An upwind local radial basis functions-finite difference (RBF-FD) method for solving compressible Euler equation with application in finite-rate Chemistry

Document Type: Research Paper

Authors

1 Amirkabir University of Technology, Tehran, Iran, Faculty of Mathematics and Computer

2 Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology,

3 Faculty of Basic Sciences, Shahid Sattari Aeronautical University of Sience and Technology, South Mehrabad

10.22052/ijmc.2017.106402.1325

Abstract

The main aim of the current paper is to propose an upwind local radial basis functions-finite
difference (RBF-FD) method for solving compressible Euler equation. The mathematical formulation of chemically reacting, inviscid, unsteady flows with species conservation equations
and finite-rate chemistry is studied. The presented technique is based on the developed idea in
[58]. For checking the ability of the new procedure, the compressible Euler equation is solved.
This equation has been classified in category of system of advection-diffusion equations. The
solutions of advection equations have some shock, thus, special numerical methods should be
applied for example discontinuous Galerkin and finite volume methods. Moreover, two problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

Keywords


  1. V. Bayona, M. Moscoso, M. Carretero and M. Kindelan, RBF-FD formulas and convergence properties, J. Comput. Phys. 229 (22) (2010) 8281–8295.
  2. V. Bayona, M. Moscoso and M. Kindelan, Optimal constant shape parameter for multiquadric based RBF-FD method, J. Comput. Phys. 230 (19) (2011) 7384–7399.
  3. V. Bayona, M. Moscoso and M. Kindelan, Gaussian RBF-FD weights and its corresponding local truncation errors, Eng. Anal. Bound. Elem. 36 (9) (2012) 1361–1369.
  4. V. Bayona, M. Moscoso and M. Kindelan, Optimal variable shape parameter for multiquadric based RBF-FD method, J. Comput. Phys. 231 (6) (2012) 2466–2481.
  5. R. Bellman, B. Kashef and J. Casti, Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, J. Comput. Phys.10 (1) (1972) 40–52.
  6. E. F. Bollig, N. Flyer and G. Erlebacher, Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs, J. Comput. Phys. 231 (21) (2012) 7133–7151.
  7. J.-C. Chassaing, X. Nogueira and S. Khelladi, Moving kriging reconstruction for high-order finite volume computation of compressible flows, Comput. Methods. Appl. Mech. Eng. 253 (2013) 463–478.
  8. S. Chaturantabut, Dimension Reduction for Unsteady Nonlinear Partial Differential Equations via Empirical Interpolation Methods, MSc Thesis, Rice University, 2009.
  9. S. Chaturantabut and D. C. Sorensen, A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J. Numer. Anal. 50 (1) (2012) 46–63.
  10. A. H. D. Cheng, Multiquadric and its shape parameter-A numerical investigation of error estimate, condition number, and round-of-error by arbitrary precision computation, Eng. Anal. Bound. Elem. 36 (2012) 220–239.
  11. B. Dai, B. Zheng, Q. Liang and L. Wang, Numerical solution of transient heat conduction problems using improved meshless local Petrov-Galerkin method, Appl. Math. Comput. 219 (2013) 10044–10052.
  12. L. Dawei, X. Xin, W. Zhi and C. Dehua, Investigation on the reynolds number simulation of supercritical airfoil, in: Digital Manufacturing and Automation (ICDMA), 2013 Fourth International Conference on, IEEE, 2013, pp. 7 40–744.
  13. M. Dehghan and A. Nikpour, Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method, Appl. Math. Model. 37 (18) (2013) 8578–8599.
  14. M. Dehghan and A. Nikpour, The solitary wave solution of coupled Klein–Gordon–Zakharov equations via two different numerical methods, Comput. Phys. Commun. 184 (9) (2013) 2145–2158.
  15. T. A. Driscoll and B. Fornberg, Interpolation in the limit of increasingly flat radial basis functions, Comput. Math. Appl. 43 (3) (2002) 413–422.
  16. J. Du, F. Fang, C. C. Pain, I. Navon, J. Zhu and D. A. Ham, Pod reduced-order unstructured mesh modeling applied to 2d and 3d fluid flow, Comput. Math. Appl. 65 (3) (2013) 362–379.
  17. X. Du, C. Corre and A. Lerat, A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids, J. Comput. Phys. 230 (11) (2011) 4201–4215.
  18. Y. T. Gu, Q. X. Wang and K. Y. Lam, A meshless local Kriging method for large deformation analyses, Comput. Methods Appl. Mech. Eng. 196 (2007) 1673–1684.
