Transport equation based wall distance calculation - Revision history

Mikemalin at 12:31, 2 August 2006

2006-08-02T12:31:48Z

← Older revision		Revision as of 12:31, 2 August 2006
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	:<math>\vec d = d{{\nabla \phi } \over {\left\| {\nabla \phi } \right\|}} </math>		:<math>\vec d = d{{\nabla \phi } \over {\left\| {\nabla \phi } \right\|}} </math>

		+	The foregoing method for evaluation of wall distances was proposed by D.B.Spalding [1994], mainly for the purpose of calculating, for arbitrary complex geometries, the wall distances required by various low-Reynolds-number turbulence closure models. Spalding's proposal was mainly intuitive, but the method has been described in a formal mathematical framework by Fares & Schroder (2002).

		+	D.B.Spalding, ‘Calculation of turbulent heat transfer in cluttered spaces’, Proc. 10th Int. Heat Transfer Conference, Brighton, UK, (1994).

		+	E.Fares & W.Schroder, ‘Differential Equation for Approximate Wall Distance' Int.J.Numer.Meth., 39:743-762, (2002).
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	<i> Return to [[Numerical methods \| Numerical Methods]] </i>		<i> Return to [[Numerical methods \| Numerical Methods]] </i>

Zxaar at 06:28, 3 October 2005

2005-10-03T06:28:22Z

← Older revision		Revision as of 06:28, 3 October 2005
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	Where as the wall distance vector could be written as: <br>		Where as the wall distance vector could be written as: <br>
	:<math>\vec d = d{{\nabla \phi } \over {\left\| {\nabla \phi } \right\|}} </math>		:<math>\vec d = d{{\nabla \phi } \over {\left\| {\nabla \phi } \right\|}} </math>
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		+	----
		+	<i> Return to [[Numerical methods \| Numerical Methods]] </i>

Zxaar at 00:33, 27 September 2005

2005-09-27T00:33:47Z

← Older revision		Revision as of 00:33, 27 September 2005
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	Where as the wall distance vector could be written as: <br>		Where as the wall distance vector could be written as: <br>
-	<math>\vec d = d{{\nabla \phi } \over {\left\| {\nabla \phi } \right\|}} </math>	+	:<math>\vec d = d{{\nabla \phi } \over {\left\| {\nabla \phi } \right\|}} </math>

Zxaar at 07:44, 15 September 2005

2005-09-15T07:44:22Z

New page

===Wall-distance variable ===

Wall distance are required for the implementation of various turbulence models.
The approaximate values of wall distance could be obtained by solving a transport equation for a variable called '''wall-distance variable''' or <math>\phi</math>. 
The transport equation for the wall distance variable could be written as: 
:<math> \int\limits_A {\nabla \phi \bullet d\vec A} = - \int\limits_\Omega {dV} </math> 
with the boundary conditions of Dirichlet at the walls as <math> \phi = 0 </math>
and Neumann at other boundaries as <math> {{\partial \phi } \over {\partial n}} = 0 </math> 

 

This transport equation could be solved with any of the approaches similar to that of Poisson's equation. 

=== Wall distance calculation ===
Wall distance from the solution of this transport equation could be easily obtained as: 
:<math> d = \sqrt {\nabla \phi \bullet \nabla \phi + 2\phi } - \left| {\nabla \phi } \right| </math> 

Where as the wall distance vector could be written as: 
<math>\vec d = d{{\nabla \phi } \over {\left| {\nabla \phi } \right|}} </math>