A DNS-based thermal second-moment closure for buoyant convection at vertical walls

Abstract

Direct numerical simulations (DNS) of natural convection in a vertical channel by Versteegh & Nieuwstadt (1998) are used for assessing the budget of the turbulent heat flux θui and the temperature variance θ2, and for modelling the transport equations governing these two properties. The analysis is confined to a simple fully developed situation in which the gravitational vector, as the sole driving force, is perpendicular to the only non-zero component of the mean temperature gradient. Despite its simplicity, the flow displays many interesting features and represents a generic case of the interaction of buoyancy-driven turbulent temperature and velocity fields. The paper discusses the near-wall variation of the second moments and their budgets, as well as possible scaling of θui and θ 2 both in the near-wall region and away from the wall. Various proposals for the Reynolds-averaged modelling are analysed and new models are proposed for these two transport equations using the term-by- term approach. An a priori test (using the DNS data for properties other than θui and θ 2) reproduced very well all terms in the transport equations, as well as their near-wall behaviours and wall limits, without the use of any wall-topology-dependent parameters. The computational effort is still comparable to that for the ‘basic model’. The new term-by-term model of the θui and θ 2 equations was then used for a full simulation in conjunction with a low-Reynolds-number second-moment velocity closure, which was earlier found to reproduce satisfactorily a variety of isothermal wall flows. Despite excellent term-by-term reproduction of thermal turbulence, the predictions with the full model show less satisfactory agreement with the DNS data than a priori validation, indicating a further need for improvement of the modelling of buoyancy effects on mechanical turbulence.

ONE-POINT CLOSURE MODELS FOR BUOYANCY-DRIVEN TURBULENT FLOWS

“…The isotropic eddy diffusivity is usually related to the eddy viscosity ν t via the turbulent Prandtl 5 ; --, Ra = 2 × 10 6 ; --, Ra = 5 × 10 6 ) (Dol et al 1999). …”

“…Inadequacy of such an assumption is illustrated in Figure 2a, which shows the DNS variation of σ t T = (u 1 u 2 dT /dx 1 )/(u 1 θdU 2 /dx 1 ) (x 1 ≡ x is horizontal direction) with wall distance and Rayleigh numbers in an infinite, side-heated vertical plane channel (Dol et al 1999, Versteegh 1998. Another frequent assumption (also used in secondmoment closures) concerns the mechanical and thermal timescales, τ = k/ε and τ th = θ 2 /ε θθ , respectively, where θ 2 is the temperature variance and ε θθ its dissipation rate.…”

“…The latter is then usually provided by assuming that the mechanical and thermal timescales are proportional to each other, i.e., R = τ θ /τ ≈ 0.5. An example of the inadequacy of such an assumption (which essentially implies the use of a single scale to model both the mechanical and thermal turbulence) is illustrated in Figure 2b, obtained from DNS of flow in a side-heated vertical plane channel) (Dol et al 1999) (discussed below).…”

Assessment of heat flux turbulence models for flows dominated by buoyancy effects

“…For these processes, the most widely used model is based on the Daly and Harlow (1970) generalized gradient diffusion hypothesis (GGDH) (3) where the empirical coefficient C θ = 0.22 is commonly used. Although not frequently used in practical applications, some available results (e.g., Dol et al, 1999) show that an extended GGDH form, which contains two more terms, and is invariant under coordinate rotation, performs better. Both forms are the result of a very strong simplification of the governing equation for the turbulent transport, ρ θ u i u k , and do not include buoyancy effects.…”

“…Their contribution to the third-moments may also be important but none of the proposed schemes has considered the issue. Other advanced modelling approaches found in the literature (e.g., Craft et al, 1996;Dol et al, 1999) are designed with greater complexity to capture the essential physics of the processes involved, and are not considered in this paper.…”