Abstract In this article we propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and in this case the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [1, 2, 3], in which the spatial domain is discretized using a face-based staggered unstructured mesh. The pressure and temperature variables are defined on the primal simplex elements, while the velocity is assigned to the dual grid. For the computation of the advection and diffusion terms, two different algorithms are presented: i) a purely Eulerian upwind-type scheme and ii) an Eulerian-Lagrangian approach. The first methodology leads to a conservative scheme whose major drawback is the time step restriction imposed by the CFL stability condition due to the explicit discretization of the convective terms. On the contrary, computational efficiency can be notably improved relying on an Eulerian-Lagrangian approach in which the Lagrangian trajectories of the flow are tracked back. This method leads to an unconditionally stable scheme if the diffusive terms are discretized implicitly. Once the advection and diffusion contributions have been computed, the pressure Poisson equation is solved and the velocity is updated. As a second model for the computation of buoyancy-driven flows, in this paper we also consider the full compressible Navier-Stokes equations. The staggered semi-implicit DG method first proposed in [4] for all Mach number flows is properly extended to account for the gravity source terms arising in the momentum and energy conservation laws. In order to assess the validity and the robustness of our novel class of staggered semi-implicit DG schemes, several classical benchmark problems are considered, showing in all cases a good agreement with available numerical reference data. Furthermore, a detailed comparison between the incompressible and the compressible solver is presented. Finally, advantages and disadvantages of the Eulerian and the Eulerian-Lagrangian methods for the discretization of the nonlinear convective terms are carefully studied.

中文翻译：

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用于自然对流问题的高效高阶精确交错半隐式不连续伽辽金方法

摘要 在本文中，我们提出了一系列新的高阶交错半隐式不连续伽辽金 (DG) 方法来模拟自然对流问题。假设温度波动很小，Boussinesq 近似是有效的，在这种情况下，流动可以简单地通过不可压缩的 Navier-Stokes 方程加上温度的传输方程和动量方程中的浮力源项来建模。我们的数值方案是从 [1, 2, 3] 中提出的工作开始的，其中空间域使用基于面的交错非结构化网格进行离散化。压力和温度变量定义在原始单纯形元素上，而速度分配给双网格。对于平流和扩散项的计算，提出了两种不同的算法：i) 纯欧拉迎风型方案和 ii) 欧拉-拉格朗日方法。第一种方法导致保守方案，其主要缺点是由于对流项的显式离散化，CFL 稳定性条件施加的时间步长限制。相反，依靠欧拉-拉格朗日方法可以显着提高计算效率，其中流的拉格朗日轨迹被回溯。如果扩散项被隐式离散化，则该方法会导致无条件稳定方案。一旦计算了对流和扩散的贡献，就可以求解压力泊松方程并更新速度。作为计算浮力驱动流的第二个模型，在本文中，我们还考虑了完全可压缩的 Navier-Stokes 方程。在 [4] 中首次提出的用于所有马赫数流的交错半隐式 DG 方法被适当扩展以解释动量和能量守恒定律中出现的重力源项。为了评估我们的新型交错半隐式 DG 方案的有效性和稳健性，考虑了几个经典的基准问题，在所有情况下都显示出与可用数值参考数据的良好一致性。此外，还给出了不可压缩和可压缩求解器之间的详细比较。最后，仔细研究了欧拉方法和欧拉-拉格朗日方法在非线性对流项离散化方面的优缺点。