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3.1.4 The VAN-Driest low-Reynolds-number mixing-length model

PHOENICS provides mixing-length expressions, for use in simple geometries with zero- and one-equation turbulence models.

These expressions are valid in the fully-turbulent parts of the relevant flows, but not in the so-called viscous sublayer and adjacent buffer layer close to the wall.

These expressions therefore need to be modified, so as to make them applicable over the entire flow region, by introduction of factors which depend (at least) on the local Reynolds Number.

One such modification was by Van Driest [1956], namely:

LM = LM0 * (1.0 - EXP(-Y+/A+))


Y+ is the relevant Reynolds Number.

The formula implies that LM / LM0 equals Y+/A+ in the immediate vicinity of the wall, where Y+ is small.

Moreover, since LM0 equals KAPPA * Y there, where KAPPA is the von Karman constant (approximately 0.4), the implication is that: LM equals (0.4/26) * Y**2 near the wall and 0.4 * Y far from it.

Other authors ( see for example Cebeci and Smith [1974] ) have proposed that the constant 26.0 should be replaced by a function of dimensionless quantities involving: - pressure gradients, - mass transfer, - the compressibility, and other factors.

Van Driest's modification has been provided in PHOENICS as an extension to the y-direction mixing-length prescriptions in subroutine GXLEN, which comprise EL1=GRND2, GRND7, GRND8 & GRND9 ( see the HELP file entry provided for EL1 ).

In view of the difficulty of prescribing the mixing length for arbitrary geometries, the Van-Driest extension has been restricted to those options which are the most commonly employed for two- dimensional wall-boundary flows, and pipe and duct flows.

These EL1 options relate the mixing length to the y-coordinate of the cell centres only, and BFC formulations are not provided. A description of these options is given under the "help-file" entry EL1.

The Van-Driest modification is activated by setting IENUTA=5 in the Q1 file.

The wall boundary conditions must be set using GRND2 wall-function PATCHes because the wall-friction velocity UTAU is retrieved in subroutine GXLEN from patchwise storage of the wall-skin-friction coefficient and relative velocity.

Since Van-Driest damping requires that the equations be integrated right down to the wall, care must be taken to ensure good numerical resolution in the near-wall region. For advice and information the user is referred to the discussion given under the Encyclopaedia entry for LOW-REynolds number turbulence models.

Several Q1 files may be found in the advanced-turbulence library which demonstrate the use of the model.

T.Cebeci and A.M.O.Smith, 'Analysis of turbulent boundary layers.', Chapter 6, Academic Press, (1974).

E.R. Van Driest, 'On turbulent flow near a wall', J.Aero.Sci., Vol.23, p1007, (1956).