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3.4.6 The K-Omega turbulence model (also referred to as KE-OMEGA)

Contents

  1. Introduction
  2. Description of the model
  3. Boundary conditions
  4. Activation of the model
  5. Sources of further information

1. Introduction

The first two-equation turbulence model was the KE-OMEGA model of Kolmogorov [1942]. This model, which is also known as the KE-F and k-w model, involves the solution of transport equations for the turbulent kinetic energy KE and the turbulence frequency F. It should be mentioned that other workers define F as the ratio EP to KE, where EP is the dissipation rate of KE.

Several different and improved versions of Kolmogrov's model have been proposed, including those of: Saiy [1974], Spalding [1979], Wilcox [1988], Speziale et al [1990] and Menter [1992].

The KE-F model of Wilcox [1988] has been selected for use in PHOENICS, mainly because it is the most extensively tested and it includes a low-Reynolds-number extension for near-wall turbulence.

Although the KE-F model is not as popular as the KE-EP model, it does have several advantages, namely that:

  1. the model is reported to perform better in transitional flows and in flows with adverse pressure gradients;
  2. the model is numerically very stable, especially the low-Re version, as it tends to produce converged solutions more rapidly than the KE-EP models; and
  3. the low-Re version is more economical and elegant than the low-Re KE-EP models, in that it does not require the calculation of wall distances, additional source terms and/or damping functions based on the friction velocity.

The main weakness of the KE-F model is that unlike the KE-EP model, it is sensitive to the free-stream boundary condition for F in free-shear flows. Modified variants exist which claim to remove this sensitivity, but Wilcox [1988] notes that it is a desirable feature for transitional applications.

2. Description of the model

The KE-F model may be summarised as follows, with t denoting differentiation with respect to time and i with respect to distance:

(RHO*KE),t + (RHO*Ui*KE - RHO*{ENUL+ENUT/PRT(KE)}*KE,i),i = RHO*(Pk - EP) (2.1)

(RHO*F),t + (RHO*Ui*F - RHO*{ENUL+ENUT/PRT(F)}*F,i ),i = RHO*F*(F1*C1F*Pk/KE - F2*C2F*F) (2.2)

ENUT = FMU*CMUCD*KE/F (2.3)

Pk = ENUT * (Ui,j + Uj,i) Ui,j (2.4)

EP = CD*F*KE (2.5)

wherein: RHO is the density; ENUL and ENUT are the laminar and turbulent kinematic viscosities; FMU, F1 and F2 are low-Re damping functions; and Pk is the volumetric production rate of KE.

The empirical constants

The following constants are normally used:

PRT(KE)=2.0, PRT(F)=2.0, CMUCD=1.0, CD=0.09, C1F=5/9, C2F=3/40

The Damping functions


The damping functions, which are set to unity in the high-Re model, are defined by:

FMU = (1/40 + RT/RK)/(1+RT/RK) (2.6)

F1 = {1/FMU}*(0.1+RT/RW)/(1+RT/RW) (2.7)

F2 = (5/18 + (RT/RB)**4)/(1.+(RT/RB)**4) (2.8)

where RB=8, RK=6.0, RW=2.7 and RT is the turbulent Reynolds number:

RT = K/(F*ENUL) (2.9)

In regions where RT is high, FMU, F1 and F2 tend to unity.

3. Boundary Conditions

Wall boundary conditions


At present, the high-Re KE-F model is restricted to using equilibrium (GRND2) wall functions, so that the following boundary conditions are applied for the turbulence variables:

KE=UTAU**2/SQRT(CD) ; F=UTAU/(SQRT(CD)*k*Y) (3.1)

where UTAU is the resultant friction velocity ( = SQRT(TAUW/RHO) ), TAUW is the wall shear stress, Y is the normal distance of the first grid point from the wall, and k is von Karman's constant.

If the low-Re version is selected, then KE=0 at the wall and the following condition is applied for F at the near-wall grid point:

F=2.*ENUL/(C2F*Y**2) (3.2)

The alternative condition of F=2.*ENUL/(CD*F2*Y**2), also proposed by Wilcox [1988], produces nearly identical results, and so it has not been coded in PHOENICS.

