Computational Methods for Unsteady Flows

  • P. G. Tucker
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 104)


Computational methods for unsteady flows are discussed. These range from standard to modern advanced approaches. An extensive overview of temporal discretizations is given. Then adaptive time stepping approaches are outlined, including adjoint based methods. Spatial schemes are discussed, including modern higher order and resolution approaches. Numerical techniques for both density and pressure-based solvers are considered. The critical issue of numerical smoothing and its control are outlined. Also, the strong relationship between grid topology and solution accuracy is considered for a range of numerical schemes. Simultaneous equations solvers and also boundary conditions are discussed. For the latter there is a strong focus on non-reflective conditions. A survey of work suggests that even though a wide range of schemes is found, just small subsets of these find practical use.


Adjoint Numerical methods Non-reflecting boundary conditions High order High resolution Grid topology Convective schemes Temporal schemes Adaptive time stepping 


  1. A. Agarwal, P.J. Morris, Direct simulation of acoustic scattering on a rotorcraft fuselage, in Proceedings of 6th AIAA/CEAS Aeroacoustics Conference, Lahaina, Hawaii, 12–14 June 2000. AIAA 2000–2030 Google Scholar
  2. Y. Allaneau, A. Jameson, Direct numerical simulations of a two-dimensional viscous flow in a shocktube using a kinetic energy preserving scheme, in Proceedings of 19th AIAA Computational Fluid Dynamics, San Antonio, Texas, 22–25 June 2010a. AIAA 2009–3797 Google Scholar
  3. Y. Allaneau, A. Jameson, Direct numerical simulations of plunging airfoils, in Proceedings of 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 4–7 January 2010b. AIAA 2010–728 Google Scholar
  4. A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1(1), 119–143 (1966) zbMATHGoogle Scholar
  5. U.M. Ascher, L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, vol. 61 (SIAM, Philadelphia, 1998) zbMATHGoogle Scholar
  6. G. Ashcroft, X. Zhang, A computational investigation of the noise radiated by flow-induced cavity oscillations. AIAA Pap. 512 (2001) Google Scholar
  7. H.L. Atkins, D.P. Lockard, A high-order method using unstructured grids for the aeroacoustic analysis of realistic aircraft configurations. AIAA Pap. 99–1945 (1999) Google Scholar
  8. I.E. Barton, R. Kirby, Finite difference scheme for the solution of fluid flow problems on non-staggered grids. Int. J. Numer. Methods Fluids 33(7), 939–959 (2000) MathSciNetzbMATHGoogle Scholar
  9. R.M. Beam, R.F. Warming, An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. J. Comput. Phys. 22(1), 87–110 (1976) MathSciNetzbMATHGoogle Scholar
  10. R.M. Beam, R.F. Warming, Implicit numerical methods for the compressible Navier-Stokes and Euler equations, in Computational Fluid Dynamics. Lecture Series, vol. 1, Von Karman Institute for Fluid Dynamics, Belgium, 29 March–2 April 1982 Google Scholar
  11. B.C. Bell, K.S. Surana, A space–time coupled p-version least-squares finite element formulation for unsteady fluid dynamics problems. Int. J. Numer. Methods Eng. 37(20), 3545–3569 (1994) MathSciNetzbMATHGoogle Scholar
  12. A. Birkefeld, C.D. Munz, Simulations of airfoil noise with the discontinuous Galerkin solver NoisSol. ERCOFTAC Bull. 90, 28–33 (2012) Google Scholar
  13. N.J. Bisek, D.P. Rizzetta, J. Poggie, Plasma control of a turbulent shock boundary-layer interaction. AIAA J. (2013). doi: 10.2514/1.J052248 Google Scholar
  14. G.A. Blaisdell, E.T. Spyropoulos, J.H. Qin, The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Numer. Math. 21(3), 207–219 (1996) MathSciNetzbMATHGoogle Scholar
