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Low Mach preconditioned density-based methods with implicit Runge–Kutta schemes in physical-time

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Abstract

As far as the authors are aware, low Mach preconditioned density-based methods found in the peer-reviewed literature only employ multi-step schemes for physical-time integration. This essentially limits the maximum achievable temporal accuracy-order of these methods to two, since the multi-step schemes of order higher than two are conditionally stable. However, the present paper shows how these methods can employ multi-stage schemes in physical-time using the same low Mach preconditioning techniques developed over the past few decades. In doing so, it opens up the rich field of Runge–Kutta time integration schemes to low Mach preconditioned density-based methods. One and two-dimensional test cases are used to demonstrate the capabilities of this novel approach. The former simulates the propagation of marginally stable entropy perturbations superposed on a uniform flow whereas the latter simulates the temporal growth of vorticity perturbations superposed on an absolutely unstable planar mixing-layer. A novel procedure is employed to generate highly accurate initial conditions for the two-dimensional test case, it minimizes receptivity regions as well as deleterious interactions with artificial boundary conditions due to the numerical error introduced by approximate initial conditions. These test cases show that second, third and fourth order multi-stage schemes with strong linear numerical stability can be successfully utilized for the physical-time integration of low Mach preconditioned density-based methods.

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References

  1. Alexander R (1977) Diagonally implicit Runge-Kutta methods for stiff O.D.E. SIAM J Numer Anal 14(6):1006–1021

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves LSB, Kelly RE, Karagozian AR (2008) Transverse jet shear layer instabilities. part ii: linear analysis for large jet-to-crossflow velocity ratios. J Fluid Mech 602:383–401

    Article  MathSciNet  MATH  Google Scholar 

  3. Alves LSB (2009) Review of numerical methods for the compressible flow equations at low mach numbers, In: XII Encontro de Modelagem Computacional, Rio de Janeiro, RJ, Brazil

  4. Alves LSB (2010) Dual time stepping with multi-stage schemes in physical time for unsteady low mach number compressible flows, In: VII Escola de Primavera de Transição e Turbulência, ABCM, Ilha Solteira, São Paulo, Brasil

  5. Teixeira RSD, Alves LSDB (2017) Minimal gain marching schemes: searching for unstable steady-states with unsteady solvers. Theo Comput Fluid Dyn 31(5–6):607–621

    Article  Google Scholar 

  6. Ascher UM, Ruuth SJ, Spiteri RJ (1997) Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl Numer Math 25:151–167

    Article  MathSciNet  MATH  Google Scholar 

  7. Bijl H, Wesseling P (1998) A unified method for computing incompressible and compressible flows in boundary-fitted coordinates. J Comput Phys 141(2):153–173

    Article  MathSciNet  MATH  Google Scholar 

  8. Bijl H, Carpenter MH, Vatsa VN, Kennedy CA (2002) Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: laminar flow. J Comput Phys 179:313–329

    Article  MATH  Google Scholar 

  9. Buelow PEO, Venkateswaran S, Merkle CL (1994) Effect of grid aspect ratio on convergence. AIAA J 32(12):2402–2408

    Article  MATH  Google Scholar 

  10. Buelow PEO, Venkateswaran S, Merkle CL (2001) Stability and convergence analysis of implicit upwind schemes. Comput Fluids 30:961–988

    Article  MathSciNet  MATH  Google Scholar 

  11. Bukhvostova A, Kuerten JGM, Geurts BJ (2015) Low Mach number algorithm for droplet-laden turbulent channel flow including phase transition. J Comput Phys 295:420–437

    Article  MathSciNet  MATH  Google Scholar 

  12. Butcher JC (2008) Numerical methods for ordinary differential equations. John Wiley & Sons Inc, England

    Book  MATH  Google Scholar 

  13. Cash JR (1979) Diagonally implicit Runge-Kutta formulae with error estimates. IMA J Appl Math 24(3):293–301

    Article  MATH  Google Scholar 

  14. Choi YH, Merkle CL (1993) The application of preconditioning in viscous flows. J Comput Phys 105:207–223

    Article  MathSciNet  MATH  Google Scholar 

  15. Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems. J Comput Phys 2:12–26

