}\biggl[ \biggr] - \frac{\partial}{\partial ~~xxx~~}\biggl[ \biggr] .		\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x - \zeta_x v_z \biggr] - \frac{\partial}{\partial y}\biggl[ \zeta_y v_z - (\zeta_z + 2\Omega) v_y \biggr] .
	</math>		</math>
	</div>		</div>

	===When <math>v_z = 0</math>===		===When <math>v_z = 0</math>===

			If we restrict our discussion to configurations that exhibit only planar flows — that is, systems in which <math>v_z = 0</math> — then the Cartesian components of <math>{\vec{A}}</math> and <math>\nabla\times\vec{A}</math> simplify somewhat to give, respectively,
			<div align="left">
			<math>
			\hat{i}: ~~~~~~ A_x = - (\zeta_z + 2\Omega) v_y ,
			</math><br />

			<math>
			\hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x ,
			</math><br />

			<math>
			\hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x ,
			</math>
			</div>

			and,
			<div align="left">
			<math>
			\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial y}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] - \frac{\partial}{\partial z}\biggl[ (\zeta_z + 2\Omega) v_x \biggr],
			</math><br />

			<math>
			\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial z}\biggl[- (\zeta_z + 2\Omega) v_y \biggr] - \frac{\partial}{\partial x}\biggl[ \zeta_x v_y - \zeta_y v_x \biggr] ,
			</math><br />

			<math>
			\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] .
			</math>
			</div>


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Tohline: /* Nonlinear Velocity Cross-Product */ Continue discussion of nonlinear vector "A"

2010-03-21T18:28:06Z

Nonlinear Velocity Cross-Product: Continue discussion of nonlinear vector "A"

← Older revision		Revision as of 18:28, 21 March 2010
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	</math>		</math>
	</div>		</div>
	Without ~~lose~~ of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system. ~~Also~~, ~~to simplify notation~~, ~~for most of~~ the ~~remainder~~ of ~~this subsection we will drop~~ the ~~subscript~~ "~~rot~~" ~~on both~~ the ~~velocity and vorticity vectors. The Cartesian components~~ of <math>{\vec{A}}</math> ~~are then,~~
			NOTE: To simplify notation, for most of the remainder of this subsection we will drop the subscript "rot" on both the velocity and vorticity vectors.

			===Align <math>{\vec\Omega}_f</math> with z-axis===
			Without loss of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system. The Cartesian components of <math>{\vec{A}}</math> are then,
			<div align="left">
			<math>
			\hat{i}: ~~~~~~ A_x = \zeta_y v_z - (\zeta_z + 2\Omega) v_y ,
			</math><br />

			<math>
			\hat{j}: ~~~~~~ A_y = (\zeta_z + 2\Omega) v_x - \zeta_x v_z ,
			</math><br />

			<math>
			\hat{k}: ~~~~~~ A_z = \zeta_x v_y - \zeta_y v_x ,
			</math>
			</div>
			where it is understood that the three Cartesian components of the vorticity vector are,
			<div align="center">
			<math>
			\zeta_x = \biggl[\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \biggr] ,
			~~~~~~ \zeta_y = \biggl[ \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \biggr] ,
			~~~~~~ \zeta_z = \biggl[ \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \biggr] .
			</math>
			</div>
			In turn, the curl of <math>\vec{A}</math> has the following three Cartesian components:
	<div align="left">		<div align="left">
	<math>		<math>
	\hat{i}: ~~~~~~		\hat{i}: ~~~~~~ [\nabla\times\vec{A}]_x = \frac{\partial}{\partial xxx}\biggl[ \biggr] - \frac{\partial}{\partial xxx}\biggl[ \biggr],
	</math><br />		</math><br />

	<math>		<math>
	\hat{j}: ~~~~~~		\hat{j}: ~~~~~~ [\nabla\times\vec{A}]_y = \frac{\partial}{\partial xxx}\biggl[ \biggr] - \frac{\partial}{\partial xxx}\biggl[ \biggr] ,
	</math><br />		</math><br />

	<math>		<math>
	\hat{k}: ~~~~~~		\hat{k}: ~~~~~~ [\nabla\times\vec{A}]_z = \frac{\partial}{\partial xxx}\biggl[ \biggr] - \frac{\partial}{\partial xxx}\biggl[ \biggr] .
	</math>		</math>
	</div>		</div>

			===When <math>v_z = 0</math>===

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Tohline: Introduce vector "A"

2010-03-21T16:04:16Z

Introduce vector "A"

← Older revision		Revision as of 16:04, 21 March 2010
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	</div>		</div>
	(see the related [http://en.wikipedia.org/wiki/Centrifugal_force#Derivation_using_vectors Wikipedia discussion]).		(see the related [http://en.wikipedia.org/wiki/Centrifugal_force#Derivation_using_vectors Wikipedia discussion]).

			==Nonlinear Velocity Cross-Product==

			In some contexts —for example, our discussion of [http://www.vistrails.org/index.php/User:Tohline/Apps/RiemannEllipsoids_Compressible Riemann ellipsoids] or the analysis by [http://www.vistrails.org/index.php/User:Tohline/Apps/Korycansky_Papaloizou_1996 Korycansky & Papaloizou (1996)] of nonaxisymmetric disk structures — it proves useful to isolate and analyze the term in the "vorticity formulation" of the Euler equation that involves a nonlinear cross-product of the rotating-frame velocity vector, namely,
			<div align="center">
			<math>
			\vec{A} \equiv ({\vec{\zeta}}_{rot}+2{\vec\Omega}_f) \times {\vec{v}}_{rot} .
			</math>
			</div>
			Without lose of generality we can set <math>{\vec\Omega}_f = \hat{k}\Omega_f</math>, that is, we can align the frame rotation axis with the z-axis of a Cartesian coordinate system. Also, to simplify notation, for most of the remainder of this subsection we will drop the subscript "rot" on both the velocity and vorticity vectors. The Cartesian components of <math>{\vec{A}}</math> are then,
			<div align="left">
			<math>
			\hat{i}: ~~~~~~
			</math><br />

			<math>
			\hat{j}: ~~~~~~
			</math><br />

			<math>
			\hat{k}: ~~~~~~
			</math>
			</div>



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