Finite Difference Methods for Incompressible and Compressible Turbulence

  • Sergio PirozzoliEmail author
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 592)


We provide a brief overview of the state of the art in numerical methods for direct numerical simulation of turbulent flows, with special reference to wall-bounded flows. Fundamentals of numerical discretization of the incompressible and compressible Navier–Stokes equations will be given in section “Numerical Methods”, which also includes practical implementation details, such as choice of the computational mesh, and suggestions for implementation on parallel computers. Selected applications of DNS will be the subject of the section “Selected DNS Applications” where the focus will be on recent results obtained for flows at high Reynolds number and on ‘non-classical’ effects associated with the formation of large rollers (Couette flow), and with compressibility effects. The behavior of passive scalars advected by the fluid phase is also discussed.


  1. Abe, H., & Antonia, R. A. (2009). Near-wall similarity between velocity and scalar fluctuations in a turbulent channel flow. Physics of Fluids, 21, 025109.zbMATHCrossRefGoogle Scholar
  2. Abe, H., Kawamura, H., & Matsuo, Y. (2004) Surface heat-flux fluctuations in a turbulent channel flow up to Re\(_{\tau }=1020\) with Pr\(= 0.025\) and \(0.71\). International Journal of Heat and Fluid Flow, 25, 404–419.Google Scholar
  3. Afzal, N., & Yajnik, K. (1973). Analysis of turbulent pipe and channel flows at moderately large Reynolds number. Journal of Fluid Mechanics, 61, 23–31.CrossRefGoogle Scholar
  4. Alfredsson, P. H., Segalini, A., & Örlü, R. (2011). A new scaling for the streamwise turbulence intensity in wall-bounded turbulent flows and what it tells us about the ‘outer’ peak. Physics of Fluids, 23, 041702.CrossRefGoogle Scholar
  5. Alfredsson, P. H., Örlü, R., & Segalini, A. (2012) A new formulation for the streamwise turbulence intensity distribution in wall-bounded turbulent flows. European Journal of Mechanics B fluids, 36, 167–175.Google Scholar
  6. Avsarkisov, V., Hoyas, S., Oberlack, M., & García-Galache, J. P. (2014). Turbulent plane Couette flow at moderately high Reynolds number. Journal of Fluid Mechanics, 751, R1.CrossRefGoogle Scholar
  7. Aydin, E. M., & Leutheusser, H. J. (1991). Plane-Couette flow between smooth and rough walls. Experiments in Fluids, 11, 302–312.CrossRefGoogle Scholar
  8. Batchelor G. K. (1959). Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity. Journal of Fluid Mechanics, 5, 113–133.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Beam, R. M., & Warming, R. F. (1978). An implicit factored scheme for the compressible navier-stokes equations. AIAA Journal, 16(4), 393–402.zbMATHCrossRefGoogle Scholar
  10. Bech, K. H., Tillmark, N., Alfredsson, P. H., & Andersson, H. I. (1995). An investigation of turbulent plane Couette flow at low Reynolds numbers. Journal of Fluid Mechanics, 286, 291–325.CrossRefGoogle Scholar
  11. Bernardini, M., Pirozzoli, S., Quadrio, M., & Orlandi, P. (2013). Turbulent channel flow simulations in convecting reference frames. Journal of Computational Physics, 232, 1–6.CrossRefGoogle Scholar
  12. Bernardini, M., Pirozzoli, S., & Orlandi, P. (2014). Velocity statistics in turbulent channel flow up to Re\(_{\tau }=4000\). Journal of Fluid Mechanics, 742, 171–191.CrossRefGoogle Scholar
