Transport equation based wall distance calculation

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 Revision as of 00:33, 27 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Latest revision as of 12:31, 2 August 2006 (view source) (One intermediate revision not shown) Line 18: Line 18: Where as the wall distance vector could be written as:
Where as the wall distance vector could be written as:
:$\vec d = d{{\nabla \phi } \over {\left| {\nabla \phi } \right|}}$ :$\vec d = d{{\nabla \phi } \over {\left| {\nabla \phi } \right|}}$ + + 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). + ---- + Return to [[Numerical methods | Numerical Methods]]

Latest revision as of 12:31, 2 August 2006

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 $\phi$.
The transport equation for the wall distance variable could be written as:

$\int\limits_A {\nabla \phi \bullet d\vec A} = - \int\limits_\Omega {dV}$

with the boundary conditions of Dirichlet at the walls as $\phi = 0$ and Neumann at other boundaries as ${{\partial \phi } \over {\partial n}} = 0$

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:

$d = \sqrt {\nabla \phi \bullet \nabla \phi + 2\phi } - \left| {\nabla \phi } \right|$

Where as the wall distance vector could be written as:

$\vec d = d{{\nabla \phi } \over {\left| {\nabla \phi } \right|}}$

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).