  19. Y. T. Gu and G. R. Liu, A local point interpolation method for static and dynamic analysis of thin beams, Comput. Methods Appl. Mech. Eng. 190 (2001) 5515–5528.
  20.  Y. T. Gu, W. Wang, L. C. Zhang and X. Q. Feng, An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields, Eng. Fract. Mech. 78 (2011) 175–190.
  21. F. Fang, C. Pain, I. Navon, G. Gorman, M. Piggott, P. Allison, P. Farrell and A. Goddard, A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows, Ocean Modelling 28 (1) (2009) 127–136.
  22. G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific, Singapore, 2007.
  23.  N. Flyer, E. Lehto, S. Blaise, G. B. Wright and A. St-Cyr, A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere, J. Comput. Phys. 231 (11) (2012) 4078–4095.
  24. B. Fornberg and E. Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys. 230 (6) (2011) 2270–2285.
  25. B. Fornberg, E. Lehto and C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Comput. Math. Appl. 65 (4) (2013) 627–637.
  26. P. Gonzalez-Rodriguez, V. Bayona, M. Moscoso and M. Kindelan, Laurent series based RBF-FD method to avoid ill-conditioning, Eng. Anal. Bound Elem. 52 (2015) 24–31.
  27. R. L. Hardy, Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res. 76 (1971) 1705–1915.
  28. G. Hauke, D. Fuster and F. Lizarraga, Variational multiscale a posteriori error estimation for systems: The Euler and Navier–Stokes equations, Comput. Methods. Appl. Mech. Eng. 283 (2015) 1493–1524.
  29. G. Hu, An adaptive finite volume method for 2D steady Euler equations with WENO reconstruction, J. Comput. Phys. 252 (2013) 591–605.
  30. G. Hu, X. Meng and N. Yi, Adjoint-based an adaptive finite volume method for steady euler equations with non-oscillatory k-exact reconstruction, Comput. Fluids 139 (2016) 174–183.
  31. G. Hu and N. Yi, An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction, J. Comput. Phys. 312 (2016) 235–251.
  32. H. Hu and O. A. Kandil, A hybrid boundary element–finite volume method for unsteady transonic airfoil flows, Eng. Anal. Bound Elem. 14 (2) (1994) 149–157.
  33. S. Isaev, P. Baranov, I. Popov, A. Sudakov and A. Usachov, Improvement of aerodynamic characteristics of a thick airfoil with a vortex cell in sub-and transonic flow, Acta Astronautica 132 (2017) 204–220.
  34. S. Jia, B. Yang, X. Zhao and J. Xu, Numerical simulation of far field acoustics of an airfoil using vortex method and 2-d fw-h equation, in: IOP Conf. Series: Materials Science and Engineering 52 (2013) 022047.
  35. E. J. Kansa, Multiquadrics–A scattered data approximation scheme with applications to computational fluid-dynamics–II, Comput. Math. Appl. 19 (1990) 127–145.
  36. E. J. Kansa, Multiquadrics A scattered data approximation scheme with applications to computational fluid dynamics - II, Comput. Math. Appl. 19 (1990) 147–161.
  37. E. J. Kansa, R. C. Aldredge and Leevan Ling, Numerical simulation of two–dimensional combustion using mesh-free methods, Eng. Anal. Bound. Elem. 33 (2009) 940–950.
  38. V. Kitsios, R. Kotapati, R. Mittal, A. Ooi, J. Soria and D. You, Numerical simulation of lift enhancement on a NACA 0015 airfoil using ZNMF jets, in: Proceedings of the Summer Program, Citeseer, 2006, p. 457.
  39. S. S. Kutanaei, N. Roshan, A. Vosoughi, S. Saghafi, A. Barari and S. Soleimani, Numerical solution of Stokes flow in a circular cavity using mesh-free local RBF-DQ, Eng. Anal. Bound Elem. 36 (5) (2012) 633–638.
  40. X. Li, Y. Liu, J. Kou and W. Zhang, Reduced-order thrust modeling for an efficiently flapping airfoil using system identification method, J. Fluids Struct. 69 (2017) 137–153.
  41. G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer Dordrecht, Berlin, Heidelberg, New York, 2005.
  42. Y. P. Marx, Numerical simulation of turbulent flows around airfoil and wing, in: Proceedings of the Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics, Springer, 1990, pp. 323–332.
  43. K. Nordanger, R. Holdahl, T. Kvamsdal, A. M. Kvarving and A. Rasheed, Simulation of airflow past a 2d NACA 0015 airfoil using an isogeometric incompressible navier–stokes solver with the spalart–allmaras turbulence model, Comput. Methods. Appl. Mech. Eng. 290 (2015) 183–208.
  44. W. Ogana, Transonic integro-differential and integral equations with artificial viscosity, Eng. Anal. Bound Elem.  6 (3) (1989) 129–135.