  13. Blaisdell, G. A., Spyropoulos, E. T., & Qin, J. H. (1996). The effect of the formulation of non-linear terms on aliasing errors in spectral methods. Applied Numerical Mathematics, 21, 207–219.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Cebeci, T., & Bradshaw, P. (1984). Physical and computational aspects of convective heat transfer. New York, NY: Springer.zbMATHCrossRefGoogle Scholar
  15. Chorin, A. J. (1969). On the convergence of discrete approximations to the Navier-Stokes equations. Mathematics of Computation, 23(106), 341–353.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Coleman, G. N., Kim, J., & Moser, R. D. (1995). A numerical study of turbulent supersonic isothermal-wall channel flow. Journal of Fluid Mechanics, 305, 159–183.zbMATHCrossRefGoogle Scholar
  17. Colonius, T., & Lele, S. K. (2004). Computational aeroacoustics: Progress on nonlinear problems of sound generation. Progress in Aerospace Sciences, 40, 345–416.CrossRefGoogle Scholar
  18. del Álamo, J. C., & Jiménez, J. (2003). Spectra of the very large anisotropic scales in turbulent channels. Physics of Fluids, 15, L41–L44.zbMATHCrossRefGoogle Scholar
  19. Ducros, F., Laporte, F., Soulères, T., Guinot, V., Moinat, P., & Caruelle, B. (2000). High-order fluxes for conservative skew-symmetric-like schemes in structures meshes: application to compressible flows. Journal of Computational Physics, 161, 114–139.MathSciNetzbMATHCrossRefGoogle Scholar
  20. El Telbany, M. M. M., & Reynolds, A. J. (1982). Velocity distributions in plane turbulent channel flows. Transactions of the ASME: Journal of Fluids Engineering, 104, 367–372.Google Scholar
  21. Feiereisen, W. J., Reynolds, W. C., & Ferziger, J. H. (1981). Numerical simulation of a compressible, homogeneous, turbulent shear flow. Report TF 13, Thermosciences Division, Mechanical Engineering, Stanford University.Google Scholar
  22. Fernholz, H. H., & Finley, P. J. (1980). A critical commentary on mean flow data for two-dimensional compressible turbulent boundary layers (Vol. 253). AGARDograph.Google Scholar
  23. Ferziger, J. H., & Peric, M. (2012). Computational methods for fluid dynamics. Berlin: Springer Science & Business Media.zbMATHGoogle Scholar
  24. Flores, O., & Jimenez, J. (2010). Hierarchy of minimal flow units in the logarithmic layer. Physics of Fluids, 22, 071704.CrossRefGoogle Scholar
  25. Garcia-Villalba, M., & Del Alamo, J. C. (2011). Turbulence modification by stable stratification in channel flow. Physics of Fluids, 23(4), 045104.Google Scholar
  26. Gowen, R. A., & Smith, J. W. (1967). The effect of the Prandtl number on temperature profiles for heat transfer in turbulent pipe flow. Chemical Engineering Science, 22, 1701–1711.CrossRefGoogle Scholar
  27. Guarini, S. E., Moser, R. D., Shariff, K., & Wray, A. (2000). Direct numerical simulation of a supersonic boundary layer at Mach 2.5. Journal of Fluid Mechanics, 414, 1–33.zbMATHCrossRefGoogle Scholar
  28. Hamilton, J. M., Kim, J., & Waleffe, F. (1995). Regeneration mechanisms of near-wall turbulent structures. Journal of Fluid Mechanics, 287, 317–348.zbMATHCrossRefGoogle Scholar
  29. Harlow, F. H., & Welch, J. E. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids, 8(12), 2182–2189.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Harten, A. (1983). On the symmetric form of systems of conservation laws with entropy. Journal of Computational Physics, 49, 151–164.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Hirsch, C. (2007). Numerical computation of internal and external flows: The fundamentals of computational fluid dynamics. Oxford: Butterworth-Heinemann.Google Scholar
