A buoyancy extension for eddy-viscosity models for the natural convection regime

International audienceThe influence of buoyancy on turbulent flows is significant in many industrial applications, in particular in the automotive industry. For instance, CFD is routinely used for dimensioning the underhood compartment, using commercial packages based on eddy-viscosity turbulence models. Although these models correctly reproduce the flow and heat transfer at cruising speed, they are not reliable in phases where natural convection dominates, i.e., when the car is stopped, mainly because buoyancy-turbulence interactions are not accounted for in a comprehensive manner. The objective of the present work is to introduce the mechanisms involved in this interaction in eddy-viscosity models in order to avoid the recourse to costly wind-tunnel experiments.The study is focused on simple-geometry flows for which the influence of the different physical processes can be isolated and the corresponding terms in the equations are available in DNS databases. Moreover, in order to ensure that the modifications introduced to account for buoyancy effects do not deteriorate predictions in the absence of buoyancy, the study encompasses forced, mixed and natural convection regimes, covering a range of Rayleigh numbers representative of underhood flow configurations: channel flows in forced, mixed and natural convection regimes, up to Ra = 1.7 × 107[2] and differentially heated cavities up to Ra = 1011 [3].Buoyancy effects are commonly introduced by adding source terms to the transport equations for the turbulent energy and the (specific) dissipation rate, in order to reproduce the influence of stratification on the dynamics of turbulence. However, this approach is ineffective in weakly stratified buoyant flows, where the temperature gradient is mainly normal to the gravity vector, as a consequence of the use of an isotropic thermal diffusivity (SGDH) to model the turbulent heat fluxes. However, the present work shows that introducing an anisotropic diffusivity (GGDH) only marginally improves the predictions. Indeed, the use of the Boussinesq constitutive relation to model the Reynolds stresses in eddy-viscosity models does not correctly represent the subtle coupling between the turbulent dynamics and the turbulent heat fluxes: the influence of buoyancy on the anisotropy of turbulence must be accounted for, in order to correctly represent the anisotropic diffusivity, which is crucial to reproduce the turbulent heat fluxes and the production mechanisms in the turbulent energy and dissipation equations.The main contribution of this work is thus the drastic improvement of the reproduction of the influence of buoyancy on turbulence in the case of weakly stratified flows in the natural convection regime, through the modification of both the turbulent heat flux model and the constitutive relation for the Reynolds stress. The former can be modeled using either an anisotropic diffusivity (GGDH) or the more sophisticate Algebraic Flux Model (AFM), and the latter is modified by introducing a buoyancy extension.The models described above have been implemented in the open-source CFD package Code_Saturne, in combination with different types of eddy-viscosity models (k-ε, k-ω, BL-v 2 /k). Detailed comparisons with DNS of the results obtained using the different hypotheses show that all the forced, mixed and natural convection regimes can be reproduced in a satisfactory manner by combining the GGDH and the buoyancy-sensitized constitutive relation (1). In particular, in natural convection, the mean velocity profiles in boundary layers close to the vertical walls are drastically improved.Detailed comparison of all the significant quantities will be presented, as well as the analysis of the influence of the different hypotheses on the reproduction of the coupling of the dynamics and heat transfer mechanisms. The results are thus very promising and this work paves the way to the improvement of CFD capabilities for the design of underhoodcompartment of automobiles, and, in general, for industrial configurations in which buoyancy has a significant influence