Turbulent Flows

  • Michel O. Deville
  • Thomas B. Gatski


One can arguably assume that the start of turbulence research as we view it within the context of conservation equations and statistical correlations began in the late nineteenth century. The seminal papers of Boussinesq (Mém Acad Sci Inst Fr 23:46–50, 1877) and Reynolds (Philos Trans R Soc Lond A 186:123–164, 1895) dealing with turbulent momentum transfer and statistical averages of turbulent fluctuations still lie at the core of the research being conducted today. Not until the middle of the last century did texts on the subject begin to appear in the form of monographs focused on either homogeneous turbulence (Batchelor, Homogeneous turbulence, 1953) where theoretical analysis was tractable or turbulent shear flows (Townsend, The structure of turbulent shear flows, 1956) where the emphasis shifted to the extensive experimental results that had been gathered up to that point. These early monographs were soon augmented with texts (Hinze, Turbulence, 1959; Lumley, Stochastic tools in turbulence, 1970a; Monin and Yaglom, Statistical fluid mechanics, vols. 1 and 2, 1971) dealing more rigorously with the (statistical) theoretical aspects. The first text directed mainly at introducing students to the subject was Tennekes and Lumley (A first course in turbulence, 1972) and this marked the beginning of an era that continues today where numerical solutions have become as, or sometimes more, important in turbulence research as physical experiments. A recent presentation including modeling and simulation considerations is given in the book by Pope (Turbulent flows, 2000). Although the primary focus in this chapter is on the incompressible flows and Newtonian fluids, there has also been much effort directed toward high-speed flows and compressible turbulence and now, within the last decade, to viscoelastic, polymeric fluids.


Large Eddy Simulation Dissipation Rate Reynolds Stress Viscoelastic Fluid Turbulent Stress 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Abid R, Speziale CG (1993) Predicting equilibrium states with Reynolds stress closures in channel flow and homogeneous shear flow. Phys Fluids A 5:1776–1782 zbMATHCrossRefGoogle Scholar
  2. Alvelius K (1999) Studies of turbulence and its modelling through large eddy- and direct numerical simulation. PhD thesis, Department of Mechanics, KTH, Stockholm, Sweden Google Scholar
  3. Bardina J, Ferziger JH, Reynolds WC (1980) Improved subgrid scale models for large eddy simulation. Paper no. 80-1357, AIAA Google Scholar
  4. Batchelor GK (1953) Homogeneous turbulence. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  5. Beris AN, Edwards BJ (1994) Thermodynamics of flowing systems with internal microstructure. Oxford University Press, New York Google Scholar
  6. Berselli LC, Iliescu T, Layton WJ (2006) Mathematics of large eddy simulation of turbulent flows. Springer, Berlin zbMATHGoogle Scholar
  7. Boussinesq J (1877) Essai sur la théorie des eaux courantes. Mém Acad Sci Inst Fr 23:46–50 Google Scholar
  8. Chou PY (1945) On velocity correlations and the solutions of the equations of turbulent fluctuation. Q Appl Math 3:38–54 zbMATHGoogle Scholar
  9. Coleman BD, Noll JM (1961) Recent results in the continuum theory of viscoelastic fluids. Ann NY Acad Sci 89:672–714 zbMATHMathSciNetCrossRefGoogle Scholar
  10. Cormack DE, Leal LG, Seinfeld JH (1978) An evaluation of mean Reynolds stress turbulence models: the triple velocity correlation. J Fluids Eng 100:47–54 CrossRefGoogle Scholar
  11. Craft TJ, Launder BE (1996) A Reynolds stress closure designed for complex geometries. Int J Heat Fluid Flow 17:245–254 CrossRefGoogle Scholar
