Acta Mechanica Sinica

, Volume 34, Issue 3, pp 431–445 | Cite as

Density enhancement mechanism of upwind schemes for low Mach number flows

Research Paper
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Abstract

Many all-speed Roe schemes have been proposed to improve performance in terms of low speeds. Among them, the F-Roe and T-D-Roe schemes have been found to get incorrect density fluctuation in low Mach flows, which is expected to be with the square of Mach number. Asymptotic analysis presents the mechanism of how the density fluctuation problem relates to the incorrect order of terms in the energy equation \({{\tilde{\rho }} {\tilde{a}} {\tilde{U}}\varDelta U}\). It is known that changing the upwind scheme coefficients of the pressure-difference dissipation term \(D^P\) and the velocity-difference dissipation term in the momentum equation \(D^{\rho U}\) to the order of \(O(c^{-1})\) and \(O(c^{0})\) can improve the level of pressure and velocity accuracy at low speeds. This paper shows that corresponding changes in energy equation can also improve the density accuracy in low speeds. We apply this modification to a recently proposed scheme, TV-MAS, to get a new scheme, TV-MAS2. Unsteady Gresho vortex flow, double shear-layer flow, low Mach number flows over the inviscid cylinder, and NACA0012 airfoil show that energy equation modification in these schemes can obtain the expected square Ma scaling of density fluctuations, which is in good agreement with corresponding asymptotic analysis. Therefore, this density correction is expected to be widely implemented into all-speed compressible flow solvers.

Keywords

Energy equation Density fluctuation Roe TV-MAS Low speeds All speeds Computational fluid dynamics 

Notes

Acknowledgements

The authors would like to acknowledge the support for this work provided by the National Natural Science Foundation of China (Grant 11402016), and all the authors are grateful to the anonymous reviewers for their constructive comments.

