Large-scale structures in a turbulent channel flow with a minimal streamwise flow unit.

*(English)*Zbl 1415.76319Summary: Direct numerical simulations are used to examine large-scale motions with a streamwise length \(2\sim 4h\) (\(h\) denotes the channel half-width) in the logarithmic and outer regions of a turbulent channel flow. We test a minimal ‘streamwise’ flow unit [S. Toh and T. Itano, ibid. 524, 249–262 (2005; Zbl 1065.76553)] (or MSU) for larger Kármán numbers (\(h^+=395\) and 1020) than in the original work. This flow unit consists of a sufficiently long \((L_x^+\approx 400)\) streamwise domain to maintain near-wall turbulence [J. Jiménez and P. Moin, ibid. 225, 213–240 (1991; Zbl 0721.76040)] and a spanwise domain which is large enough to represent the spanwise behaviour of inner and outer structures correctly; as \(h^+\) increases, the streamwise extent of the MSU domain decreases with respect to \(h\). Particular attention is given to whether the spanwise organization of the large-scale structures may be represented properly in this simplified system at sufficiently large \(h^+\) and how these structures are associated with the mean streamwise velocity \(\overline{U}\). It is shown that, in the MSU, the large-scale structures become approximately two-dimensional at \(h^+=1020\). In this case, the streamwise velocity fluctuation \(u\) is energized, whereas the spanwise velocity fluctuation \(w\) is weakened significantly. Indeed, there is a reduced energy redistribution arising from the impaired global nature of the pressure, which is linked to the reduced linear-nonlinear interaction in the Poisson equation (i.e. the rapid pressure). The logarithmic dependence of \(\overline{ww}\) is also more evident due to the reduced large-scale spanwise meandering. On the other hand, the spanwise organization of the large-scale \(u\) structures is essentially identical for the MSU and large streamwise domain (LSD). One discernible difference, relative to the LSD, is that the large-scale structures in the MSU are more energized in the outer region due to a reduced turbulent diffusion. In this region, there is a tight coupling between neighbouring structures, which yields antisymmetric pairs (with respect to centreline) of large-scale structures with a spanwise spacing of approximately \(3h\); this is intrinsically identical with the outer energetic mode in the optimal transient growth of perturbations [J. C. Del Álamo et al., ibid. 561, 329–358 (2006; Zbl 1157.76346)].

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##### References:

[1] | Abe, H.; Kawamura, H.; Matsuo, Y., Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence, ASME J. Fluids Engng, 123, 382-393, (2001) |

[2] | Abe, H.; Kawamura, H.; Matsuo, Y., Surface heat-flux fluctuations in a turbulent channel flow up to Re_𝜏 = 1020 with Pr = 0. 025 and 0.71, Intl J. Heat Fluid Flow, 25, 404-419, (2004) |

[3] | Abe, H.; Kawamura, H.; Choi, H., Very large-scale structures and their effects on the wall shear-stress fluctuations in a turbulent channel flow up to Re_𝜏 = 640, ASME J. Fluids Engng, 126, 835-843, (2004) |

[4] | Abe, H., Matsuo, Y. & Kawamura, H.2005A DNS study of Reynolds-number dependence on pressure fluctuations in a turbulent channel flow. In Proc. of the 4th International Symposium on Turbulence and Shear Flow Phenomena, Williamsburg, VA, USA (ed. Humphrey, J. A. C., Gatski, T. B., Eaton, J. K., Friedrich, R., Kasagi, N. & Leschziner, M. A.), vol. 1, pp. 189-194. |

[5] | Abe, H., Kawamura, H., Toh, S. & Itano, T.2007Effects of the streamwise computational domain size on DNS of a turbulent channel flow at high Reynolds number. In Advances in Turbulence XI, Proc. of the 11th EUROMECH European Turbulence Conference, Porto, Portugal, June 25-28, 2007 (ed. Palma, J. M. L. M. & Lopes, A. S.), pp. 233-235. Springer. |