  12. Craft TJ, Launder BE (2002) Closure modelling near the two-component limit. In: Launder B, Sandham N (eds) Closure strategies for turbulent and transitional flows. Cambridge University Press, Cambridge, pp 102–126 Google Scholar
  13. Craft TJ, Launder BE, Suga K (1996) Development and application of a cubic eddy-viscosity model of turbulence. Int J Heat Fluid Flow 17:108–115 CrossRefGoogle Scholar
  14. Craft TJ, Launder BE, Suga K (1997) Prediction of turbulent transitional phenomena with a nonlinear eddy-viscosity model. Int J Heat Fluid Flow 18:15–28 CrossRefGoogle Scholar
  15. Crow SC (1967) Visco-elastic character of fine-grained isotropic turbulence. Phys Fluids 10:1587–1589 CrossRefGoogle Scholar
  16. Crow SC (1968) Visco-elastic properties of fine-grained incompressible turbulence. J Fluid Mech 33:1–20 zbMATHCrossRefGoogle Scholar
  17. Cruz DOA, Pinho FT (2003) Modeling the new stress for improved drag reduction predictions of viscoelastic pipe flow. J Non-Newton Fluid Mech 114:109–148 zbMATHCrossRefGoogle Scholar
  18. Cruz DOA, Pinho FT, Resende PR (2004) Modeling the new stress for improved drag reduction predictions of viscoelastic pipe flow. J Non-Newton Fluid Mech 121:127–141 zbMATHCrossRefGoogle Scholar
  19. Daly BJ, Harlow FH (1970) Transport equations of turbulence. Phys Fluids 13:2634–2649 CrossRefGoogle Scholar
  20. De Angelis E, Casciola CM, Piva R (2002) DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput Fluids 31:495–507 zbMATHCrossRefGoogle Scholar
  21. Deardorff JW (1970) Turbulence measurements in supersonic two-dimensional wake. J Fluid Mech 41:453–480 zbMATHCrossRefGoogle Scholar
  22. Dimitropoulos CD, Sureshkumar R, Beris AN (1998) Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters. J Non-Newton Fluid Mech 79:433–468 zbMATHCrossRefGoogle Scholar
  23. Dimitropoulos CD, Sureshkumar R, Beris AN, Handler RA (2001) Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow. Phys Fluids 13(4):1016–1024 CrossRefGoogle Scholar
  24. Dimitropoulos CD, Dubief Y, Shakfeh ESG, Moin P, Lele SK (2005) Direct numerical simulation of polymer-induced drag reduction in turbulent boundary layer flow. Phys Fluids 17:011705 CrossRefGoogle Scholar
  25. Dimitropoulos CD, Dubief Y, Shakfeh ESG, Moin P (2006) Direct numerical simulation of polymer-induced drag reduction in turbulent boundary layer flow of inhomogeneous polymer solutions. J Fluid Mech 566:153–162 zbMATHCrossRefGoogle Scholar
  26. Dubief Y, White CM, Terrapon VE, Shakfeh ESG, Moin P, Lele SK (2004) On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J Fluid Mech 514:271–280 zbMATHCrossRefGoogle Scholar
  27. Durbin PA (1991) Near-wall turbulence closure modeling without ‘damping functions’. Theor Comput Fluid Dyn 3:1–13 zbMATHGoogle Scholar
  28. Durbin PA (1993) A Reynolds stress model for near-wall turbulence. J Fluid Mech 249:465–498 CrossRefGoogle Scholar
  29. Durbin PA, Petterson Reif BA (2010) Statistical theory and modeling for turbulent flows, 2nd edn. Wiley, New York CrossRefGoogle Scholar
  30. Eyink L (1994) The renormalization group method in statistical hydrodynamics. Phys Fluids 6:3063–3078 zbMATHMathSciNetCrossRefGoogle Scholar
  31. Fasel HF, von Terzi DA, Sandberg RD (2006) A methodology for simulating compressible turbulent flows. J Appl Mech 73:405–412 zbMATHCrossRefGoogle Scholar
  32. Fu S, Launder BE, Tselepidakis DP (1987) Accommodating the effects of high strain rates in modelling the pressure-strain correlation. Technical report TFD/87/5, UMIST Google Scholar
  33. Gatski TB (1996) Turbulent flows: model equations and solution methodology. In: Peyret R (ed) Handbook of computational fluid dynamics. Academic Press, London, pp 339–415 Google Scholar
  34. Gatski TB, Jongen T (2000) Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog Aerosp Sci 36:655–682 CrossRefGoogle Scholar