    Article  MathSciNet  MATH  Google Scholar 

  16. Chorin AJ (1968) Numerical solution of the Navier-Stokes equations. Math Comput 22(104):745–762

    Article  MathSciNet  MATH  Google Scholar 

  17. Dahlquist G (1963) A special stability problem for linear multistep methods. BIT Numer Math 3:27–43

    Article  MathSciNet  MATH  Google Scholar 

  18. Darmofal DL, Schmid PJ (1996) The importance of eigenvectors for local preconditioners of the Euler equations. J Comput Phys 127:346–362

    Article  MathSciNet  MATH  Google Scholar 

  19. Denaro FM (2003) On the application of the helmholtzñhodge decomposition in projection methods for incompressible flows with general boundary conditions. Int J Numer Methods Fluids 43:43–69

    Article  MathSciNet  MATH  Google Scholar 

  20. Ferracina L, Spijker MN (2008) Strong stability of singly-diagonally-implicit runge-kutta methods. Appl Numer Math 58:1675–1686

    Article  MathSciNet  MATH  Google Scholar 

  21. Folkner D, Katz A, Sankaran V (2015) An unsteady preconditioning scheme based on convective-upwind split-pressure artificial dissipation. Int J Numer Methods Fluids 78(1):1–16

    Article  MathSciNet  Google Scholar 

  22. Frochte J, Heinrichs W (2009) A splitting technique of higher order for the Navier-Stokes equations. J Comput Appl Math 228:373–390

    Article  MathSciNet  MATH  Google Scholar 

  23. Germanos RAC, Souza LF, Medeiros MAF (2009) Numerical investigation of the three-dimensional secondary instabilities in the time-developing compressible mixing layer. J Brazil Soc Mech Sci Eng 31(2):125–136

    Article  Google Scholar 

  24. Greene PT, Eldredge JD, Zhong X, Kim J (2016) A high-order multi-zone cut-stencil method for numerical simulations of high-speed flows over complex geometries. J Comput Phys 316:652–681

    Article  MathSciNet  MATH  Google Scholar 

  25. Guerra J, Gustafsson B (1986) A numerical method for incompressible and compressible flow problems with smooth solutions. J Comput Phys 63:377–397

    Article  MathSciNet  MATH  Google Scholar 

  26. Guermond JL, Minev P, Shen J (2006) An overview of projection methods for incompressible flows. Comput Methods Appl Mech Eng 195:6011–6045

    Article  MathSciNet  MATH  Google Scholar 

  27. Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressible flow of fluids with free surface. Phys Fluids 8:2182–2189

    Article  MathSciNet  MATH  Google Scholar 

  28. Harlow FH, Amsden AA (1971) A numerical fluid dynamics calculation method for all flow speeds. J Comput Phys 8(2):197–213

    Article  MATH  Google Scholar 

  29. Hejranfar K, Kamali-Moghadam R (2012) Assessment of three preconditioning schemes for solution of the two-dimensional euler equations at low mach number flows. Int J Numer Methods Eng 89:20–52

    Article  MathSciNet  MATH  Google Scholar 

  30. van der Heul DR, Vuik C, Wesseling P (2003) A conservative pressure-correction method for flow at all speeds. Comput Fluids 32:1113–1132

    Article  MathSciNet  MATH  Google Scholar 

  31. Ho CM, Huerre P (1984) Perturbed free shear layers. Ann Rev Fluid Mech 16:365–424

    Article  Google Scholar 

  32. Hosangadi A, Merkle CL, Turns SR (1990) Analysis of forced combusting jets. AIAA J 28(8):1473–1480

    Article  Google Scholar 

  33. Hou Y, Mahesh K (2005) A robust, colocated, implicit algorithm for direct numerical simulation of compressible, turbulent flows. J Comput Phys 205:205–221

    Article  MATH  Google Scholar 

  34. Issa RI, Gosman AD, Watkins AP (1986) The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme. J Comput Phys 62(1):66–82

    Article  MathSciNet  MATH  Google Scholar 

  35. Jothiprasad G, Mavriplis DJ, Caughey DA (2003) Higher-order time integration schemes for the unsteady Navier-Stokes equations on unstructured meshes. J Comput Phys 191(2):542–566

