Advertisement

A Brief History of Simple Invariant Solutions in Turbulence

  • Lennaert van VeenEmail author
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)

Abstract

When studying fluid mechanics in terms of instability, bifurcation and invariant solutions one quickly finds out how little can be done by pen and paper. For flows on sufficiently simple domains and under sufficiently simple boundary conditions, one may be able to predict the parameter values at which the base flow becomes unstable and the basic properties of the secondary flow. On more complicated domains and under more realistic boundary conditions, such questions can usually only be addressed by numerical means. Moreover, for a wide class of elementary parallel shear flows the base flow remains stable in the presence of sustained turbulent motion. In such flows, secondary solutions often appear with finite amplitude and completely unconnected to the base flow. Only using techniques from computational dynamical systems can such behaviour be explained. Many of these techniques, such as for the detection and classification of bifurcations and for the continuation in parameters of equilibria and time-periodic solutions, were developed in the late 1970s for dynamical systems with few degrees of freedom. The application to fluid dynamics or, to be more precise, to spatially discretized Navier–Stokes flow, is far from straightforward. In this historical review chapter, we follow the development of this field of research from the valiant naivety of the early 1980s to the open challenges of today.

Notes

Acknowledgements

I would like to thank Sebastian Altmeyer, Andrew Hazel, Björn Hof, Genta Kawahara, Rich Kerswell, Masato Nagata and Fabian Waleffe for sharing their ideas and memories.