  15. C. Bogey, C. Bailly, A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J. Comput. Phys. 194(1), 194–214 (2004) zbMATHGoogle Scholar
  16. D.L. Book, J.P. Boris, K. Hain, Flux-corrected transport II: generalizations of the method. J. Comput. Phys. 18(3), 248–283 (1975) zbMATHGoogle Scholar
  17. J.P. Boris, D.L. Book, Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11(1), 38–69 (1973) zbMATHGoogle Scholar
  18. A. Brandt, Multilevel adaptive computations in fluid dynamics. AIAA J. 18(10), 1165–1172 (1980) MathSciNetzbMATHGoogle Scholar
  19. K.E. Brenan, S.L.V. Campbell, L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (SIAM, Philadelphia, 1996) zbMATHGoogle Scholar
  20. W. Briley, H. McDonald, Solution of the three-dimensional compressible Navier-Stokes equations by an implicit technique, in Proceedings of the 4th International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics (Springer, Berlin, 1975), pp. 105–110 Google Scholar
  21. T. Broeckhoven, J. Ramboer, S. Smirnov, C. Lacor, Numerical methods, in Large-Eddy Simulation for Acoustics, ed. by C. Wagner, T. Huttl, P. Sagaut (Cambridge University Press, Cambridge, 2007) Google Scholar
  22. J. Burmeister, G. Horton, Time-parallel multigrid solution of the Navier-Stokes equations. Multigrid Methods III 98, 155–166 (1991) MathSciNetGoogle Scholar
  23. T.D. Butler, LINC method extensions, in Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics (Springer, Berlin, 1971), pp. 435–440 Google Scholar
  24. M.S. Campobasso, M.B. Giles, Effects of flow instabilities on the linear analysis of turbomachinery aeroelasticity. J. Propuls. Power 19(2), 250–259 (2003) Google Scholar
  25. M.S. Campobasso, M.B. Giles, Stabilization of linear flow solver for turbomachinery aeroelasticity using recursive projection method. AIAA J. 42(9), 1765–1774 (2004) Google Scholar
  26. M. Chapman, FRAM—nonlinear damping algorithms for the continuity equation. J. Comput. Phys. 44(1), 84–103 (1981) zbMATHGoogle Scholar
  27. G. Chesshire, W.D. Henshaw, Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys. 90(1), 1–64 (1990) MathSciNetzbMATHGoogle Scholar
  28. Y. Choi, C. Merkle, Time-derivative preconditioning for viscous flows, in 22nd AIAA, Fluid Dynamics, Plasma Dynamics and Lasers Conference (1991). AIAA–91–1652 Google Scholar
  29. A.J. Chorin, A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2(1), 12–26 (1967) zbMATHGoogle Scholar
  30. F.K. Chow, P. Moin, A further study of numerical errors in large-eddy simulations. J. Comput. Phys. 184(2), 366–380 (2003) zbMATHGoogle Scholar
  31. Y.M. Chung, P.G. Tucker, Accuracy of higher-order finite difference schemes on nonuniform grids. AIAA J. 41(8), 1609–1611 (2003) Google Scholar
  32. T. Colonius, S.K. Lele, Computational aeroacoustics: progress on nonlinear problems of sound generation. Prog. Aerosp. Sci. 40(6), 345–416 (2004) Google Scholar
  33. W.P. Crowley, Second-order numerical advection. J. Comput. Phys. 1(4), 471–484 (1967) zbMATHGoogle Scholar
  34. F. Daude, J. Berland, T. Emmert, P. Lafon, F. Crouzet, C. Bailly, A high-order finite-difference algorithm for direct computation of aerodynamic sound. Comput. Fluids 61, 46–63 (2012) MathSciNetGoogle Scholar
  35. R.W. Davis, E.F. Moore, A numerical study of vortex shedding from rectangles. J. Fluid Mech. 116(3), 475–506 (1982) zbMATHGoogle Scholar
  36. C.C. de Wiart, K. Hillewaert, P. Geuzaine, DNS of a low pressure turbine blade computed with the discontinuous Galerkin method, in Proceedings of the ASME Turbo Expo, Copenhagen, Denmark, 11–15 June 2012. GT2012–68900 Google Scholar