  32. Honein, A. E., & Moin, P. (2004). Higher entropy conservation and numerical stability of compressible turbulence simulations. Journal of Computational Physics, 201, 531–545.zbMATHCrossRefGoogle Scholar
  33. Howarth, L. (1948). Concerning the effect of compressibility on laminar boundary layers and their separation. Proceedings of the Royal Society of London Series A, 194(1036), 16–42.MathSciNetzbMATHGoogle Scholar
  34. Hoyas, S., & Jiménez, J. (2006). Scaling of velocity fluctuations in turbulent channels up to \({R}e_{\tau } = 2003\). Physics of Fluids, 18, 011702.CrossRefGoogle Scholar
  35. Hoyas, S., & Jiménez, J. (2008). Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Physics of Fluids, 20, 101511.zbMATHCrossRefGoogle Scholar
  36. Huang, P. G., & Coleman, G. N. (1994). van Driest transformation and compressible wall-bounded flows. AIAA Journal, 32(10), 2110–2113.CrossRefGoogle Scholar
  37. Huang, P. G., Coleman, G. N., & Bradshaw, P. (1995). Compressible turbulent channel flows: DNS results and modeling. Journal of Fluid Mechanics, 305, 185–218.zbMATHCrossRefGoogle Scholar
  38. Hultmark, M., Vallikivi, M., Bailey, S. C. C., & Smits, A. J. (2012). Turbulent pipe flow at extreme Reynolds numbers. Physical Review Letters, 108, 094501.CrossRefGoogle Scholar
  39. Hunt, J. C. R., & Morrison, J. F. (2001). Eddy structure in turbulent boundary layers. European Journal of Mechanics-B/Fluids, 19, 673–694.zbMATHCrossRefGoogle Scholar
  40. Hutchins, N., & Marusic, I. (2007). Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. Journal of Fluid Mechanics, 579, 1–28.zbMATHCrossRefGoogle Scholar
  41. Hutchins, N., Nickels, T. B., Marusic, I., & Chong, M. S. (2009). Hot-wire spatial resolution issues in wall-bounded turbulence. Journal of Fluid Mechanics, 635, 103–136.zbMATHCrossRefGoogle Scholar
  42. Hwang, Y., & Cossu, C. (2010). Amplification of coherent streaks in the turbulent Couette flow: an input-output analysis at low Reynolds number. Journal of Fluid Mechanics, 643, 333–348.zbMATHCrossRefGoogle Scholar
  43. Jiménez, J., & Moin, P. (1991). The minimal flow unit in near-wall turbulence. Journal of Fluid Mechanics, 225, 213–240.zbMATHCrossRefGoogle Scholar
  44. Jiménez, J., Wray, A. A., Saffman, P. G., & Rogallo, R. S. (1993). The structure of intense vorticity in isotropic turbulence. Journal of Fluid Mechanics, 255, 65–90.MathSciNetzbMATHCrossRefGoogle Scholar
  45. Kader, B. A. (1981). Temperature and concentration profiles in fully turbulent boundary layers. International Journal of Heat and Mass Transfer, 24, 1541–1544.CrossRefGoogle Scholar
  46. Kawamura, H., Abe, H., & Matsuo, Y. (1999). DNS of turbulent heat transfer in channel flow with respect to Reynolds and Prandtl number effects. International Journal of Heat and Fluid Flow, 20, 196–207.CrossRefGoogle Scholar
  47. Kennedy, C. A., & Gruber, A. (2008). Reduced aliasing formulations of the convective terms within the Navier-Stokes equations. Journal of Computational Physics, 227, 1676–1700.MathSciNetzbMATHCrossRefGoogle Scholar
  48. Kim, J., Moin, P., & Moser, R. D. (1987). Turbulence statistics in fully developed channel flow at low Reynolds number. Journal of Fluid Mechanics, 177, 133–166.zbMATHCrossRefGoogle Scholar
  49. Kim, K. C., & Adrian, R. J. (1999). Very large-scale motion in the outer layer. Physics of Fluids, 11, 417–422.MathSciNetzbMATHCrossRefGoogle Scholar
