and the dissipation rate equation

where

In the implementation of this model the Kolmogorov - Prandtl expression for the turbulent viscosity is used

together with equation (4.2.11) for the eddy diffusivity. In the above , , , and are all taken to be constants and are given respectively the values[96] 0.09, 1.0, 1.3, 1.44 and 1.92.

The methods used in the discretisation of the above equations are the
same as those defined in previous chapters with the term on the right of
each of (4.2.13) and (4.2.14) being treated as source terms.
The continuity equation remains unchanged whilst the only differences in
the momentum and enthalpy equations is the change in the diffusion
coefficient. The laminar viscosity
is replaced by
and the conductivity *k* in the enthalpy
equation is replaced by
).

Handling of the boundary conditions concerned with turbulent flows is the
same as for laminar flows except in the case of walls. The k - epsilon
model provides accurate solutions only for fully turbulent
flows. In the part of the flow near to walls there exist regions in which
the local Reynolds number of turbulence,
,
is so small
that the viscous effects become more significant than the turbulent ones.
In this viscous sublayer very steep gradients occur so for accurate
modelling many grid points would be required in this region. Fortunately
it is not necessary to discretise the *k* and
conservation
equations over this region as there exist suitable laws which relate
the wall conditions to values of the dependent variables just outside the
viscous sublayer. In the region just outside the
sublayer[97], such that

where

where is the wall shear stress. The wall skin friction factor,

in which

where , the Kaman constant, is equal to 0.435,

For a node positioned just outside the viscous sublayer the effective viscosity is set to the maximum of the laminar dynamic viscosity and

As previously stated in this region the production of turbulent kinetic energy is equal to its dissipation. This sets

The value of the gradient term is calculated by substituting (4.2.19) into equation (4.2.20) to remove

(4.2.24) |

Substitution of this into equation (4.2.23) and using (4.2.18) gives

The source terms in the conservation equations (4.2.13) and (4.2.14) include the quantity

but using the divergence theorem

By making the usual approximations to the right hand side of (4.2.27) and equating to (4.2.26) the following estimate for the derivative's value at P is obtained.

This formula only requires the estimation of a representative value of

The source terms, particularly those in the dissipation rate equation, are
non linear. Three methods of linearisation are available within the code
in line with those used by PHOENICS[60]. Assuming that the
linearised form
of the sources can be written in the form *S*_{C} -
then the
following gives the formulae for the coefficients for the various methods.

METHOD 1 | |

k source terms :- | |

source terms :- | |

METHOD 2 | |

k source terms :- | |

source terms :- | |

METHOD 3 | |

k source terms :- | |

source terms :- | |