  37. J.W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41(02), 453–480 (1970) zbMATHGoogle Scholar
  38. I. Demirdžić, M. Perić, Space conservation law in finite volume calculations of fluid flow. Int. J. Numer. Methods Fluids 8(9), 1037–1050 (1988) zbMATHGoogle Scholar
  39. J. Douglas, J.E. Gunn, A general formulation of alternating direction methods. Numer. Math. 6(1), 428–453 (1964) MathSciNetzbMATHGoogle Scholar
  40. D. Drikakis, W. Rider, High-Resolution Methods for Incompressible and Low-Speed Flows (Springer, Berlin, 2004) Google Scholar
  41. F. Ducros, V. Ferrand, F. Nicoud, C. Weber, D. Darracq, C. Gacherieu, T. Poinsot, Large-eddy simulation of the shock/turbulence interaction. J. Comput. Phys. 152(2), 517–549 (1999) zbMATHGoogle Scholar
  42. F. Ducros, F. Laporte, T. Souleres, V. Guinot, P. Moinat, B. Caruelle, High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: application to compressible flows. J. Comput. Phys. 161(1), 114–139 (2000) MathSciNetzbMATHGoogle Scholar
  43. J.K. Dukowicz, J.D. Ramshaw, Tensor viscosity method for convection in numerical fluid dynamics. J. Comput. Phys. 32(1), 71–79 (1979) zbMATHGoogle Scholar
  44. M.S. Engelman, R.L. Sani, Finite element simulation of incompressible fluid flows with a free/moving surface. Comput. Tech. Fluid. Flow 47, 47–74 (1986) MathSciNetGoogle Scholar
  45. C.A.J. Fletcher, Fundamental and General Techniques. Computational Techniques for Fluid Dynamics, vol. 1 (Springer, Berlin, 1997) Google Scholar
  46. G. Fritsch, M. Giles, Second-order effects of unsteadiness on the performance of turbomachines, in 37th International Gas Turbine and Aeroengine Congress and Exposition (1992). GT–92–389 Google Scholar
  47. L. Gamet, F. Ducros, F. Nicoud, T. Poinsot, Compact finite difference schemes on non-uniform meshes. application to direct numerical simulations of compressible flows. Int. J. Numer. Methods Fluids 29(2), 159–191 (1999) MathSciNetzbMATHGoogle Scholar
  48. B.J. Geurts, D.D. Holm, Regularization modeling for large-eddy simulation. Phys. Fluids 15(1), L13–L16 (2003) MathSciNetGoogle Scholar
  49. S. Ghosal, An analysis of numerical errors in large-eddy simulations of turbulence. J. Comput. Phys. 125(1), 187–206 (1996) MathSciNetzbMATHGoogle Scholar
  50. M.B. Giles, Nonreflecting boundary conditions for Euler equation calculations. AIAA J. 28(12), 2050–2058 (1990) Google Scholar
  51. M. Giles, The HYDRA user’s guide. Version 0.06 (2004) Google Scholar
  52. W. Glanfield, Personal communication (2000) Google Scholar
  53. J. Glass, W. Rodi, A higher order numerical scheme for scalar transport. Comput. Methods Appl. Mech. Eng. 31(3), 337–358 (1982) zbMATHGoogle Scholar
  54. P.M. Gresho, R.L. Lee, R.L. Sani, On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions. Recent Adv. Numer. Methods Fluids 1, 27–79 (1980) Google Scholar
  55. P.M. Gresho, S.T. Chan, R.L. Lee, C.D. Upson, A modified finite element method for solving the time-dependent, incompressible Navier-Stokes equations. Part 1: theory. Int. J. Numer. Methods Fluids 4(6), 557–598 (1984) zbMATHGoogle Scholar
  56. A. Hadjadj, Large eddy simulation of shock/boundary layer interaction. AIAA J. 50(12), 2919–2927 (2012) Google Scholar
  57. T. Haga, H. Gao, Z.J. Wang, A high-order unifying discontinuous formulation for the Navier-Stokes equations on 3d mixed grids. Math. Model. Nat. Phenom. 6(3), 28–56 (2011) MathSciNetzbMATHGoogle Scholar
  58. F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182–2189 (1965) zbMATHGoogle Scholar
  59. A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983) MathSciNetzbMATHGoogle Scholar
  60. A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71(2), 231–303 (1987) MathSciNetzbMATHGoogle Scholar
