Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T10:07:46.788Z Has data issue: false hasContentIssue false
  • English
  • Français

TRANSITION TO TURBULENCE FROM PLANE COUETTE FLOW

Part of:TurbulenceNumerical linear algebra

Published online by Cambridge University Press:  22 September 2015

L. K. FORBES
L. K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Hobart, 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.

Modelling fluid turbulence is perhaps one of the hardest problems in Applied Mathematics. In a recent paper, the author argued that the classical Navier–Stokes equation is not sufficient to describe the transition to turbulence, but that a Reiner–Rivlin type equation is needed instead. This is explored here for the simplest of all viscous fluid flows, the Couette flow, which is a simple shear between two moving plates. It is found that at high wavenumbers, the transition to unstable flow at the critical Reynolds number is characterized by a large number of eigenvalues of the Orr–Sommerfeld equation moving into the unstable zone essentially simultaneously. This would generate high-dimensional chaos almost immediately, and is a suggested mechanism for the transition to turbulence. Stability zones are illustrated for the flow, and a simple asymptotic solution confirms some of the features of these numerical results.

Keywords

Couette flowinstabilitynonNewtonian effectsturbulence

MSC classification

Primary:76F20: Dynamical systems approach to turbulence
Secondary:65F15: Eigenvalues, eigenvectors

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 2 , October 2015, pp. 89 - 113
Copyright
© 2015 Australian Mathematical Society 

References

Aris, R., Vectors, tensors, and the basic equations of fluid mechanics (Dover Publications, New York, 1962).Google Scholar
Bayly, B. J., Orszag, S. A. and Herbert, T., “Instability mechanisms in shear-flow transition”, Annu. Rev. Fluid Mech. 20 (1988) 359–391; doi:10.1146/annurev.fl.20.010188.002043.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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).CrossRefGoogle Scholar
Eckhardt, B., “Turbulence transition in shear flows: chaos in high-dimensional spaces”, Procedia IUTAM 5 (2012) 165–168; doi:10.1016/j.piutam.2012.06.021.CrossRefGoogle Scholar
Faranda, D., Lucarini, V., Manneville, P. and Wouters, J., “On using extreme values to detect global stability thresholds in multi-stable systems: the case of transitional plane Couette flow”, Chaos Solitons & Fractals 64 (2014) 26–35; doi:10.1016/j.chaos.2014.01.008.CrossRefGoogle Scholar
Forbes, L. K., “On turbulence modelling and the transition from laminar to turbulent flow”, ANZIAM J. 56 (2014) 28–47; doi:10.1017/S1446181114000224.Google Scholar
Frederiksen, J. S., “Singular vectors, finite-time normal modes, and error growth during blocking”, J. Atmos. Sci. 57 (2000) 312–333; doi:10.1175/1520-0469(2000)057<0312:SVFTNM>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Graebel, W. P., “Stability of a Stokesian fluid in Couette flow”, Phys. Fluids 4 (1961) 362–368; doi:10.1063/1.1706334.CrossRefGoogle Scholar
Groisman, A. and Steinberg, V., “Elastic turbulence in a polymer solution flow”, Nature 405 (2000) 53–55; doi:10.1038/35011019.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Jiménez, J., “Turbulence”, in: Perspectives in fluid dynamics. A collective introduction to current research (eds Batchelor, G. K., Moffatt, H. K. and Worster, M. G.), (Cambridge University Press, Cambridge, 2000) 231–288.Google Scholar
Kitsios, V., Cordier, L., Bonnet, J.-P., Ooi, A. and Soria, J., “Development of a nonlinear eddy-viscosity closure for the triple-decomposition stability analysis of a turbulent channel”, J. Fluid Mech. 664 (2010) 74–107; doi:10.1017/S0022112010003617.CrossRefGoogle Scholar
Larson, R. G., “Fluid dynamics – turbulence without inertia”, Nature 405 (2000) 27–28 doi:10.1038/35011172.CrossRefGoogle ScholarPubMed
Lee, K.-C. and Finlayson, B. A., “Stability of plane Poiseuille and Couette flow of a Maxwell fluid”, J. Non-Newtonian Fluid Mech. 21 (1986) 65–78; doi:10.1016/0377-0257(86)80063-5.CrossRefGoogle Scholar
Mase, G. E., “Continuum mechanics”, in: Schaum’s outline series (McGraw-Hill, New York, 1970).Google Scholar
McComb, W. D., “Theory of turbulence”, Rep. Prog. Phys. 58 (1995) 1117–1206 doi:10.1088/0034-4885/58/10/001.CrossRefGoogle Scholar
Pan, L., Morozov, A., Wagner, C. and Arratia, P. E., “Nonlinear elastic instability in channel flows at low Reynolds numbers”, Phys. Rev. Lett. 110 (2013) 174502; 5 pages; doi:10.1103/PhysRevLett.110.174502.CrossRefGoogle ScholarPubMed
Poole, R. J., “The Deborah and Weissenberg numbers. The British Society of Rheology”, Rheology Bulletin 53 (2012) 32–39; http://pcwww.liv.ac.uk/∼robpoole/PAPERS/POOLE_45.pdf.Google Scholar
Rabin, S. M. E., Caulfield, C. P. and Kerswell, R. R., “Triggering turbulence efficiently in plane Couette flow”, J. Fluid Mech. 712 (2012) 244–272; doi:10.1017/jfm.2012.417.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J. and Henningson, D. S., “Pseudospectra of the Orr-Sommerfeld operator”, SIAM J. Appl. Math. 53 (1993) 15–47; doi:10.1137/0153002.CrossRefGoogle Scholar
Reiner, M., “A mathematical theory of dilatancy”, Amer. J. Math. 67 (1945) 350–362http://www.jstor.org/stable/2371950.CrossRefGoogle 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) 123–164http://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
Rott, N., “Note on the history of the Reynolds number”, Annu. Rev. Fluid Mech. 22 (1990) 1–11; doi:10.1146/annurev.fl.22.010190.000245.CrossRefGoogle Scholar
Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A. N., Wagner, C. and Hof, B., “Elasto-inertial turbulence”, Proc. Natl. Acad. Sci. USA 110 (2013) 10557–10562 doi:10.1073/pnas.1219666110.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Speziale, C. G., “On nonlinear $K$–$\ell$ and $K$–${\it\epsilon}$ models of turbulence”, J. Fluid Mech. 178 (1987) 459–475; doi:10.1017/S0022112087001319.CrossRefGoogle 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
Thompson, J. M. T. and Stewart, H. B., Nonlinear dynamics and chaos (Wiley Publications, New York, 1986).Google 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.CrossRefGoogle ScholarPubMed
Valério, J. V., Carvalho, M. S. and Tomei, C., “Efficient computation of the spectrum of viscoelastic flows”, J. Comput. Phys. 228 (2009) 1172–1187; doi:10.1016/j.jcp.2008.10.018.CrossRefGoogle Scholar
Waleffe, F., “Exact coherent structures in channel flow”, J. Fluid Mech. 435 (2001) 93–102; doi:10.1017/S0022112001004189.CrossRefGoogle 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.CrossRefGoogle Scholar