    Article  MATH  Google Scholar 

  36. Karki KC, Patankar SV (1989) Pressure-based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J 27(9):1167–1174

    Article  Google Scholar 

  37. Kelly RE, Alves LSB (2008) A uniformly valid asymptotic solution for the transverse jet and its linear stability analysis, Philosophical Transactions of the Royal Society of London. Series A Math Phys Sci 366:2729–2744

    MATH  Google Scholar 

  38. Kennedy CA, Carpenter MH (2003) Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl Numer Math 44:139–181

    Article  MathSciNet  MATH  Google Scholar 

  39. Keshtiban IJ, Belblidia F, Webster MF (2004) Compressible flow solvers for low Mach number flows: a review, Department of computer science at university of wales, swansea. Institute of Non-Newtonian Fluid Mechanics, u.k.

    Google Scholar 

  40. Klainerman S, Majda A (1982) Compressible and incompressible fluids. Commun Pure Appl Math 35:629–651

    Article  MathSciNet  MATH  Google Scholar 

  41. Laney CB (1998) Computational gas dynamics. Cambridge University Press, United Kingdom

    MATH  Google Scholar 

  42. Lardjane N, Fedioun I, Gokalp I (2004) Accurate initial conditions for the direct numerical simulation of temporal compressible binary shear layers with high density ratio. Comput Fluids 33:549–576

    Article  MATH  Google Scholar 

  43. Lee SH (2005) Convergence characteristics of preconditioned Euler equations. J Comput Phys 208(1):266–288

    Article  MATH  Google Scholar 

  44. Lee SH (2007) Cancellation problem of preconditioning method at low Mach numbers. J Comput Phys 225(2):1199–1210

    Article  MATH  Google Scholar 

  45. Lomax H, Pulliam TH, Zingg DW (2001) Fudamentals od computational fluid dynamics. Scientific computation. Springer & Verlag, Berlin

    MATH  Google Scholar 

  46. Merkle CL, Athavale M (1987) A time accurate unsteady incompressible algorithm based on artificial compressibility. AIAA Conf Paper 87–1137:397–407

    Google Scholar 

  47. Merkle CL, Choi YH (1987) Computation of low-speed flow with heat addition. AIAA J 25:831–838

    Article  Google Scholar 

  48. Merkle CL, Choi YH (1988) Computation of low speed compressible flows with time-marching methods. Int J Numer Methods Eng 25:293–311

    Article  MATH  Google Scholar 

  49. Müller B (1999) Low Mach Number Asymptotics of the Navier-Stokes Equations and Numerical Implications, lecture series 1999-03 Edition, von Karman Institute for Fluid Dynamics

  50. Panton RL (2003) Incompressible flow, 4th edn. Wiley & Sons, Hoboken

    MATH  Google Scholar 

  51. Pareschi L, Russo G (2005) Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J Sci Comput 25(1):129–155

    Article  MathSciNet  MATH  Google Scholar 

  52. Patankar SV, Spalding DB (1972) A calculation procedure for heat, mass, and momentum transfer in three-dimensional parabolic flow. Int J Heat Mass Trans 15:1787–1806

    Article  MATH  Google Scholar 

  53. Paolucci S (1982) On the filtering of sound from the Navier-Stokes equations, Tech. Rep. Technical Report SAND 82-8257, Sandia National Laboratories

  54. Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Numerical recipes in FORTRAN 77: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  55. Rai MM (1987) Navier-stokes simulations of blade-vortex interaction using high-order accurate upwind schemes. AIAA Conf Paper 87–0543:1–25

    Google Scholar 

  56. Rossow CC (2003) A blended pressure/density based method for the computation of incompressible and compressible flows. J Comput Phys 185(2):375–398

    Article  MathSciNet  MATH  Google Scholar 

  57. Sandham ND, Reynolds WC (1989) Compressible mixing layer: linear theory and direct simulation. AIAA J 28(4):618–624

    Article  Google Scholar 

  58. Sandham ND, Reynolds WC (1991) Three-dimensional simulations of large eddies in the compressible mixing layer. J Fluid Mech 224:133–158

    Article  MATH  Google Scholar 

  59. Schwer DA (1999) Numerical study of unsteadiness in non-reacting and reacting mixing layers, Ph.D. thesis, Pennsylvania State University

