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，我们将给出本项目的主要研究问题的摘要，以及我们打算如何解决这些问题。三维可压缩流问题联合收割机结合了不可压缩流的不稳定性，如紊流，在亚音速范围内的声学效应，以及在超音速范围内的非线性波，如激波、接触间断和稀疏效应。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限于标量问题的渐近估计。部分原因是由于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激波捕获型方案仍然知之甚少。如果对精确解作了额外的假设，例如解可以分解为有限个由不连续性隔开的光滑部分，而这些不连续性的支撑具有某些有利的性质，那么是否可以进行更完整的数值分析？3.实际上，计算效率对于大规模计算至关重要。数值稳定性方面和使用显式时间步进的可能性，使间断伽辽金方法流行的可压缩流计算。然而，在二维空间中，分段仿射不连续逼近的自由度（dofs）是标准连续有限元的六倍。这个数字在三维中增加到20倍。不连续Galerkin方法还依赖于昂贵的Riemann求解器，这些求解器可能并不总是鲁棒的，请参阅[Abg 17 a]。因此，我们问是否可以设计一种连续有限元方法，该方法可以使用更少的dofs来结合不连续Galerkin的优点[Gue 16 a，Abg 17 b]？本项目的主要目的是解决这些问题，利用我们最近的结果，有限元方法（FEM）近似湍流近似不可压缩Navier-Stokes方程[Mou 22]，局部估计稳定的FEM标量线性传输问题[Bu 22 a]和整体估计标量线性传输问题离散非线性稳定化[Bu 22 b]。本建议的基石是：1。发展了非线性波的保不变激波捕捉方法;2.发展控制二次振荡的线性稳定方法;3.显式和隐式-显式时间离散格式的开发和数值分析;4.质量矩阵的载荷平衡区域分解方法，用于显式时间步进;5.可压缩紊流的高性能三维计算。参考文献：[Abg 17 a] Abgrall，R.黎曼解算器的一些故障。双曲问题数值方法手册，2017年18月。[Abg17b] Abgrall，R.使用全局连续逼近和避免质量矩阵的双曲型问题的高阶格式。《科学杂志》Comput.（2017年）。[Bu22a] Burman，E.用连续有限元离散的瞬态输运问题的加权误差估计及梯度跳跃上的内罚镇定。Vietnam J. Math.（2022）. [Bu22b] Burman，E.双曲型守恒律连续有限元法线性与非线性稳定化相互作用的一些观察。出现在SIAM J. Sci. Comput. [Gue16a] Guermond，J.- L.的;波波夫，B。非线性标量守恒方程一阶拉格朗日有限元方法的误差估计。SIAM J.编号Anal.（2016年）。项[Mou 22]莫拉、R. C.的; Cassinelli，A.; da Silva，A.; Burman，E.; Sherwin，S.欠分辨湍流模拟谱/hp元离散的梯度跳跃惩罚稳定。CMAME（2022）