The k-є method – A typical two-equation method

Probably the most widely used method for calculating turbulent flows is the k—є model which is incorporated into most commercial CFD software. It was independently developed at Los Alamos* and at Imperial College London.* ♦Baldwin, B. S. and Lomax, H. (1978) Thin layer approximation and algebraic model for separated turbulent flows. AIAA Paper 78-257.

* Harlow, F. H. and Nakayama, P. I. (1968) Transport of turbulence energy decay rate. Univ. of California, Los Alamos Science Lab. Rep. LA-3854.

* Jones, W. P. and Launder, B. E. (1972) The prediction of laminarization with a two-equation model of turbulence, Int. J. Heat Mass Transfer, 15, 301-314.

The basis of the k—s and most other two-equation models is an eddy-viscosity formula based on dimensional reasoning and taking the form:

sT = C^H Сц is an empirical const.

Note that the kinetic energy per unit mass, k = (ua + v’2 + vv’2)/2. Some previous two-equation models derived a transport equation for the length-scale £. This seemed rather unphysical so, based on dimensional reasoning, the k—s model took

Є = kV2/s

where є is the viscous dissipation rate per unit mass and should not be confused with the eddy viscosity, sT. A transport equation for s was then derived from the Navier – Stokes equations.

Both this equations for є and the turbulence kinetic energy к contain terms involving additional unknown dependent variables. These terms must be modelled semi-empirically. For flows at high Reynolds number the transport equations for к and є are modelled as follows:

Turbulence energy:

Energy dissipation:

De d (dst ds є /ди2 _ рє2

pW< = + – C2X

These equations contain 5 empirical constants that are usually assigned the following values:

where ak and aE are often termed effective turbulence Prandtl numbers. Further modification of Eqns (7.154 and 7.155) is required to deal with relatively low Reynolds numbers. See Wilcox (1993) for details of this and the choice of wall boundary conditions.

The k—e model is intended for computational calculations of general turbulent flows. It is questionable whether it performs any better than, or even as well as, the zero-equation models described in Section 7.11.5 for boundary layers. But it can be used for more complex flows, although the results should be viewed with caution. A common misconception amongst practising engineers who use commercial CFD packages containing the k—s model is that they are solving the exact Navier-Stokes equations. They are, in fact, solving a system of equations that contains several approximate semi-empirical formulae, including the eddy-viscosity model described above. Real turbulent flows are highly unsteady and three-dimensional. The best one can expect when using the k—s, or any other similar turbulence model, is an approximate result that gives guidance to some of the features of the real turbulent flow. At worst, the results can be completely misleading, for an example, see the discussion in Wilcox (1993) of the round-jet/plane-jet anomaly.

For a full description and discussion of two-equation turbulence models and other more advanced turbulence models see Wilcox (1993), Pope (2000) and other special­ized books.