Skip to main content
SpringerLink
Log in

Turbulent Flows

  • Chapter
Mathematical Modeling for Complex Fluids and Flows

  • 2419 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that throughout this chapter the density field is assumed constant and has been assimilated into the pressure field and the viscous stress tensor. It, therefore, will not appear explicitly in any of the differential formulations.

  2. 2.

    The reader will notice that, throughout the chapter, index notation will be predominantly used in the governing equations to be discussed. This notation is more common in the turbulent flow literature.

  3. 3.

    Figures  and reprinted with permission from: T. Jongen, T.B. Gatski (1999) A unified analysis of planar homogeneous turbulence using single-point closure equations. J Fluid Mech 399:117–150. Copyright Cambridge University Press.

  4. 4.

    Reprinted with permission from: T.B. Gatski (2004) Constitutive equations for turbulent flows. Theor Comput Fluid Dyn 18:345–369. Copyright Springer 2004.

  5. 5.

    Figure  reprinted with permission from: T.B. Gatski, T. Jongen (2000) Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog Aerosp Sci 36:655–682. Copyright Elsevier 2000.

  6. 6.

    In general, the inertial frame of relevance in this curved flow case may not necessarily be the one from which Eq. (6.143) was obtained originally. For example, a fixed relative rotation between the two inertial frames can occur. Nevertheless, since the essence of the discussion here is the proper choice of equilibrium condition and appropriate measure of effect of curvature, the transformation back to the inertial frame is simply based on the orthogonal transformation tensor Q.

  7. 7.

    For the two-dimensional case, the basis tensor Φ 3 is reduced to the form

    $$\boldsymbol{\varPhi}_3 = \frac{1}{3}\boldsymbol{I}-\frac {1}{2}\boldsymbol{I}^{(2)}\ ,$$

    by the Cayley-Hamilton theorem where I (2) is the two-dimensional Kronecker delta.

References

  • 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

    Article  MATH  Google Scholar 

  • 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 

  • Bardina J, Ferziger JH, Reynolds WC (1980) Improved subgrid scale models for large eddy simulation. Paper no. 80-1357, AIAA

    Google Scholar 

  • Batchelor GK (1953) Homogeneous turbulence. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Beris AN, Edwards BJ (1994) Thermodynamics of flowing systems with internal microstructure. Oxford University Press, New York

    Google Scholar 

  • Berselli LC, Iliescu T, Layton WJ (2006) Mathematics of large eddy simulation of turbulent flows. Springer, Berlin

    MATH  Google Scholar 

  • Boussinesq J (1877) Essai sur la théorie des eaux courantes. Mém Acad Sci Inst Fr 23:46–50

    Google Scholar 

  • Chou PY (1945) On velocity correlations and the solutions of the equations of turbulent fluctuation. Q Appl Math 3:38–54

    MATH  Google Scholar 

  • Coleman BD, Noll JM (1961) Recent results in the continuum theory of viscoelastic fluids. Ann NY Acad Sci 89:672–714

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Craft TJ, Launder BE (1996) A Reynolds stress closure designed for complex geometries. Int J Heat Fluid Flow 17:245–254

    Article  Google Scholar 

  • 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 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Crow SC (1967) Visco-elastic character of fine-grained isotropic turbulence. Phys Fluids 10:1587–1589

    Article  Google Scholar 

  • Crow SC (1968) Visco-elastic properties of fine-grained incompressible turbulence. J Fluid Mech 33:1–20

    Article  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Daly BJ, Harlow FH (1970) Transport equations of turbulence. Phys Fluids 13:2634–2649

    Article  Google Scholar 

  • De Angelis E, Casciola CM, Piva R (2002) DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput Fluids 31:495–507

    Article  MATH  Google Scholar 

  • Deardorff JW (1970) Turbulence measurements in supersonic two-dimensional wake. J Fluid Mech 41:453–480

