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Abstract

This research originally was aimed at modeling all flows (except free-molecular) by systems of hyperbolic-relaxation equations (moments of the Boltzmann equation), and developing efficient numerical methods for these. Such systems have many potential numerical advantages, mainly because there are no second or higher derivatives to be approximated. This avoids accuracy problems on adaptive unstructured grids, and the source terms, though often stiff, are only local; the compact stencils facilitate code parallelization. A single code could simulate flows up to intermediate Knudsen numbers, and be hybridized with DSMC where needed. In this project, one major problem arose that we have not yet solved: the accurate representation of shock structures. This makes the methodology currently unsuited for, e.g., re-entry flows. We have validated it for subsonic and transonic flows and are concentrating on applications to MEMS-related flows. We discuss the challenges of our approach, present numerical algorithms and results based on the 10-moment model, and report progress in our latest research topic: formulating accurate solid-boundary conditions.

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References

  1. Allègre J., Raffin M., Lengrand J.C.: Experimental flowfields around NACA 0012 airfoils located in subsonic and supersonic rarefied air streams. In: Bristeau, M.O., Glowinski, R., Periaux, J., Viviand, H. (eds) Numerical Simulation of Compressible Navier–Stokes Flows: A GAMM-Workshop, Notes on Numerical Fluid Mechanics, 18, pp. 59–68. Friedrick Vieweg & Sohn, Braunschweig (1987)

    Google Scholar 

  2. Bassi F., Rebay S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. 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. Phys. Rev. 94(3), 511–525 (1954)

    Article  MATH  ADS  Google Scholar 

  4. Brown, S.L.: Approximate Riemann solvers for moment models of dilute gases. PhD thesis, The University of Michigan (1996)

  5. Brown, S.L., Roe, P.L., Groth, C.P.T.: Numerical solution of a 10-moment model for nonequilibrium gasdynamics. In: 12th AIAA Computational Fluid Dynamics Conference, San Diego, California; USA, 19–22 June 1995. (AIAA Paper 1995-1677)

  6. Chapman S., Cowling T.G.: The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, third edition. Cambridge University Press, Cambridge (1970)

    Google Scholar 

  7. Chen G.-Q., Levermore C.D., Liu T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47(6), 787–830 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fan J., Boyd I.D., Cai C.-P., Hennighausen K., Candler G.V.: Computation of rarefied gas flows around a NACA 0012 airfoil. AIAA J. 39(4), 618–625 (2001)

    Article  ADS  Google Scholar 

  9. Fletcher, C.A.J.: Computational Techniques for Fluid Dynamics, vol. 1: Fundamental and General Techniques. Springer Series in Computational Physics, 2nd edn. Springer-Verlag, Berlin (1991)

  10. Gad-el-Hak M.: The fluid mechanics of microdevices—The Freeman scholar lecture. J. Fluids Eng. 121(1), 5–33 (1999)

    Article  Google Scholar 

  11. Gombosi T.I.: Gaskinetic Theory. Cambridge Atmospheric and Space Science Series. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  12. Grad H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331–407 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  13. Groth, C.P.T.: Numerical modeling of non-equilibrium micron-scale flows using the Gaussian moment closure. In: 8th Annual Conference of the CFD society of Canada, Montreal (2000)

  14. Groth, C.P.T., Roe, P.L., Gombosi, T.I., Brown, S.L.: On the nonstationary wave structure of a 35-moment closure for rarefied gas dynamics. In: 26th AIAA Fluid Dynamics Conference, San Diego, California; USA, 19–22 June 1995. (AIAA Paper 1995-2312)

  15. Harten A., Lax P.D., Van Leer B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hittinger, J.A.: Foundations for the generalization of the Godunov method to hyperbolic systems with stiff relaxation source terms. PhD thesis, The University of Michigan (2000)

  17. Holway L.H. Jr.: Kinetic theory of shock structure using an ellipsoidal distribution function. In: Leeuw, J.H. (eds) Rarefied Gas Dynamics, Proceedings of the Fourth International Symposium on Rarefied Gas Dynamics, vol. 1, pp. 193–215. Academic Press, New York (1965)

    Google Scholar 

  18. Huynh H.T.: An upwind moment scheme for conservation laws. In: Groth, C., Zingg, D.W. (eds) Computational Fluid Dynamics 2004: Proceedings of the Third International Conference on Computational Fluid Dynamics, ICCFD3, Toronto, 12–16 July 2004, pp. 761–766. Springer-Verlag, Berlin (2006)