[6] | Abe, H.; Antonia, R. A.; Kawamura, H., Correlation between small-scale velocity and scalar fluctuations in a turbulent channel flow, J. Fluid Mech., 627, 1-32, (2009) · Zbl 1171.76390 |

[7] | Abe, H.; Antonia, R. A., Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow, J. Fluid Mech., 798, 140-164, (2016) |

[8] | Abe, H.; Antonia, R. A., Relationship between the heat transfer law and the scalar dissipation function in a turbulent channel flow, J. Fluid Mech., 830, 300-325, (2017) · Zbl 1421.76114 |

[9] | Adrian, R. J.; Meinhart, C. D.; Tomkins, C. D., Vortex organization in the outer region of the turbulent boundary layer, J. Fluid Mech., 422, 1-54, (2000) · Zbl 0959.76503 |

[10] | Ahn, J.; Lee, J. H.; Lee, J. L.; Kang, J.-H.; Sung, H. J., Direct numerical simulation of a 30R long turbulent pipe flow at Re_𝜏 = 3008, Phys. Fluids, 27, (2015) |

[11] | Antonia, R. A.; Abe, H.; Kawamura, H., Analogy between velocity and scalar fields in a turbulent channel flow, J. Fluid Mech., 628, 241-268, (2009) · Zbl 1181.76078 |

[12] | Balakumar, B. J.; Adrian, R. J., Large- and very-large-scale motions in channel and boundary-layer flows, Phil. Trans. R. Soc. A, 365, 665-681, (2007) · Zbl 1152.76369 |

[13] | Bernardini, M.; Pirozzoli, S.; Orlandi, P., Velocity statistics in turbulent channel flow up to Re_𝜏 = 4000, J. Fluid Mech., 742, 171-191, (2014) |

[14] | Bradshaw, B., ‘Inactive’ motion and pressure fluctuations in turbulent boundary layers, J. Fluid Mech., 30, 241-258, (1967) |

[15] | Bradshaw, P.; Koh, Y. M., A note on Poisson’s equation for pressure in a turbulent flow, Phys. Fluids, 24, 777, (1981) |

[16] | Brown, G. L.; Thomas, A. S. W., Large structure in a turbulent boundary layer, Phys. Fluids, 20, S243-S252, (1977) |

[17] | Butler, K. M.; Farrell, B. F., Three-dimensional optimal perturbations in viscous shear flow, Phys. Fluids A, 4, 8, 1637-1650, (1992) |

[18] | Christensen, K. T.; Adrian, R. J., Statistical evidence of hairpin vortex packets in wall turbulence, J. Fluid Mech., 431, 433-443, (2001) · Zbl 1008.76029 |

[19] | Del Álamo, J. C.; Jiménez, J., Spectra of the very large anisotropic scales in turbulent channels, Phys. Fluids, 15, 6, L41-L44, (2003) · Zbl 1186.76136 |

[20] | Del Álamo, J. C.; Jiménez, J., Linear energy amplification in turbulent channels, J. Fluid Mech., 559, 205-213, (2006) · Zbl 1095.76021 |

[21] | Del Álamo, J. C.; Jiménez, J.; Zandonade, P.; Moser, R. D., Scaling of the energy spectra of turbulent channels, J. Fluid Mech., 500, 135-144, (2004) · Zbl 1059.76031 |

[22] | Del Álamo, J. C.; Jiménez, J.; Zandonade, P.; Moser, R. D., Self-similar vortex clusters in the logarithmic region, J. Fluid Mech., 561, 329-358, (2006) · Zbl 1157.76346 |

[23] | Flores, O.; Jiménez, J., Hierarchy of minimal flow units in the logarithmic layer, Phys. Fluids, 22, (2010) |

[24] | Hamilton, J. M.; Kim, J.; Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287, 317-348, (1995) · Zbl 0867.76032 |

[25] | Hoyas, S.; Jiménez, J., Reynolds number effects on the Reynolds-stress budgets in turbulent channels, Phys. Fluids, 20, (2008) · Zbl 1182.76330 |