  35. Gatski TB, Speziale CG (1993) On algebraic stress models for complex turbulent flows. J Fluid Mech 254:59–78 zbMATHMathSciNetCrossRefGoogle Scholar
  36. Gatski TB, Wallin S (2004) Extending the weak-equilibrium condition for algebraic Reynolds stress models to rotating and curved flows. J Fluid Mech 518:147–155 zbMATHCrossRefGoogle Scholar
  37. Gatski TB, Rumsey CL, Manceau R (2007) Current trends in modelling research for turbulent aerodynamic flows. Philos Trans R Soc A 365:2389–2418 MathSciNetCrossRefGoogle Scholar
  38. Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 3:1760–1765 zbMATHCrossRefGoogle Scholar
  39. Geurts BJ (2004) Elements of direct and large-eddy simulation. Edwards, Philadelphia Google Scholar
  40. Girimaji SS (1996) Fully explicit and self-consistent algebraic Reynolds stress model. Theor Comput Fluid Dyn 8:387–402 zbMATHGoogle Scholar
  41. Hallbäck M, Groth J, Johansson AV (1990) An algebraic model for nonisotropic turbulent dissipation rate in Reynolds stress closures. Phys Fluids A 2:1859–1866 zbMATHCrossRefGoogle Scholar
  42. Hanjalić K (1994) Advanced turbulence closure models: a view of current status and future prospects. Int J Heat Fluid Flow 15:178–203 CrossRefGoogle Scholar
  43. Hanjalić K, Jakirlić S (2002) Second-moment turbulence closure modelling. In: Launder B, Sandham N (eds) Closure strategies for turbulent and transitional flows. Cambridge University Press, Cambridge, pp 47–101 Google Scholar
  44. Hanjalić K, Launder BE (1972) A Reynolds stress model of turbulence and its application to thin shear flows. J Fluid Mech 52:609–638 zbMATHCrossRefGoogle Scholar
  45. Hanjalić K, Launder BE (1976) Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence. J Fluid Mech 74:593–610 zbMATHCrossRefGoogle Scholar
  46. Hanjalić K, Launder BE (2011) Modelling turbulence in engineering and the environment: second-moment routes to closure. Cambridge University Press, Cambridge Google Scholar
  47. Haworth DC, Pope SB (1986) A generalized Langevin model for turbulent flows. Phys Fluids 29:387–405 zbMATHMathSciNetCrossRefGoogle Scholar
  48. Hinze JO (1959) Turbulence. McGraw-Hill, New York Google Scholar
  49. Housiadas KD, Beris AN, Handler RA (2005) Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow. Phys Fluids 17:035106 CrossRefGoogle Scholar
  50. Jakirlić S, Hanjalić K (2002) A new approach to modelling near-wall turbulence energy and stress dissipation. J Fluid Mech 459:139–166 zbMATHCrossRefGoogle Scholar
  51. Johansson AV, Hallbäck M (1994) Modelling of rapid pressure-strain in Reynolds-stress closures. J Fluid Mech 269:143–168 zbMATHCrossRefGoogle Scholar
  52. Jongen T, Gatski TB (1999) A unified analysis of planar homogeneous turbulence using single-point closure equations. J Fluid Mech 399:117–150 zbMATHCrossRefGoogle Scholar
  53. Kampé de Fériet J, Betchov R (1951) Theoretical and experimental averages of turbulent functions. Proc K Ned Akad Wet 53:389–398 Google Scholar
  54. Kim J, Moin P, Moser RD (1982) Numerical investigation of turbulent channel flow. J Fluid Mech 118:341–377 zbMATHCrossRefGoogle Scholar
  55. Kim J, Moin P, Moser RD (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–186 zbMATHCrossRefGoogle Scholar
  56. Kolmogorov AN (1941a) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl Akad Nauk SSSR 30:301–305 Google Scholar
  57. Kolmogorov AN (1941b) On degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl Akad Nauk SSSR 31:538–541 Google Scholar
  58. Kolmogorov AN (1942) Equations of turbulent motion in an incompressible fluid. Izv Akad Nauk SSSR, Ser Fiz 6:56–58 Google Scholar
  59. Kraichnan RH (1964) Kolmogorov’s hypothesis and Eulerian turbulence theory. Phys Fluids 7:1723–1734 zbMATHMathSciNetCrossRefGoogle Scholar
  60. Kreuzinger J, Friedrich R, Gatski TB (2006) Compressibility effects in the solenoidal dissipation rate equation: a priori assessment and modeling. Int J Heat Fluid Flow 27:696–706 CrossRefGoogle Scholar
