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3.4.8 The Realisable k-ε model

Contents

  1. Introduction
  2. Description of the model
  3. Activation of the model
  4. References

1. Introduction

During the last decade the realisable k-ε model [1] has become increasingly popular in the CFD community due to its improved performance over the standard k-ε model when applied to flows involving boundary layers in strong adverse pressure gradients, streamwise curvature, separation and recirculation zones. The model is also reported to improve significantly the predicted spreading rates of round jets. The model is a two-equation high-Reynolds-number turbulence model that differs from the standard k-ε model in two respects. First, the model employs a different formulation of the transport equation for the dissipation rate that is derived from the transport equation for the mean-square vorticity fluctuations. Secondly, the model uses a different eddy-viscosity formulation which is based on several realisability constraints for the turbulent Reynolds stresses. In practice this means that the eddy-viscosity coefficient Cμ is a function of local flow parameters, rather than a constant, as in the standard k-ε model.

2. Description of the model

The realisable k-ε model is defined by the following equations:

∂/∂t (ρ*k) + ∇.(ρ*u*k)= ∇.(ρ*{νl+νt/σt,k}*∇ k ) + ρ*(Pk - ε)  

∂/∂t (ρ*ε) + ∇.(ρ*u*ε)= ∇.(ρ*{νl +νt/σt,ε}*∇ ε ) + ρ*(C1ε*S*ε - C2ε*ε2/{k+√(νl*ε)})

νt = Cμ*k2/ε

Pk = νt*S2

S=√(2*Sij*Sij)

Sij= 0.5*(∂ui/∂xj  + ∂uj/∂xi )

(1)

(2)

(3)

(4)

(5)

(6)

 

The model coefficients C1ε and Cμ are computed from the following equations:

C1ε=max[0.43, η/(η+5)]

η=S*k/ε

Cμ=1./[A0 + As*k*Û/ε]

Û=√(Sij*Sij+Ωij*Ωij)

Ωij= 0.5*(∂ui/∂xj  - ∂uj/∂xi )

As=√6*cosφ

φ=cos-1[ max(-1, min[ √6*W,1] ) ]

W=Sij*Sjk*Ski/Š3

Š=√(Sij*Sij)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

 

The remaining empirical constants are defined by:

σt,k=1.0, σt,ε=1.2, A0=4.04, C2ε=1.9.

The model presented above is applicable only in regions where the turbulence Reynolds number is high.

3. Activation of the model

The Realisable k-ε model can be activated from the VR Menu, or alternatively is by inserting the PIL command TURMOD(KEREAL) in the Q1 file, which is equivalent to the following PIL commands: 

  TURMOD(KEMODL);SOLVE(KE,EP);STORE(CMU)
  STORE(DUDX,DUDY,DUDZ,DVDX,DVDY,DVDZ,DWDX,DWDY,DWDZ)
  ENUT=GRND5;IENUTA=14
  PRT(EP)=1.2;C2E=1.9;SPEDAT(SET,KECONST,C2E,R,1.9)
  FIINIT(CMU)=0.09;KELIN=3
  COVAL(KESOURCE,EP,0.0,0.0)      
  PATCH(KESOURCE,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
  COVAL(KESOURCE,KE,GRND4,GRND4)
  PATCH(REKESO,PHASEM,1,NX,1,NY,1,NZ,1,LSTEP)
  COVAL(REKESO,EP,GRND4,GRND4)

The turbulence-model coefficients Cμ and C1ε are variable, and provision has been made to store these variables whole field, not only for output purposes, but also, if necessary, to aid convergence by limiting and linearly relaxing their values during the course of the CFD simulation. The coefficient Cμ is stored whole field by default, and similar storage can be invoked for C1ε by inserting STORE(C1E) in the Q1 input file.

4. References

  1. T.H. Shih, W.W.Liou, A.Shabbir, Z.Yang,Z. & J.Zhu, "A New k-ε Eddy-Viscosity Model for High Reynolds Number Turbulent Flows - Model Development and Validation. Computers Fluids, 24(3):227-238, (1995).