  60. Shuen JS, Chen KH, Choi YH (1993) A coupled implicit method for chemical non-equilibrium flows at all speeds. J Comput Phys 106:306–318

    Article  MATH  Google Scholar 

  61. Shukla RK, Zhong X (2005) Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation. J Comput Phys 204:404–429

    Article  MathSciNet  MATH  Google Scholar 

  62. Suh J, Frankel SH, Mongeau L, Plesniak MW (2006) Compressible large eddy simulations of wall-bounded turbulent flows using a semi-implicit numerical scheme for low Mach number aeroacoustics. J Comput Phys 215(2):526–551

    Article  MathSciNet  MATH  Google Scholar 

  63. Teixeira RS, Alves LSB (2012) Modeling far field entrainment in compressible flows. Int J Comput Fluid Dyn 26:67–78

    Article  MathSciNet  Google Scholar 

  64. Teixeira PC, Alves LSB (2015) Modeling supercritical heat transfer in compressible fluids. Int J Therm Sci 88:267–278

    Article  Google Scholar 

  65. Turkel E (1987) Preconditioned methods for solving the incompressible and low speed compressible equations. J Comput Phys 72(2):277–298

    Article  MATH  Google Scholar 

  66. Turkel E (1993) Review of preconditioning methods for fluid dynamics. Appl Numer Math 12:257–284

    Article  MathSciNet  MATH  Google Scholar 

  67. Turkel E (1999) Preconditioning techniques in computational fluid dynamics. Ann Rev Fluid Mech 31:385–416

    Article  MathSciNet  Google Scholar 

  68. Turkel E, Vatsa VN (2005) Local preconditioners for steady and unsteady flow applications. ESAIM Math Model Numer Anal 39(3):515–535

    Article  MathSciNet  MATH  Google Scholar 

  69. Venkateswaran S, Merkle CL (1999) Analysis of Preconditioning Methods for the Euler and Navier-Stokes Equations, Von Karman Institute Lecture Series

  70. Wang L, Mavriplis DJ (2007) Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations. J Comput Phys 225(2):1994–2015

    Article  MathSciNet  MATH  Google Scholar 

  71. Weiss JM, Smith WA (1995) Preconditioning applied to variable and constant density flows. AIAA J 33(11):2050–2057

    Article  MATH  Google Scholar 

  72. Wolfram S (2003) The mathematica book, 5th edn. Cambridge University Press, New York, Wolfram Media

    MATH  Google Scholar 

  73. Yoh JJ, Zhong X (2004) New hybrid Runge-Kutta methods for unsteady reactive flow simulation. AIAA J 42(8):1593–1600

    Article  Google Scholar 

  74. Yoh JJ, Zhong X (2004) New hybrid Runge-Kutta methods for unsteady reactive flow simulation: applications. AIAA J 42(8):1601–1611

    Article  Google Scholar 

  75. Zhong X (1996) Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows. J Comput Phys 128:19–31

    Article  MathSciNet  MATH  Google Scholar 

  76. Zhong X, Tatineni M (2003) High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition. J Comput Phys 190(2):419–458

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author would like to thank CNPq for the financial support through grants 480599/2007-6, 305031/ 2010-4 and 312255/2013-6 as well as FAPERJ through grant E-26/103.254/2011. Some of the data in this paper has been presented at the AIAA Aviation conference (2014-3085).

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Authors and Affiliations

  1. PGMEC/TEM/UFF, Rua Passo da Pátria 156, Bloco E, Sala 211, São Domingos, Niterói, RJ, 24210-240, Brazil

    Leonardo Santos de Brito Alves & Ricardo Dias dos Santos

  2. PGMEC/UFSM, Cidade Universitária - Camobi, Av. Roraima 1000, Santa Maria, RS, 97105-900, Brazil

    Carlos Eduardo Guex Falcão

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  1. Leonardo Santos de Brito Alves
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  2. Ricardo Dias dos Santos
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Correspondence to Leonardo Santos de Brito Alves.

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Alves, L.S.B., Santos, R.D. & Falcão, C.E.G. Low Mach preconditioned density-based methods with implicit Runge–Kutta schemes in physical-time. J Braz. Soc. Mech. Sci. Eng. 43, 341 (2021). https://doi.org/10.1007/s40430-021-03055-9

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