  45. R. Qin, L. Krivodonova, A discontinuous Galerkin method for solutions of the Euler equations on cartesian grids with embedded geometries, J. Comput. Sci. 4 (1-2) (2013) 24–35.
  46. H. Rahimi, W. Medjroubi, B. Stoevesandt and J. Peinke, 2D numerical investigation of the laminar and turbulent flow over different airfoils using openfoam, in: J. Phys.: Conf. Ser. 555 (2014) 012070.
  47. S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, J. Sci. Comput. 15 (4) (2000) 457–478.
  48. S. S. Ravindran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Int. J. Numer. Methods Fluids 34 (5) (2000) 425–448.
  49. P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (2) (1981) 357–372.
  50. A. Saadatmandi, N. Nafar and S. P. Toufighi, Numerical study on the reaction cum diffusion process in a spherical biocatalyst, Iranian J. Math. Chem. 5 (1) (2014) 47–61.
  51. S. A. Sarra, A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains, Appl. Math. Comput. 218 (19) (2012) 9853–9865.
  52. S. A. Sarra, Regularized symmetric positive definite matrix factorizations for linear systems arising from RBF interpolation and differentiation, Eng. Anal. Bound. Elem. 44 (2014) 76–86.
  53. S. A. Sarra, Radial basis function approximation methods with extended precision floating point arithmetic, Eng. Anal. Bound. Elem. 35 (2011) 68–76.
  54. T. K. Sengupta, A. Bhole and N. Sreejith, Direct numerical simulation of 2d transonic flows around airfoils, Comput. Fluids 88 (2013) 19–37.
  55. V. Shankar, G. B. Wright, R. M. Kirby and A. L. Fogelson, A radial basis function (RBF)-finite difference (FD) method for diffusion and reaction–diffusion equations on surfaces, J. Sci. Comput. 63 (3) (2015) 745–768.
  56. A. Shokri and M. Dehghan, Meshless method using radial basis functions for the numerical solution of two-dimensional complex Ginzburg-Landau equation, Comput. Model. Eng. Sci. 34 (2012) 333–358 .
  57. C. Shu, Differential Quadrature and its Application in Engineering, Springer Science & Business Media, 2012.
  58. C. Shu, H. Ding, H. Chen and T. Wang, An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput. Methods. Appl. Mech. Eng. 194 (18) (2005) 2001–2017.
  59. C. Shu, H. Ding and K. Yeo, Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations, Comput. Methods. Appl. Mech. Eng. 192 (7) (2003) 941–954.
  60. C. Shu, H. Ding and K. Yeo, Solution of partial differential equations by a global radial basis function-based differential quadrature method, Eng. Anal. Bound. Elem. 28 (10) (2004) 1217–1226.
  61. R. tef nescu and I. M. Navon, POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model, J. Comput. Phys. 237 (2013) 95–114.
  62. A. I. Tolstykh, On using RBF-based differencing formulas for unstructured and mixed structured-unstructured grid calculations, in: Proceedings of the 16th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation, Lausanne, Switzerland, 2000, p. 6.
  63. F. Tornabene, N. Fantuzzi, M. Bacciocchi, A. M. Neves and A. J. Ferreira, MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells, Composites Part B: Engineering 99 (2016) 30–47.
  64. Q. Wang, Y.-X. Ren and W. Li, Compact high order finite volume method on unstructured grids ii: Extension to two-dimensional euler equations, J. Comput. Phys. 314 (2016) 883–908.
  65. A.-M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer Science & Business Media, 2010.
  66. H. Wendland, Scattered Data Approximation, in: Cambridge Monograph on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
  67. B. Xie, X. Deng, Z. Sun and F. Xiao, A hybrid pressure–density-based mach uniform algorithm for 2D Euler equations on unstructured grids by using multi-moment finite volume method, J. Comput. Phys. 335 (2017) 637–663.
  68. L. W. Zhang, The IMLS-Ritz analysis of laminated CNT-reinforced composite quadrilateral plates subjected to a sudden transverse dynamic load, Composite Structures 180 (2017) 638–646.
  69. G. Zhang, L. Ji and X. Hu, Vortex-induced vibration for an isolated circular cylinder under the wake interference of an oscillating airfoil: Part ii. single degree of freedom, Acta Astronautica 133 (2017) 311–323.
  70. M. Zhao, M. Zhang and J. Xu, Numerical simulation of flow characteristics behind the aerodynamic performances on an airfoil with leading edge protuberances, Eng. Appl. Comput. Fluid Mech. 11 (1) (2017) 193–209.