Curvilinear Coordinates - Revision history

Robertson at 00:51, 18 May 2016

2016-05-18T00:51:51Z

← Older revision		Revision as of 00:51, 18 May 2016
Line 1:		Line 1:
	<div class="title">Curvilinear Coordinates</div>		<div class="title">Curvilinear Coordinates</div>
	~~<wikitex>~~
	~~The requirement for a boundary-following coordinate system and for a~~
	~~laterally variable grid resolution can both be met (for suitably~~
	~~smooth domains) by introducing an appropriate orthogonal coordinate~~
	~~transformation in the horizontal. Let the new coordinates be~~
	~~$\xi(x,y)$ and $\eta(x,y)$ where the relationship of horizontal arc~~
	~~length to the differential distance is given by:~~

	$$(ds)_{\xi~~} =~~ \~~left~~( ~~{1 \over m} \right~~) ~~d \xi$$~~		The requirement for a boundary-following coordinate system and for a laterally variable grid resolution can both be met (for suitably
			smooth domains) by introducing an appropriate orthogonal coordinate transformation in the horizontal. Let the new coordinates be <math>\xi(x,y)</math> and <math>\eta(x,y)</math> where the relationship of horizontal arc length to the differential distance is given by:

	$$(ds)_{\~~eta~~} = \left( {1 \over n} \right) d \~~eta$$~~		<math display="block">(ds)_{\xi} = \left( {1 \over m} \right) d \xi</math>

	~~Here, $m~~(\~~xi,~~\~~eta)$ and $n~~(\~~xi,~~\~~eta~~)~~$ are the scale factors which~~		<math display="block">(ds)_{\eta} = \left( {1 \over n} \right) d \eta</math>
	~~relate the differential distances $(\Delta \xi,\Delta~~ \eta~~)$ to the~~
	~~actual (physical) arc lengths.~~

	It is helpful to write the equations in vector notation and to use		Here, <math>m(\xi,\eta)</math> and <math>n(\xi,\eta)</math> are the scale factors which relate the differential distances <math>(\Delta \xi,\Delta \eta)</math> to the actual (physical) arc lengths.
	the formulas for div, grad, and curl in curvilinear coordinates (see
			It is helpful to write the equations in vector notation and to use the formulas for div, grad, and curl in curvilinear coordinates (see
	[[Bibliography#Batchelor67 \|Batchelor, Appendix 2]]):		[[Bibliography#Batchelor67 \|Batchelor, Appendix 2]]):

	$$\nabla \phi = \hat{\xi} m {\partial \phi \over \partial \xi} +		<math display="block">\nabla \phi = \hat{\xi} m {\partial \phi \over \partial \xi} +
	\hat{\eta} n {\partial \phi \over \partial \eta}$$		\hat{\eta} n {\partial \phi \over \partial \eta}</math>

	$$\nabla \cdot \vec{a} = mn \left[		<math display="block">\nabla \cdot \vec{a} = mn \left[
	{\partial \over \partial \xi} \!\! \left( {a \over n} \right) +		{\partial \over \partial \xi} \!\! \left( {a \over n} \right) +
	{\partial \over \partial \eta} \!\! \left( {b \over m} \right)		{\partial \over \partial \eta} \!\! \left( {b \over m} \right)
	\right]$$		\right]</math>

	$$		<math display="block">\nabla \times \vec{a} = mn \left\| \begin{matrix}
	\nabla \times \vec{a} = mn \left\| \matrix{		{\hat{\xi}_1 \over m} & {\hat{\xi}_2 \over n} & \hat{k} \\
	{\hat{\xi}_1 \over m} & {\hat{\xi}_2 \over n} & \hat{k} \cr
	{\partial \over \partial \xi} &		{\partial \over \partial \xi} &
	{\partial \over \partial \eta} &		{\partial \over \partial \eta} &
	{\partial \over \partial z} \cr		{\partial \over \partial z} \\
	{a \over m} & {b \over n} & c		{a \over m} & {b \over n} & c
	} \right\|		\end{matrix} \right\|</math>
	$$

	$$\nabla^2 \phi = \nabla \cdot \nabla \phi = mn \left[		<math display="block">\nabla^2 \phi = \nabla \cdot \nabla \phi = mn \left[
	{\partial \over \partial \xi} \!\! \left( {m \over n}		{\partial \over \partial \xi} \!\! \left( {m \over n}
	{\partial \phi \over \partial \xi} \right) +		{\partial \phi \over \partial \xi} \right) +
	{\partial \over \partial \eta} \!\! \left( {n \over m}		{\partial \over \partial \eta} \!\! \left( {n \over m}
	{\partial \phi \over \partial \eta} \right) \right]$$		{\partial \phi \over \partial \eta} \right) \right]</math>

	where $\phi$ is a scalar and $\vec{a}$ is a vector with components		where <math>\phi</math> is a scalar and <math>\vec{a}</math> is a vector with components <math>a</math>, <math>b</math>, and <math>c</math>.
	$a$, $b$, and $c$.
	</~~wikitex~~>

Kate: Change to wikitex, whew!

