An efficient and simplified Gay‐Lussac approach in secondary variables form for the non‐Boussinesq simulation of free convection problems. (29th July 2021) - Mayeli, Peyman; Sheard, Gregory J.
Abstract: The Gay‐Lussac (GL) approach is an incompressible‐based strategy for non‐Boussinesq treatment of the governing equations for free convection problems that is established based on extending the density variations beyond the gravity term. Such a strategy leads to emerging the GL parameter as a non‐Boussinesq prefactor of different terms in the governing equations. In this article, the GL approach is expressed/discussed in terms of the secondary variables, that is, vorticity and stream‐function, for the first time and a simplified version of this approach is proposed by removing density variations from the continuity equation. The difference of results under the simplified and traditional GL approach ranges within a maximum of 1% for different parameters. The lower computational cost of numerical solution of governing equations in the secondary variables formula and the corresponding convergence rate is scrutinized for the simplified GL approach showing around 25% lower computational cost. The performance of this approach is evaluated at high relative temperature differences against the low Mach number scheme and the Boussinesq approximations. In this respect, natural convection in an annulus cavity is numerically simulated using a CVFEM solver under the aforementioned approximations up to Rayleigh number Ra = 10 5 at Prandtl number Pr = 1 and high relative temperature differences ( ϵ = 0.15 and 0.3). The largest deviations found for either the simplified GL or Boussinesq methods from the low Mach number scheme solution are less than 20% for velocity magnitude, 14% for stream function, 6% for vorticity, and 5% for temperature. Results under the three approximations are also analyzed in terms of the skin friction and local and average Nusselt number, indicating that the Gay‐Lussac approach requires some revisions to act more accurately than the classical Boussinesq approximation at high relative temperature differences in natural convection problems, especially within the convection dominated regime. Abstract : An efficient form of the Gay‐Lussac approach in the context of the secondary variables, that is, vorticity stream‐function is presented for buoyancy driven flows in which the density variations are extended to the advection and convection terms of the momentum and energy equations, respectively. In the proposed formulation, the density ratio is removed from the continuity equation under the traditional Gay‐Lussac approach. It is shown that, the proposed simplified Gay‐Lussac approach gives identical results to the traditional one with a reduced computational cost. The figures show the difference for different parameters between the traditional and simplified Gay‐Lussac approaches in a steady state at Ra  = 10 5 that ranges within 1% of the considered parameters values.