  50. Kitoh, O., Nakabayashi, K., & Nishimura, F. (2005). Experimental study on mean velocity and turbulence characteristics of plane Couette flow: Low-Reynolds-number effects and large longitudinal vortical structure. Journal of Fluid Mechanics, 539, 199–227.zbMATHCrossRefGoogle Scholar
  51. Klein, M., Sadiki, A., & Janicka, J. (2003). A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. Journal of Computational Physics, 186, 652–665.zbMATHCrossRefGoogle Scholar
  52. Komminaho, J., Lundbladh, A., & Johansson, A. V. (1996). Very large structures in plane turbulent Couette flow. Journal of Fluid Mechanics, 320, 259–285.zbMATHCrossRefGoogle Scholar
  53. Kravchenko, A. G., & Moin, P. (1997). On the effect of numerical errors in large eddy simulations of turbulent flows. Journal of Computational Physics, 131, 310–322.zbMATHCrossRefGoogle Scholar
  54. Lax, P. D. (1973). Hyperbolic systems of conservation laws and the mathematical theory of shock waves., Regional Conference Series in Applied Mathematics Providence: SIAM.zbMATHCrossRefGoogle Scholar
  55. Lee, M., & Moser, R. D. (2015). Direct simulation of turbulent channel flow layer up to Re\(_{\tau } = 5200\). Journal of Fluid Mechanics, 774, 395–415.CrossRefGoogle Scholar
  56. Lee, M. J., & Kim, J. (1991). The structure of turbulence in a simulated plane Couette flow. In Proceedings 8th Symposium Turbulent Shear Flows (pp. 5.3.1–5.3.6). MunichGoogle Scholar
  57. Lele, S. K. (1992). Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103, 16–42.MathSciNetzbMATHCrossRefGoogle Scholar
  58. LeVecque, R. (1990). Numerical methods for conservation laws. Basel: Birkhauser-Verlag.CrossRefGoogle Scholar
  59. Lilly, D. K. (1965). On the computational stability of numerical solutions of time-dependent non-linear geophysical fluid dynamics problems. Journal of Computational Physics, 93, 11–26.Google Scholar
  60. Mahesh, K., Constantinescu, G., & Moin, P. (2004). A numerical method for large-eddy simulation in complex geometries. Journal of Computational Physics, 197(1), 215–240.zbMATHCrossRefGoogle Scholar
  61. Majda, A. (1984). Compressible fluid flow and systems of conservation laws in several space variables (Vol. 53)., Applied Mathematical Sciences Berlin: Springer.zbMATHGoogle Scholar
  62. Mansour, N. N., Moin, P., Reynolds, W. C., & Ferziger, J. H. (1979). Improved methods for large eddy simulations of turbulence. In B. F. Launder, F. W. Schmidt, & H. H. Whitelaw (Eds.), Turbulent Shear Flows I (pp. 386–401). Berlin: Springer.CrossRefGoogle Scholar
  63. Martín, M. P. (2007). Direct numerical simulation of hypersonic turbulent boundary layers. Part 1. Initialization and comparison with experiments. Journal of Fluid Mechanics, 570, 347–364.zbMATHCrossRefGoogle Scholar
  64. Modesti, D., & Pirozzoli, S. (2016). Reynolds and Mach number effects in compressible turbulent channel flow. International Journal of Heat and Fluid Flow, 59, 33–49.CrossRefGoogle Scholar
  65. Monin, A. S., & Yaglom, A. M. (1971). Statistical fluid mechanics: Mechanics of turbulence (Vol. 1). Cambridge MA: MIT Press.Google Scholar
  66. Monty, J. P., Hutchins, N., Ng, H. C. H., Marusic, I., & Chong, M. S. (2009). A comparison of turbulent pipe, channel and boundary layer flows. Journal of Fluid Mechanics, 632, 431–442.zbMATHCrossRefGoogle Scholar
  67. Morinishi, Y. (2010). Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows. Journal of Computational Physics, 229, 276–300.MathSciNetzbMATHCrossRefGoogle Scholar
