Acta Mechanica Sinica

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

Density enhancement mechanism of upwind schemes for low Mach number flows

  • Bo-Xi Lin
  • Chao YanEmail author
  • Shu-Sheng Chen
Research Paper


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.


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



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.


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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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