    Article  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Durbin PA (1991) Near-wall turbulence closure modeling without ‘damping functions’. Theor Comput Fluid Dyn 3:1–13

    MATH  Google Scholar 

  • Durbin PA (1993) A Reynolds stress model for near-wall turbulence. J Fluid Mech 249:465–498

    Article  Google Scholar 

  • Durbin PA, Petterson Reif BA (2010) Statistical theory and modeling for turbulent flows, 2nd edn. Wiley, New York

    Book  Google Scholar 

  • Eyink L (1994) The renormalization group method in statistical hydrodynamics. Phys Fluids 6:3063–3078

    Article  MATH  MathSciNet  Google Scholar 

  • Fasel HF, von Terzi DA, Sandberg RD (2006) A methodology for simulating compressible turbulent flows. J Appl Mech 73:405–412

    Article  MATH  Google Scholar 

  • 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 

  • 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 

  • Gatski TB, Jongen T (2000) Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog Aerosp Sci 36:655–682

    Article  Google Scholar 

  • Gatski TB, Speziale CG (1993) On algebraic stress models for complex turbulent flows. J Fluid Mech 254:59–78

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Gatski TB, Rumsey CL, Manceau R (2007) Current trends in modelling research for turbulent aerodynamic flows. Philos Trans R Soc A 365:2389–2418

    Article  MathSciNet  Google Scholar 

  • Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 3:1760–1765

    Article  MATH  Google Scholar 

  • Geurts BJ (2004) Elements of direct and large-eddy simulation. Edwards, Philadelphia

    Google Scholar 

  • Girimaji SS (1996) Fully explicit and self-consistent algebraic Reynolds stress model. Theor Comput Fluid Dyn 8:387–402

    MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Hanjalić K (1994) Advanced turbulence closure models: a view of current status and future prospects. Int J Heat Fluid Flow 15:178–203

    Article  Google Scholar 

  • 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 

  • Hanjalić K, Launder BE (1972) A Reynolds stress model of turbulence and its application to thin shear flows. J Fluid Mech 52:609–638

    Article  MATH  Google Scholar 

  • Hanjalić K, Launder BE (1976) Contribution towards a Reynolds-stress closure for low-Reynolds-number turbulence. J Fluid Mech 74:593–610

    Article  MATH  Google Scholar 

  • Hanjalić K, Launder BE (2011) Modelling turbulence in engineering and the environment: second-moment routes to closure. Cambridge University Press, Cambridge

    Google Scholar 

  • Haworth DC, Pope SB (1986) A generalized Langevin model for turbulent flows. Phys Fluids 29:387–405

    Article  MATH  MathSciNet  Google Scholar 

  • Hinze JO (1959) Turbulence. McGraw-Hill, New York

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Jakirlić S, Hanjalić K (2002) A new approach to modelling near-wall turbulence energy and stress dissipation. J Fluid Mech 459:139–166

    Article  MATH  Google Scholar 

  • Johansson AV, Hallbäck M (1994) Modelling of rapid pressure-strain in Reynolds-stress closures. J Fluid Mech 269:143–168

    Article  MATH  Google Scholar 

  • Jongen T, Gatski TB (1999) A unified analysis of planar homogeneous turbulence using single-point closure equations. J Fluid Mech 399:117–150

    Article  MATH  Google Scholar 

  • Kampé de Fériet J, Betchov R (1951) Theoretical and experimental averages of turbulent functions. Proc K Ned Akad Wet 53:389–398

    Google Scholar 

  • Kim J, Moin P, Moser RD (1982) Numerical investigation of turbulent channel flow. J Fluid Mech 118:341–377

    Article  MATH  Google Scholar 

  • Kim J, Moin P, Moser RD (1987) Turbulence statistics in fully developed channel flow at low Reynolds number. J Fluid Mech 177:133–186

    Article  MATH  Google Scholar 

  • 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 

  • Kolmogorov AN (1941b) On degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl Akad Nauk SSSR 31:538–541