  68. Morinishi, Y., Lund, T. S., Vasiliev, O. V., & Moin, P. (1998). Fully conservative higher order finite difference schemes for incompressible flow. Journal of Computational Physics, 143, 90–124.MathSciNetzbMATHCrossRefGoogle Scholar
  69. Morinishi, Y., Tamano, S., & Nakabayashi, K. (2004). Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls. Journal of Fluid Mechanics, 502, 273–308.zbMATHCrossRefGoogle Scholar
  70. Morkovin, M. V. (1961). Effects of compressibility on turbulent flows. In A. Favre (Ed.), Mécanique de la Turbulence (pp. 367–380). Paris: CNRS.Google Scholar
  71. Nagano, Y., & Tagawa, M. (1988). Statistical characteristics of wall turbulence with a passive scalar. Journal of Fluid Mechanics, 196, 157–185.zbMATHCrossRefGoogle Scholar
  72. Nagib, H. M., & Chauhan, K. A. (2008). Variations of von Kármán coefficient in canonical flows. Physics of Fluids, 20, 101518.zbMATHCrossRefGoogle Scholar
  73. Nagib, H. M., Chauhan, K. A., & Monkewitz, P. A. (2007). Approach to an asymptotic state of zero pressure gradient turbulent boundary layers. Philosophical Transactions of the Royal Society of London A, 365, 755–770.zbMATHCrossRefGoogle Scholar
  74. Oliver, T. A., Malaya, N., Ulerich, R., & Moser, R. D. (2014). Estimating uncertainties in statistics computed from direct numerical simulation. Physics of Fluids, 26(3), 035101.CrossRefGoogle Scholar
  75. Orlandi, P. (1998). Numerical solution of 3D flows periodic in one direction and with complex geometries in 2D. Center for Turbulence Research: Annual research briefs.Google Scholar
  76. Orlandi, P. (2000). Fluid flow phenomena: A numerical toolkit. Dordrecht: Kluwer.zbMATHCrossRefGoogle Scholar
  77. Orlandi, P., Bernardini, M., & Pirozzoli, S. (2015). Poiseuille and Couette flows in the transitional and fully turbulent regime. Journal of Fluid Mechanics, 770, 424–441.CrossRefGoogle Scholar
  78. Papavassiliou, D. V., & Hanratty, T. J. (1997). Interpretation of large-scale structures observed in a turbulent planet Couette flow. International Journal of Heat and Fluid Flow, 18, 55–69.CrossRefGoogle Scholar
  79. Perry, A. E., & Li, J. D. (1990). Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers. Journal of Fluid Mechanics, 218, 405–438.CrossRefGoogle Scholar
  80. Peyret, R., & Taylor, T. D. (2012). Computational methods for fluid flow. Berlin: Springer Science & Business Media.zbMATHGoogle Scholar
  81. Phillips, N. A. (1959). An example of nonlinear computational instability. The atmosphere and the sea in motion (pp. 501–504). New York: Rockefeller Institute Press and Oxford University Press.Google Scholar
  82. Pirozzoli, S. (2007). Performance analysis and optimization of finite difference schemes for wave propagation problems. Journal of Computational Physics, 222, 809–831.MathSciNetzbMATHCrossRefGoogle Scholar
  83. Pirozzoli, S. (2010). Generalized conservative approximations of split convective derivative operators. Journal of Computational Physics, 229, 7180–7190.MathSciNetzbMATHCrossRefGoogle Scholar
  84. Pirozzoli, S. (2014). Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction. Journal of Fluid Mechanics, 745, 378–397.MathSciNetCrossRefGoogle Scholar
  85. Pirozzoli, S., & Bernardini, M. (2011). Turbulence in supersonic boundary layers at moderate Reynolds number. Journal of Fluid Mechanics, 688, 120–168.MathSciNetzbMATHCrossRefGoogle Scholar
  86. Pirozzoli, S., & Bernardini, M. (2013). Probing high-Reynolds-number effects in numerical boundary layers. Physics of Fluids, 25, 021704.CrossRefGoogle Scholar
