About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity

François G. Schmitt

doi:10.1016/j.crme.2007.08.004

Unsteady flows, turbulent flows

About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity

Presented by: Presented by

François G. Schmitt ¹

¹ CNRS, FRE 2816 ELICO, Wimereux Marine Station, Université des sciences et technologies de Lille – Lille 1, 28, avenue Foch, 62930 Wimereux, France

Comptes Rendus. Mécanique, Volume 335 (2007) no. 9-10, pp. 617-627.

Abstract
Résumé

Boussinesq's hypothesis is at the heart of eddy viscosity models, which are used in many different fields to model turbulent flows. In its present time formulation, this hypothesis corresponds to an alignment between the Reynolds stress and mean strain tensors. We begin with historical remarks on Boussinesq's results and recall that he introduced a local averaging twenty years before Reynolds, but using an approach that prevented him from discovering Reynolds' stress tensor. We then introduce an indicator that characterizes the validity of this hypothesis. For experimental and numerical databases, when the tensors are known, this can be used to directly estimate the validity of this hypothesis. We show, using several different databases, that this hypothesis is almost never verified. We address, in conclusion, the analogy with kinetic theory, and the reason why this analogy cannot be applied, in general, for turbulent flows.

L'hypothèse de Boussinesq est au coeur des modèles de viscosité, utilisés dans un grand nombre de contextes pour modéliser des écoulements turbulents. Dans sa formulation moderne, cette hypothèse correspond à un alignement entre tenseur de contrainte de Reynolds et tenseur de déformation moyen. Nous rappelons le contexte historique de l'énoncé de cette hypothèse, en soulignant que Boussinesq avait introduit une moyenne locale vingt ans avant Reynolds, mais en effectuant une erreur qui l'a privé de la mise en évidence du tenseur de Reynolds. Nous introduisons ensuite un indicateur, compris entre 0 et 1, indiquant le degré de validité de cette hypothèse. Pour des bases de données expérimentales et numériques, lorsque les différents tenseurs sont connus, ceci permet de tester directement, a priori, cette hypothèse. Nous montrons ainsi, utilisant différentes bases de données d'écoulements turbulents, que l'hypothèse n'est presque jamais vérifiée. Nous discutons en conclusion de la théorie cinétique des gaz et de la raison pour laquelle cette analogie est discutable pour les écoulements turbulents.

Received: 2006-12-18
Accepted: 2007-04-04
Published online: 2007-09-01

DOI: 10.1016/j.crme.2007.08.004

Keywords: Fluid Mechanics, Turbulence, Constitutive equation
Mot clés : Mécanique des fluides, Turbulence, Equation constitutive

Author's affiliations:

François G. Schmitt ¹

¹ CNRS, FRE 2816 ELICO, Wimereux Marine Station, Université des sciences et technologies de Lille – Lille 1, 28, avenue Foch, 62930 Wimereux, France

@article{CRMECA_2007__335_9-10_617_0,
     author = {Fran\c{c}ois G. Schmitt},
     title = {About {Boussinesq's} turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {617--627},
     publisher = {Elsevier},
     volume = {335},
     number = {9-10},
     year = {2007},
     doi = {10.1016/j.crme.2007.08.004},
     language = {en},
}

TY  - JOUR
AU  - François G. Schmitt
TI  - About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity
JO  - Comptes Rendus. Mécanique
PY  - 2007
SP  - 617
EP  - 627
VL  - 335
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crme.2007.08.004
LA  - en
ID  - CRMECA_2007__335_9-10_617_0
ER  -

%0 Journal Article
%A François G. Schmitt
%T About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity
%J Comptes Rendus. Mécanique
%D 2007
%P 617-627
%V 335
%N 9-10
%I Elsevier
%R 10.1016/j.crme.2007.08.004
%G en
%F CRMECA_2007__335_9-10_617_0

François G. Schmitt. About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity. Comptes Rendus. Mécanique, Volume 335 (2007) no. 9-10, pp. 617-627. doi : 10.1016/j.crme.2007.08.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.08.004/

References
Cited by

[1] H. Tennekes; J.L. Lumley A First Course in Turbulence, MIT Press, Cambridge, 1972

[2] D.C. Wilcox Turbulent Modeling for CFD, DCW Industries, La Canada, 1998

[3] S.B. Pope Turbulent Flows, Cambridge University Press, Cambridge, 2000

[4] J. Mathieu; J. Scott An Introduction to Turbulent Flow, Cambridge University Press, Cambridge, 2000

[5] P.S. Bernard; J.M. Wallace Turbulent Flow: Analysis, Measurement, and Prediction, John Wiley and Sons, Hoboken, 2002

[6] O. Reynolds On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. R. Soc. London A, Volume 186 (1895), pp. 123-164

[7] Proc. R. Soc. London A, 451 (1995), pp. 1-318

[8] U. Frisch Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995

[9] J. Boussinesq Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l'Académie des Sciences, Volume XXIII (1877) no. 1, pp. 1-680

[10] L. Prandtl Bericht über Untersuchungen zur ausgebildeten Turbulenz, Z. Angew. Math. Mech., Volume 5 (1925), pp. 136-139

[11] Saint-Venant Rapport sur un mémoire de M. Boussinesq, Mémoires présentés par divers savants à l'Académie des Sciences, Volume XXIII (1877) no. 1, p. I-XXII