  61. L. He, D.X. Wang, Concurrent blade aerodynamic-aero-elastic design optimization using adjoint method. J. Turbomach. 133, 011021 (2011) Google Scholar
  62. L.S. Hedges, A.K. Travin, P.R. Spalart, Detached-eddy simulations over a simplified landing gear. J. Fluids Eng. 124(2), 413–423 (2002) Google Scholar
  63. R.A.W.M. Henkes, Natural-convection boundary layers. PhD thesis, Technical University Delft (1990) Google Scholar
  64. B.P. Hignett, A.A. White, R.D. Carter, W.D.N. Jackson, R.M. Small, A comparison of laboratory measurements and numerical simulations of baroclinic wave flows in a rotating cylindrical annulus. Q. J. R. Meteorol. Soc. 111(467), 131–154 (1985) Google Scholar
  65. C.W. Hirt, A.A. Amsden, J.L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974) zbMATHGoogle Scholar
  66. R. Hixon, Prefactored small-stencil compact schemes. J. Comput. Phys. 165(2), 522–541 (2000) MathSciNetzbMATHGoogle Scholar
  67. K. Horiuti, T. Itami, Truncation error analysis of the rotational form for the convective terms in the Navier–Stokes equation. J. Comput. Phys. 145(2), 671–692 (1998) zbMATHGoogle Scholar
  68. F.Q. Hu, A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables. J. Comput. Phys. 173(2), 455–480 (2001) MathSciNetzbMATHGoogle Scholar
  69. F.Q. Hu, M.Y. Hussaini, J.L. Manthey, Low-dissipation and low-dispersion Runge–Kutta schemes for computational acoustics. J. Comput. Phys. 124(1), 177–191 (1996) MathSciNetzbMATHGoogle Scholar
  70. A. Hujeirat, R. Rannacher, A method for computing compressible, highly stratified flows in astrophysics based on operator splitting. Int. J. Numer. Methods Fluids 28(1), 1–22 (1998) MathSciNetzbMATHGoogle Scholar
  71. H.G. Im, Numerical studies of transient opposed-flow flames using adaptive time integration. KSME Int. J. 14(1), 103–112 (2000) Google Scholar
  72. A. Iserles, Generalized leapfrog methods. IMA J. Numer. Anal. 6(4), 381–392 (1986) MathSciNetzbMATHGoogle Scholar
  73. R.I. Issa, Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62(1), 40–65 (1986) MathSciNetzbMATHGoogle Scholar
  74. A. Jameson, Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA Pap. 1596, 1991 (1991) Google Scholar
  75. A. Jameson, The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. J. Sci. Comput. 34(2), 152–187 (2008a) MathSciNetzbMATHGoogle Scholar
  76. A. Jameson, Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. J. Sci. Comput. 34(2), 188–208 (2008b) MathSciNetzbMATHGoogle Scholar
  77. A. Jameson, W. Schmidt, E. Turkel, Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA Pap. 81(125), 9 (1981) Google Scholar
  78. R.J. Jefferson-Loveday, Numerical simulations of unsteady impinging jet flows. PhD thesis, Swansea University (2008) Google Scholar
  79. W.P. Jones, A.J. Marquis, Calculation of axisymmetric re-circulating flows with a second order turbulence model, in Proc. of the 5th Symp. on Turbulent Shear Flows, Cornel University (1985), pp. 20.1–20.11 Google Scholar
  80. J. Joo, P. Durbin, Simulation of turbine blade trailing edge cooling. J. Fluids Eng. 131, 021102 (2009) Google Scholar
  81. S.A. Karabasov, V.M. Goloviznin, New efficient high-resolution method for nonlinear problems in aeroacoustics. AIAA J. 45(12), 2861–2871 (2007) Google Scholar
  82. S.A. Karabasov, V.M. Goloviznin, Compact accurately boundary-adjusting high-resolution technique for fluid dynamics. J. Comput. Phys. 228(19), 7426–7451 (2009) MathSciNetzbMATHGoogle Scholar
  83. Z. Khatir, Discrete vortex modelling of near-wall flow structure in turbulent boundary layers. PhD thesis, Fluid Dynamics Research Centre, The University of Warwick (2000) Google Scholar