  61. Launder BE, Reece GJ, Rodi W (1975) Progress in the development of a Reynolds-stress turbulence closure. J Fluid Mech 68:537–566 zbMATHCrossRefGoogle Scholar
  62. Lesieur M (2008) Turbulence in fluids, 4th edn. Springer, Dordrecht zbMATHCrossRefGoogle Scholar
  63. Lesieur M, Métais O, Comte P (2005) Large-eddy simulations of turbulence. Cambridge University Press, Cambridge zbMATHCrossRefGoogle Scholar
  64. Li CF, Gupta VK, Sureshkumar R, Khomami B (2006) Turbulent channel flow of dilute polymeric solutions: drag reduction scaling and an eddy viscosity model. J Non-Newton Fluid Mech 139:177–189 zbMATHCrossRefGoogle Scholar
  65. Lien FS, Leschziner MA (1995) Modelling 2D separation from a high lift aerofoil with a non-linear eddy-viscosity model and second-moment closure. Aeronaut J 99:125–144 Google Scholar
  66. Lilly DK (1966) On the application of the eddy-viscosity concept in the inertial sub-range of turbulence. Manuscript 123, NCAR Google Scholar
  67. Lilly DK (1992) A proposed modification of the Germano subgrid-scale closure method. Phys Fluids A 4:633–635 CrossRefGoogle Scholar
  68. Lumley JL (1967) Rational approach to relations between motions of differing scales in turbulent flows. Phys Fluids 10:1405–1408 zbMATHMathSciNetCrossRefGoogle Scholar
  69. Lumley JL (1970a) Stochastic tools in turbulence. Academic Press, New York zbMATHGoogle Scholar
  70. Lumley JL (1970b) Toward a turbulent constitutive relation. J Fluid Mech 41:413–434 CrossRefGoogle Scholar
  71. Lumley JL (1978) Computational modeling of turbulent flows. Adv Appl Mech 18:123–176 zbMATHMathSciNetCrossRefGoogle Scholar
  72. Lumley JL, Khajeh-Nouri B (1974) Computational modeling of turbulent transport. In: Turbulent diffusion in environmental pollution. Academic Press, New York, pp 237–248 Google Scholar
  73. Lumley JL, Newman GR (1977) The return to isotropy of homogeneous turbulence. J Fluid Mech 82:161–178 zbMATHMathSciNetCrossRefGoogle Scholar
  74. Magnaudet J (1993) Modelling of inhomogeneous turbulence in the absence of mean velocity gradients. Appl Sci Res 51:525–531 zbMATHCrossRefGoogle Scholar
  75. Manceau R, Hanjalić K (2000a) Elliptic blending model: a new near-wall Reynolds stress turbulence closure. Phys Fluids 14:744–754 CrossRefGoogle Scholar
  76. Manceau R, Hanjalić K (2000b) A new form of the elliptic relaxation equation to account for wall effects in RANS modeling. Phys Fluids 12:2345–2351 CrossRefGoogle Scholar
  77. Manceau R, Carlson JR, Gatski TB (2002) A rescaled elliptic relaxation approach: neutralizing the effect on the log layer. Phys Fluids 14:3868–3879 CrossRefGoogle Scholar
  78. McComb WD (1990) The physics of fluid turbulence. Clarendon, Oxford Google Scholar
  79. Mellor GL, Herring HJ (1973) A survey of the mean turbulent field closure models. AIAA J 11:590–599 zbMATHCrossRefGoogle Scholar
  80. Menter F (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 32:1598–1605 CrossRefGoogle Scholar
  81. Min T, Yoo JY, Choi H, Joseph DD (2003) Drag reduction by polymer additives in a turbulent channel flow. J Fluid Mech 486:213–238 zbMATHCrossRefGoogle Scholar
  82. Monin AS, Yaglom AM (1971) Statistical fluid mechanics, vols 1 and 2. MIT Press, Boston Google Scholar
  83. Moser RD, Kim J, Mansour NN (1999) Direct numerical simulation of turbulent channel flow up to Re τ=590. Phys Fluids 11:943–945 zbMATHCrossRefGoogle Scholar
  84. Myong HK, Kasagi N (1990) Prediction of anisotropy of the near-wall turbulence with an anisotropic low-Reynolds-number kε turbulence model. J Fluids Eng 112:521–524 CrossRefGoogle Scholar
  85. Nisizima S, Yoshizawa A (1987) Turbulent channel and Couette flows using an anisotropic kε model. AIAA J 25:414–420 zbMATHCrossRefGoogle Scholar
