A space-time outlook on CFD: Spatial-temporally correlated models and spatial-temporally coupled algorithms

LI Jiequan

doi:10.7638/kqdlxxb-2020.0162

English Version Journal Login

您当前的位置：

首页 >

文章列表页 >

A space-time outlook on CFD: Spatial-temporally correlated models and spatial-temporally coupled algorithms

更新时间：2026-05-28

- A space-time outlook on CFD: Spatial-temporally correlated models and spatial-temporally coupled algorithms
- Vol. 39, Issue 1, Pages: 92-110(2021)
- 作者机构：
  
  1. 北京应用物理与计算数学研究所计算物理重点实验室,北京,100088
  2. 北京大学应用物理与技术中心,北京,100871
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2020.0162
  CLC： V211.3
- Published：2021
- 稿件说明：
移动端阅览
李杰权. 计算流体力学的时空观：模型的时空关联性及算法的时空耦合性[J]. 空气动力学学报, 2021, 39(1): 92-110.

LI Jiequan. A space-time outlook on CFD: Spatial-temporally correlated models and spatial-temporally coupled algorithms[J]. 2021, 39(1): 92-110.
李杰权. 计算流体力学的时空观：模型的时空关联性及算法的时空耦合性[J]. 空气动力学学报, 2021, 39(1): 92-110. DOI： 10.7638/kqdlxxb-2020.0162.

LI Jiequan. A space-time outlook on CFD: Spatial-temporally correlated models and spatial-temporally coupled algorithms[J]. 2021, 39(1): 92-110. DOI： 10.7638/kqdlxxb-2020.0162.

摘要

流体力学中波的有限传播、粒子的碰撞、各种力之间相互作用，无不体现时空关联效应。本文从计算方法的视角探讨计算流体力学的时空观，即流体力学模型的时空关联性和计算方法的时空耦合性。从流体力学微团法建模出发，明确模型时空关联性的涵义，建立有限体积格式的基本原理，阐述算法时空耦合的必要性，实现流体力学基本控制方程物理建模与有限体积格式数学原理的统一。在实践中，给出时空耦合高精度数值方法设计思路，利用算例比较它与时空解耦方法的差别。期望通过时空观的建立，对未来计算流体力学的算法研究提供帮助。

Abstract

A space-time outlook on Computational Fluid Dynamics (CFD) is advocated: models in fluid mechanics often have the spatial-temporally correlated property

which should be inherited and preserved in the corresponding numerical algorithms. Starting from the fundamental formulas of fluid mechanics under continuum hypothesis

this paper defines the meaning of the spatial-temporal correlation of the models

establishes the fundamental principles of finite volume schemes

expounds the necessity of spatial-temporal coupling of algorithms

as well as realizes the physical and mathematical unification of basic governing equations of fluid mechanics and finite volume schemes. In practice

the design methodology of spatial-temporally coupled high-order numerical algorithms is presented

and the difference from spatial-temporally decoupled methods is compared. It is expected that this space-time outlook could guide the development of CFD algorithms in the near future.

关键词

Keywords

references

华罗庚, 大哉数学之为用[N]. 人民日报, 1959年5月28日.

BEN-ARTZI M, LI J Q. Consistency of finite volume approximations to nonlinear hyperbolic balance laws[J]. Math Comp, 2021, 90: 141-169.

LI J Q. Two-stage fourth order: Temporal-spatial coupling in computational fluid dynamics (CFD)[J]. Advances in Aerodynamics, 2019, 1: 3. https: //doi. org/10.1186/s42774-019-0004-9

BEN-ARTZI M, FALCOVITZ J. A second-order Godunov-type scheme for compressible fluid dynamics[J]. J Comput Phys, 1984, 55: 1-32.

BEN-ARTZI M, LI J Q, WARNECKE G. A direct Eulerian GRP scheme for compressible fluid flows[J]. J Comput Phys, 2006, 218(1): 19-43.

LI J Q, WANG Y. Thermodynamical effects and high resolution methods for compressible fluid flows[J]. J Comput Phys, 2017, 343: 340-354.

CHANG S C. The method of spacetime conservation element and solution element-A new approach for solving the Navier-Stokes and Euler equations[J]. J Comput Phys, 1995, 119: 295-324.

XU K. A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method[J]. J Comput Phys, 2001, 171: 289-335.

XU K, HUANG J C. A unified gas-kinetic scheme for continuum and rarefied flows[J]. J Comput Phys, 2010, 229: 7747-7764.

CAUCHY A L. Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques or non élastiques[J]. Bull Soc Philomathique, 1823, 10(2): 9-13.

CAUCHY A L. Da la pression ou tension dans un corps solide[J]. Exercises de Matehématiques, 1827, 2(2): 42-56.

FEDERER H. The Gauss-Green theorem[J]. Trans Am Math Soc, 1945, 58: 44-76.

FEDERER H. Geometric measure theory[M]. Springer-Verlag, 1969.