    Google Scholar 

  19. Jin S.: Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21(2), 441–454 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Karniadakis, G. Beskok, A. Aluru, N.: Microflows and Nanoflows: Fundamentals and Simulation. Interdisciplinary Applied Mathematics, 1st edn. Springer (2005)

  21. Khieu, L., Suzuki, Y., Van Leer, B.: An analysis of a space-time discontinuous-Galerkin method for moment equations and its solid-boundary treatment. 22–25 June 2009. (AIAA Paper 2009-3874)

  22. Le Tallec P., Perlat J.P.: Boundary conditions and existence results for Levermore’s moments system. Math. Models Methods Appl. Sci. 10(1), 127–152 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Levermore C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83(5-6), 1021–1065 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Levermore C.D., Morokoff W.J.: The Gaussian moment closure for gas dynamics. SIAM J. Appl. Math. 59(1), 72–96 (1998)

    Article  MathSciNet  Google Scholar 

  25. Levermore C.D., Morokoff W.J., Nadiga B.T.: Moment realizability and the validity of the Navier–Stokes equations for rarefied gas dynamics. Phys. Fluids 10(12), 3214–3226 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Linde T.: A practical, general-purpose, two-state HLL Riemann solver for hyperbolic conservation laws. Int. J. Numer. Methods Fluids 40(3–4), 391–402 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. Linde, T.J.: A three-dimensional adaptive multifluid MHD model of the heliosphere. PhD thesis, The University of Michigan (1998)

  28. Liu T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108(1), 153–175 (1987)

    Article  MATH  ADS  Google Scholar 

  29. Lockerby D.A., Reese J.M., Emerson D.R., Barber R.W.: Velocity boundary condition at solid walls in rarefied gas calculations. Phys. Rev. E 70, 017303 (2004)

    Article  ADS  Google Scholar 

  30. Maxwell J.C.: On stresses in rarefied gases arising form inequalities of temperature. Philos. Trans. R. Soc. Lond. 170, 231–256 (1879)

    Article  Google Scholar 

  31. McDonald, J.G., Groth, C.P.T.: Numerical modeling of micron-scale flows using the Gaussian moment closure. In: 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Ontario; Canada, 6–9 June 2005. (AIAA Paper 2005-5035)

  32. McDonald, J.G., Groth, C.P.T.: Extended fluid-dynamic model for micron-scale flows based on Gaussian moment closure. In: 46th AIAA Aerospace Science Meeting and Exhibit, Reno, Nevada; USA, 7–10 Jan 2008. (AIAA Paper 2008-691)

  33. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37, 2nd edn. Springer-Verlag, New York (1998)

  34. Struchtrup H.: Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory. Interaction of Mechanics and Mathematics Series. Springer-Verlag, Berlin (2005)

    Google Scholar 

  35. Struchtrup, H.: Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory. Interaction of Mechanics and Mathematics, 1st edn. Springer (2005)

  36. Sun Q., Boyd I.D.: A direct simulation method for subsonic, microscale gas flows. J. Comput. Phys. 179(2), 400–425 (2002)

    Article  MATH  ADS  Google Scholar 

  37. Sun Y., Wang Z.J., Liu Y.: Spectral (finite) volume method for conservation laws on unstructured grids VI: Extension to viscous flow. J. Comput. Phys. 215(1), 41–58 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. Suzuki, Y.: Discontinuous Galerkin methods for extended hydrodynamics. PhD thesis, The University of Michigan (2008)

  39. Suzuki, Y., Van Leer, B.: Application of the 10-moment model to MEMS flows. In: 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada; USA, 10–13 Jan 2005. (AIAA Paper 2005-1398)

  40. Torrilhon M.: Characteristic waves and dissipation in the 13-moment-case. Continuum Mech. Thermodyn. 12(5), 289–301 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. Vincenti W.G., Kruger C.H. Jr: Introduction to Physical Gas Dynamics. Krieger Publishing Company, Malabar, Florida (1986)

    Google Scholar 

  42. White F.M.: Viscous Fluid Flow. McGraw-Hill Series in Mechanical Engineering. second edition. McGraw-Hill, New York (1991)

    Google Scholar 

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Correspondence to Yoshifumi Suzuki.

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Communicated by M. Torrilhon.

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Suzuki, Y., Khieu, L. & van Leer, B. CFD by first order PDEs. Continuum Mech. Thermodyn. 21, 445–465 (2009). https://doi.org/10.1007/s00161-009-0124-2

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