Direct investigation of the K-transport equation for a complex turbulent flow

Abstract

Many turbulence models, including one- and two-equation eddy viscosity models, involve a turbulent kinetic energy (K-) transport equation. Together with the constitutive equation and (for two-equation models) the transport equation for another turbulence quantity such as the dissipation rate ε, this equation is the basic building block of these turbulence models. Yet, compared to other investigations, the K-transport equation has received little attention. It is often (hastily) considered as the most reliable part of turbulence models, since it rests mainly on a single hypothesis, the gradient diffusion hypothesis for the fluctuating kinetic energy. In this paper this equation is directly tested using large-eddy simulation data for the flow past a square cylinder. All the components of the pressure flux and kinetic energy flux have been estimated, together with the gradient of K. The results obtained are the following: (i) the pressure flux is not negligible compared to the kinetic energy flux; (ii) the gradient diffusion hypothesis is not confirmed, especially in complex flow regions, where there is no alignment between the total flux and the gradient of K; (iii) the turbulent Prandtl number σ _k , which is constant and is 1 for the K–ε model and 2 for the K–ω model, has been directly estimated from the data; the results show its huge variability.

This shows that the transport equation for K should be reconsidered. It has already been suggested that the output of eddy viscosity turbulence models is not very sensitive to this transport equation. In order to check this we have simulated the above-mentioned flow with a classical K–ε model, with several different constant σ _k -values between 0.5 and 4. The streamline patterns (including the recirculation length behind the cylinder) vary greatly from one simulation to the other. Also differences are observed in the patterns of turbulent kinetic energy. This illustrates that an adequate modelling of the K transport equation is still an important step in the design of a satisfactory turbulence model.