Yakhot and Orszag ( hereafter denoted YO ) [1986] derived a KE-EP model based on Renormalization Group (RNG) methods. In this approach, RNG techniques are used to develop a theory for the large scales in which the effects of the small scales are represented by modified transport coefficients.

For example, a modified eddy viscosity is the parameter appearing in the momentum equations through which the high-wave-number modes (i.e.small scales) affect the retained larger scales.

The RNG procedure employs a universal random force which drives the small-scale velocity fluctuations and represents the effect of the large scales (including initial and boundary conditions ) on the eddies in the inertial range ( see for example Hinze [1959] ).

This force is chosen in such a way that the global properties of the resulting flow field are the same as those in the flow driven by the mean strain.

The equations of motion for the large-scale field are derived by averaging over an infinitesimal band of small scales to remove them from explicit consideration. The removal process is iterated until the infinitesimal corrections to the equations accumulate to give finite changes.

The procedure retains only the modifications of the viscosity, and, after the elimination of the small scales is complete, the random force is dropped from the resulting equations and the initial and boundary conditions are restored.

The RNG procedure of YO gives rise to a set of equations having the attractive feature of no undetermined constants, and the presence of built-in corrections that allow use of the model in both high- and low-Reynolds-number regions of the flow.

At high turbulence Reynolds numbers the RNG KE-EP model of YO is of the same general form as the standard KE-EP model, except that the model constants are calculated explicitly from the RNG analysis and assume somewhat different values.

Quite recently Smith and Reynolds [1992] identified several problems with YO's original derivation of the EP-equation.

This led Yakhot and Smith [1992] to reformulate the derivation of this equation, which resulted in:

- a re-evaluation of the constant controlling the production of EP; and
- the discovery of an additional production term in the EP- equation which becomes significant in rapidly-distorted flows and flows removed from equilibrium.

Although RNG methods were unable to close the additional production term, Yakhot et al [1992] developed a model for the closure of this term.

The resulting high-Reynolds-number form of the RNG KE-EP model proved successful for the calculation of a number of separated flows, and it is this version of the model that has been provided in PHOENICS. However, the user is advised that the model results in substantial deterioration in the prediction of plane and round free jets in stagnant surroundings.

The RNG KE-EP model differs from the standard high-Reynolds-form of the KE-EP model in that:

(a) the following model constants take different values: PRT(KE) = 0.7194 ; PRT(EP) = 0.7194 ; C1E = 1.42 C2E = 1.68 ; CMUCD = 0.0845 ; and

(b) the dissipation-rate transport equation includes an additional source term per unit volume:

S,EP = - RHO1*ALF*EP**2/KE (2.1)

where

ALF = CMUCD*ETA**3*(1-ETA/ETA0)/(1+BETA*ETA**3), (2.2)

ETA0 = 4.38, and BETA = 0.012.

The dimensionless parameter ETA is defined by:

ETA = S*KE/EP (2.3) where S**2 = 2.*Sij*Sij (2.4) and Sij = 0.5 * (DUi/DXj + DUj/DXi) (2.5)

In PHOENICS terms, it may be noted that S is simply the square root of the generation function, LGEN1.

The additional source term (2.1) becomes significant for flows with large strain rates, i.e. when ETA >> ETA0. The parameter ETA is a measure of the ratio of the turbulent to mean time scale.

In the limit of weak strain where S and ETA tends to zero, the additional source term tends to zero and the original form of the k-e model is recovered.

In the limit of strong strain where S and ETA tend to infinity, the additional source term becomes:

S,EP = - RHO1*CMUCD*ETA*EP**2 --------------------- BETA*ETA0*KE

ETA0 is the fixed point for homogeneously-strained turbulent flows and BETA is a constant evaluated to yield a von Karman constant of about 0.41 ( see Yakhot et al [1992] ).

The low-Re corrections to the RNG model have not been provided, as a correct statement of these has yet to appear in the literature.

The RNG KE-EP model is activated by setting TURMOD(KERNG) in the Q1 file, which is equivalent to TURMOD(KEMODL) plus the following PIL statements:

IENUTA=1;PRT(KE)=0.7194;PRT(EP)=PRT(KE) PATCH(RNGMKE,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP) COVAL(RNGMKE,EP,GRND4,GRND4)When STORE(ALF,ETA) appears in the Q1 file, these parameters as defined by equations (2.2) and (2.3), may be printed in the RESULT file or viewed via PHOTON and AUTOPLOT.

The FORTRAN coding sequences for the RNG model may be found in subroutine GXRNGM which is called from Group 1, Section 1 of GREX3 and also from Group 13 of GREX3.

Finally, a number of Q1's may be found in the advanced-turbulence- model library which both validate and demonstrate the use of the model.

J.O.Hinze, 'Turbulence', McGraw Hill Book Company, Chapter 3, p181-190,(1959).

L.M.Smith and W.C.Reynolds, 'On the Yakhot-Orszag Renormalization group method for deriving turbulence statistics and models', Phys. Fluids A, Vol.4, No.2, p364, (1992).

V.Yakhot and S.A.Orszag, 'Renormalization group analysis of turbulence', J.Sci.Comput., Vol.1, p3, (1986).

V.Yakhot, S.A.Orszag, S.Thangam, T.B.Gatski and C.G.Speziale, 'Development of turbulence models for shear flows by a double expansion technique', Phys.Fluids A, Vol.4, No.7, (1992).

V.Yakhot and L.M.Smith, 'The Renormalization group, the eps- expansion and derivation of turbulence models', J.Sci.Comput., Vol.7, No.1, (1992).

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