In the last decade, there has been a lot of interest in developing high-order methods as a viable option for unsteady scale-resolving-simulations which are increasingly important in the industrial design process. High-order methods offer the advantages of low numerical dissipation, high efficiency on modern architectures and quasi mesh-independence. Despite significant advance in high-order solution methods, the general CFD workflow (geometry, CAD preparation, meshing, solution, post-processing) has largely remained unchanged, with mesh generation being a significant bottleneck and often determining the overall quality of the solution. In this work, we aim to combine the numerical advantages of the high-order Flux Reconstruction (FR) method and the simplicity of the mesh generation (or lack thereof) of the Immersed Boundary Method (IBM) for steady and unsteady problems over moving geometries. The volume penalization (penalty-IBM) method is selected for its ease of implementation and robustness. Detailed discussions about numerical implementation, including the boundary representation, mask function, data reconstruction, and selection of the penalization parameter are given. Advantages of combining volume penalization in the high-order framework are shown by various numerical test cases. The approach is firstly demonstrated for the linear advection-diffusion equation by investigating the numerical convergence for the coupled FR-IBM approach. Thereafter, the accuracy of the approach is demonstrated for canonical (static) test cases in 2D and 3D when compared to a standard body-fitted unstructured simulation. Finally, the efficiency of the method to handle moving geometries is demonstrated for the flow around an airfoil with pitching and plunging motions.

在过去的十年中，人们对开发高阶方法作为非定常尺度解析模拟的可行选择产生了很大的兴趣，这在工业设计过程中变得越来越重要。高阶方法提供了低数值耗散、现代架构上的高效率和准网格独立性的优点。尽管高阶求解方法取得了重大进展，但一般的 CFD 工作流程（几何、CAD 准备、网格划分、求解、后处理）在很大程度上保持不变，网格生成是一个重要的瓶颈，通常决定了解决方案的整体质量。在这项工作中，我们的目标是结合高阶通量重建 (FR) 方法的数值优势和浸入边界方法 (IBM) 的网格生成（或缺乏）的简单性，以解决移动几何上的稳态和非稳态问题。选择体积惩罚（penalty-IBM）方法是因为它易于实现和健壮性。详细讨论了数值实现，包括边界表示、掩码函数、数据重建和惩罚参数的选择。各种数值测试用例显示了在高阶框架中结合体积惩罚的优点。该方法首先通过研究耦合 FR-IBM 方法的数值收敛性来证明线性对流-扩散方程。此后，与标准的贴身非结构化模拟相比，该方法的准确性在 2D 和 3D 中的规范（静态）测试用例中得到了证明。最后，该方法处理移动几何形状的效率在机翼周围具有俯仰和俯冲运动的流动中得到了证明。