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First Turbulence Models for Shear Flows

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Nonlinear, Nonlocal and Fractional Turbulence

Abstract

In daily life, turbulent motions constitute ubiquitous fluid mechanical elements, which can be observed in various forms, e.g., in wind gusts and surface water flows, in rivers, lakes, and the oceans.

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Change history

  • 25 July 2021

    A correction has been published.

Notes

  1. 1.

    Be aware that this moderate success mainly stems also in these cases from the fact of adjusting free parameters to the problem under consideration.

  2. 2.

    These are the coefficients of the eigenvalue equation of \( \overline{\mathbf{D}} \).

  3. 3.

    Note that Eq. (115a) is an identity, but Eq. (115b) is an equation by replacing \( \overline{\mathbf{D}} \) by the Reynolds stress constitutive postulate (Eq. 96b).

  4. 4.

    This is the end of the calculation by Frank Obermeier .

References

  • Afzal, N.: Power law and log law velocity profiles in turbulent boundary-layer flow: equivalent relations at large Reynolds numbers. Acta Mech. 151, 195 (2001)

    MATH  Google Scholar 

  • Arrowsmith, D.K., Place, C.M.: Dynamische Systeme. Academic Edition Spectrum, Heidelberg (1990). ISBN 3-86025-308-5 (in German)

    MATH  Google Scholar 

  • Barenblatt, G.I.: Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis. J. Fluid Mech. 248, 513 (1993a)

    MathSciNet  MATH  ADS  Google Scholar 

  • Barenblatt, G.I.: Intermediate asymptotics, scaling laws and renormalization group in continuum mechanics. Meccanica. 28(3), 177 (1993b)

    MATH  Google Scholar 

  • Barenblatt, G.I., Chorin, A.J., Prostokishin, V.M.: Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers. J. Fluid Mech. 410, 263 (2000)

    MathSciNet  MATH  ADS  Google Scholar 

  • Batchelor, G.K.: Diffusion in a field of homogeneous turbulence. Aust. J. Sci. Res. A. 2, 437 (1949)

    Google Scholar 

  • Beer, F.P., Johnston Jr., E.R., DeWolf, J.T.: Mechanics of Materials. McGraw-Hill, New York (1981). ISBN 13-978-0073398235

    Google Scholar 

  • Bernard, P.S., Handler, R.A.: Reynolds stress and the physics of turbulent momentum transport. J. Fluid Mech. 220, 99 (1990)

    Google Scholar 

  • Bodenschatz, E., Eckert, M.: “Prandtl and the Göttingen School”, in Davidson et al. (see in this reference list and notice the remarks: Bodenschatz and Eckert report, on the basis of correspondences of Prandtl with a number of involved scientists, that the 1/7th law and the struggle on it had been around between 1913 and 1921, say. In a letter of Prandtl to von Kármán in 1921 Prandtl stated that he already knew that the velocity distribution was proportional to y1/7, where y is the distance from the wall. Prandtl stated in this letter that he had known this “already for a pretty long time, say since 1913”. It appears (see Bodenschatz and Eckert, 2011) that Prandtl was too much overloaded in that period with other work, to formally publish his derivation before the 1920s. However, there is evidence that Prandtl’s statement is correct (more details in Bodenschatz and Eckerts) (2011))

    Google Scholar 

  • Boussinesq, J.: Essai sur la théorie des eaux courants. Mémoires présentés par divers savants à l’Académie des Sciences. 23(1), (1877). (in French)

    Google Scholar 

  • Boussinesq, J.: Théorie de l’écoulement tourbillonnant et tumulteux des liquids. Gaulthier-Villars et fils, Paris (1897)

    MATH  Google Scholar 

  • Bradshaw, P.: Turbulence: the chief outstanding difficulty of our subject. Exp. Fluids. 16, 203 (1994)

    Google Scholar 

  • Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: An analytical solution of the advection-diffusion equation considering non-local turbulence closure. Environ. Fluid Mech. 7, 43 (2007)

    Google Scholar 

  • Canuto, V.M.: Turbulent convection: old and new models. Astrophys. J. 467, 385 (1996). https://doi.org/10.1086/177613

    Article  ADS  Google Scholar 

  • Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge University Press, Cambridge, UK (1970). ISBN 10-0521075777

