Analysis of destruction term in transport equation for turbulent energy dissipation rate

Abstract

The K–ε model used in turbulence simulations involves the transport equations for the turbulent kinetic energy and its dissipation rate. In contrast to the turbulent energy equation derived from the Navier–Stokes equation, the transport equation for the dissipation rate has been modeled empirically and has less of a theoretical grounding. An analysis of the dependence of terms in the exact transport equation on the Reynolds number Re has suggested that the two dominant O(Re 1 / 2 ) terms cancel out at the leading order and their O(1) difference yields the terms in the model equation. The two-scale direct interaction approximation (TSDIA) can be used to derive the model equation for inhomogeneous turbulence. In this study, as a first step toward deriving all the terms in the model equation, the two dominant terms in the exact transport equation were investigated theoretically and numerically for the case of homogeneous isotropic turbulence, in order to derive the destruction term. The two terms were first analyzed by using the TSDIA under the assumption of a simple energy spectrum form. However, the finite-width effect of the inertial range of the energy spectrum did not give the expected O(Re - 1 / 2 ) corrections to the leading-order terms. Then, the model equations of the Markovianized Lagrangian renormalized approximation (MLRA) were numerically solved to obtain the accurate energy spectrum profile of a decaying homogeneous isotropic turbulence. Thereby, it was shown that the deviation of the energy spectrum from the - 5 / 3 law was responsible for the O(Re - 1 / 2 ) corrections. By assuming the energy spectrum suggested by the MLRA results, the two dominant terms were analyzed again by using the TSDIA to successfully derive the destruction term in the model equation.