2009-09-23T00:46:31Z

Change to wikitex, whew!

← Older revision		Revision as of 00:46, 23 September 2009
Line 1:		Line 1:
	<div class="title">Curvilinear Coordinates</div>		<div class="title">Curvilinear Coordinates</div>
			<wikitex>
	The requirement for a boundary-following coordinate system and for a		The requirement for a boundary-following coordinate system and for a
	laterally variable grid resolution can both be met (for suitably		laterally variable grid resolution can both be met (for suitably
	smooth domains) by introducing an appropriate orthogonal coordinate		smooth domains) by introducing an appropriate orthogonal coordinate
	transformation in the horizontal. Let the new coordinates be		transformation in the horizontal. Let the new coordinates be
	~~<math>~~\xi(x,y)~~\!\,</math>~~ and ~~<math>~~\eta(x,y)~~\!\,</math>~~ where the relationship of horizontal arc		$\xi(x,y)$ and $\eta(x,y)$ where the relationship of horizontal arc
	length to the differential distance is given by:		length to the differential distance is given by:

	~~:<math>~~(ds)_{\xi} = \left( {1 \over m} \right) d \xi~~</math>~~		$$(ds)_{\xi} = \left( {1 \over m} \right) d \xi$$

	~~:<math>~~(ds)_{\eta} = \left( {1 \over n} \right) d \eta~~</math>~~		$$(ds)_{\eta} = \left( {1 \over n} \right) d \eta$$

	Here, ~~<math>~~m(\xi,\eta)~~\!\,</math>~~ and ~~<math>~~n(\xi,\eta)~~\!\,</math>~~ are the scale factors which		Here, $m(\xi,\eta)$ and $n(\xi,\eta)$ are the scale factors which
	relate the differential distances ~~<math>~~(\Delta \xi,\Delta \eta)~~\!\,</math>~~ to the		relate the differential distances $(\Delta \xi,\Delta \eta)$ to the
	actual (physical) arc lengths.		actual (physical) arc lengths.

	It is helpful to write the equations in vector notation and to use		It is helpful to write the equations in vector notation and to use
	the formulas for div, grad, and curl in curvilinear coordinates (see		the formulas for div, grad, and curl in curvilinear coordinates (see
	Batchelor, Appendix 2):		[[Bibliography#Batchelor67 \|Batchelor, Appendix 2]]):

	~~:<math>~~\nabla \phi = \hat{\xi} m {\partial \phi \over \partial \xi} +		$$\nabla \phi = \hat{\xi} m {\partial \phi \over \partial \xi} +
	\hat{\eta} n {\partial \phi \over \partial \eta}~~</math>~~		\hat{\eta} n {\partial \phi \over \partial \eta}$$

	~~:<math>~~\nabla \cdot \vec{a} = mn \left[		$$\nabla \cdot \vec{a} = mn \left[
	{\partial \over \partial \xi} \!\! \left( {a \over n} \right) +		{\partial \over \partial \xi} \!\! \left( {a \over n} \right) +
	{\partial \over \partial \eta} \!\! \left( {b \over m} \right)		{\partial \over \partial \eta} \!\! \left( {b \over m} \right)
	\right]~~</math>~~		\right]$$

	~~:<math>~~\nabla \times \vec{a} = mn \left\| \~~begin{array}~~{~~ccc}~~		$$
	~~\vspace{1 mm}~~		\nabla \times \vec{a} = mn \left\| \matrix{
	{\hat{\xi}_1 \over m} & {\hat{\xi}_2 \over n} & \hat{k} \\		{\hat{\xi}_1 \over m} & {\hat{\xi}_2 \over n} & \hat{k} \cr
	~~\vspace{1 mm}~~
	{\partial \over \partial \xi} &		{\partial \over \partial \xi} &
	{\partial \over \partial \eta} &		{\partial \over \partial \eta} &
	{\partial \over \partial z} \\		{\partial \over \partial z} \cr
	{a \over m} & {b \over n} & c		{a \over m} & {b \over n} & c
	~~\end{array~~} \right\|~~</math>~~		} \right\|
			$$

