Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T05:11:56.076Z Has data issue: false hasContentIssue false
  • English
  • Français

ON TURBULENCE MODELLING AND THE TRANSITION FROM LAMINAR TO TURBULENT FLOW

Part of: Turbulence Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  09 October 2014

LAWRENCE K. FORBES
LAWRENCE K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Tasmania 7004, Australia email Larry.Forbes@utas.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Fluid turbulence is often modelled using equations derived from the Navier–Stokes equations, perhaps with some semi-heuristic closure model for the turbulent viscosity. This paper considers a possible alternative hypothesis. It is argued that regarding turbulence as a manifestation of non-Newtonian behaviour may be a viewpoint of at least comparable validity. For a general description of nonlinear viscosity in a Stokes fluid, it is shown that the flow patterns are indistinguishable from those predicted by the Navier–Stokes equation in one- or two-dimensional geometry, but that fully three-dimensional flows differ markedly. The stability of linearized plane Poiseuille flow to three-dimensional disturbances is then considered, in a Tollmien–Schlichting formulation. It is demonstrated that the flow may become unstable at significantly lower Reynolds numbers than those expected from Navier–Stokes theory. Although similar results are known in sections of the rheological literature, the present work attempts to advance the philosophical viewpoint that turbulence might always be regarded as a non-Newtonian effect, to a degree that is dependent only on the particular fluid in question. Such an approach could give a more satisfactory account of the underlying physics.

Keywords

fluid turbulenceflow stabilitynonlinear viscositynon-Newtonian behaviourturbulence modelling