Inlet conditions


At mass-inflow boundaries, the inlet values of KE and F are usually unknown, and one needs to take guidance from experimental data for similar flows. The simplest practice is to assume uniform values of KE and F computed from:

KE = (I *U)**2; F = EP/(CD*KE) ; EP=0.1643*KE**1.5/LM (3.3)

where U is the bulk inlet velocity, I is the turbulent intensity (typically in the range 0.01.lt.I.lt 0.05) and the mixing length LM ~ 0.1H, where H is a characteristic inlet dimension, say the hydraulic radius of the inlet pipe.

Free-stream conditions


At free (entrainment) boundaries, where a fixed-pressure condition is employed, it is necessary to prescribe free stream values for KE and F. If the ambient stream is assumed to be free of turbulence, then KE and EP can be set to negligibly small values and F can then be calculated from equation (2.5).

It should be mentioned that when using F=0 in the free stream, the KE-F model consistently predicts spreading rates of free-shear layers that exceed measured values by more than 20%. As was noted earlier and discussed by Wilcox [1993], these solutions are in fact quite sensitive to the free-stream value of F.

Speziale et al [1990] and Menter [1992] have proposed the inclusion of cross-diffusion source terms in the F equation which remove the sensitivity to free-stream conditions, but these have not been coded in PHOENICS.

4. Activation of the model

The high-Re form of KE-F model is activated by inserting the PIL command TURMOD(KOMODL) in the Q1 file, which is equivalent to the following PIL commands: 

  
  SOLVE(KE,OMEG);ENUT=GRND7;EL1=GRND5;KELIN=0;EL1A=0.6
  PATCH(KESOURCE,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
  COVAL(KESOURCE,KE,GRND4,GRND4)
  COVAL(KESOURCE,OMEG,GRND4,GRND4);GENK=T;IENUTA=10
  PRT(KE)=2.0;PRT(EP)=2.0
  TERMS(KE,N,Y,Y,Y,Y,N);TERMS(OMEG,N,Y,Y,Y,Y,N)
The sources for KE and OMEG are calculated and inserted in the subroutine GXKESO called from GROUP 13 of GREX. The generation rate used in the source terms can be stored by the command STORE(GENK).

The low-Re form is activated by the setting TURMOD(KOMODL-LOWRE), which is equivalent to TURMOD(KOMODL) but with IENUTA=11.

The WALL and CONPOR commands automatically create the required COVALs for wall boundaries, i.e.

COVAL(WALLN,KE,GRND2,GRND2); COVAL(WALLN,OMEG,GRND2,GRND2)

for the high-Re version, and

COVAL(WALLN,KE,1.0,0.0); COVAL(WALLN,OMEG,GRND2,GRND2)

for the low-Re version.

Further information on wall functions can be found in Section 8 below.

Information and advice on the use of low-Re models in general can be found in Section 3.4.4 on the Lam-Bremhorst k-e model.

5. Sources of further information

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

A.N.Kolmogorov, 'Equations of turbulent motion of an incompressible fluid', Izv Akad Nauk SSR Ser Phys, 6, Vol 1/2, 56, (1942).

F.R.Menter, 'Improved two-equation k-w turbulence model for aerodynamic flows', NASA TM-103975, (1992).

M.Saiy, 'Turbulent mixing of gas streams', PhD Thesis, Imperial College, University of London, (1974).

D.B.Spalding, 'Mathematical models of turbulent transport processes' HTS/79/2, Imperial College, Mech.Eng.Dept., (1979).

C.G.Speziale, R.Abid and E.C.Anderson, 'A critical evaluation of two-equation turbulence models for near-wall turbulence', AIAA Paper 90-1481, (1990).

D.C.Wilcox, 'Reassessment of the scale determiing equation for advanced turbulence models', AIAA J., Vol.26, No.11, p1299, (1988).

D.C.Wilcox, 'Turbulence modelling for CFD' DCW Industries, La Canada, California, USA, (1993).

See also the Instruction Course lectures on Turbulence Modelling.

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