Continuous finite element methods for under resolved turbulence in compressible flow

可压缩流中未解析湍流的连续有限元方法

• 批准号: EP/X042650/1
• 负责人: Erik Burman
• 金额: $ 60.4万
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX042650%2F1
• 关键词: Continuous; finite; element; methods; under

Here we will give a summary of the main research questions of the present project and how we intend to address them.1. Three dimensional compressible flow problems combine the instabilities known from incompressible flows such as turbulence with the effects of acoustics in the subsonic regime and nonlinear waves such as shocks, contact discontinuities and rarefactions in the supersonic regime. How can a method be designed that handles all these phenomena in a unified way while remaining computationally efficient?2. The analysis of numerical methods for compressible flow is typically restricted to asymptotic estimates for scalar problems. In part due to the lack of theoretical understanding of the continuous equations, but even for linear model problems the non-linear shock capturing type schemes necessary to suppress local oscillations, remain poorly understood. Can a more complete numerical analysis be carried out if additional assumptions are made on the exact solution, for instance that the solution can be decomposed in a finite number of smooth parts, separated by discontinuities, where the support of these discontinuities has some favourable properties?3. In practice computational efficiency is of essence for large-scale computations. Aspects of numerical stability and the possibility to use explicit time-stepping have made the discontinuous Galerkin method popular for compressible flow computations. However, in two space dimensions piecewise affine discontinuous approximation has six times as many degrees of freedom (dofs) as standard continuous FEM. This number increases to 20 times in three dimensions. The discontinuous Galerkin method also relies on expensive Riemann solvers that may not always be robust, see [Abg17a].Therefore, we ask if a continuous finite element method can be designed that incorporates the advantages of the discontinuous Galerkin using substantially fewer dofs [Gue16a,Abg17b]?The main aim of the present project is to address these questions drawing on our recent results on finite element methods (FEM) for approximating of turbulent flows in the approximation of the incompressible Navier-Stokes' equations [Mou22], local estimates for stabilized FEM for scalar linear transport problems [Bu22a] and global estimates for scalar linear transport problems discretized with nonlinear stabilization [Bu22b]. The cornerstones of the present proposal are:1. development of invariant preserving shock capturing methods for nonlinear waves;2. development of linear stabilisation methods for the control of secondary oscillations;3. development and numerical analysis of explicit and implicit-explicit time discretisation schemes;4. load balanced domain decomposition methods for the mass matrix, for explicit time-stepping;5. High performance three dimensional computations of compressible turbulent flows.References:[Abg17a] Abgrall, R. Some failures of Riemann solvers. Handbook of numerical methods for hyperbolic problems, 18 2017. [Abg17b] Abgrall, R. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. J. Sci. Comput. (2017).[Bu22a] Burman, E. Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps. Vietnam J. Math. (2022). [Bu22b] Burman, E. Some observations on the interaction between linear and nonlinear stabilization for continuous finite element methods applied to hyperbolic conservation laws. To appear in SIAM J. Sci. Comput. [Gue16a] Guermond, J.-L.; Popov, B. Error estimates of a first-order Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal. (2016).\item[Mou22] Moura, R. C.; Cassinelli, A.; da Silva, A.; Burman, E.; Sherwin, S. Gradient jump penalty stabilisation of spectral/hp element discretisation for under-resolved turbulence simulations. CMAME (2022)

在这里，我们将总结本项目的主要研究问题以及我们打算如何解决这些问题。 1．三维可压缩流动问题将不可压缩流动（例如湍流）中已知的不稳定性与亚音速范围内的声学效应和非线性波（例如超音速范围内的冲击、接触不连续性和稀疏性）相结合。如何设计一种方法，以统一的方式处理所有这些现象，同时保持计算效率？2．可压缩流数值方法的分析通常仅限于标量问题的渐近估计。部分原因是缺乏对连续方程的理论理解，但即使对于线性模型问题，抑制局部振荡所需的非线性冲击捕获类型方案仍然知之甚少。如果对精确解做出额外的假设，例如解可以分解为有限数量的由不连续性分隔的平滑部分，其中这些不连续性的支持具有一些有利的属性，是否可以进行更完整的数值分析？ 3.在实践中，计算效率对于大规模计算至关重要。数值稳定性和使用显式时间步长的可能性使得间断伽辽金方法在可压缩流计算中广受欢迎。然而，在两个空间维度中，分段仿射不连续近似的自由度 (dof) 是标准连续 FEM 的六倍。这个数字在三个维度上增加到 20 倍。不连续伽辽金方法还依赖于昂贵的黎曼求解器，这些求解器可能并不总是鲁棒的，请参见[Abg17a]。因此，我们询问是否可以设计一种连续有限元方法，以结合使用更少自由度的不连续伽辽金方法的优点[Gue16a， Abg17b]？本项目的主要目的是利用我们最近关于近似湍流的有限元方法 (FEM) 的结果来解决这些问题在不可压缩纳维-斯托克斯方程 [Mou22] 的近似中，标量线性输运问题的稳定 FEM 的局部估计 [Bu22a] 以及非线性稳定离散化的标量线性输运问题的全局估计 [Bu22b]。本提案的基石是： 1．非线性波保不变激波捕捉方法的发展；2．开发用于控制二次振荡的线性稳定方法；3．显式和隐式-显式时间离散方案的开发和数值分析；4．质量矩阵的负载平衡域分解方法，用于显式时间步进；5．可压缩湍流的高性能三维计算。参考文献：[Abg17a] Abgrall, R. 黎曼求解器的一些失败。双曲问题数值方法手册，2017 年 18 月。[Abg17b] Abgrall, R。使用全局连续逼近并避免质量矩阵的双曲问题高阶方案。 J. 科学。计算。 (2017).[Bu22a] Burman, E. 使用连续有限元对梯度跳跃进行内罚稳定离散化瞬态传输问题的加权误差估计。越南 J. 数学。 （2022）。 [Bu22b] Burman，E。关于应用于双曲守恒定律的连续有限元方法的线性和非线性稳定之间相互作用的一些观察。发表于 SIAM J. Sci。计算。 [Gue16a] 格尔蒙德，J.-L.； Popov，B。非线性标量守恒方程的一阶拉格朗日有限元技术的误差估计。 SIAM J. 数字。肛门。 （2016）。\item[Mou22] Moura，R.C.；卡西内利，A.；达席尔瓦，A.；布尔曼，E.； Sherwin, S. 解析不足的湍流模拟的光谱/hp 元素离散化的梯度跳跃罚分稳定性。 CMAME (2022)