In this section we shall discuss two variants k - ε turbulence models. The RNG based k- ε turbulence model follows the two equation turbulence modeling framework and has been derived from the original governing equations for fluid flow using mathematical techniques called Renormalization Group (RNG) method due to Yakhot and Orszag (1986). The RNG method is applicable to scale invariant phenomena lacking externally imposed length and time scales. For turbulence, this signifies that the method can describe the small scales which should be statistically independent of the external initial conditions and dynamical forces that create them through different instability phenomena. The RNG method gives a theory of the Kolmogorov equilibrium range of turbulence, comprising the so-called inertial range of small-scale eddies whose energy spectrum follows the famous Kolmogorov law E(k)~k^-5/3 . Figure 31.1 shows the scales of effective excitation in turbulence. They range from the low wave number k₀ =2π /L (large scale eddies) to high wavenumber viscous cutoff (corresponding to smallest energy containing eddies). The RNG method removes a narrow band of modes near by representing these modes in terms of lower modes in the interval k₀ < k < ε ^-l \ ( l <<1). When this narrow band of modes is removed, the resulting equations of motion for the remaining modes is a modified system of Navier-Stokes equations. The equations are dictated by a modified viscosity. The first band of modes is removed from the dynamics and the process of removal of degrees of freedom is repeated (also see Choudhury, 1993). In this way the RNG method enables computation of Navier-Stokes equations on relatively coarser grids at high Reynolds numbers.