  84. J.W. Kim, D.J. Lee, Optimized compact finite difference schemes with maximum resolution. AIAA J. 34(5), 887–893 (1996) zbMATHGoogle Scholar
  85. J. Kim, P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59(2), 308–323 (1985) MathSciNetzbMATHGoogle Scholar
  86. M.H. Kobayashi, J.C.F. Pereira, Culation of incompressible laminar flows on a nonstaggered, nonorthogonal grid. Numer. Heat Transf., Part B, Fundam. 19(2), 243–262 (1991) Google Scholar
  87. J.A. Krakos, D.L. Darmofal, Effect of small-scale output unsteadiness on adjoint-based sensitivity. AIAA J. 48(11), 2611–2623 (2010) Google Scholar
  88. C. Lacor, Industrial Computational Fluid Dynamics. Lecture Series 1999-06 (von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, 1999) Google Scholar
  89. S. Laizet, E. Lamballais, High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy. J. Comput. Phys. 228(16), 5989–6015 (2009) MathSciNetzbMATHGoogle Scholar
  90. K.R. Lee, J.H. Park, K.H. Kim, High-order interpolation method for overset grid based on finite volume method. AIAA J. 49(7), 1387–1398 (2011) Google Scholar
  91. C.E. Leith, Numerical simulation of the earth’s atmosphere. University of California, Lawrence Radiation Laboratory (1964) Google Scholar
  92. S.K. Lele, Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992) MathSciNetzbMATHGoogle Scholar
  93. B.P. Leonard, A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Eng. 19(1), 59–98 (1979) zbMATHGoogle Scholar
  94. M.S. Liou, C.J. Steffen, A new flux splitting scheme. J. Comput. Phys. 107(1), 23–39 (1993) MathSciNetzbMATHGoogle Scholar
  95. X.D. Liu, S. Osher, T. Chan, Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994) MathSciNetzbMATHGoogle Scholar
  96. Y. Liu, M. Vinokur, Z.J. Wang, Spectral (finite) volume method for conservation laws on unstructured grids v: extension to three-dimensional systems. J. Comput. Phys. 212(2), 454–472 (2006) MathSciNetzbMATHGoogle Scholar
  97. D.P. Lockard, K.S. Brentner, H.L. Atkins, High-accuracy algorithms for computational aeroacoustics. AIAA J. 33, 246–251 (1995) zbMATHGoogle Scholar
  98. S. Loiodice, P.G. Tucker, J. Watson, Coupled open rotor engine intake simulations, in Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, 4–7 January 2010. AIAA 2010–840 Google Scholar
  99. Y. Lu, W.N. Dawes, X. Yuan, Investigation of 3D internal flow using new flux-reconstruction higher order method, in Proceedings of the ASME Turbo Expo, Copenhagen, Denmark, 11–15 June 2012. GT2012–69270 Google Scholar
  100. K. Mani, D.J. Mavriplis, Discrete adjoint based time-step adaptation and error reduction in unsteady flow problems, in 18th AIAA Computational Fluid Dynamics Conference, Miami, FL, 25–28 June 2007. AIAA 2007–3944 Google Scholar
  101. K. Mani, D.J. Mavriplis, Spatially non-uniform time-step adaptation for functional outputs in unsteady flow problems, in 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 4–7 January 2010. AIAA 2010–121 Google Scholar
  102. E. Manoha, B. Troff, P. Sagaut, Trailing-edge noise prediction using large-eddy simulation and acoustic analogy. AIAA J. 38(4), 575–583 (2000) Google Scholar
  103. J.C. Marongiu, F. Leboeuf, E. Parkinson, Numerical simulation of the flow in a Pelton turbine using the meshless method smoothed particle hydrodynamics: a new simple solid boundary treatment. Proc. Inst. Mech. Eng. A, J. Power Energy 221(6), 849–856 (2007) Google Scholar
  104. J.C. Marongiu, F. Leboeuf, J.Ë. Caro, E. Parkinson, Free surface flows simulations in Pelton turbines using an hybrid SPH-ALE method. J. Hydraul. Res. 48(S1), 40–49 (2010) Google Scholar
  105. I. Mary, P. Sagaut, Large eddy simulation of flow around an airfoil near stall. AIAA J. 40(6), 1139–1145 (2002) Google Scholar