    Google Scholar 

  • Kolmogorov AN (1942) Equations of turbulent motion in an incompressible fluid. Izv Akad Nauk SSSR, Ser Fiz 6:56–58

    Google Scholar 

  • Kraichnan RH (1964) Kolmogorov’s hypothesis and Eulerian turbulence theory. Phys Fluids 7:1723–1734

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Launder BE, Reece GJ, Rodi W (1975) Progress in the development of a Reynolds-stress turbulence closure. J Fluid Mech 68:537–566

    Article  MATH  Google Scholar 

  • Lesieur M (2008) Turbulence in fluids, 4th edn. Springer, Dordrecht

    Book  MATH  Google Scholar 

  • Lesieur M, Métais O, Comte P (2005) Large-eddy simulations of turbulence. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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 

  • Lilly DK (1966) On the application of the eddy-viscosity concept in the inertial sub-range of turbulence. Manuscript 123, NCAR

    Google Scholar 

  • Lilly DK (1992) A proposed modification of the Germano subgrid-scale closure method. Phys Fluids A 4:633–635

    Article  Google Scholar 

  • Lumley JL (1967) Rational approach to relations between motions of differing scales in turbulent flows. Phys Fluids 10:1405–1408

    Article  MATH  MathSciNet  Google Scholar 

  • Lumley JL (1970a) Stochastic tools in turbulence. Academic Press, New York

    MATH  Google Scholar 

  • Lumley JL (1970b) Toward a turbulent constitutive relation. J Fluid Mech 41:413–434

    Article  Google Scholar 

  • Lumley JL (1978) Computational modeling of turbulent flows. Adv Appl Mech 18:123–176

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Lumley JL, Newman GR (1977) The return to isotropy of homogeneous turbulence. J Fluid Mech 82:161–178

    Article  MATH  MathSciNet  Google Scholar 

  • Magnaudet J (1993) Modelling of inhomogeneous turbulence in the absence of mean velocity gradients. Appl Sci Res 51:525–531

    Article  MATH  Google Scholar 

  • Manceau R, Hanjalić K (2000a) Elliptic blending model: a new near-wall Reynolds stress turbulence closure. Phys Fluids 14:744–754

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Manceau R, Carlson JR, Gatski TB (2002) A rescaled elliptic relaxation approach: neutralizing the effect on the log layer. Phys Fluids 14:3868–3879

    Article  Google Scholar 

  • McComb WD (1990) The physics of fluid turbulence. Clarendon, Oxford

    Google Scholar 

  • Mellor GL, Herring HJ (1973) A survey of the mean turbulent field closure models. AIAA J 11:590–599

    Article  MATH  Google Scholar 

  • Menter F (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 32:1598–1605

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Monin AS, Yaglom AM (1971) Statistical fluid mechanics, vols 1 and 2. MIT Press, Boston

    Google Scholar 

  • Moser RD, Kim J, Mansour NN (1999) Direct numerical simulation of turbulent channel flow up to Re τ =590. Phys Fluids 11:943–945

    Article  MATH  Google Scholar 

  • Myong HK, Kasagi N (1990) Prediction of anisotropy of the near-wall turbulence with an anisotropic low-Reynolds-number k–ε turbulence model. J Fluids Eng 112:521–524

    Article  Google Scholar 

  • Nisizima S, Yoshizawa A (1987) Turbulent channel and Couette flows using an anisotropic k–ε model. AIAA J 25:414–420

    Article  MATH  Google Scholar 

  • Oberlack M (1997) Non-isotropic dissipation in non-homogeneous turbulence. J Fluid Mech 350:351–374

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • Orszag SA (1970) Analytical theories of turbulence. J Fluid Mech 41:59–82

    Article  Google Scholar 

  • Pennisi S (1992) On third order tensor-valued isotropic functions. Int J Eng Sci 30:679–692