  87. Pirozzoli, S., Grasso, F., & Gatski, T. B. (2004). Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at \(M=2.25\). Physics of Fluids, 16(3), 530–545.zbMATHCrossRefGoogle Scholar
  88. Pirozzoli, S., Bernardini, M., & Grasso, F. (2008). Characterization of coherent vortical structures in a supersonic turbulent boundary layer. Journal of Fluid Mechanics, 613, 205–231.zbMATHCrossRefGoogle Scholar
  89. Pirozzoli, S., Bernardini, M., & Grasso, F. (2010). Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation. J. Fluid Mech., 657, 361–393.zbMATHCrossRefGoogle Scholar
  90. Pirozzoli, S., Bernardini, M., & Orlandi, P. (2011). Large-scale organization and inner-outer layer interactions in turbulent Couette-Poiseuille flows. Journal of Fluid Mechanics, 680, 534–563.zbMATHCrossRefGoogle Scholar
  91. Pirozzoli, S., Bernardini, M., & Orlandi, P. (2014). Turbulence statistics in Couette flow at high Reynolds number. Journal of Fluid Mechanics, 758, 327–343.MathSciNetCrossRefGoogle Scholar
  92. Pirozzoli, S., Bernardini, M., & Orlandi, P. (2016). Passive scalars in turbulent channel flow at high Reynolds number. Journal of Fluid Mechanics, 788, 614–639.MathSciNetzbMATHCrossRefGoogle Scholar
  93. Pope, S. B. (2000). Turbulent flows. Cambridge: Cambridge University Press.zbMATHCrossRefGoogle Scholar
  94. Quadrio, M., Frohnapfel, B., & Hasegawa, Y. (2016). Does the choice of the forcing term affect flow statistics in dns of turbulent channel flow? European Journal of Mechanics-B/Fluids, 55, 286–293.MathSciNetzbMATHCrossRefGoogle Scholar
  95. Rai, M. M., & Moin, P. (1993). Direct numerical simulation of transition and turbulence in a spatially evolving boundary layer. Journal of Computational Physics, 109, 169–192.zbMATHCrossRefGoogle Scholar
  96. Rai, M. M., & Moin, P. (1991). Direct simulations of turbulent flow using finite-difference schemes. Journal of Computational Physics, 96, 15–53.zbMATHCrossRefGoogle Scholar
  97. Reichardt, H. (1956). Über die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couetteströmung. Zeitschrift für Angewandte Mathematik und Mechanik, 36, 26–29.CrossRefGoogle Scholar
  98. Robertson, J. M. (1959). On turbulent plane Couette flow. In Proceedings of Sixth Midwestern Conference on Fluid Mechanics (pp. 169–182). University of Texas: Austin.Google Scholar
  99. Schlatter, P., & Örlü, R. (2010). Assessment of direct numerical simulation data of turbulent boundary layers. Journal of Fluid Mechanics, 659, 116–126.zbMATHCrossRefGoogle Scholar
  100. Schlichting, H., & Gersten, K. (2000). Boundary layer theory (8th ed.). Berlin: Springer.zbMATHCrossRefGoogle Scholar
  101. Sengupta, T. K., Ganeriwal, G., & De, S. (2003). Analysis of central and upwind compact schemes. Journal of Computational Physics, 192(2), 677–694.zbMATHCrossRefGoogle Scholar
  102. Sengupta, T. K., Sircar, S. K., & Dipankar, A. (2006). High accuracy schemes for DNS and acoustics. Journal of Scientific Computing, 26, 151–193.MathSciNetzbMATHCrossRefGoogle Scholar
  103. Sillero, J., Jiménez, J., Moser, R. D., & Malaya, N. P. (2011). Direct simulation of a zero-pressure-gradient turbulent boundary layer up to Re\(_{\theta } = 6650\). Journal of Physics: Conference Series, 318(022023),Google Scholar
  104. Simens, M. P., Jimenez, J., Hoyas, S., & Mizuno, Y. (2009). A high-resolution code for turbulent boundary layers. Journal of Computational Physics, 228, 4218–4231.zbMATHCrossRefGoogle Scholar
