https://www.cfd-online.com/W/index.php?title=Special:Contributions/Mikemalin&feed=atom&limit=50&target=Mikemalin&year=&month=CFD-Wiki - User contributions [en]2017-07-21T04:20:58ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Transport_equation_based_wall_distance_calculationTransport equation based wall distance calculation2006-08-02T12:31:48Z<p>Mikemalin: </p>
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<div>===Wall-distance variable ===<br />
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Wall distance are required for the implementation of various turbulence models. <br />
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>. <br><br />
The transport equation for the wall distance variable could be written as: <br><br />
:<math> \int\limits_A {\nabla \phi \bullet d\vec A} = - \int\limits_\Omega {dV} </math> <br><br />
with the boundary conditions of Dirichlet at the walls as <math> \phi = 0 </math> <br />
and Neumann at other boundaries as <math> {{\partial \phi } \over {\partial n}} = 0 </math> <br><br />
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This transport equation could be solved with any of the approaches similar to that of Poisson's equation. <br><br />
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=== Wall distance calculation ===<br />
Wall distance from the solution of this transport equation could be easily obtained as: <br><br />
:<math> d = \sqrt {\nabla \phi \bullet \nabla \phi + 2\phi } - \left| {\nabla \phi } \right| </math> <br><br />
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Where as the wall distance vector could be written as: <br><br />
:<math>\vec d = d{{\nabla \phi } \over {\left| {\nabla \phi } \right|}} </math><br />
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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). <br />
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D.B.Spalding, ‘Calculation of turbulent heat transfer in cluttered spaces’, Proc. 10th Int. Heat Transfer Conference, Brighton, UK, (1994).<br />
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E.Fares & W.Schroder, ‘Differential Equation for Approximate Wall Distance' Int.J.Numer.Meth., 39:743-762, (2002).<br />
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