GURTIN M E, MARTINS L C. Cauchy's theorem in classical physics[J]. Arch Rat Mech Anal, 1975, 60/76: 305-324.

ŠILHAV M. The existence of the flux vector and the divergence theorem for general Cauchy fluxes[J]. Arch Rat Mech Anal, 1985, 90: 195-212.

CHEN G Q, COMI G E, TORRES M. Cauchy fluxes and Gauss-Green formulas for divergence measure fields over general open sets[J]. Arch Rat Mech Anal, 2019, 233: 87-166.

DAFERMOS C M. Hyperbolic conservation laws in continuum physics[M]. Fourth Edition. Grundlehren der Mathematischen Wissenschaften, vol. 325, Springer, 2016.

HOPF E. The partial differential equation u t + uu x = μu xx [J ] . Comm Pure Appl Math, 1950, 3: 201-230.

BEC J, IHANIN K. Burgers turbulence[J]. Physics Reports, 2007, 447: 1-66.

TANG T, TENG Z H. The sharpness of Kuznetsov's O( $$ \sqrt {\Delta x} $$ ) L 1 -error estimate for monotone difference schemes[J ] . Math Comp, 1995, 64: 581-589.

DU Z F, LI J Q. A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws[J]. J Comput Phys, 2018, 355: 385-396.

LAX P D, WENDROFF B. Systems of conservation laws[J]. Comm Pure Appl Math, 1960, 13: 217-237.

EYMARD R, GALLOUET T, HERBIN R. Finite volume methods, in handbook of numerical analysis[M]. Vol. Ⅶ, Eds. CIARLET P G, LIONS J L. North-Holland, 2000: 713-1020.

ELLING V. A Lax-Wendroff type theorem for unstructured quasiuniform grids[J]. Math Comp, 2007, 76: 251-272.

GODUNOV S K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[J]. Mat Sb (N.S.), 47(89): 3(1959), 271-306.

LI J Q, ZHANG T, YANG S L. The two-dimensional Riemann problem in gas dynamics[M]. Pitman Monographs and Surveys in Pure and Applied Mathematics, 98. Longman, Harlow, 1998.

MAIRE P H, ABGRALL R, BREIL J, et al. A cell-centered Lagrangian scheme for two-dimensional compressible flow problems[J]. SIAM J Sci Comput, 2007, 29: 1781-1824.

LI J Q, DU Z F. A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers I. Hyperbolic conservation laws[J]. SIAM J Sci Comput, 2016, 38: A3046-A3069.

GUO Z L, XU K, WANG R J. Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case[J]. Phys Rev E, 2013, 88: 033305.

EVANS L. Partial differential equations[M]. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.

BEN-ARTZI M, LI J Q. Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem[J]. Numer Math, 2007, 106(3): 369-425.

TORO E, TITAREV V. Derivative Riemann solvers for systems of conservation laws and ADER methods[J]. J Comput Phys, 2006, 212(1): 150-165.

QIAN J Z, LI J Q, WANG S H. The generalized Riemann problems for compressible fluid flows: towards high order[J]. J Comput Phys, 2014, 259: 358-389.

MONTECINOS G, CASTRO C E, DUMBSER M, et al. Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms[J]. J Comput Phys, 2012, 231(19): 6472-6494.

BEN-ARtzi M, FALCOVITZ J. Generalized Riemann problems in computational fluid dynamics[M]. Cambridge Monographs on Applied and Computational Mathematics, 11. Cambridge University Press, Cambridge, 2003.

BEN-ARTZI M, LI J Q. Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem[J]. Numer Math, 2007, 106(3): 369-425.

LEI X, LI J Q. Transversal effects of high order numerical schemes for compressible fluid flows[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40: 343-354.

LI J Q. On the two-dimensional gas expansion for compressible Euler equations[J]. SIAM J Appl Math, 2001/2002, 62: 831-852.

LI J Q, ZHENG Y X. Interaction of rarefaction waves of the two-dimensional self-similar Euler equations[J]. Arch Rat Mech Anal, 2009, 193: 623-657.

LI J Q, ZHENG Y X. Interaction of four rarefaction waves in the bi-symmetric class of the two-dimensional Euler equations[J]. Comm Math Phys, 2010, 296: 303-321.

BALSARA D, NKONGA B. Multidimensional Riemann problem with self-similar internal structure. part Ⅲ. A multidimensional analogue of the HLLI Riemann solver for conservative hyperbolic systems[J]. J Comput Phys, 2017, 346: 25-48.

LI J Q, SUN Z F. Remark on the generalized Riemann problem method for compressible fluid flows[J]. J Comput Phys, 2007, 222: 796-808.

BAER M R, NUNZIATO J W. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials[J]. Int J Multiphase Flow, 1986, 12(6): 861-889.

LEI X, LI J Q. A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows[J]. Physics of Fluids, 2018, 30: 006891.

LEI X, DU Z F, LI J Q. The simulation of compressible multi-fluid flows by a GRP-based energy-splitting method[J]. Computers Fluids, 2019, 181: 416-428.