[12] E. Picard Discours et Notices, Gauthier–Villars, Paris, 1936

[13] J. Boussinesq Mémoire sur l'influence des frottements dans les mouvements réguliers des fluides, J. Math. Pures Appl. Sér. II, Volume 13 (1868), pp. 377-423

[14] O. Reynolds An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Phil. Trans. R. Soc. London A, Volume 174 (1883), pp. 935-982

[15] Lord Kelvin (W. Thomson) On the propagation of laminar motion through a turbulently moving inviscid liquid, Phil. Mag., Volume 24 (1887), pp. 342-353

[16] J. Boussinesq Théorie de l'écoulement tourbillonnant et tumultueux des liquides, Gauthier–Villars et fils, Paris, 1897

[17] B.E. Launder; R. Spalding The numerical computation of turbulent flows, Comput. Meth. Appl. Mech. Eng., Volume 3 (1974), pp. 269-289

[18] F.G. Schmitt; C. Hirsch Experimental study of the constitutive equation for an axisymmetric complex turbulent flow, Zeit. Angew. Math. Mech., Volume 80 (2000), pp. 815-825

[19] F.G. Schmitt Direct test of a nonlinear constitutive equation for simple turbulent shear flows using DNS data, Comm. Nonlinear Sci. Numer. Simul., Volume 12 (2007), pp. 1251-1264

[20] K.H. Bech et al. An investigation of turbulent plane Couette flow at low Reynolds numbers, J. Fluid Mech., Volume 286 (1995), pp. 291-325

[21] J. Kim; P. Moin; R. Moser Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., Volume 177 (1987), pp. 133-166

[22] R.D. Moser; J. Kim; N.N. Mansour Direct numerical simulation of turbulent channel flow up to $R e_{τ} = 590$ , Phys. Fluids, Volume 11 (1999), pp. 943-945

[23] P. Spalart Direct simulation of a turbulent boundary layer up to $R t = 1410$ , J. Fluid Mech., Volume 187 (1988), pp. 61-98

[24] M. Quadrio; P. Luchini Direct numerical simulation of the turbulent flow in a pipe with annular cross-section, Eur. J. Mech. Fluid Ser., Volume 21 (2002), pp. 413-427

[25] S. Nisizima; A. Yoshizawa Turbulent channel and Couette flows using a anisotropic $k – ϵ$ model, AIAA J., Volume 25 (1986), pp. 414-420

[26] C.G. Speziale On nonlinear $k – ℓ$ and $k – ϵ$ models of turbulence, J. Fluid Mech., Volume 178 (1987), p. 458

[27] W. Rodi; J.H. Ferziger; M. Breuer; M. Pourquié Status of large eddy simulation: Results of a workshop, Trans. ASME: J. Fluid Engrg., Volume 119 (1997), pp. 248-262

[28] F.G. Schmitt; B. Merci; E. Dick; C. Hirsch Direct investigation of the K-transport equation for a complex turbulent flow, J. Turbulence, Volume 3 (2003), p. 021

[29] F. Schmitt; B.K. Hazarika; C. Hirsch LDV measurements of the flow field in the nozzle region of a confined double annular burner, Trans. ASME: J. Fluid Engrg., Volume 123 (2001), pp. 228-236

[30] S.P. Pope A more general effective-viscosity hypothesis, J. Fluid Mech., Volume 72 (1975), pp. 331-340

[31] A.J. Spencer Theory of invariants (A.C. Eringen, ed.), Continuum Physics, vol. 1, Academic Press, New York, 1971, pp. 239-353

[32] A. Yoshizawa Statistical analysis of the deviation of the Reynolds stress from its eddy-viscosity representation, Phys. Fluids, Volume 27 (1984), pp. 1377-1387

[33] J.L. Lumley Toward a turbulent constitutive relation, J. Fluid Mech., Volume 41 (1970), pp. 413-434

[34] S. Corrsin Limitations of gradient transport models in random walks and turbulence, Adv. Geophys., Volume 18A (1974), pp. 25-60

[35] P.W. Egolf Difference-quotient model: a generalization of Prandtl's mixing-length theory, Phys. Rev. E, Volume 49 (1994), pp. 1260-1268

[36] J.O. Hinze; R.E. Sonnenberg; P.J.H. Builtjes Memory effect in a turbulent boundary-layer flow due to a relatively strong axial variation of the mean-velocity gradient, Appl. Sci. Res., Volume 29 (1974), pp. 1-13

[37] P. Bernard; R. Handler Reynolds stress and the physics of turbulent momentum transport, J. Fluid Mech., Volume 220 (1990), pp. 99-124

[38] S. Tavoularis; S.S. Corrsin Experiments in nearly homogeneous turbulent shear flow with a uniform mean temperature gradient, part I, J. Fluid Mech., Volume 104 (1981), pp. 311-347

[39] Y.N. Huang On modelling the Reynolds stress in the context of continuum mechanics, Comm. Nonlinear. Sci. Numer. Simul., Volume 9 (2004), pp. 543-559

[40] F. Hamba Nonlocal analysis of the Reynolds stress in turbulent shear flow, Phys. Fluids, Volume 17 (2005), p. 115102

[41] R.A. Handler; P.S. Bernard On the role of accelerating fluid particles in the generation of Reynolds stress, Phys. Fluids A, Volume 4 (1992), pp. 1317-1319

Cited by Sources:

Comments - Policy