  106. P.J. Morris, L.N. Long, A. Bangalore, Q. Wang, A parallel three-dimensional computational aeroacoustics method using nonlinear disturbance equations. J. Comput. Phys. 133(1), 56–74 (1997) zbMATHGoogle Scholar
  107. C. Moulinec, S. Benhamadouche, D. Laurence, M. Peric, LES in a U-bend pipe meshed by polyhedral cells. Eng. Turbul. Model. Exp. 6, 237–246 (2005) Google Scholar
  108. B. Muhlbauer, B. Noll, M. Aigner, Numerical investigation of entropy noise and its acoustic sources in aero-engine, in Proceedings of the ASME Turbo Expo, Berlin, Germany, 9–13 June 2008. GT2008–50321 Google Scholar
  109. T. Muramatsu, H. Ninokata, Thermal striping temperature fluctuation analysis using the algebraic stress turbulence model in water and sodium. JSME Int. J., Ser. 2, Fluids Eng. Heat Transf. Power Combust. Thermophys. Prop. 35(4), 486–496 (1992) Google Scholar
  110. K. Nakahashi, F. Togashi, Unstructured overset grid method for flow simulation of complex multiple body problems, in ICAS 2000 Congress, (2000). ICAS 0263 Google Scholar
  111. A.G.F.A. Nasser, M.A. Leschziner, Computation of transient recirculating flow using spline approximations and time-space characteristics, in Proc. 4th Int. Conf. on Numerical Methods in Laminar and Turbulent Flow, Swansea (1985), pp. 480–491 Google Scholar
  112. P.D. Orkwis, M.G. Turner, J.W. Barter, Linear deterministic source terms for hot streak simulations. J. Propuls. Power 18(2), 383–389 (2002) Google Scholar
  113. S.A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50(4), 689–703 (1971) zbMATHGoogle Scholar
  114. Y. Ozyoruk, L.N. Long, Multigrid acceleration of a high-resolution computational aeroacoustics scheme. AIAA J. 35(3), 428–433 (1997) Google Scholar
  115. U. Paliath, H. Shen, R. Avancha, C. Shieh, Large eddy simulation for jets from chevron and dual nozzles, in Proceedings of 17th AIAA/CEAS Aeroacoustics Conference, Portland, Oregon, 5–8 June 2011. AIAA 2011–2881 Google Scholar
  116. A.T. Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984) MathSciNetzbMATHGoogle Scholar
  117. L.L. Pauley, P. Moin, W.C. Reynolds, The structure of two-dimensional separation. J. Fluid Mech. 220(1), 397–411 (1990) Google Scholar
  118. D.W. Peaceman, H.H. Rachford, The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955) MathSciNetzbMATHGoogle Scholar
  119. A. Pinelli, I.Z. Naqavi, U. Piomelli, J. Favier, Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers. J. Comput. Phys. 229(24), 9073–9091 (2010) MathSciNetzbMATHGoogle Scholar
  120. H. Pitsch, Large-eddy simulation of turbulent combustion. Annu. Rev. Fluid Mech. 38, 453–482 (2006) MathSciNetGoogle Scholar
  121. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes: The Art of Scientific Computing (FORTRAN) (Cambridge University Press, Cambridge, 1989) zbMATHGoogle Scholar
  122. D. Rayner, A numerical study into the heat transfer beneath the stator blade of an axial compressor. PhD thesis, School of Engineering, University of Sussex (1993) Google Scholar
  123. D.T. Reindl, W.A. Beckman, J.W. Mitchell, C.J. Rutland, Benchmarking transient natural convection in an enclosure, in National Heat Transfer Conference, vol. 91 (1991), pp. 1–7 Google Scholar
  124. C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21(11), 1525–1532 (1983) zbMATHGoogle Scholar
  125. D.P. Rizzetta, M.R. Visbal, P.E. Morgan, A high-order compact finite-difference scheme for large-eddy simulation of active flow control. Prog. Aerosp. Sci. 44(6), 397–426 (2008) Google Scholar
  126. P.J. Roache, A flux-based modified method of characteristics. Int. J. Numer. Methods Fluids 15(11), 1259–1275 (1992) MathSciNetzbMATHGoogle Scholar