    Article  MathSciNet  Google Scholar 

  • Pinho FT (2003) A GNF framework for turbulent flow models of drag reducing fluids and proposal for a k–ε type closure. J Non-Newton Fluid Mech 121:127–141

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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 

  • Pope SB (1975) A more general effective viscosity hypothesis. J Fluid Mech 72:331–340

    Article  MATH  Google Scholar 

  • Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • 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 

  • Pruett CD (2000) On Eulerian times-domain filtering for spatial large-eddy simulation. AIAA J 38:1634–1642

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Resende PR, Kim K, Younis BA, Sureshkumar R, Pinho FT (2011) A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction. J Non-Newton Fluid Mech 166:639–660

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Reynolds WC (1976) Computation of turbulent flows. In: Annual review of fluid mechanics. Annual Reviews, New York, pp 183–208

    Google Scholar 

  • Rivlin RS (1957) The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids. Q Appl Math 15:212–215

    MATH  MathSciNet  Google Scholar 

  • Rivlin RS, Ericksen JL (1955) Stress-deformation relations for isotropic materials. Arch Ration Mech Anal 4:323–425

    MATH  MathSciNet  Google Scholar 

  • 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 

  • Rodi W (1976) A new algebraic relation for calculating the Reynolds stresses. Z Angew Math Mech 56:219–221

    Article  MathSciNet  Google Scholar 

  • Rotta JC (1951) Statistiche theorie nichthomogener turbulenz. Z Phys 129:547–572

    Article  MATH  MathSciNet  Google Scholar 

  • Rubinstein R, Barton JH (1990) Nonlinear Reynolds stress models and the renormalization group. Phys Fluids A 2:1472–1476

    Article  MATH  Google Scholar 

  • Saffman PG (1970) A model for inhomogeneous turbulence. Proc R Soc Lond Ser A 317:417–433

    Article  MATH  Google Scholar 

  • Sagaut P (2006) Large eddy simulation for incompressible flows. Springer, Berlin

    MATH  Google Scholar 

  • Sagaut P, Deck S, Terracol M (2006) Multiscale and multiresolution approaches in turbulence. Imperial College Press, London

    Book  Google Scholar 

  • Sarkar S, Speziale CG (1990) A simple nonlinear model for the return to isotropy in turbulence. Phys Fluids 29:84–93

    MathSciNet  Google Scholar 

  • Schumann U (1977) Realizability of Reynolds stress models. Phys Fluids 20:721–725

    Article  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Shih T-H, Zhu J, Lumley JL (1995) A new Reynolds stress algebraic equation model. Comput Methods Appl Mech Eng 125:287–302

    Article  Google Scholar 

  • Shir CC (1973) A preliminary numerical study of atmospheric turbulent flows in the idealized planetary boundary layer. J Atmos Sci 30:1327–1339

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Simonsen AJ, Krogstad P-Å(2005) Turbulent stress invariant analysis: clarification of existing terminology. Phys Fluids 17:088103

    Article  Google Scholar 

  • Sinha K, Candler GV (2003) Turbulent dissipation-rate equation for compressible flows. AIAA J 41:1017–1021

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Smagorinsky I (1963) General circulation experiments with the primitive equations. I. The basic experiment. Mon Weather Rev 91:99–164

    Article  Google Scholar 

  • Smith GF, Younis BA (2004) Isotropic tensor-valued polynomial function of second-order and third-order tensors. Int J Eng Sci 43:447–456

    Article  MathSciNet  Google Scholar 

  • So RMC, Lai YG, Zhang HS, Hwang BC (1991) Second-order near-wall turbulence closures: a review. AIAA J 29:1819–1835

    Article  MATH  Google Scholar 

  • So RMC, Aksoy H, Yuan SP, Sommer TP (1996) Modeling Reynolds-number effects in wall-bounded turbulent flows. J Fluids Eng 118:260–267