References

  1. 1.
    Benjamin, T.B.: Bifurcation phenomena in steady flows of a viscous fluid. i. theory. Proc. R. Soc. A 359, 1–26 (1978a)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benjamin, T.B.: Bifurcation phenomena in steady flows of a viscous fluid. ii. experiments. Proc. R. Soc. A 359, 27–43 (1978b)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benjamin, T.B., Mullin, T.: Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A: Math. Phys. 377, 221–249 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Busse, F.H., Clever, R.M.: Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319–335 (1979)CrossRefGoogle Scholar
  5. 5.
    Orszag, S.A., Patterson, G.S.: Numerical simulation of 3-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76 (1972)CrossRefGoogle Scholar
  6. 6.
    Kim, J., Moin, P., Moser, R.D.: Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166 (1987)CrossRefGoogle Scholar
  7. 7.
    Doedel, E.: AUTO: a program for the automatic bifurcation analysis for autonomous systems. Congr. Numer. 30, 265–284 (1981)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Classics in applied mathematics. SIAM (1979)Google Scholar
  9. 9.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  10. 10.
    Bomans, L., Roose, D.: Benchmarking the ipcs/2 hypercube multiprocessor. Concurr. Pract. Ex 1, 3–18 (1989)CrossRefGoogle Scholar
  11. 11.
    Saltzman, B.: Finite amplitude free convection as an initial value problem i. J Atmos Sci 19, 329–342 (1962)CrossRefGoogle Scholar
  12. 12.
    Osinga, H.: Interview with K Andrew Cliffe, University of Nottingham, UK for DSWebmagazine (2010). https://dsweb.siam.org/The-Magazine/Article/welcome-to-academia-after-30-years-in-industry-1
  13. 13.
    Riley, D.: Professor Andrew Cliffe (eulogy) (2014). https://www.nottingham.ac.uk/mathematics/news/professor-andrew-cliffe.aspx
  14. 14.
    Cliffe, K.A., Mullin, T.: A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech. 153, 243–258 (1985)CrossRefGoogle Scholar
  15. 15.
    Cliffe KA (1996) ENTWIFE (release 6.3) reference manual. Oxford, UK: Harwell Laboratory. http://ww.swmath.org/software/4255
  16. 16.
    Brindley, J., Kaas-Petersen, C., Spence, A.: Path-following methods in bifurcation problems. Physica D 34, 456–461 (1989)CrossRefGoogle Scholar
  17. 17.
    van Veen, L.: Interview with Masato Nagata for DSWebmagazine (2017). https://dsweb.siam.org/The-Magazine/The-Magazine/All-Issues/interview-with-masato-nagata
  18. 18.
    Nagata, M.: Three-dimensional tertiary motions in a plane shear layer. J. Fluid Mech. 135, 1–26 (1983)CrossRefGoogle Scholar
  19. 19.
    Nagata, M.: On wavy instabilities of Taylor vortex flow between corotating cylinders. J. Fluid Mech. 188, 585–598 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nagata, M.: Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Waleffe, F.: Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143 (1998)CrossRefGoogle Scholar
  22. 22.
    Waleffe, F., Ma, C., Potter, S.: Exact coherent states. In: Proceedings of the 2011 Program in Geophysical Fluid Dynamics. Woods Hole Oceanographic Institution (2011)Google Scholar
  23. 23.
    Chantry, M., Tuckerman, L.S., Barkley, D.: Turbulent-laminar patterns in shear flows without walls. J. Fluid Mech. 791, R8 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kawahara, G., Kida, S.: Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300 (2001)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Faisst, H., Eckhardt, B.: Traveling waves in pipe flow. Phys. Rev. Lett. 91(224), 502 (2003)Google Scholar
  26. 26.
    Wedin, H., Kerswell, R.R.: Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hof, B., van Doorne, C.W.H., Westerweel, J., Nieuwstadt, F.T.M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R.R., Waleffe, F.: Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305, 1594–1598 (2004)CrossRefGoogle Scholar
  28. 28.
    Sánchez, J., Net, M., García-Archilla, B., Simó, C.: Newton-Krylov continuation of periodic orbits for Navier-Stokes flows. J. Comput. Phys. 201(1), 13–33 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    de Lozar, A., Mellibovski, F., Avila, M., Hof, B.: Edge states in pipe flow experiments. Phys. Rev. Lett. 108(214), 502 (2012)Google Scholar
  30. 30.
    Saad, Y., Schultz, M.H.: GMRES - a generalized minimal residual algorithm for solving nonsymmetric linear-systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Krylov, A.N.: On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined. Izvestija AN SSSR (News of Academy of Sciences of the USSR), Otdel mat i estest nauk 7(4), 491–539 (1931). see also A. N. Krylov: a short biography by Mike Botchev at http://www.staff.science.uu.nl/~vorst102/kryl.html
  32. 32.
    Tuckerman, L.S.: Steady state solving via Stokes preconditioning – recursion relations for elliptic operators. In: Proceedings of the Eleventh International Conference on Numerical Methods in Fluid Dynamics, pp. 573–577. Springer, Berlin (1989)Google Scholar
  33. 33.
    Sánchez Umbría, J., Net, M.: Numerical continuation methods for large-scale dissipative dynamical systems. Eur. Phys. J-ST 225, 2465 (2016)CrossRefGoogle Scholar
  34. 34.
    Constantin, P., Foias, P., Manley, O.P., Temam, R.: Determining modes and fractal dimension of turbulent flows. J. Fluid Mech. 150, 427–440 (1985)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Viswanath, D.: Recurrent motions within plane couette turbulence. J. Fluid Mech. 580, 339–358 (2007)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Gibson, J.F.: Channelflow: A spectral Navier-Stokes simulator in C++. Technical report, U. New Hampshire, Channelflow.org (2014)Google Scholar
  37. 37.
    Willis, A.P.: The Openpipeflow Navier–Stokes solver. SoftwareX 6, 124–127 (2017). http://arxiv.org/abs/1705.03838CrossRefGoogle Scholar
  38. 38.
    Eckhardt, B., Scheider, T.M., Hof, B., Westerweel, J.: Turbulence transition in pipe flow. Ann. Rev. Fluid Mech. 39, 447–468 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Kawahara, G., Uhlmann, M., van Veen, L.: The significance of simple invariant solutions in turbulent flows. Ann. Rev. Fluid Mech. 44, 203–222 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Salinger, A., Romero, L., Pawlowski, R., Wilkes, E., Lehoucq, R., Burroughs, B., Bou-Rabee, N.: LOCA: Library of Continuation Algorithms (2001). http://www.cs.sandia.gov/loca/
  41. 41.
    Pawlowski, R.P., Salinger, A.G., Shadid, J.N., Mountziaris, T.J.: Bifurcation and stability analysis of laminar isothermal counterflowing jets. J. Fluid Mech. 551, 117–139 (2006)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Mullin, T., Kerswell, R.R. (eds.): Laminar–Turbulent Transition and Finite-Amplitude Solutions Fluid Mechanics and its Applications. Fluid mechanics and its applications (2005). https://www.newton.ac.uk/event/hrt, https://perso.limsi.fr/duguet/Cargese/master.pdf, https://www.kitp.ucsb.edu/activities/transturb17
  43. 43.
    Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B., Henningson, D.S.: Edge states as mediators of bypass transition in boundary-layer flows. J. Fluid Mech. 801, R2 (2016)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Suri, B., Tithof, J., Grigoriev, R.O., Schatz, M.F.: Forecasting fluid flows using the geometry of turbulence. Phys. Rev. Lett. 118(114), 501 (2017)Google Scholar
  45. 45.
    Hwang, Y., Willis, A.P., Cossu, C.: Invariant solutions of minimal large-scale structures in turbulent channel flow for \(re_{\tau }\) up to 1000. J. Fluid Mech. 802, R1 (2016)Google Scholar
  46. 46.
    Sasaki, E., Kawahara, G., Sekimoto, A., Jiménez, J.: Unstable periodic orbits in plane Couette flow with the Smagorinsky model. J. Phys. Conf. Ser. 708(012), 003 (2016)Google Scholar
  47. 47.
    Sekimoto, A., Jiménez, J.: Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow. J. Fluid Mech. 827, 225–249 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of Ontario Institute of TechnologyOntarioCanada

Personalised recommendations