Higher-order dissipation in the theory of homogeneous isotropic turbulence

Norbert Peters,Michael Gauding,Reginald J Hill,Jonas Boschung,Jens Henrik Goebbert,Heinz Pitsch

doi:10.1017/jfm.2016.489

Abstract

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

Highlights

Fully developed Navier–Stokes turbulence at high Reynolds numbers is characterized by a large range of length scales, varying from the geometrical lengths of the flow over the so-called integral length scale L, at which large velocity fluctuations occur on average, down to the Kolmogorov or dissipation scale η, at which kinetic energy is dissipated
He introduced the dissipation scale η, which essentially subdivides the range of small-scale turbulence into two subranges, a dissipative subrange for small r, where both the viscosity ν and the dissipation ε determine the solution of the second-order structure function equation, and an inertial range for large r, where only ε remains as a scaling parameter
In addition to ν and ε, all higher-order moments of the dissipation distribution function appear as dissipation parameters in the extended system of two-point equations for small-scale turbulence

Summary

Introduction

Developed Navier–Stokes turbulence at high Reynolds numbers is characterized by a large range of length scales, varying from the geometrical lengths of the flow over the so-called integral length scale L, at which large velocity fluctuations occur on average, down to the Kolmogorov or dissipation scale η, at which kinetic energy is dissipated. In Kolmogorov (1941b) he described velocity fluctuations separated by a distance r as two-point velocity differences or velocity increments, the statistical moments of which are known as structure functions He introduced the dissipation scale η, which essentially subdivides the range of small-scale turbulence into two subranges, a dissipative subrange for small r, where both the viscosity ν and the dissipation ε determine the solution of the second-order structure function equation, and an inertial range for large r, where only ε remains as a scaling parameter. Kurien & Sreenivasan (2001) discussed the Yakhot (2001) paper and the models presented therein in detail They used high-Reynolds-number experimental data from the atmospheric boundary layer to compute the pressure terms from Yakhot’s model and balance the terms of the transverse and mixed fourth-order structure function equations in the inertial range.

The pressure source term is given by

Conclusion

Reλ ν

Pi The