    Article  Google Scholar 

  • Spalart PR, Allmaras SR (1994) A one-equation turbulence model for aerodynamic flows. Rech Aérosp 1:5–21

    Google Scholar 

  • 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 

  • Speziale CG (1987) On nonlinear k–l and k–ε models of turbulence. J Fluid Mech 178:459–475

    Article  MATH  Google Scholar 

  • 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 

  • Speziale CG (1998a) A consistency condition for non-linear algebraic Reynolds stress models in turbulence. Int J Non-Linear Mech 33:579–584

    Article  MATH  MathSciNet  Google Scholar 

  • Speziale CG (1998c) Turbulence modeling for time-dependent RANS and VLES: a review. AIAA J 36:173–184

    Article  MATH  Google Scholar 

  • Speziale CG, Gatski TB (1997) Analysis and modelling of anisotropies in the dissipation rate of turbulence. J Fluid Mech 344:155–180

    Article  MATH  MathSciNet  Google Scholar 

  • 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

    Article  Google Scholar 

  • Stolz S, Adams NA (1999) An approximate deconvolution procedure for large-eddy simulation. Phys Fluids 11:1699–1701

    Article  MATH  Google Scholar 

  • 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 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Straatman AG (1999) A modified model for diffusion in second-moment turbulence closures. J Fluids Eng 121:747–756

    Article  Google Scholar 

  • Straatman AG, Stubley GD, Raithby GD (1998) Examination of diffusion modeling using zero-mean-shear turbulence. AIAA J 36:929–935

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Sureshkumar R, Beris AN, Handler RA (1997) Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys Fluids 9:743–755

    Article  Google Scholar 

  • Taulbee DB (1992) An improved algebraic Reynolds stress model and corresponding nonlinear stress model. Phys Fluids A 4:2555–2561

    Article  MATH  Google Scholar 

  • Tennekes H, Lumley JL (1972) A first course in turbulence. MIT Press, Boston

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Townsend AA (1956) The structure of turbulent shear flows. Cambridge University Press, Cambridge

    Google Scholar 

  • van Driest ER (1956) On turbulent flow near a wall. J Aeronaut Sci 23:1007–1011

    MATH  Google Scholar 

  • Wallin S, Johansson AV (2000) An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. J Fluid Mech 403:89–132

    Article  MATH  MathSciNet  Google Scholar 

  • Wilcox DC (2006) Turbulence modeling for CFD, 3rd edn. DCW Industries, La Cañada

    Google Scholar 

  • Wilson KG (1971) Renormalization group and critical phenomena. 1. Renormalization group and Kadanoff scaling picture. Phys Rev B 4:3174–3183

    Article  MATH  Google Scholar 

  • Yakhot A, Orszag SA, Yakhot V, Israeli M (1989) Renormalisation group formulation for large-eddy simulation. J Sci Comput 4:139–159

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • Yakhot V, Orszag SA (1986) Renormalisation group analysis of turbulence. I. Basic theory. J Sci Comput 1:3–51

    Article  MATH  MathSciNet  Google Scholar 

  • Ying R, Canuto VM (1996) Turbulence modeling over two-dimensional hills using an algebraic Reynolds stress expression. Bound-Layer Meteorol 77:69–99

    Article  Google Scholar 

  • Yoshizawa A (1984) Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation. Phys Fluids 27:1377–1387

    Article  MATH  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanical Engineering, Swiss Federal Institute of Technology, EPFL, Lausanne, Switzerland

    Michel O. Deville

  2. Institute PPRIME, CNRS-Université de Poitiers-ENSMA, Futuroscope Chasseneuil, France

    Thomas B. Gatski

  3. Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, USA

    Thomas B. Gatski

Authors
  1. Michel O. Deville
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas B. Gatski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michel O. Deville .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Deville, M.O., Gatski, T.B. (2012). Turbulent Flows. In: Mathematical Modeling for Complex Fluids and Flows. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25295-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation