Abstract
The characteristic boundary conditions are applied and assessed for the solution of incompressible inviscid flows. The two-dimensional incompressible Euler equations based on the artificial compressibility method are considered and then the characteristic boundary conditions are formulated in the generalized curvilinear coordinates and implemented on both the far-field and wall boundaries. A fourth-order compact finite-difference scheme is used to discretize the resulting system of equations. The solution methodology adopted is more suitable for this assessment because the Euler equations and the high-accurate numerical scheme applied are quite sensitive to the treatment of boundary conditions. Two benchmark test cases are computed to investigate the accuracy and performance of the characteristic boundary conditions implemented compared to the simplified boundary conditions. The sensitivity of the solution obtained by applying the characteristic boundary conditions to the different numerical parameters is also studied. Indications are that the characteristic boundary conditions applied improve the accuracy and the convergence rate of the solution compared to the simplified boundary conditions.
Acknowledgements
The authors would like to thank Sharif University of Technology for the support of this research.
References
[1] Y. Adam, A Hermitian finite difference method for the solution of parabolic equations Comput. Math. Appl. 1 (1975), 393–406. 10.1016/0898-1221(75)90041-3Search in Google Scholar
[2] T. Amro, A. Arnold and M. Ehrhardt, Optimized far field boundary conditions for hyperbolic systems, preprint BUW-IMACM 12.02 (2012). Search in Google Scholar
[3] A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. 33 (1980), no. 6, 707–725. 10.1002/cpa.3160330603Search in Google Scholar
[4] E. Bécache and A. Prieto, Remarks on the stability of Cartesian PMLs in corners, Appl. Numer. Math. 62 (2012), no. 11, 1639–1653. 10.1016/j.apnum.2012.05.003Search in Google Scholar
[5] J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114 (1994), no. 2, 185–200. 10.1006/jcph.1994.1159Search in Google Scholar
[6] M. H. Carpenter, D. Gottlieb and S. Abarbanel, Stable and accurate boundary treatments for compact, high-order finite-difference schemes, Appl. Numer. Math. 12 (1993), no. 1–3, 55–87. 10.1016/0168-9274(93)90112-5Search in Google Scholar
[7] S. R. Chakravarthy, Euler equations-implicit schemes and boundary conditions, AIAA J. 21 (1983), no. 5, 699–706. 10.2514/6.1982-228Search in Google Scholar
[8] A. J. Chorin, A numerical method for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967), no. 1, 12–26. 10.1016/0021-9991(67)90037-XSearch in Google Scholar
[9] T. Colonius, Modeling artificial boundary conditions for compressible flow, Annu. Rev. Fluid Mech. 36 2004, 315–345. 10.1146/annurev.fluid.36.050802.121930Search in Google Scholar
[10] R. Corral, A. Escribano, F. Gisbert, A. Serrano and C. Vasco, Validation of a linear multigrid accelerated unstructured Navier–Stokes solver for the computation of turbine blades on hybrid grids, AIAA Paper 3326 (2003), no. 9. 10.2514/6.2003-3326Search in Google Scholar
[11] J. R. Dea, An experimental adaptation of Higdon-type non-reflecting boundary conditions to linear first-order systems, J. Comput. Appl. Math. 235 (2011), no. 5, 1354–1366. 10.1016/j.cam.2010.08.023Search in Google Scholar
[12] D. Drikakis, P. A. Govatsos and D. E. Papantonis, A characteristic-based method for incompressible flows, Internat. J. Numer. Methods Fluids 19 (1994), no. 8, 667–685. 10.1002/fld.1650190803Search in Google Scholar
[13] K. Duru and G. Kreiss, On the accuracy and stability of the perfectly matched layer in transient waveguides, J. Sci. Comput. 53 (2012), no. 3, 642–671. 10.1007/s10915-012-9594-7Search in Google Scholar
[14] B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math. 4 (1988), no. 1, 21–45. 10.1016/S0168-9274(88)80004-7Search in Google Scholar
[15] B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Natl. Acad. Sci. USA 74 (1977), no. 5, 1765–1766. 10.1073/pnas.74.5.1765Search in Google Scholar
[16] B. Engquist and A. Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math. 32 (1979), no. 3, 314–358. 10.1002/cpa.3160320303Search in Google Scholar
[17] D. V. Gaitonde, J. Shang and J. L. Young, Practical aspects of higher-order numerical schemes for wave propagation phenomena, Internat. J. Numer. Methods Engrg. 45 (1999), no. 12, 1849–1869. 10.1002/(SICI)1097-0207(19990830)45:12<1849::AID-NME657>3.0.CO;2-4Search in Google Scholar
[18] A. Gelb, Z. Jackiewicz and B. D. Welfert, Absorbing boundary conditions of the second order for the pseudospectral Chebyshev methods for wave propagation, J. Sci. Comput. 17 (2002), no. 1–4, 501–512. 10.1023/A:1015158227243Search in Google Scholar
[19] M. B. Giles, Nonreflecting boundary conditions for Euler equation calculations, AIAA J. 28 (1990), no. 12, 2050–2058. 10.2514/3.10521Search in Google Scholar
[20] D. Givoli and B. Neta, High-order nonreflecting boundary conditions for the dispersive shallow water equations, J. Comput. Appl. Math. 158 (2003), no. 1, 49–60. 10.1016/S0377-0427(03)00462-XSearch in Google Scholar
[21] D. Givoli and B. Neta, High-order nonreflecting boundary scheme for time-dependent waves, J. Comput. Phys. 186 (2003), no. 1, 24–46. 10.1016/S0021-9991(03)00005-6Search in Google Scholar
[22] T. Hagstrom and S. I. Hariharan, A formulation of asymptotic and exact boundary conditions using local operators, Appl. Numer. Math. 27 (1998), no. 4, 403–416. 10.1016/S0168-9274(98)00022-1Search in Google Scholar
[23] M. Y. Hashemi and K. Zamzamian, Efficient and non-reflecting far-field boundary conditions for incompressible flow calculations, Appl. Math. Comput. 230 (2014), 248–258. 10.1016/j.amc.2013.12.089Search in Google Scholar
[24] M. E. Hayder, F. Q. Hu and M. Y. Hussaini, Toward perfectly absorbing boundary conditions for Euler equations, AIAA J. 37 (1999), no. 8, 912–918. 10.2514/6.1997-2075Search in Google Scholar
[25] G. W. Hedstrom, Nonreflecting boundary conditions for nonlinear hyperbolic systems, J. Comput. Phys. 30 (1979), no. 2, 222–237. 10.1016/0021-9991(79)90100-1Search in Google Scholar
[26] K. Hejranfar and R. Kamali-Moghadam, Preconditioned characteristic boundary conditions for solution of the preconditioned Euler equations at low Mach number flows, J. Comput. Phys. 231 (2012), no. 12, 4384–4402. 10.1016/j.jcp.2012.01.040Search in Google Scholar
[27] D. Heubes, A. Bartel and M. Ehrhardt, Characteristic boundary conditions in the lattice Boltzmann method for fluid and gas dynamics, J. Comput. Appl. Math. 262 (2014), 51–61. 10.1016/j.cam.2013.09.019Search in Google Scholar
[28] R. L. Higdon, Radiation boundary conditions for dispersive waves, SIAM J. Numer. Anal. 31 (1994), no. 1, 64–100. 10.1137/0731004Search in Google Scholar
[29] R. S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19 (1975), no. 1, 90–109. 10.1016/0021-9991(75)90118-7Search in Google Scholar
[30] K. A. Hoffmann and S. T. Chiang, Computational Fluid Dynamics. Vol. 1, Engineering Education System, Wichita, 2000. Search in Google Scholar
[31] F. Q. Hu, Absorbing boundary conditions, Int. J. Comput. Fluid Dyn. 18 (2004), no. 6, 513–522. 10.1080/10618560410001673524Search in Google Scholar
[32] M. Huet, One-dimensional characteristic boundary conditions using nonlinear invariants, J. Comput. Phys. 283 (2015), 312–328. 10.1016/j.jcp.2014.12.010Search in Google Scholar
[33] A. Jameson, W. Schmidt and E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge–Kutta time-stepping schemes, AIAA Paper 1259 (1981), urlhttps://doi.org/10.2514/6.1981-1259. 10.2514/6.1981-1259Search in Google Scholar
[34] J. W. Kim and D. J. Lee, Generalized characteristic boundary conditions for computational aeroacoustics, AIAA J. 38 (2000), no. 11, 2040–2049. 10.2514/2.891Search in Google Scholar
[35] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), no. 1, 16–42. 10.1016/0021-9991(92)90324-RSearch in Google Scholar
[36] K. Mahesh, A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys. 145 (1998), no. 1, 332–358. 10.1006/jcph.1998.6022Search in Google Scholar
[37] M. Meinke, W. Schröder, E. Krause and T. Rister, A comparison of second-and sixth-order methods for large-eddy simulations, Comput. & Fluids 31 (2002), no. 4, 695–718. 10.1016/S0045-7930(01)00073-1Search in Google Scholar
[38] W. Polifke, C. Wall and P. Moin, Partially reflecting and non-reflecting boundary conditions for simulation of compressible viscous flow, J. Comput. Phys. 213 (2006), no. 1, 437–449. 10.1016/j.jcp.2005.08.016Search in Google Scholar
[39] S. E. Rogers and D. Kwak, An upwind differencing scheme for the incompressible Navier–Stokes equations, Appl. Numer. Math. 8 (1991), no. 1, 43–64. 10.1016/0168-9274(91)90097-JSearch in Google Scholar
[40] Z.-S. Sun, Y.-X. Ren, B.-L. Zha and S.-Y. Zhang, High order boundary conditions for high order finite difference schemes on curvilinear coordinates solving compressible flows, J. Sci. Comput. 65 (2015), no. 2, 790–820. 10.1007/s10915-015-9988-4Search in Google Scholar
[41] K. W. Thompson, Time-dependent boundary conditions for hyperbolic systems, J. Comput. Phys. 68 (1987), no. 1, 1–24. 10.1016/0021-9991(87)90041-6Search in Google Scholar
[42] K. W. Thompson, Time-dependent boundary conditions for hyperbolic systems. II, J. Comput. Phys. 89 (1990), no. 2, 439–461. 10.1016/0021-9991(90)90152-QSearch in Google Scholar
[43] S. V. Tsynkov and V. N. Vatsa, Improved treatment of external boundary conditions for three-dimensional flow computations, AIAA J. 36 (1998), no. 11, 1998–2004. 10.2514/2.327Search in Google Scholar
[44] M. R. Visbal and D. V. Gaitonde, High-order-accurate methods for complex unsteady subsonic flows, AIAA J. 37 (1999), no. 10, 1231–1239. 10.2514/2.591Search in Google Scholar
[45] D. L. Whitfield and L. K. Taylor, Numerical solution of the two-dimensional time-dependent incompressible Euler equations, Technical Report NASA-CR-195775, 1994. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