  86. Oberlack M (1997) Non-isotropic dissipation in non-homogeneous turbulence. J Fluid Mech 350:351–374 zbMATHMathSciNetCrossRefGoogle Scholar
  87. Oceni AG, Manceau R, Gatski TB (2010) Introduction of wall effects in explicit algebraic stress models through elliptic blending. In: Stanislas M, Jimenez J, Marusic I (eds) Progress in wall turbulence: understanding and modelling. Springer, Heidelberg, pp 275–283 Google Scholar
  88. Orszag SA (1970) Analytical theories of turbulence. J Fluid Mech 41:59–82 CrossRefGoogle Scholar
  89. Pennisi S (1992) On third order tensor-valued isotropic functions. Int J Eng Sci 30:679–692 MathSciNetCrossRefGoogle Scholar
  90. Pinho FT (2003) A GNF framework for turbulent flow models of drag reducing fluids and proposal for a kε type closure. J Non-Newton Fluid Mech 121:127–141 Google Scholar
  91. Pinho FT, Li CF, Younis BA, Sureshkumar R (2008a) A low Reynolds number turbulence closure for viscoelastic fluids. J Non-Newton Fluid Mech 154:89–108 CrossRefGoogle Scholar
  92. Pinho FT, Sadanandan B, Sureshkumar R (2008b) One equation model for turbulent channel flow with second order viscoelastic corrections. Flow Turbul. Combust. doi: 10.1007/s10494-008-9134-6 Google Scholar
  93. Pope SB (1975) A more general effective viscosity hypothesis. J Fluid Mech 72:331–340 zbMATHCrossRefGoogle Scholar
  94. Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  95. Prandtl L (1945) Über ein neues formelsystem für die ausgebildete turbulenz. In: Nachr. Akad. Wiss. Göttingen. van den Loerck und Ruprecht, Göttingen, pp 6–19 Google Scholar
  96. Pruett CD (2000) On Eulerian times-domain filtering for spatial large-eddy simulation. AIAA J 38:1634–1642 CrossRefGoogle Scholar
  97. Pruett CD, Gatski TB, Grosch CE, Thacker WD (2003) The temporally filtered Navier-Stokes equations: properties of the residual stress. Phys Fluids 15:2127–2140 CrossRefGoogle Scholar
  98. Ptasinski PK, Boersma BJ, Nieuwstadt FTM, Hulsen MA, Van den Brule HAA, Hunt JCR (2003) Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms. J Fluid Mech 490:251–291 zbMATHCrossRefGoogle Scholar
  99. Resende PR, Kim K, Younis BA, Sureshkumar R, Pinho FT (2011) A FENE-P kε turbulence model for low and intermediate regimes of polymer-induced drag reduction. J Non-Newton Fluid Mech 166:639–660 CrossRefGoogle Scholar
  100. Reynolds O (1895) On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Philos Trans R Soc Lond A 186:123–164 zbMATHCrossRefGoogle Scholar
  101. Reynolds WC (1976) Computation of turbulent flows. In: Annual review of fluid mechanics. Annual Reviews, New York, pp 183–208 Google Scholar
  102. Rivlin RS (1957) The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids. Q Appl Math 15:212–215 zbMATHMathSciNetGoogle Scholar
  103. Rivlin RS, Ericksen JL (1955) Stress-deformation relations for isotropic materials. Arch Ration Mech Anal 4:323–425 zbMATHMathSciNetGoogle Scholar
  104. Rodi W (1972) The prediction of free turbulent boundary layers by use of a two-equation model of turbulence. PhD thesis, University of London, London Google Scholar
  105. Rodi W (1976) A new algebraic relation for calculating the Reynolds stresses. Z Angew Math Mech 56:219–221 MathSciNetCrossRefGoogle Scholar
  106. Rotta JC (1951) Statistiche theorie nichthomogener turbulenz. Z Phys 129:547–572 zbMATHMathSciNetCrossRefGoogle Scholar
  107. Rubinstein R, Barton JH (1990) Nonlinear Reynolds stress models and the renormalization group. Phys Fluids A 2:1472–1476 zbMATHCrossRefGoogle Scholar
  108. Saffman PG (1970) A model for inhomogeneous turbulence. Proc R Soc Lond Ser A 317:417–433 zbMATHCrossRefGoogle Scholar
  109. Sagaut P (2006) Large eddy simulation for incompressible flows. Springer, Berlin zbMATHGoogle Scholar