    MATH  Google Scholar 

  • Cipra, B.: A new theory of turbulence causes a stir among experts. Science. 272(5264), 951 (1996). https://doi.org/10.1126/science.272.5264.951

    Article  ADS  Google Scholar 

  • Corrsin, S.: Transport model limitations in turbulence. Adv. Geophys. 18A, 25 (1974)

    ADS  Google Scholar 

  • Courant, R., Hilbert, D.: Methods of Mathematical Physics, Differential Equations, vol. I. Wiley, New York (1989). ISBN 978-0-471-50439-9

    MATH  Google Scholar 

  • Davidson, P.A., Kaneda, Y., Moffat, K., Sreenivasan, K.R.: A Voyage Through Turbulence. Cambridge University Press, Cambridge, UK (2011). ISBN 978-0-521-19868-4

    MATH  Google Scholar 

  • Deardoff, J.W.: The counter-gradient heat flux in the lower atmosphere and in the laboratory. J. Atmos. Sci. 23, 503 (1966)

    ADS  Google Scholar 

  • Deardoff, J.W.: Numerical investigation of neutral and unstable planetary boundary layer. J. Atmos. Sci. 29, 91 (1972)

    ADS  Google Scholar 

  • Deardoff, J.W., Willis, G.E.: A parameterization of diffusion into the mixed layer. J. Appl. Meteorol. 14, 1451 (1975)

    ADS  Google Scholar 

  • Doering, C.R., Constantin, P.: Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E. 49(5), 4087 (1994)

    MathSciNet  ADS  Google Scholar 

  • Egolf, P.W.: Difference-quotient turbulence model: a generalization of Prandtl’s mixing-length theory. Phys. Rev. E. 49(2), 1260 (1994)

    MathSciNet  ADS  Google Scholar 

  • Egolf, P.W.: Lévy statistics and beta model: a new solution of “wall” turbulence with a critical phenomenon. Int. J. Refrig. 32, 1815 (2009)

    Google Scholar 

  • Egolf, P.W., Weiss, D.A.: Difference-quotient turbulence model: analytical solutions for the core region of plane Poiseuille flow. Phys Rev E. 62(1-A), 553 (2000)

    ADS  Google Scholar 

  • Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 17, 549 (1905). (in German)

    MATH  Google Scholar 

  • Eisner, F.: Reibungswiderstand. Werft, Reederei, Hafen. 13, 207 (1932). (in German)

    Google Scholar 

  • Ertel, H.: Der vertikale Turbulenz-Wärmestrom in der Atmosphäre. Meteorol. Z. 59, 250 (1942). (in German)

    MATH  Google Scholar 

  • Frisch, U.: Turbulence – The Legacy of A.N. Kolmogorov, 1st edn. Cambridge University Press, Cambridge, UK (1995). ISBN 0-85-403-441-2

    MATH  Google Scholar 

  • Gad-el-Hak, M.: Flow Control: Passive, Active and Research Flow Management. Cambridge University Press, Cambridge, UK (2000). ISBN 13-978-0521036719

    MATH  Google Scholar 

  • George, W.K.: Is there a universal log law for turbulent wall-bounded flows? Phil. Trans. R. Soc. A. 365, 789 (2007)

    MathSciNet  MATH  ADS  Google Scholar 

  • George, W.K.: Lectures in Turbulence for the 21st Century. www.turbulence-online.com (2013)

  • George, W.K., Castillo, L.: Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50(11), 689 (1997)

    ADS  Google Scholar 

  • Goldstein, S.: Modern Developments in Fluid Dynamics. Dover Press, New York (1965). ISBN 13-978-0486613574

    Google Scholar 

  • Görtler, H.: Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen Näherungsansatzes. ZAMM. 22, 241 (1942). (in German)

    MATH  ADS  Google Scholar 

  • Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill, New York (1975). ISBN 0-07-029037-7

    Google Scholar 

  • Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004). ISBN 13-978-364-205-831-8

    MATH  Google Scholar 

  • Hutter, K., Wang, Y.: Fluid and thermodynamics. In: Basic Fluid Mechanics, vol. 1. Springer, Berlin (2016a). ISBN 978-3-319-33632-9

    MATH  Google Scholar 

  • Hutter, K., Wang, Y.: Fluid and thermodynamics. In: Advanced Fluid Mechanics and Thermodynamic Fundamentals, vol. 2. Springer, Berlin (2016b). ISBN 978-3-319-33635-0

