A 2D½ model for low Prandtl number convection in an enclosure

• We present an efficient numerical model for low Prandtl number natural convection in a slender enclosure.
• Governing equations are integrated along the small dimension to account for the stick condition on side-walls in a 2D model.
• The originality of our model is that the assumed velocity profile is a ‘uniform bulk + boundary layers’ profile.
• The averaged deviation between the obtained temperature field and that issued from 3D simulations is typically less than 2%.
• A computed longitudinal temperature profile at mid-height of the cavity is in good agreement with to an experimental one.

Numerical simulations of stationary thermal convection in a differentially heated enclosure corresponding to the AFRODITE solidification benchmark experiment [1], [2], [3], [4] and [5] are presented. The cavity of relative dimensions 10:6:1 (length:height:width) is characterized by a small transverse width. The Prandtl number Pr is varied within the range [0.0045, 0.03], typical of liquid metals, whereas the Grashof number, defined as Gr = gβ(Δθ/L)H4/ν2, is varied within the range [1.3 × 106, 1.6 × 107]. As shown by the reference 3D simulations, the temperature field in these situations is 2D (independent of the transverse direction); 2D simulations are, however, not able to catch the physics of the flow and the resulting temperature results are also erroneous. To improve these 2D simulation results while keeping reasonable computational times, a 2D½ model is developed, which will take into account the no-slip condition at the side walls. This model is obtained by averaging the governing equations over the width of the domain, with a transverse profile for the velocity featuring a uniform central part and two boundary layers of size δ (δ is fixed for the whole domain). The relative deviation of the temperature field between the 2D½ and 3D computations is investigated as a function of the Prandtl number, the Grashof number and the chosen boundary layer thickness. It is shown that an optimum value exists for δ, which gives a mean deviation in the middle plane of less than 2%, whereas the choice of a more usual parabolic profile would lead to a twice larger deviation. Good comparisons are also obtained with the original experimental results reported at the end of the paper. The 2D½ model is thus able to give results which compare well with fully 3D results. It can then be used for extensive parametric studies at a reasonable cost.