  105. Smith, M. W., & Smits, A. J. (1995). Visualization of the structure of supersonic turbulent boundary layers. Experiments in Fluids, 18, 288–302.CrossRefGoogle Scholar
  106. Smith, R. W. (1994) Effect of reynolds number on the structure of turbulent boundary layers. Ph.D. thesis, Department of Mechanical and Aerospace Engineering, Princeton University.Google Scholar
  107. Smits, A. J., & Dussauge, J.-P. (1996). Turbulent shear layers in supersonic flow (2nd ed.). New York: American Institute of Physics.Google Scholar
  108. Smits, A. J., & Dussauge, J.-P. (2006). Turbulent shear layers in supersonic flow (2nd ed.). New York: American Institute of Physics.Google Scholar
  109. Smits, A. J., Matheson, N., & Joubert, P. N. (1983). Low-Reynolds-number turbulent boundary layers in zero and favourable pressure gradients. Journal of Ship Research, 147–157.Google Scholar
  110. Spina, E. F., Smits, A. J., & Robinson, S. K. (1994). The physics of supersonic turbulent boundary layers. Annual Review of Fluid Mechanics, 26, 287–319.CrossRefGoogle Scholar
  111. Steger, J. L., & Warming, R. F. (1981). Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. Journal of Computational Physics, 40, 263–293.MathSciNetzbMATHCrossRefGoogle Scholar
  112. Strand, B. (1994). Summation by parts for finite difference approximations for d/dx. Journal of Computational Physics, 110, 47–67.MathSciNetzbMATHCrossRefGoogle Scholar
  113. Subramanian, C. S., & Antonia, R. A. (1981). Effect of Reynolds number on a slightly heated turbulent boundary layer. International Journal of Heat and Mass Transfer, 24, 1833–1846.CrossRefGoogle Scholar
  114. Tillmark, N., & Alfredsson, P. H. (1992). Experiments on transition in plane Couette flow. Journal of Fluid Mechanics, 235, 89–102.CrossRefGoogle Scholar
  115. Townsend, A. A. (1976). The structure of turbulent shear flow (2nd ed.). Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  116. Trettel, S., & Larsson, J. (2016). Mean velocity scaling for compressible wall turbulence with heat transfer. Physics of Fluids, 28(2), 026102.CrossRefGoogle Scholar
  117. Tsukahara, T., Kawamura, H., & Shingai, K. (2006). DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. Journal of Turbulence, 7, 1–16.MathSciNetCrossRefGoogle Scholar
  118. van der Poel, E. P., Ostilla-Mónico, R., Donners, J., & Verzicco, R. (2015). A pencil distributed finite difference code for strongly turbulent wall-bounded flows. Computers and Fluids, 116, 10–16.MathSciNetzbMATHCrossRefGoogle Scholar
  119. van Driest, E. R. (1951). Turbulent boundary layer in compressible fluids. Journal of the Aeronautical Sciences, 18, 145–160.MathSciNetzbMATHCrossRefGoogle Scholar
  120. van Driest, E. R. (1956). The problem of aerodynamic heating. Aeronautical Engineering Review, 15, 26–41.Google Scholar
  121. Verzicco, R., & Orlandi, P. (1996). A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. Journal of Computational Physics, 123(2), 402–414.MathSciNetzbMATHCrossRefGoogle Scholar
  122. Vichnevetsky, R., & Bowles, J. B. (1982). Fourier analysis of numerical approximations of hyperbolic equations. Philadelphia: SIAM.zbMATHCrossRefGoogle Scholar
  123. Waleffe, F. (1997). On a self-sustaining process in shear flows. Physics of Fluids, 9, 883–900.CrossRefGoogle Scholar

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© CISM International Centre for Mechanical Sciences 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace Engineering‘Sapienza’ University of RomeRomeItaly

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