    MATH  Google Scholar 

  • Hutter, K., Wang, Y.: Fluid and thermodynamics. In: Structured and Multiphase Fluids, vol. 3. Springer, Berlin (2018). ISBN 978-3-319-77745-0

    Google Scholar 

  • Jakob, M., Erk, S.: Der Druckabfall in glatten Rohren und die Durchflussziffer von Normaldüsen. In: Forschungsarbeiten auf dem Gebiet des Ingenieurwesens. Verein der Deutschen Ingenieure (V.D.I.), Berlin (1924). (in German)

    Google Scholar 

  • Jongen, T., Gatski, T.B.: General explicit algebraic stress relations and best approximation for three-dimensional flows. Int. J. Eng. Sci. 36, 739 (1998)

    Google Scholar 

  • Kaneda, Y., Ishihara, T.: High-resolution direct numerical simulation of turbulence. J. Turbul. 7, 20 (2006)

    ADS  Google Scholar 

  • Kevlahan, N.K.R., Farge, M.: Vorticity filaments in two-dimensional turbulence: creation, stability and effect. J. Fluid Mech. 346, 49 (1997)

    Google Scholar 

  • Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics. Springer, New York (2010). ISBN 978-1-4419-2812-2

    MATH  Google Scholar 

  • Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid with very large Reynolds numbers. Dokl. Akad. Nauk. SSSR Seria fizichka. 30, 301 (1941a). (in Russian). English translation in Proc. R. Soc. Lond. A 434, 9

    MathSciNet  ADS  Google Scholar 

  • Kolmogorov, A.N.: Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR. 32, 16 (1941b). (in Russian). English translation in Proc. R. Soc. Lond. A 434, 15

    MathSciNet  MATH  ADS  Google Scholar 

  • Launder, B., Spalding, D.B.: The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 3, 269 (1974)

    MATH  ADS  Google Scholar 

  • Lesieur, M.: Turbulence in Fluids: Stochastic and Numerical Modeling, 2nd edn. Kluwer Academic, Dordrecht (1990). ISBN 0-7923-0645-7-00

    MATH  Google Scholar 

  • Libby, P.A.: Introduction to Turbulence Combustion, An International Series. Taylor & Francis, Abingdon, UK (1996). ISBN 1-56032-100-8

    Google Scholar 

  • Millikan, C.B.: A critical discussion of turbulent flows in channels and circular tubes. In: Den Hartog, J.P., Peters, H. (eds.) Proceedings of the Fifth International Congress for Applied Mechanics, Massachusetts (1938)

    Google Scholar 

  • Moore, J.G., Schorn, S.A., Moore, J.: Methods of classical mechanics applied to turbulence stresses in a tip leakage vortex. ASME J. Turbomach. 118(4), 622 (1996)

    Google Scholar 

  • Nicuradse, J.: Untersuchungen über die Geschwindigkeitsverteilung in turbulenten Strömungen, PhD thesis Göttingen, VDI Res. J. 281 (1926)

    Google Scholar 

  • Oberlack, M.: Symmetries and scaling‐laws in turbulence. ZAMM. 79, 123 (1999)

    MATH  ADS  Google Scholar 

  • Oberlack, M.: A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299 (2001)

    MATH  ADS  Google Scholar 

  • Obermeier, F.: Letter of Frank Obermeier (Technical University, Bergakademie Freiberg, Germany) to Daniel Weiss with a copy to the first author on 22nd of January (1996)

    Google Scholar 

  • Obermeier, F.: Prandtl’s mixing length model – revisited, PAMM. Proc. Appl. Math. Mech. 6, 577 (2006)

    Google Scholar 

  • Penrose, R.: The Road to Reality. Vintage Books, New York (2007). ISBN 978-0-679-77631-4

    Google Scholar 

  • Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, UK (2000). ISBN 978-0-521-59125-6

    MATH  Google Scholar 

  • Prandtl, L.: Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In: Proceeding of Third International Mathematical Congress, Heidelberg (1904). (in German), 484 (Engl. transl. in NACA Tech. Memo. 452)

    Google Scholar 

  • Prandtl, L.: Approximative time of discovery of 1/7-power law of the wall by Prandtl (see also Davidson et al., (2011), p. 53) (1913)

