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.

摘要

在浮力驱动流的 Boussinesq 近似下，密度变化仅限于重力项。相比之下，Gay-Lussac (GL) 方法是基于考虑出现密度的 Navier-Stokes 方程的任何项中的密度变化而开发的。在这两种不可压缩方法中，采用线性密度状态方程将密度变化与温差联系起来。在本文中，提出了一种计算成本降低的简化 Gay-Lussac (SGL) 方法，其中从连续性方程中省略了密度变化。结果表明，SGL 方法在瞬态和稳态下都给出了与传统 GL 方法相同的结果。然后，SGL 方法在高相对温差下的性能高达$\epsilon =0.3$根据低马赫数方案和 Boussinesq 近似值进行评估。在这方面，方腔基准问题中三种不同倾角下的自然对流（$\gamma =0$ 和 $\pm \pi /6$) 被模拟到 $\mathrm{电阻}\mathrm{一种}={10}^7$ 在 $磷r=1$并根据局部和平均努塞尔数以及皮肤摩擦系数对结果进行分析。比较模拟的计算成本$\mathrm{电阻}\mathrm{一种}={10}^7$表明与传统的 GL 方法相比，引入的 SGL 方法使用逆风和中心方案的计算成本分别降低了 17% 和 11%，而收敛速度不受提议的简化的影响。比较 Nusselt 数表明，SGL 和 Boussinesq 近似值在高相对温差下的差异可以忽略不计，两者都偏离了低马赫数方案。最后，通过将 SGL 方法获得的摩擦系数结果与弱可压缩方法进行比较，得出的结论是，GL 系列方法需要认真修改才能胜过 Boussinesq 近似作为具有高相对温差的浮力驱动流动的不可压缩方法。