7.1 The k-epsilon turbulence model

The k model (k-epsilon) was the first turbulence model to be widely adopted for a variety of flows in CFD.2 It is one of a family of two equation models which solve two transport equations, one usually for eqn and one for another variable, often a dissipation rate. These models are the industry standard for CFD.

The k model combines the transport equations for eqn and , Eq. (6.28 ) and Eq. (6.32 ) respectively, which are reproduced below assuming eqn = constant and replacing material derivatives in conservative form as in Eq. (2.26 ).

@k 2 @t-+ r (uk) r (Dkrk) = G 3k(r u)

The standard model coefficients are

Dk = + t= k; D
The choice of coefficients in Eq. (7.3 ) originates from a large series of validation simulations of free shear flows.3

The model can be deployed in a steady flow solution, setting the local time derivatives to zero, e.g. eqn. It can also form part of a transient solution to capture unsteady features such as vortex shedding.

The model uses a momentum equation in ensemble-averaged form, e.g. Eq. (6.26 ), using eqn calculated from updated eqn and eqn according to Eq. (6.27 ).

The model equations are added to the end of the main loop in the steady and transient algorithms from Sec. 5.12 or Sec. 5.21 , respectively. The figure below shows the additions to the transient algorithm, starting from the momentum corrector step.

PICT\relax \special {t4ht=

Following the momentum and flux correctors, eqn can be updated, e.g. for non-Newtonian models where eqn. eqn and eqn are calculated and stored for source terms in both the eqn and equations. Those equations are then solved, followed by an update to eqn, and subsequently eqn.

2William Jones and Brian Launder, The prediction of laminarization with a two-equation model of turbulence, 1972.
3Brian Launder, Alan Morse, Wolfgang Rodi and Brian Spalding, Prediction of free shear flows — A comparison of the performance of six turbulence models, NASA Conference on Free Shear Flows, Langley, 1972.

Notes on CFD: General Principles - 7.1 The k-epsilon turbulence model