    Google Scholar 

  • Prandtl, L.: Letter of Prandtl to von Kármán on 16 February 1921, MPGA, Abt. III, rep. 61, no. 792 (see also in Davidson et al. (2011)), p. 53 (1921a)

    Google Scholar 

  • Prandtl, L.: Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen. Bericht, Oldenbourg (1921b). (in German)

    MATH  Google Scholar 

  • Prandtl, L.: Bericht über Untersuchungen zur ausgebildeten Turbulenz. ZAMM. 5(2), 136 (1925). (in German)

    MATH  ADS  Google Scholar 

  • Prandtl, L.: Neuere Ergebnisse der Turbulenzforschung. Zeitschr. VDI. 77, 105 (1933). (in German)

    MATH  Google Scholar 

  • Prandtl, L.: Bemerkungen zur Theorie der freien Turbulenz. ZAMM. 22(5), 241 (1942). (in German)

    MathSciNet  MATH  ADS  Google Scholar 

  • Prandtl, L.: Führer durch die Strömungslehre, 3rd edn. Friedr. Vieweg & Sohn, Braunschweig (1949). (in German)

    MATH  Google Scholar 

  • Reichardt, H.: Über eine neue Theorie der freien Turbulenz. ZAMM. 21, 257 (1941). (in German)

    MATH  ADS  Google Scholar 

  • Reynolds, O.: On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Phil. Trans. R. Soc. Lond. A. 186, 123 (1894)

    MATH  ADS  Google Scholar 

  • Reynolds, A.J.: Turbulent Flows in Engineering. Wiley, London (1974). ISBN 10-0471717827

    Google Scholar 

  • Richardson, L.F.: Weather Prediction by Numerical Methods. Cambridge University Press, Cambridge, UK (1922)

    MATH  Google Scholar 

  • Rodi, W.: In: Launder, B.E. (ed.) A Review of Experimental Data of Uniform Density Free Turbulent Boundary Layers, Studies in Convection, vol. 1. Academic, London (1976a)

    Google Scholar 

  • Rodi, W.: A new algebraic relation for calculating the Reynolds stresses. ZAMM. 56, T 219 (1976b)

    MATH  ADS  Google Scholar 

  • Schlichting, H.: Boundary-Layer Theory. McGraw-Hill, New York (1979). ISBN 0-07-055-334-3

    MATH  Google Scholar 

  • Schmitt, F.G.: Direct test of a nonlinear constitutive equation for simple turbulent shear flows using DNS data. Commun. Nonlinear Sci. Numer. Simul. 12, 1251 (2007a)

    MATH  ADS  Google Scholar 

  • Schmitt, F.G.: About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity. C.R. Mécanique. 335, 617 (2007b)

    MATH  ADS  Google Scholar 

  • Schmitt, F.G., Hirsch, C.: Experimental study of the constitutive equation for an axisymmetric complex turbulent flow. ZAMM. 80, 815 (2000)

    MATH  ADS  Google Scholar 

  • Schmitt, F.G., Merci, B., Dick, E., Hirsch, C.: Direct investigation of the K-transport equation for a complex turbulent flow. J. Turbul. 4, 21 (2003)

    ADS  Google Scholar 

  • Smagorinski, J.: General circulation model of the atmosphere. Mon. Weather Rev. 91, 99 (1963)

    ADS  Google Scholar 

  • Smith, A.M.O., Cebeci, T.: Numerical Solution of the Turbulent Boundary Layer Equations. Douglas Aircraft Division Report DAC 33735, USA (1967)

    Google Scholar 

  • Spalart, P.R.: Philosophies and fallacies in turbulence modeling. Prog. Aerosp. Sci. 74, 1 (2015)

    Google Scholar 

  • Spalart, P.R., Allmaras, S.R.: A one-equation turbulence model for aerodynamic flows. Recherche Aero-spatiale. 1, 5–21 (1994)

    Google Scholar 

  • Speziale, C.G.: A consistency condition for non-linear algebraic Reynolds stress models in turbulence. Int. J. Non-Linear Mech. 33, 579 (1998)

    MathSciNet  MATH  ADS  Google Scholar 

  • Stanišić, M.M.: The Mathematical Theory of Turbulence, 2nd edn. Springer, Berlin (1988). ISBN 0-387-96685-4