	~~:<math>~~\nabla^2 \phi = \nabla \cdot \nabla \phi = mn \left[		$$\nabla^2 \phi = \nabla \cdot \nabla \phi = mn \left[
	{\partial \over \partial \xi} \!\! \left( {m \over n}		{\partial \over \partial \xi} \!\! \left( {m \over n}
	{\partial \phi \over \partial \xi} \right) +		{\partial \phi \over \partial \xi} \right) +
	{\partial \over \partial \eta} \!\! \left( {n \over m}		{\partial \over \partial \eta} \!\! \left( {n \over m}
	{\partial \phi \over \partial \eta} \right) \right]~~</math>~~		{\partial \phi \over \partial \eta} \right) \right]$$

	where ~~<math>~~\phi~~\!\,</math>~~ is a scalar and ~~<math>~~\vec{a}~~\!\,</math>~~ is a vector with components		where $\phi$ is a scalar and $\vec{a}$ is a vector with components
	~~<math>~~a~~\!\~~,~~</math>, <math>~~b~~\!\,</math>~~, and ~~<math>~~c~~\!\,~~</~~math~~>.		$a$, $b$, and $c$.
			</wikitex>

Robertson at 18:28, 8 December 2006

2006-12-08T18:28:53Z

← Older revision		Revision as of 18:28, 8 December 2006
Line 1:		Line 1:
			<div class="title">Curvilinear Coordinates</div>

	The requirement for a boundary-following coordinate system and for a		The requirement for a boundary-following coordinate system and for a
	laterally variable grid resolution can both be met (for suitably		laterally variable grid resolution can both be met (for suitably

Kate: First whack and I don't know how to fix the error on the 3x3 matrix.

2006-11-08T18:18:29Z

First whack and I don't know how to fix the error on the 3x3 matrix.

New page

The requirement for a boundary-following coordinate system and for a
laterally variable grid resolution can both be met (for suitably
smooth domains) by introducing an appropriate orthogonal coordinate
transformation in the horizontal. Let the new coordinates be
<math>\xi(x,y)\!\,</math> and <math>\eta(x,y)\!\,</math> where the relationship of horizontal arc
length to the differential distance is given by:

:<math>(ds)_{\xi} = \left( {1 \over m} \right) d \xi</math>

:<math>(ds)_{\eta} = \left( {1 \over n} \right) d \eta</math>

Here, <math>m(\xi,\eta)\!\,</math> and <math>n(\xi,\eta)\!\,</math> are the scale factors which
relate the differential distances <math>(\Delta \xi,\Delta \eta)\!\,</math> to the
actual (physical) arc lengths.

It is helpful to write the equations in vector notation and to use
the formulas for div, grad, and curl in curvilinear coordinates (see
Batchelor, Appendix 2):

:<math>\nabla \phi = \hat{\xi} m {\partial \phi \over \partial \xi} +
\hat{\eta} n {\partial \phi \over \partial \eta}</math>

:<math>\nabla \cdot \vec{a} = mn \left[
{\partial \over \partial \xi} \!\! \left( {a \over n} \right) +
{\partial \over \partial \eta} \!\! \left( {b \over m} \right)
\right]</math>

:<math>\nabla \times \vec{a} = mn \left| \begin{array}{ccc}
\vspace{1 mm}
{\hat{\xi}_1 \over m} & {\hat{\xi}_2 \over n} & \hat{k} \\
\vspace{1 mm}
{\partial \over \partial \xi} &
{\partial \over \partial \eta} &
{\partial \over \partial z} \\
{a \over m} & {b \over n} & c
\end{array} \right|</math>

:<math>\nabla^2 \phi = \nabla \cdot \nabla \phi = mn \left[
{\partial \over \partial \xi} \!\! \left( {m \over n}
{\partial \phi \over \partial \xi} \right) +
{\partial \over \partial \eta} \!\! \left( {n \over m}
{\partial \phi \over \partial \eta} \right) \right]</math>

where <math>\phi\!\,</math> is a scalar and <math>\vec{a}\!\,</math> is a vector with components
<math>a\!\,</math>, <math>b\!\,</math>, and <math>c\!\,</math>.