References

  1. 1.
    Zheng, X., Zhou, S., Hou, A., et al.: Separation control using synthetic vortex generator jets in axial compressor cascade. Acta Mech. Sin. 22, 521–527 (2006)CrossRefGoogle Scholar
  2. 2.
    Xu, G., Jiang, X., Liu, G.: Delayed detached eddy simulations of fighter aircraft at high angle of attack. Acta Mech. Sin. 32, 588–603 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Zheng, W., Yan, C., Liu, H., et al.: Comparative assessment of SAS and DES turbulence modeling for massively separated flows. Acta Mech. Sin. 32, 12–21 (2016)CrossRefGoogle Scholar
  4. 4.
    Fang, J., Lu, L.-P., Shao, L.: Heat transport mechanisms of low Mach number turbulent channel flow with spanwise wall oscillation. Acta Mech. Sin. 26, 391–399 (2010)CrossRefMATHGoogle Scholar
  5. 5.
    Turkel, E.: Preconditioning techniques in computational fluid dynamics. Annu. Rev. Fluid Mech. 31, 385–416 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Weiss, J., Smith, W.: Preconditioning applied to variable and constant density flows. AIAA J. 33, 2050–2057 (1995)CrossRefMATHGoogle Scholar
  7. 7.
    Guillard, H., Viozat, C.: On the behaviour of upwind schemes in the low Mach number limit. Comput. Fluids 28, 63–86 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Roe, P.L., Pike, J.: Efficient construction and utilisation of approximate Riemann solutions. In: Computing Methods in Applied Sciences and Engineering, VI, North Holland, 499–518 (1984)Google Scholar
  9. 9.
    Boniface, J.-C.: Rescaling of the Roe scheme in low Mach-number flow regions. J. Comput. Phys. 328, 177–199 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, X.-S., Gu, C.-W.: Mechanism of Roe-type schemes for all-speed flows and its application. Comput. Fluids 86, 56–70 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Li, X., Gu, C.: An all-speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour. J. Comput. Phys. 227, 5144–5159 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, X.-S., Gu, C.-W., Xu, J.-Z.: Development of Roe-type scheme for all-speed flows based on preconditioning method. Comput. Fluids 38, 810–817 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Thornber, B.J.R., Drikakis, D.: Numerical dissipation of upwind schemes in low Mach flow. Int. J. Numer. Methods Fluids 56, 1535–1541 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rieper, F.: A low-Mach number fix for Roe’s approximate Riemann solver. J. Comput. Phys. 230, 5263–5287 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fillion, P., Chanoine, A., Dellacherie, S., et al.: FLICA-OVAP: a new platform for core thermalhydraulic studies. Nucl. Eng. Des. 241, 4348–4358 (2011)CrossRefGoogle Scholar
  16. 16.
    Li, X.-S.: Uniform algorithm for all-speed shock-capturing schemes. Int. J. Comput. Fluid Dyn. 28, 329–338 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Qu, F., Yan, C., Sun, D., et al.: A new Roe-type scheme for all speeds. Comput. Fluids 121, 11–25 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Sun, D., Yan, C., Qu, F., et al.: A robust flux splitting method with low dissipation for all-speed flows. Int. J. Numer. Methods Fluids 84, 3–18 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liou, M.-S., Steffen, C.J.: A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liou, M.S.: A sequel to \(\{\text{ AUSM: } \text{ AUSM }\}^+\). J. Computat. Phys. 129, 364–382 (1996)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Liou, M.: A sequel to AUSM, part II: AUSM+-up for all speeds. J. Comput. Phys. 214, 137–170 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shima, E., Kitamura, K.: Parameter-free simple low-dissipation AUSM-family scheme for all speeds. AIAA J. 49, 1693–1709 (2011)CrossRefGoogle Scholar
  23. 23.
    Kitamura, K., Shima, E., Fujimoto, K., et al.: Performance of low-dissipation euler fluxes and preconditioned LU-SGS at low speeds. Commun. Comput. Phys. 10, 90–119 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Shima, E., Kitamura, K.: New approaches for computation of low Mach number flows. Comput. Fluids 85, 143–152 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Yao, S.B., Sun, Z.X., Guo, D.L., et al.: Numerical study on wake characteristics of high-speed trains. Acta Mech. Sin. 29, 811–822 (2013)Google Scholar
  26. 26.
    Guo, D., Shang, K., Zhang, Y., et al.: Influences of affiliated components and train length on the train wind. Acta Mech. Sin. 32, 191–205 (2016)Google Scholar
  27. 27.
    Xiao, Z., Fu, S.: Studies of the unsteady supersonic base flows around three afterbodies. Acta Mech. Sin. 25, 471–479 (2009)CrossRefMATHGoogle Scholar
  28. 28.
    Qu, F., Yan, C., Sun, D.: Investigation into the influences of the low speed’s accuracy on the hypersonic heating computations. Int. Commun. Heat Mass Transf. 70, 53–58 (2016)CrossRefGoogle Scholar
  29. 29.
    Qu, F., Sun, D., Shi, Y., et al.: Investigation into the influences of the low speeds’ accuracy on RANS simulations. In: 21st AIAA International Space Planes and Hypersonics Technologies Conference, Xiamen, China, 1–14 (2017)Google Scholar
  30. 30.
    Zha, G., Bilgen, E.: Numerical solutions of Euler equations by using a new flux vector splitting scheme. Int. J. Numer. Methods Fluids 17, 115–144 (1993)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Toro, E.F., Vazquez-Cendon, M.E.: Flux splitting schemes for the Euler equations. Comput. Fluids 70, 1–12 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Qu, F., Yan, C., Yu, J., et al.: A new flux splitting scheme for the Euler equations. Comput. Fluids 102, 203–214 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kapen, P.T., Tchuen, G.: An extension of the TV-HLL scheme for multi-dimensional compressible flows. Int. J. Comput. Fluid Dyn. 29, 303–312 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Xie, W., Li, H., Tian, Z., et al.: A low diffusion flux splitting method for inviscid compressible flows. Comput. Fluids 112, 83–93 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  36. 36.
    Sun, M., Takayama, K.: An artificially upstream flux vector splitting scheme for the Euler equations. J. Comput. Phys. 189, 305–329 (2003)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Tong, B.G., Kong, X.Y., Deng, G.H.: Gas Dynamics, 2nd edn. Higher Education Press, Beijing (2012). (in Chinese)Google Scholar
  38. 38.
    Gresho, P.M.: On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: theory. Int. J. Numer. Methods Fluids 11, 620–687 (1990)Google Scholar
  39. 39.
    Gresho, P.M., Chan, S.T.: On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 2: implementation. Int. J. Numer. Methods Fluids 11, 621–659 (1990)CrossRefMATHGoogle Scholar
  40. 40.
    Gottlieb, S.: On high order strong stability preserving Runge–Kutta and multi step time discretizations. J. Sci. Comput. 25, 105–128 (2005)MathSciNetMATHGoogle Scholar
  41. 41.
    Ishiko, K., Ohnishi, N., Sawada, K.: Implicit LES for Two-Dimensional Turbulence Using Shock Capturing Monotone Scheme. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 1–12 (2006)Google Scholar
  42. 42.
    Kitamura, K., Hashimoto, A.: Reduced dissipation AUSM-family fluxes: HR-SLAU2 and HR-AUSM+-up for high resolution unsteady flow simulations. Comput. Fluids 126, 41–57 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Yoon, S., Jamesont, A.: Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations. AIAA J. 26, 1025–1026 (1988)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Aeronautic Science and EngineeringBeihang UniversityBeijingChina

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