    MATH  Google Scholar 

  • Stull, R.B.: An Introduction to Boundary Layer Meteorology, Atmospheric Sciences Library. Kluwer Academic, Dordrecht (1988). ISBN 90-277-2769-4

    MATH  Google Scholar 

  • Taylor, G.I.: The transport of vorticity and heat through fluids in turbulent motion. Proc. R. Soc. Lond. 135A, 685 (1932)

    MATH  ADS  Google Scholar 

  • Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge, MA (1972). ISBN 0-262-200-19-8

    MATH  Google Scholar 

  • Truesdell, C.A., Muncaster, R.G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monoatomic Gas: Treated as a Branch of Rational Mechanics. Academic, Cambridge, MA (1980). ISBN 0-12-701350-4

    Google Scholar 

  • Tsinober, A.: An Informal Conceptual Introduction to Turbulence. Springer, Cham (2009). ISBN 978-90-481-3174-7

    MATH  Google Scholar 

  • Tsinober, A.: The Essence of Turbulence as a Physical Phenomenon with Emphasis on Issues of Paradigmatic Nature. Springer, Dordrecht (2014). ISBN 978-94-007-7179-6

    MATH  Google Scholar 

  • van Dyke, M.: Higher approximations in boundary-layer theory part 3. Parabola in uniform stream. J. Fluid Mech. 19, 145 (1964)

    MathSciNet  MATH  ADS  Google Scholar 

  • Vilhena, M.T., Cost, C.P., Moreira, D.M., Tirabassi, T.: A semi-analytical solution for the three-dimensional advection–diffusion equation considering non-local turbulence closure. Atmos. Res. 90, 63 (2008)

    Google Scholar 

  • von Kármán, T.: Über laminare und turbulente Reibung. ZAMM. 1(4), 233 (1921). (in German)

    MATH  ADS  Google Scholar 

  • von Kármán, T.: Mechanische Ähnlichkeit und Turbulenz. Nachr. Akad. Wiss. Göttingen, Math-Phys. Kl., 58 (1930). (in German). Proc. 3. Intern. Congr. Appl. Mech. Stockholm 1930. Also in: Collected Works, vol. 2, pp. 322–346. London, Butterworth (1956)

    Google Scholar 

  • von Kármán, T.: Mechanical similitude and turbulence. In: Proceedings of the Third International Congress of Applied Mechanics, vol. 1, p. 85, Stockholm (1931)

    Google Scholar 

  • Wallace, J.M.: Twenty years of experimental and direct numerical simulation access to the velocity gradient tensor: What have we learned about turbulence? Phys. Fluids. 21(2), 021301 (2009)

    MathSciNet  MATH  ADS  Google Scholar 

  • Wieghardt, K.: Turbulente Grenzschichten. Göttinger Monographie, Part B, 5 Germany (in German) (1946)

    Google Scholar 

  • Wilcox, D.C.: Turbulent Modeling for CFD, 2nd edn. DCW Industries, La Canada (1998)

    Google Scholar 

  • Zagarola, M.V., Smits, A.J., Orszag, S.A., Yakot, V. : Experiments in high Reynolds number turbulent pipe flow. AIAA paper No. 96-0654, Reno Nevada (1996)

    Google Scholar 

  • Zagarola, M.V., Perry, A.E., Smits, A.J.: Log laws or power laws: the scaling in the overlap region. Phys. Fluids. 9(7), 2094 (1997)

    MathSciNet  MATH  ADS  Google Scholar 

  • Zagarola, M.V., Perry, A.E., Smits, A.J.: http://www.princeton.edu/~gasdyn/#superpipe_data (2015)

  • Zilitinkevich, S., Gryanik, V.M., Lykossov, V.N., Mironov, D.V.: Third-order transport and nonlocal turbulence closure for convective boundary layers. J. Atmos. Sci. 56, 3463 (1999)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Institute of Theoretical Turbulence Research, Niederlenz, Aargau, Switzerland

    Peter William Egolf

  2. VAW, ETHZ, Zürich, Switzerland

    Kolumban Hutter

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Egolf, P.W., Hutter, K. (2020). First Turbulence Models for Shear Flows. In: Nonlinear, Nonlocal and Fractional Turbulence. Springer, Cham. https://doi.org/10.1007/978-3-030-26033-0_5

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