  110. Sagaut P, Deck S, Terracol M (2006) Multiscale and multiresolution approaches in turbulence. Imperial College Press, London CrossRefGoogle Scholar
  111. Sarkar S, Speziale CG (1990) A simple nonlinear model for the return to isotropy in turbulence. Phys Fluids 29:84–93 MathSciNetGoogle Scholar
  112. Schumann U (1977) Realizability of Reynolds stress models. Phys Fluids 20:721–725 zbMATHCrossRefGoogle Scholar
  113. Schwarz WR, Bradshaw P (1994) Term-by-term tests of stress-transport turbulence models in a three-dimensional boundary layer. Phys Fluids 6:986–998 CrossRefGoogle Scholar
  114. Shih T-H, Zhu J, Lumley JL (1995) A new Reynolds stress algebraic equation model. Comput Methods Appl Mech Eng 125:287–302 CrossRefGoogle Scholar
  115. Shir CC (1973) A preliminary numerical study of atmospheric turbulent flows in the idealized planetary boundary layer. J Atmos Sci 30:1327–1339 CrossRefGoogle Scholar
  116. Shur ML, Spalart PR, Strelets MK, Travin AK (2008) A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities. Int J Heat Fluid Flow 29:1638–1649 CrossRefGoogle Scholar
  117. Simonsen AJ, Krogstad P-Å(2005) Turbulent stress invariant analysis: clarification of existing terminology. Phys Fluids 17:088103 CrossRefGoogle Scholar
  118. Sinha K, Candler GV (2003) Turbulent dissipation-rate equation for compressible flows. AIAA J 41:1017–1021 CrossRefGoogle Scholar
  119. Sjögren T, Johansson AV (2000) Development and calibration of algebraic nonlinear models for terms in the Reynolds stress transport equations. Phys Fluids 12:1554–1572 zbMATHCrossRefGoogle Scholar
  120. Smagorinsky I (1963) General circulation experiments with the primitive equations. I. The basic experiment. Mon Weather Rev 91:99–164 CrossRefGoogle Scholar
  121. Smith GF, Younis BA (2004) Isotropic tensor-valued polynomial function of second-order and third-order tensors. Int J Eng Sci 43:447–456 MathSciNetCrossRefGoogle Scholar
  122. So RMC, Lai YG, Zhang HS, Hwang BC (1991) Second-order near-wall turbulence closures: a review. AIAA J 29:1819–1835 zbMATHCrossRefGoogle Scholar
  123. So RMC, Aksoy H, Yuan SP, Sommer TP (1996) Modeling Reynolds-number effects in wall-bounded turbulent flows. J Fluids Eng 118:260–267 CrossRefGoogle Scholar
  124. Spalart PR, Allmaras SR (1994) A one-equation turbulence model for aerodynamic flows. Rech Aérosp 1:5–21 Google Scholar
  125. Spalart PR, Jou W-H, Strelets M, Allmaras SR (1997) Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. In: Liu C, Liu Z (eds) Advances in DNS/LES. Greyden Press, Columbus, pp 137–147 Google Scholar
  126. Speziale CG (1987) On nonlinear kl and kε models of turbulence. J Fluid Mech 178:459–475 zbMATHCrossRefGoogle Scholar
  127. Speziale CG (1991) Analytical methods for the development of Reynolds-stress closures in turbulence. In: Annual review of fluid mechanics. Annual Reviews, New York, pp 107–157 Google Scholar
  128. Speziale CG (1998a) A consistency condition for non-linear algebraic Reynolds stress models in turbulence. Int J Non-Linear Mech 33:579–584 zbMATHMathSciNetCrossRefGoogle Scholar
  129. Speziale CG (1998c) Turbulence modeling for time-dependent RANS and VLES: a review. AIAA J 36:173–184 zbMATHCrossRefGoogle Scholar
  130. Speziale CG, Gatski TB (1997) Analysis and modelling of anisotropies in the dissipation rate of turbulence. J Fluid Mech 344:155–180 zbMATHMathSciNetCrossRefGoogle Scholar
  131. Speziale CG, Sarkar S, Gatski TB (1991) Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach. J Fluid Mech 227:245-27 CrossRefGoogle Scholar
  132. Stolz S, Adams NA (1999) An approximate deconvolution procedure for large-eddy simulation. Phys Fluids 11:1699–1701 zbMATHCrossRefGoogle Scholar
  133. Stolz S, Adams NA, Kleiser L (1999) The approximate deconvolution model applied to turbulent channel flow. In: Voke P, Sandham N, Kleiser L (eds) Direct and large eddy simulation III. Kluwer Academic, Dordrecht, pp 163–174 Google Scholar