WANG Y. Two relaxation methods for fluid dynamical systems[D].[Ph. D Thesis], Beiijng Normal University, 2012.

MONECINOS G, TORO E. Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes[J]. J Comput Phys, 2014, 275: 415-442.

BHATNAGAR P L, GROSS E P, KROOK M. A model for collision processes in gases I: Small amplitude processes in charged and neutral one-component systems[J]. Phys Rev, 1954, 94: 511.

POINSOT T J, LEE S K. Boundary conditions for direct simulations of compressible viscous flows[J]. J Comput Phys, 1992, 101: 104-129.

LU J, FANG J, TAN S, et al. Inverse Lax-Wendroff procedure for numerical boundary conditions of convection-diffusion equations[J]. J Comput Phys, 2016, 317: 276-300.

LI J Q, ZHANG Q L. One-sided GRP solver and high order numerical boundary conditions for compressible fluid flows[J]. Preprint, 2020.

DU Z F, LI J Q. Accelerated piston problem and high order moving boundary tracking method for compressible fluid flows[J]. SIAM J Sci Comput, 2020, 42(3): A1558-A1581.

DU Z F, LI J Q. A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solver, Ⅱ: High order numerical boundary conditions[J]. J Comput Phys, 2018, 369: 125-147.

LI J Q, ZHANG Y J. Numerical boundary treatment over unstructured meshes for compressible fluid flows[J]. Work in Preparation, 2020.

CHENG J, DU Z F, LEI X, et al. A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible Euler equations[J]. Computers and Fluids, 2019, 181: 248-258.

ZHAO F X, JI X, SHYY W, et al. Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations[J]. Adv Aerodyn 2019, 1: 13. https: //doi. org/10.1186/s42774-019-0015-6

PAN L, XU K, LI Q B, et al. An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations[J]. J Comput Phys, 2016, 326: 197-221.

CHEN Y, CHEN C, XIAO F, et al. A two-stage fourth-order multi-moment global shallow water model on cubed sphere[J]. Mon Wea Rev, 2020: 1-40. https: //doi. org/10.1175/MWR-D-20-0004.1

ZHANG C, LI Q B, WANG Z J, et al. A two-stage fourth-order gas-kinetic CPR method for Navier-Stokes equations on triangular meshes[J]. Preprint, 2020.

SHU C W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes[J]. Acta Numerica, 2020: 1-63.

QIU J X, SHU C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case[J]. J Comput Phys, 2003, 193: 115-135.

PAN L, LI J Q, XU K, A Few benchmark test cases for higher-order Euler solvers[J]. Numerical Mathematics: Theory, Methods and Applications, 2017, 10: 489-514.

TANG H Z, LIU T G. A note on the conservative schemes for the Euler equations[J]. J Comput Phys, 2006, 218: 451-459.

TITAREV V A, TORO E F. Finite volume WENO schemes for three-dimensional conservation laws[J]. J Comput Phys, 2014, 201: 238-260.

QUIRK J J, KARNI S. On the dynamics of a shock-bubble interaction[J]. J Fluid Mech, 1996, 318: 129-163.

CAO G Y, PAN L, XU K, Three dimensional high-order gas-kinetic scheme for supersonic isotropic turbulence I: Criterion for direct numerical simulation[J]. Computers Fluids, 2019, 192: 104273.

SAMTANEY R, PULLIN D I, KOSOVIC B. Direct numerical simulation of decaying compressible turbulence and shocklet statistics[J]. Physics of Fluids, 2001, 13: 1415-1430.

GUO Z L, LI J Q, XU K. On unified preserving properties of kinetic schemes[J]. Preprint, 2020, arXiv: 1909. 04923.

HE G W, JIN G D, YANG Y. Spacetime correlations and dynamic coupling in turbulent flows[J]. Annu Rev Fluid Mech, 2017, 49: 51-70.

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

An investigation of thermochemical reaction and aerodynamic ablation model on leading edges of reentry vehicles considering rarefaction effect

Impact of inter-wing spacing on aerodynamic noise in multi-flapping wing configurations

Research on helicity modification turbulence model based on the AENS simulation platform

Effect of pitch angle on the flapping flight and sound generation

Numerical simulation of the effect of transition position on static and dynamic aerodynamic characteristics of cones

Related Author

Chen Song

Zhang Yonghao

Xiao Tianbai

Wang Xu

Meng Xueguang

Chen Zengshuang

Xie Yuxin

LUO Pengzhong

Related Institution

International Innovation Institute, Beihang University, Hangzhou

School of Aeronautic Science and Engineering, Beihang University, Beijing

School of Engineering Science, University of Chinese Academy of Sciences, Beijing

School of Future Technology, University of Chinese Academy of Sciences, Beijing

Centre for Interdisciplinary Research in Fluids, Institute of Mechanics, Chinese Academy of Sciences, Beijing

AI问答

⁰