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On modelling transitional turbulent flows using under-resolved direct numerical simulations: the case of plane Couette flow. (English) Zbl 1272.76140
Summary: Direct numerical simulations have proven of inestimable help to our understanding of the transition to turbulence in wall-bounded flows. While the dynamics of the transition from laminar flow to turbulence via localised spots can be investigated with reasonable computing resources in the domains of limited extent, the study of the decay of turbulence in conditions approaching those in the laboratory requires the consideration of domains so wide as to exclude the recourse to fully resolved simulations. Using Gibson’s C++ code Channel-Flow, we scrutinise the effects of a controlled lowering of the numerical resolution on the decay of turbulence in-plane Couette flow at a quantitative level. We show that the number of Chebyshev polynomials describing the cross-stream dependence can be drastically decreased while preserving all the qualitative features of the solution. In particular, the oblique turbulent band regime experimentally observed in the upper part of the transitional range is extremely robust. In terms of Reynolds numbers, the resolution lowering is seen to yield a regular downward shift of the upper and lower thresholds \(R_t\) and \(R_g\) where the bands appear and break down. The study is illustrated with the results of two preliminary experiments.

76F06 Transition to turbulence
76F65 Direct numerical and large eddy simulation of turbulence
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[1] Prigent A. et al.: Long-wavelength modulation of turbulent shear flows. Physica D 174, 100–113 (2003) · Zbl 1036.76023 · doi:10.1016/S0167-2789(02)00685-1
[2] Andereck C.D., Liu S.S., Swinney H.L.: Flow regimes in a circular Couette flow system with independently rotating cylinders. J. Fluid Mech. 164, 155–183 (1986) · doi:10.1017/S0022112086002513
[3] Jimenez J., Moin P.: The minimal flow unit in near wall turbulence. J. Fluid Mech. 225, 213–240 (1991) · Zbl 0721.76040 · doi:10.1017/S0022112091002033
[4] Hamilton J.M., Kim J., Waleffe F.: Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348 (1995) · Zbl 0867.76032 · doi:10.1017/S0022112095000978
[5] Eckhardt B., Faisst H., Schmiegel A., Schneider T.M.: Dynamical systems and the transition to turbulence in linearly stable shear flows. Philos. Trans. R. Soc. A 366, 1297–1315 (2008) · doi:10.1098/rsta.2007.2132
[6] Komminaho J., Lundbladh A., Johansson A.J.: Very large structures in plane turbulent Couette flow. J. Fluid Mech. 320, 259–285 (1996) · Zbl 0875.76160 · doi:10.1017/S0022112096007537
[7] McKeon, B.J., Sreenivasan, K.R. (eds.): Scaling and structure in high Reynolds number wall-bounded flows. Focus Issue, Philos. Trans. R. Soc. A 365 (2007) · Zbl 1152.76378
[8] Barkley D., Tuckerman L.S.: Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett. 94, 014502 (2005) · Zbl 1124.76018 · doi:10.1103/PhysRevLett.94.014502
[9] Barkley D., Tuckerman L.S.: Turbulent-laminar patterns in plane Couette flow. In: Mullin, T., Kerswell, R. (eds) IUTAM symposium on laminar-turbulent transition and finite amplitude solutions, Springer, Dordrecht (2005)
[10] Barkley D., Tuckerman L.S.: Mean flow of turbulent-laminar patterns in plane Couette flow. J. Fluid Mech. 576, 109–137 (2007) · Zbl 1124.76018 · doi:10.1017/S002211200600454X
[11] Duguet Y., Schlatter Ph., Henningson D.S.: Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech. 650, 119–129 (2010) · Zbl 1189.76254 · doi:10.1017/S0022112010000297
[12] Tsukahara T., Kawaguchi Y., Kawamura H.: DNS of turbulent plane Couette flow with emphasis on turbulent stripes. In: Eckhardt, B. (eds) Advances in turbulence 12, pp. 71–74. Springer, Dordrecht (2009)
[13] Lagha M., Manneville P.: Modeling transitional plane Couette flow. Eur. Phys. J. B 58, 433–447 (2007) · Zbl 1182.76422 · doi:10.1140/epjb/e2007-00243-y
[14] Manneville, P.: Spatiotemporal perspective on the decay of turbulence in wall-bounded flows. Phys. Rev. E 79 (2009) 025301 [R]; 039904 [E]
[15] Waleffe F.: On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900 (1997) · doi:10.1063/1.869185
[16] Willis A., Kerswell R.R.: Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised ’edge’ state. J. Fluid Mech. 619, 213–233 (2009) · Zbl 1156.76395 · doi:10.1017/S0022112008004618
[17] Pomeau Y.: Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 23, 3–11 (1986) · doi:10.1016/0167-2789(86)90104-1
[18] Gibson, J.: http://www.channelflow.org/
[19] Duguet Y., Schlatter Ph., Henningson D.S.: Localized edge states in plane Couette flow Phys. Fluid 21, 111701 (2009) · Zbl 1183.76187 · doi:10.1063/1.3265962
[20] Bottin S., Dauchot O., Daviaud F., Manneville P.: Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys. Lett. 43, 171–176 (1998) · doi:10.1209/epl/i1998-00336-3
[21] Bottin S., Chaté H.: Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B 6, 143–155 (1998) · doi:10.1007/s100510050536
[22] Rolland, J., Manneville, P.: Oblique turbulent bands in plane Couette flow: from visual to quantitative data. In: Ji Hantao, Smits, L. (eds.) 16th International Couette–Taylor Workshop, Princeton University, Sept 2009. http://www.princeton.edu/\(\sim\)asmits/ICTW/PROGRAM-Sep06.pdf
[23] Tuckerman, L.S., Barkley, D., Dauchot, O.: Instability of uniform turbulent plane Couette flow: spectra, probability distribution functions and K closure model. Schlatter, Ph., Henningson, D.S. (eds.) 7th IUTAM Symposium on laminar-turbulent transition, Stockholm, June 2009. Springer (2010)
[24] Shimizu M., Kida S.: A driving mechanism of a turbulent puff in pipe flow. Fluid Dyn. Res. 41, 045501 (2009) · Zbl 1422.76107 · doi:10.1088/0169-5983/41/4/045501
[25] Pomeau Y.: Chapitre IV, Transition vers la turbulence dans les écoulements parallèles. In: Bergé, P., Pomeau, Y., Vidal, Ch. (eds) L’espace chaotique, Hermann, Paris (1998)
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