Modelling turbulent dissipation correlations for Rayleigh-Benard convection based on two-point correlation technique and invariant theory

Abstract

The two-point correlation technique and the invariant theory are used to study the turbulence closure for dissipation correlations in turbulent natural convection. Analytically, an equation for the dissipation rate of turbulence kinetic energy is derived. The structure of this equation is analysed by using direct numerical simulation data (DNS) of turbulent Rayleigh-B enard convection in air and sodium and of internally heated natural convection. The local homogeneity assumption is found to hold for the two-point velocity correlations and is used to simplify the analytical equation. For the two-point temperature-velocity correlation, on the other hand, it is shown that this assumption is not valid for Rayleigh-B enard convection. Using the DNS data, the anisotropy of the Reynolds stresses in natural convection is quanti ed in the anisotropy invariant map. A new model is proposed for the sink term in the dynamic equation for the dissipation rate which is based on the invariants of the anisotropy tensor of the Reynolds stresses. A model is also proposed for the buoyancy production term, which is of special importance in pure natural convection ows. For the dissipation correlation in the equation for the turbulent heat ux a model is proposed which is consistent with that for the buoyancy production term in the dissipation equation. Both models developed do properly account for wall e ects without any explicit wall corrections, but involving only the uid Prandtl number and the ratio of turbulence thermal to mechanical time scale.