A simplified and efficient Gay-Lussac approach for non-Boussinesq treatment of natural convection problems

Abstract

Under the Boussinesq approximation for buoyancy driven flows, density variations are restricted to the gravity term. In contrast, the Gay-Lussac (GL) approach is developed based on considering density variations in any term of the Navier—Stokes equations in which density appears. In both incompressible approaches, a linear density state equation is adopted to relate density variations to temperature differences. In this article, a simplified Gay-Lussac (SGL) approach with a reduced computational cost is proposed in which density variations are omitted from the continuity equation. It is shown that the SGL approach gives identical results to the traditional GL approach in both transient and steady states. Then, performance of the SGL approach at high relative temperature differences up to $\epsilon =0.3$ is evaluated against the low Mach number scheme and the Boussinesq approximations. In this respect, natural convection in square cavity benchmark problem at three different inclination angles ($\gamma =0$ and $\pm \pi /6$) is simulated up to $Ra={10}^7$ at $Pr=1$ and results are analyzed in terms of the local and average Nusselt number, and the skin friction coefficient. Comparing computational cost of simulations at $Ra={10}^7$ indicates the introduced SGL approach has 17% and 11% less computational cost using upwind and central schemes, respectively, compared to the traditional GL approach, while the convergence rate is not affected by the proposed simplification. Comparing the Nusselt number shows a negligible difference between the SGL and the Boussinesq approximations at high relative temperature differences, both deviating from the low Mach number scheme. Finally, by comparing the friction coefficient results obtained by the SGL approach against the weakly compressible approach it is concluded that the GL family approaches require serious revisions to outperform the Boussinesq approximation as an incompressible approach for buoyancy driven flows with high relative temperature differences.

Additional information

Funding

This research was supported by the Australian Research Council through Discovery Project DP180102647. P. M. is supported by a Monash Graduate Scholarship and a Monash International Postgraduate Research Scholarship. The authors are also supported by time allocations on the National Computational Infrastructure (NCI) peak facility and the Pawsey Supercomputing Centre through NCMAS grants. NCI is supported by the Australian Government.