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 C_{μ} is a function of local flow
parameters, rather than a constant, as in the standard k-ε model.

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

∂/∂t (ρ*k) + **∇.**(ρ***u***k)=
**∇.**(ρ*{ν_{l}+ν_{t}/σ_{t,k}}***∇** k )
+ ρ*(P_{k} - ε)

∂/∂t (ρ*ε) + **∇.**(ρ***u***ε)=
**∇.**(ρ*{ν_{l}
+ν_{t}/σ_{t,ε}}***∇** ε )
+ ρ*(C_{1ε}*S*ε -
C_{2ε}*ε^{2}/{k+√(ν_{l}*ε)})

ν_{t} = C_{μ}*k^{2}/ε

P_{k} = ν_{t}*S^{2}

S=√(2*S_{ij}*S_{ij})

S_{ij}= 0.5*(∂u_{i}/∂x_{j}
+ ∂u_{j}/∂x_{i} )

(1)

(2)

(3)

(4)

(5)

(6)

The model coefficients C_{1ε} and C_{μ} are
computed from the following equations:

C_{1ε}=max[0.43, η/(η+5)]

η=S*k/ε

C_{μ}=1./[A_{0} + A_{s}*k*Û/ε]

Û=√(S_{ij}*S_{ij}+Ω_{ij}*Ω_{ij})

Ω_{ij}= 0.5*(∂u_{i}/∂x_{j}
- ∂u_{j}/∂x_{i} )

A_{s}=√6*cosφ

φ=cos^{-1}[ max(-1, min[ √6*W,1] ) ]

W=S_{ij}*S_{jk}*S_{ki}/Š^{3}

Š=√(S_{ij}*S_{ij})

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

The remaining empirical constants are defined by:

σ_{t,k}=1.0, σ_{t,ε}=1.2,
A_{0}=4.04, C_{2ε}=1.9.

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

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 C_{μ} and
C_{1ε} 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 C_{μ} is stored whole field by
default, and similar storage can be invoked for C_{1ε} by
inserting STORE(C1E) in the Q1 input file.

- 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).