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ON MODELLING THE TRANSITION TO TURBULENCE IN PIPE FLOW

Part of: Turbulence

Published online by Cambridge University Press:  11 July 2017

LAWRENCE K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Tasmania, Australia email [email protected] and [email protected]
MICHAEL A. BRIDESON
Affiliation:
School of Mathematics and Physics, University of Tasmania, Tasmania, Australia email [email protected] and [email protected]
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Abstract

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As a possible model for fluid turbulence, a Reiner–Rivlin-type equation is used to study Poiseuille–Couette flow of a viscous fluid in a rotating cylindrical pipe. The equations of motion are derived in cylindrical coordinates, and small-amplitude perturbations are considered in full generality, involving all three velocity components. A new matrix-based numerical technique is proposed for the linearized problem, from which the stability is determined using a generalized eigenvalue approach. New results are obtained in this cylindrical geometry, which confirm and generalize the predictions of previous recent studies. A possible mechanism for the transition to turbulent flow is discussed.

MSC classification

Type
Research Article
Copyright
© 2017 Australian Mathematical Society 

References

Abramowitz, M. and Stegun, I. A., Handbook of mathematical functions (Dover, New York, NY, 1972).Google Scholar
Aris, R., Vectors, tensors and the basic equations of fluid mechanics (Dover, New York, NY, 1962).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) 153 ; 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) 031404 , 11 pages; 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 pipe flow: some open questions”, Nonlinearity 21 (2008) T1T11 ; doi:10.1088/0951-7715/21/1/T01.CrossRefGoogle Scholar
Eckhardt, B., “Turbulence transition in shear flows: chaos in high-dimensional spaces”, Procedia IUTAM 5 (2012) 165168 ; doi:10.1016/j.piutam.2012.06.021.CrossRefGoogle Scholar
Forbes, L. K., “On turbulence modelling and the transition from laminar to turbulent flow”, ANZIAM J. 56 (2014) 2847 ; doi:10.1017/S1446181114000224.Google Scholar
Forbes, L. K., “Transition to turbulence from plane Couette flow”, ANZIAM J. 57 (2015) 89113 ; doi:10.1017/S1446181115000176.CrossRefGoogle Scholar
George, W. K., Lectures in turbulence for the 21st century (Turbulence Research Lab., Chalmers University, Gothenburg, Sweden, 2013) http://www.turbulence-online.com.Google Scholar
Harman, T. L., Dabney, J. and Richert, N., Advanced engineering mathematics with MATLAB, 2nd edn (Brooks & Cole, Pacific Grove, CA, 2000).Google Scholar
Kelly, P. A., Foundations of continuum mechanics (The University of Auckland, Auckland, New Zealand, 2015) http://homepages.engineering.auckland.ac.nz/∼pkel015/SolidMechanicsBooks/Part_III/index.html (Last updated: 24 February 2015).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) 74107 ; doi:10.1017/S0022112010003617.CrossRefGoogle Scholar
Larson, R. G., “Fluid dynamics—turbulence without inertia”, Nature 405 (2000) 2728 ; doi:10.1038/35011172.CrossRefGoogle ScholarPubMed
McComb, W. D., “Theory of turbulence”, Rep. Progr. Phys. 58 (1995) 11171206 ; doi:10.1088/0034-4885/58/10/001.CrossRefGoogle Scholar
Morozov, A. N. and van Saarloos, W., “Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids”, Phys. Rev. Lett. 95 (2005) 024501 , 4 pages; doi:10.1103/PhysRevLett.95.024501.CrossRefGoogle ScholarPubMed
Orszag, S. A., “Accurate solution of the Orr–Sommerfeld stability equation”, J. Fluid Mech. 50 (1971) 689703 ; doi:10.1017/S0022112071002842.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
Pelton, M., Chakraborty, D., Malachosky, E., Guyot-Sionnest, P. and Sader, J. E., “Viscoelastic flows in simple liquids generated by vibrating nanostructures”, Phys. Rev. Lett. 111 (2013) 244502 , 5 pages; doi:10.1103/PhysRevLett.111.244502.CrossRefGoogle ScholarPubMed
Poole, R. J., “The Deborah and Weissenberg numbers. The British Society of Rheology”, Rheology Bulletin 53 (2012) 3239 ; http://pcwww.liv.ac.uk/∼robpoole/PAPERS/POOLE_45.pdf.Google Scholar
Reiner, M., “A mathematical theory of dilatancy”, Amer. J. Math. 67 (1945) 350362 ;http://www.jstor.org/stable/2371950.CrossRefGoogle 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) 123164 ;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) 260281 ; 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) 212215 ; http://www.jstor.org/stable/43634450.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) 1055710562 ; doi:10.1073/pnas.1219666110.CrossRefGoogle ScholarPubMed
Sano, M. and Tamai, K., “A universal transition to turbulence in channel flow”, Nat. Phys. 12 (2016) 249253 ; doi:10.1038/nphys3659.CrossRefGoogle Scholar
Shaqfeh, E. S. G., “Purely elastic instabilities in viscometric flows”, Annu. Rev. Fluid Mech. 28 (1996) 129185 ; doi:10.1146/annurev.fl.28.010196.001021.CrossRefGoogle Scholar
Speziale, C. G., “On nonlinear $K-\ell$ and $K-\unicode[STIX]{x1D700}$ models of turbulence”, J. Fluid Mech. 178 (1987) 459475 ; 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) 621628 ; doi:10.1098/rspa.1933.0193.Google Scholar
Thompson, J. M. T. and Stewart, H. B., Nonlinear dynamics and chaos (Wiley, 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) 578584 ; 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) 11721187 ; doi:10.1016/j.jcp.2008.10.018.CrossRefGoogle Scholar
Waleffe, F., “Exact coherent structures in channel flow”, J. Fluid Mech. 435 (2001) 93102 ; doi:10.1017/S0022112001004189.CrossRefGoogle Scholar