  134. Stolz S, Adams NA, Kleiser L (2001a) An approximate deconvolution model for large-eddy simulation with application to wall-bounded incompressible flows. Phys Fluids 13:997–1015 CrossRefGoogle Scholar
  135. Stolz S, Adams NA, Kleiser L (2001b) An approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction. Phys Fluids 13:2985–3001 CrossRefGoogle Scholar
  136. Straatman AG (1999) A modified model for diffusion in second-moment turbulence closures. J Fluids Eng 121:747–756 CrossRefGoogle Scholar
  137. Straatman AG, Stubley GD, Raithby GD (1998) Examination of diffusion modeling using zero-mean-shear turbulence. AIAA J 36:929–935 CrossRefGoogle Scholar
  138. Sureshkumar R, Beris AN (1995) Effect of artificial stress diffusivity on the stability of numerical calculations and the dynamics of time-dependent viscoelastic flows. J Non-Newton Fluid Mech 60:53–80 CrossRefGoogle Scholar
  139. Sureshkumar R, Beris AN, Handler RA (1997) Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys Fluids 9:743–755 CrossRefGoogle Scholar
  140. Taulbee DB (1992) An improved algebraic Reynolds stress model and corresponding nonlinear stress model. Phys Fluids A 4:2555–2561 zbMATHCrossRefGoogle Scholar
  141. Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, Boston Google Scholar
  142. Thais L, Tejada-Martínez AE, Gatski TB, Mompean G (2010) Direct and temporal large eddy numerical simulations of FENE-P drag reduction flows. Phys Fluids 22:013103. doi: 10.1063/1.3294574 CrossRefGoogle Scholar
  143. Thais L, Tejada-Martínez AE, Gatski TB, Mompean G (2011) A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow. Comput Fluids 43:134–142 zbMATHMathSciNetCrossRefGoogle Scholar
  144. Townsend AA (1956) The structure of turbulent shear flows. Cambridge University Press, Cambridge Google Scholar
  145. van Driest ER (1956) On turbulent flow near a wall. J Aeronaut Sci 23:1007–1011 zbMATHGoogle Scholar
  146. Wallin S, Johansson AV (2000) An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. J Fluid Mech 403:89–132 zbMATHMathSciNetCrossRefGoogle Scholar
  147. Wilcox DC (2006) Turbulence modeling for CFD, 3rd edn. DCW Industries, La Cañada Google Scholar
  148. Wilson KG (1971) Renormalization group and critical phenomena. 1. Renormalization group and Kadanoff scaling picture. Phys Rev B 4:3174–3183 zbMATHCrossRefGoogle Scholar
  149. Yakhot A, Orszag SA, Yakhot V, Israeli M (1989) Renormalisation group formulation for large-eddy simulation. J Sci Comput 4:139–159 MathSciNetCrossRefGoogle Scholar
  150. Yakhot A, Orszag SA, TB Gatski ST, Speziale CG (1992) Development of turbulence models for shear flows by a double expansion technique. Phys Fluids A 4:1510–1520 zbMATHMathSciNetCrossRefGoogle Scholar
  151. Yakhot V, Orszag SA (1986) Renormalisation group analysis of turbulence. I. Basic theory. J Sci Comput 1:3–51 zbMATHMathSciNetCrossRefGoogle Scholar
  152. Ying R, Canuto VM (1996) Turbulence modeling over two-dimensional hills using an algebraic Reynolds stress expression. Bound-Layer Meteorol 77:69–99 CrossRefGoogle Scholar
  153. Yoshizawa A (1984) Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation. Phys Fluids 27:1377–1387 zbMATHCrossRefGoogle Scholar
  154. Younis BA, Gatski TB, Speziale CG (2000) Towards a rational model for the triple velocity correlations of turbulence. Proc R Soc Lond A 456:909–920 zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Mechanical EngineeringSwiss Federal Institute of Technology, EPFLLausanneSwitzerland
  2. 2.Institute PPRIMECNRS-Université de Poitiers-ENSMAFuturoscope ChasseneuilFrance
  3. 3.Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric SciencesOld Dominion UniversityNorfolkUSA

Personalised recommendations