Table of Content

• Numerical Simulation of 1-D Sod Shock Tube
  • Information
  • Description
    • Main Program
    • Attached Function Modules
    • Exact Riemann Solution: Referred Functions by Gogol (2021)
  • Reference

Numerical Simulation of 1-D Sod Shock Tube

The MATLAB codes for the realization of the numerical simulation of 1-D Sod Shock Tube (v1.0)

Information

Author: pkufzh (Small Shrimp)

Course: Fundamentals of Computational Fluid Dynamics (CFD)

Submit: 2021/12/28

Version: v1.0

Description

All the developed MATLAB codes are saved under the Codes folder.

Main Program

• Program_Sod_Shock_Tube_Main
  • Main Program: Numerical simulation of 1-D compressible flow (Sod Shock Tube)

Attached Function Modules

Important Note: Please ensure the following files are placed in the same folder with the main program!

• Flux_Vect_Split_Common.m

  • Flux Vector Splitting (FVS) with different methods (Optional)
    • Steger-Warming (S-W)
    • Lax-Friedrichs (L-F)
    • van Leer
    • Liou-Steffen (Advection Upstream Splitting Method, AUSM)

• Flux_Diff_Split_Common.m

  • Flux Difference Splitting (FDS) with different methods (Optional)
    • Roe Scheme

• Diff_Cons_Common.m

• Calculate the flux difference $\frac{\partial \mathbf{F}}{\partial x}$ from positive flux $\frac{\partial \mathbf{F}^{+}}{\partial x}$ and negative flux $\frac{\partial \mathbf{F}^{-}}{\partial x}$ with the conservation form through FVS or FDS, i.e.

$$ \mathbf{F}{j + \frac{1}{2}} = \mathbf{F}{j + \frac{1}{2} L}^{+} + \mathbf{F}_{j + \frac{1}{2} R}^{-} $$

• Shock Capturing Methods (Optional)

  • (TVD) Total Variation Diminishing Scheme with van Leer Limiter
  • (NND, H. X. Zhang) Non-oscillatory, Non-free-parameters Dissipative Difference Scheme
  • (Original WENO, 5 order, Jiang & Shu) Weighted Essentially Non-oscillatory Scheme Scheme

• First Level Upwind Schemes (Optional)

  • 1 order (2 points)
  • 2 order (3 points)
  • 3 order (4 points with bias)
  • 5 order (6 points with bias)

• Note: All the upwind schemes used in this program had been converted into the conservative form.

• Cal_Minmod.m

  • Calculate minmod(a, b)
  • Sign Definition

  $$ \operatorname{minmod}(a,b) = \frac{1}{2}\left[\operatorname{sgn}(a) + \operatorname{sgn}(b)\right]\cdot\operatorname{min}\left(\left|a\right|, \left|b\right|\right) $$

​ where $a$, $b$ is 1-Dimensional array with same length.

• Plot_Props.m
  • Plot the properties of fluid with preset and uniform axis coordinates

Exact Riemann Solution: Referred Functions by Gogol (2021)

Reference: Gogol (2021). Sod Shock Tube Problem Solver Click to the Website, From MATLAB Central File Exchange. Retrieved December 28, 2021. The main codes were developed by the original author.

• analytic_sod.m

  • Solve Sod's Shock Tube problem using exact Riemann solution
  • Reference Page

• sod_func.m

  • Define functions to be used in analytic_sod.m
  • Initial conditions

• sod_demo.m

  • A demo script file to show the use of analytic_sod.m

Note:

• The above MATLAB codes were tested and passed on MATLAB R2021b, Windows 64-bit system.
• For the vector images saved in the Paper.pdf are large, loading may be slow. Thanks for your patient waiting!!!

Reference

1. Gogol (2021). Sod Shock Tube Problem Solver (Click to the Website), MATLAB Central File Exchange. Retrieved December 28, 2021.
2. Steger, J. L., & Warming, R. F. (1981). Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. Journal of computational physics, 40(2), 263-293.
3. Van Leer, B. (1997). Flux-vector splitting for the Euler equation. In Upwind and high-resolution schemes (pp. 80-89). Springer, Berlin, Heidelberg.
4. Liou, M. S., & Steffen Jr, C. J. (1993). A new flux splitting scheme. Journal of Computational physics, 107(1), 23-39.
5. Roe, P. L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of computational physics, 43(2), 357-372.
6. Godunov, S., & Bohachevsky, I. (1959). Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematičeskij sbornik, 47(3), 271-306.
7. Jennings, G. (1974). Discrete shocks. Communications on pure and applied mathematics, 27(1), 25-37.
8. Van Leer, B. (1979). Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method. Journal of computational Physics, 32(1), 101-136.
9. Yee, H. C., Warming, R. F., & Harten, A. (1985). Implicit total variation diminishing (TVD) schemes for steady-state calculations. Journal of Computational Physics, 57(3), 327-360.
10. Sweby, P. K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM journal on numerical analysis, 21(5), 995-1011.
11. Fu, D., & Ma, Y. (1997). A high order accurate difference scheme for complex flow fields. Journal of Computational physics, 134(1), 1-15.
12. 马延文, & 傅德薰. (1992). 计算空气动力学中一个新的激波捕捉法——耗散比拟法. 中国科学(A辑 数学 物理学 天文学 技术科学), 35(3), 263-271.
13. 张涵信. (1984). 差分计算中激波上、下游解出现波动的探讨. 空气动力学学报(01), 14-21.
14. 张涵信. (1988). 无波动,无自由参数的耗散差分格式. 空气动力学学报(2).
15. ZHUANG, F., & ZHANG, H. (1987). Computational fluid dynamics in China. In 8th Computational Fluid Dynamics Conference (p. 1134).
16. Harten, A., Engquist, B., Osher, S., & Chakravarthy, S. R. (1987). Uniformly high order accurate essentially non-oscillatory schemes, III. In Upwind and high-resolution schemes (pp. 218-290). Springer, Berlin, Heidelberg.
17. Shu, C. W., & Osher, S. (1988). Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of computational physics, 77(2), 439-471.
18. Chakravarthy, S. R. (1990). Some Aspects of Essentially Nonoscillatory (ENO) Formulations for the Euler Equations. National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Division.
19. Jiang, G. S., & Shu, C. W. (1996). Efficient implementation of weighted ENO schemes. Journal of computational physics, 126(1), 202-228.
20. 刘儒勋, & 舒其望. (2003). 计算流体力学的若干新方法. 科学出版社.
21. 吴望一,蔡庆东.(2000).时间空间均为二阶的新型NND差分格式. 应用数学和力学 (06),561-572.
22. Liu, X. D., Osher, S., & Chan, T. (1994). Weighted essentially non-oscillatory schemes. Journal of computational physics, 115(1), 200-212.

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Developed or Finished by pkufzh (Small Shrimp) on 2022/01/20.

Contact me:

Github Page: https://github.com/pkufzh

ResearchGate: https://www.researchgate.net/profile/Zhenghao-Feng

Bilibili Space: https://space.bilibili.com/167343763

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