  127. P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981) MathSciNetzbMATHGoogle Scholar
  128. P.L. Roe, Characteristic-based schemes for the Euler equations. Annu. Rev. Fluid Mech. 18(1), 337–365 (1986) MathSciNetGoogle Scholar
  129. S.E. Rogers, D. Kwak, Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA J. 28, 253–262 (1990) zbMATHGoogle Scholar
  130. S.E. Rogers, D. Kwak, J.L.C. Chang, Numerical solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates. NASA STI/Recon Tech. Rep. N 87, 11964 (1986) Google Scholar
  131. M.P. Rumpfkeil, D.W. Zingg, A general framework for the optimal control of unsteady flows with applications, in Proceedings of the 45th AIAA Aerospace Meeting and Exhibit. Reno, Nevada, 8–11 January 2007. AIAA 2010–1128 Google Scholar
  132. N.D. Sandham, Q. Li, H.C. Yee, Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178(2), 307–322 (2002) zbMATHGoogle Scholar
  133. L.J. Segerlind, Applied Finite Element Analysis (Wiley, New York, 1976) zbMATHGoogle Scholar
  134. V. Seidl, M. Peric, M. Schmidt, Space-and time-parallel Navier-Stokes solver for 3d block-adaptive Cartesian grids, in Proceedings of the Parallel Computational Fluid Dynamics, vol. 95 (1995), pp. 557–584 Google Scholar
  135. J.A. Sethian, Fast marching methods. SIAM Rev. 41(2), 199–235 (1999) MathSciNetzbMATHGoogle Scholar
  136. M.L. Shur, P.R. Spalart, M.K. Strelets, A.K. Travin, Towards the prediction of noise from jet engines. Int. J. Heat Fluid Flow 24(4), 551–561 (2003) Google Scholar
  137. S. Skelboe, The control of order and steplength for backward differentiation methods. BIT Numer. Math. 17(1), 91–107 (1977) MathSciNetzbMATHGoogle Scholar
  138. P. Spalart, L. Hedges, M. Shur, A. Travin, Simulation of active flow control on a stalled airfoil. Flow Turbul. Combust. 71(1), 361–373 (2003) zbMATHGoogle Scholar
  139. E.T. Spyropoulos, G.A. Blaisdell, Large-eddy simulation of a spatially evolving supersonic turbulent boundary-layer flow. AIAA J. 36(11), 1983–1990 (1998) Google Scholar
  140. D. Stanescu, M.Y. Hussaini, F. Farassat, Aircraft engine noise scattering—a discontinuous spectral element approach. AIAA Pap. 800, 2002 (2002) Google Scholar
  141. A. Staniforth, J. Côté, Semi-Lagrangian integration schemes for atmospheric models—a review. Mon. Weather Rev. 119(9), 2206–2223 (1991) Google Scholar
  142. J.L. Steger, P. Kutler, Implicit finite-difference procedures for the computation of vortex wakes. AIAA J. 15(4), 581–590 (1977) MathSciNetzbMATHGoogle Scholar
  143. J.L. Steger, R.F. Warming, Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J. Comput. Phys. 40(2), 263–293 (1981) MathSciNetzbMATHGoogle Scholar
  144. J. Steinhoff, D. Underhill, Modification of the Euler equations for “vorticity confinement”: application to the computation of interacting vortex rings. Phys. Fluids 6, 2738–2744 (1994) zbMATHGoogle Scholar
  145. M. Strelets, Detached eddy simulation of massively separated flows, in Proceedings of 39th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 8–11 January 2001. AIAA 2001–0879 Google Scholar
  146. R.C. Swanson, E. Turkel, On central-difference and upwind schemes. J. Comput. Phys. 101(2), 292–306 (1992) MathSciNetzbMATHGoogle Scholar
  147. T. Talha, A numerical investigation of three-dimensional unsteady turbulent channel flow subjected to temporal acceleration. PhD thesis, School of Engineering, University of Warwick (2012) Google Scholar
  148. C.K.W. Tam, Advances in numerical boundary conditions for computational aeroacoustics. J. Comput. Acoust. 6(4), 377–402 (1998) Google Scholar
  149. C.K.W. Tam, H. Shen, Direct computation of nonlinear acoustic pulses using high order finite difference schemes. AIAA Pap. 4325 (1993) Google Scholar