MSC classification

Secondary: 76F02: Fundamentals 76A02: Foundations of fluid mechanics
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 1 , July 2014 , pp. 28 - 47
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Aris, R., Vectors, tensors, and the basic equations of fluid mechanics (Dover, New York, 1962).Google Scholar
Batchelor, G. K., The theory of homogeneous turbulence (Cambridge University Press, Cambridge, 1953).Google Scholar
Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
Cambon, C. and Scott, J. F., “Linear and nonlinear models of anisotropic turbulence”, Annu. Rev. Fluid Mech. 31 (1999) 1–53; doi:10.1146/annurev.fluid.31.1.1.Google Scholar
Cherubini, S., De Palma, P., Robinet, J.-Ch. and Bottaro, A., “A purely nonlinear route to transition approaching the edge of chaos in a boundary layer”, Fluid Dyn. Res. 44 (2012) 1–11; doi:10.1088/0169-5983/44/3/031404.Google Scholar
Davidson, P. A., Turbulence. An introduction for scientists and engineers (Oxford University Press, Oxford, 2004).Google Scholar
Drazin, P. G. and Reid, W. H., Hydrodynamic stability, 2nd edn (Cambridge University Press, Cambridge, 2004).Google Scholar
Eckhardt, B., “Turbulence transition in pipe flow: some open questions”, Nonlinearity 21 (2008) T1–T11; doi:10.1088/0951-7715/21/1/T01.Google Scholar
Frederiksen, J. S., “Statistical dynamical closures and subgrid modeling for inhomogeneous QG and 3D turbulence”, Entropy 14 (2012) 32–57; doi:10.3390/e14010032.CrossRefGoogle Scholar
Frederiksen, J. S. and O’Kane, T. J., “Entropy, closures and subgrid modeling”, Entropy 10 (2008) 635–683; doi:10.3390/e10040635.Google Scholar
Golubitsky, M. and Dellnitz, M., Linear algebra and differential equations using MATLAB (Brooks/Cole, Pacific Grove, CA, 1999).Google Scholar
Hale, J. K. and Koçak, H., Dynamics and bifurcations (Springer, New York, 1991).Google Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. and Waleffe, F., “Experimental observation of nonlinear traveling waves in turbulent pipe flow”, Science 305 (2004) 1594–1598; doi:10.1126/science.1100393.Google Scholar
Jensen, R. V., “Functional integral approach to classical statistical dynamics”, J. Stat. Phys. 25 (1981) 183–210; doi:10.1007/BF01022182.Google Scholar
Jerome, J. J. S. and Chomaz, J.-M., “Extended Squire’s transformation and its consequences for transient growth in a confined shear flow”, J. Fluid Mech. 744 (2014) 430–456; doi:10.1017/jfm.2014.83.Google Scholar
Kraichnan, R. H., “The structure of isotropic turbulence at very high Reynolds numbers”, J. Fluid Mech. 5 (1959) 497–543; doi:10.1017/S0022112059000362.Google Scholar
Larson, R. G., “Instabilities in viscoelastic flows”, Rheol. Acta 3 (1992) 213–263; doi:10.1007/BF00366504.Google Scholar
Larson, R. G., “Fluid dynamics – Turbulence without inertia”, Nature 405 (2000) 27–28; doi:10.1038/35011172.Google Scholar
Martin, P. C., Siggia, E. D. and Rose, H. A., “Statistical dynamics of classical systems”, Phys. Rev. A 8 (1973) 423–437; doi:10.1103/PhysRevA.8.423.Google Scholar
Mase, G. E., Continuum mechanics, Schaum’s Outline Series (McGraw-Hill, New York, 1970).Google Scholar
McComb, W. D., “Theory of turbulence”, Rep. Progr. Phys. 58 (1995) 1117–1206; doi:10.1088/0034-4885/58/10/001.Google Scholar
Orszag, S. A., “Accurate solution of the Orr–Sommerfeld stability equation”, J. Fluid Mech. 50 (1971) 689–703; doi:10.1017/S0022112071002842.Google Scholar
Poole, R. J., “The Deborah and Weissenberg numbers”, Rheol. Bull. 53 (2012) 32–39; http://pcwww.liv.ac.uk/∼robpoole/PAPERS/POOLE_45.pdf.Google Scholar
Reiner, M., “A mathematical theory of dilatancy”, Amer. J. Math. 67 (1945) 350–362; http://www.jstor.org/stable/2371950.Google Scholar
Renardy, M. and Renardy, Y., “Linear stability of plane Couette flow of an upper convected Maxwell fluid”, J. Non-Newtonian Fluid Mech. 22 (1986) 23–33; doi:10.1016/0377-0257(86)80002-7.Google Scholar
Reynolds, O., “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”, Proc. R. Soc. Lond. 35 (1883) 84–99; doi:10.1098/rspl.1883.0018.Google Scholar
Reynolds, O., “On the dynamical theory of incompressible viscous fluids and the determination of the criterion”, Philos. Trans. R. Soc. Lond. A 186 (1895) 123–164; http://www.jstor.org/stable/90643.Google Scholar
Rivlin, R. S., “The hydrodynamics of non-Newtonian fluids. I”, Proc. R. Soc. Lond. A 193 (1948) 260–281; http://www.jstor.org/stable/97992.Google Scholar
Rivlin, R. S., “The relation between the flow of non-Newtonian fluids and turbulent Newtonian fluids”, Quart. Appl. Math. 15 (1957) 212–215.Google Scholar
Shaqfeh, E. S. G., “Purely elastic instabilities in viscometric flows”, Annu. Rev. Fluid Mech. 28 (1996) 129–185; doi:10.1146/annurev.fl.28.010196.001021.Google Scholar
Speziale, C. G., “On nonlinear $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$–$\ell $ and $K$–$\epsilon $ models of turbulence”, J. Fluid Mech. 178 (1987) 459–475; doi:10.1017/S0022112087001319.Google Scholar
Spiegel, E. A. (ed.), The theory of turbulence. Subrahmanyan Chandrasekhar’s 1954 lectures, Volume 810 of Lecture Notes in Physics (Springer, Dordrecht, 2011); doi:10.1007/978-94-007-0117-5.Google Scholar
Squire, H. B., “On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls”, Proc. R. Soc. Lond. A 142 (1933) 621–628; doi:10.1098/rspa.1933.0193.Google Scholar
Sreenivasan, K. R., “Fluid turbulence”, Rev. Modern. Phys. 71 (1999) S383–S395; doi:10.1103/RevModPhys.71.S383.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. and Driscoll, T. A., “Hydrodynamic stability without eigenvalues”, Science 261 (1993) 578–584; doi:10.1126/science.261.5121.578.Google Scholar
Waleffe, F., “Exact coherent structures in channel flow”, J. Fluid Mech. 435 (2001) 93–102; doi:10.1017/S0022112001004189.Google Scholar
Wilson, H. J., Renardy, M. and Renardy, Y., “Structure of the spectrum in zero Reynolds number flow of the UCM and Oldroyd-B liquids”, J. Non-Newtonian Fluid Mech. 80 (1999) 251–268; doi:10.1016/S0377-0257(98)00087-1.Google Scholar