Abstract Accurate methods to solve the Reynolds-Averaged Navier-Stokes (RANS) equations coupled to turbulence models are still of great interest, as this is often the only computationally feasible approach to simulate complex turbulent flows in large engineering applications. In this work, we present a novel discontinuous Galerkin (DG) solver for the RANS equations coupled to the k − ϵ model (in logarithmic form, to ensure positivity of the turbulence quantities). We investigate the possibility of modeling walls with a wall function approach in combination with DG. The solver features an algebraic pressure correction scheme to solve the coupled RANS system, implicit backward differentiation formulae for time discretization, and adopts the Symmetric Interior Penalty method and the Lax-Friedrichs flux to discretize diffusive and convective terms respectively. We pay special attention to the choice of polynomial order for any transported scalar quantity and show it has to be the same as the pressure order to avoid numerical instability. A manufactured solution is used to verify that the solution converges with the expected order of accuracy in space and time. We then simulate a stationary flow over a backward-facing step and a Von Karman vortex street in the wake of a square cylinder to validate our approach.

中文翻译：

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耦合到 k−ϵ 湍流模型的不可压缩 RANS 方程的高阶不连续 Galerkin 求解器

摘要 求解耦合到湍流模型的雷诺平均纳维-斯托克斯 (RANS) 方程的准确方法仍然很受关注，因为这通常是在大型工程应用中模拟复杂湍流的唯一计算可行的方法。在这项工作中，我们为耦合到 k − ϵ 模型（以对数形式，以确保湍流量的正性）的 RANS 方程提出了一种新颖的不连续伽辽金 (DG) 求解器。我们研究了结合 DG 使用壁函数方法对壁进行建模的可能性。求解器采用代数压力校正方案来求解耦合 RANS 系统，时间离散的隐式后向微分公式，并采用对称内部惩罚方法和 Lax-Friedrichs 通量分别离散化扩散项和对流项。我们特别注意任何传输标量的多项式阶数的选择，并表明它必须与压力阶数相同以避免数值不稳定。制造的解决方案用于验证该解决方案是否在空间和时间上以预期的精度顺序收敛。然后，我们模拟了在方形圆柱体后面的反向台阶和 Von Karman 涡街上的静止流，以验证我们的方法。制造的解决方案用于验证该解决方案是否在空间和时间上以预期的精度顺序收敛。然后，我们模拟了在方形圆柱体后面的反向台阶和 Von Karman 涡街上的静止流，以验证我们的方法。制造的解决方案用于验证该解决方案是否在空间和时间上以预期的精度顺序收敛。然后，我们模拟了在方形圆柱体后面的反向台阶和 Von Karman 涡街上的静止流，以验证我们的方法。