  150. C.K.W. Tam, J.C. Webb, Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107(2), 262–281 (1993) MathSciNetzbMATHGoogle Scholar
  151. C.K.W. Tam, C. Jay, D. Zhong, A study of the short wave components in computational acoustics. J. Comput. Acoust. 1(01), 1–30 (1993) MathSciNetGoogle Scholar
  152. L. Tang, J.D. Baeder, Uniformly accurate finite difference schemes for p-refinement. SIAM J. Sci. Comput. 20(3), 1115–1131 (1998) MathSciNetGoogle Scholar
  153. P.D. Thomas, C.K. Lombard, Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17, 1030–1037 (1979) MathSciNetzbMATHGoogle Scholar
  154. C. Tu, M. Deville, L. Dheur, L. Vanderschuren, Finite element simulation of pulsatile flow through arterial stenosis. J. Biomech. 25(10), 1141–1152 (1992) Google Scholar
  155. P.G. Tucker, Numerical precision and dissipation errors in rotating flows. Int. J. Numer. Methods Heat Fluid Flow 7(7), 647–658 (1997) zbMATHGoogle Scholar
  156. P.G. Tucker, Computation of Unsteady Internal Flows: Fundamental Methods with Case Studies (Kluwer Academic, Dordrecht, 2001) Google Scholar
  157. P.G. Tucker, Novel multigrid orientated solution adaptive time-step approaches. Int. J. Numer. Methods Fluids 40(3–4), 507–519 (2002a) zbMATHGoogle Scholar
  158. P.G. Tucker, Temporal behavior of flow in rotating cavities. Numer. Heat Transf., Part A, Appl. 41(6–7), 611–627 (2002b) Google Scholar
  159. P.G. Tucker, Novel MILES computations for jet flows and noise. Int. J. Heat Fluid Flow 25(4), 625–635 (2004) Google Scholar
  160. P.G. Tucker, Turbulence modelling of problem aerospace flows. Int. J. Numer. Methods Fluids 51(3), 261–283 (2006) zbMATHGoogle Scholar
  161. P.G. Tucker, The LES model’s role in jet noise. Prog. Aerosp. Sci. 44(6), 427–436 (2008) Google Scholar
  162. A. Uzun, M.Y. Hussaini, Simulation of noise generation in near-nozzle region of a chevron nozzle jet. AIAA J. 47(8), 1793–1810 (2009) Google Scholar
  163. G.D. Van Albada, B. Van Leer, W.W. Roberts Jr., A comparative study of computational methods in cosmic gas dynamics. Astron. Astrophys. 108, 76–84 (1982) zbMATHGoogle Scholar
  164. B. Van Leer, Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow. J. Comput. Phys. 23(3), 263–275 (1977) zbMATHGoogle Scholar
  165. B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979) Google Scholar
  166. B.C. Vermeire, J.S. Cagnone, S. Nadarajah, Iles using the correction procedure via reconstruction scheme, in Proceedings of 51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Grapevine, Dallas/Ft. Worth Region, Texas, 7–10 January 2013. AIAA 2013–1001 Google Scholar
  167. M.R. Visbal, D.V. Gaitonde, On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181(1), 155–185 (2002) MathSciNetzbMATHGoogle Scholar
  168. S.G. Wallis, J.R. Manson, Accurate numerical simulation of advection using large time steps. Int. J. Numer. Methods Fluids 24(2), 127–139 (1997) zbMATHGoogle Scholar
  169. Z.J. Wang, Y. Liu, G. May, A. Jameson, Spectral difference method for unstructured grids ii: extension to the Euler equations. J. Sci. Comput. 32(1), 45–71 (2007) MathSciNetzbMATHGoogle Scholar
  170. J.M. Weiss, W.A. Smith, Preconditioning applied to variable and constant density flows. AIAA J. 33(11), 2050–2057 (1995) zbMATHGoogle Scholar
  171. G. Yang, D.M. Causon, D.M. Ingram, R. Saunders, P. Batten, A Cartesian cut cell method for compressible flows. Part B: moving body problems. Aeronaut. J. 101(1002), 57–65 (1997) Google Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • P. G. Tucker
    • 1
  1. 1.Department of Engineering, Whittle LaboratoryUniversity of CambridgeCambridgeUK

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