[26] | Hunt, J. C. R.; Morrison, J. F., Eddy structure in turbulent boundary layers, Eur. J. Mech. (B/Fluids), 19, 673-694, (2001) · Zbl 1005.76035 |

[27] | Hutchins, N.; Marusic, I., Evidence of very long meandering features in the logarithmic region of turbulent boundary layers, J. Fluid Mech., 579, 467-477, (2007) · Zbl 1113.76004 |

[28] | Hutchins, N.; Marusic, I., Large-scale influences in near-wall turbulence, Phil. Trans. R. Soc. Lond. A, 365, 647-664, (2007) · Zbl 1152.76421 |

[29] | Hwang, J.; Lee, J.; Sung, H. J.; Zaki, T. A., Inner-outer interactions of large-scale structures in turbulent channel flow, J. Fluid Mech., 790, 128-157, (2016) · Zbl 1382.76124 |

[30] | Jiménez, J.; Moin, P., The minimal flow unit in near-wall turbulence, J. Fluid Mech., 225, 213-240, (1991) · Zbl 0721.76040 |

[31] | Jiménez, J.; Pinelli, A., The autonomous cycle of near wall turbulence, J. Fluid Mech., 389, 335-359, (1999) · Zbl 0948.76025 |

[32] | Jiménez, J.; Hoyas, S., Turbulent fluctuations above the buffer layer of wall-bounded flows, J. Fluid Mech., 611, 215-236, (2008) · Zbl 1151.76512 |

[33] | Jiménez, J. & Kawahara, G.2011Dynamics of wall-bounded turbulence. In Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 221-268. Cambridge University Press. |

[34] | Kim, J., On the structure of pressure fluctuations in simulated turbulent channel flow, J. Fluid Mech., 205, 421-451, (1989) |

[35] | Kim, J.; Hussain, F., Propagation velocity of perturbations in turbulent channel flow, Phys. Fluids A, 5, 695-706, (1993) |

[36] | Kline, S. J.; Reynolds, W. C.; Schraub, F. A.; Runstadler, P. W., The structure of turbulent boundary layers, J. Fluid Mech., 30, 741-773, (1967) |

[37] | Kawamura, H., Abe, H. & Matsuo, Y.2004 Very large-scale structure observed in DNS of turbulent channel flow with passive scalar transport. In Proc. of 15th Australasian Fluid Mech. Conf. The University of Sydney. |

[38] | Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166, (1987) · Zbl 0616.76071 |

[39] | Kim, K. C.; Adrian, R. J., Very large-scale motion in the outer layer, Phys. Fluids, 11, 417-422, (1999) · Zbl 1147.76430 |

[40] | Lee, J. H.; Sung, H. J., Very-large-scale motions in a turbulent boundary layer, J. Fluid Mech., 673, 80-120, (2011) · Zbl 1225.76162 |

[41] | Lee, M.; Moser, R. D., Direct numerical simulation of turbulent channel flow up to Re_𝜏 ≈ 5200, J. Fluid Mech., 774, 395-415, (2015) |

[42] | Lozano-Durán, A.; Jiménez, J., Effect of the computational domain on direct simulations of turbulent channels up to Re_𝜏 = 4200, Phys. Fluids, 26, (2014) |

[43] | Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J.2013On the logarithmic region in wall turbulence. J. Fluid Mech.716, R3 (11 pages). doi:10.1017/jfm.2012.511 · Zbl 1284.76206 |

[44] | Moin, P.; Mahesh, K., Direct numerical simulation: a tool in turbulence research, Annu. Rev. Fluid Mech., 30, 539-578, (1998) · Zbl 1398.76073 |

[45] | Monty, J. P.; Stewart, J. A.; Williams, R. C.; Chong, M. S., Large-scale features in turbulent pipe and channel flows, J. Fluid Mech., 589, 147-156, (2007) · Zbl 1141.76316 |

[46] | Monty, J.; Chong, M. S., Turbulent channel flow: comparison of streamwise velocity data from experiments and direct numerical simulation, J. Fluid Mech., 633, 461-474, (2009) · Zbl 1183.76035 |

[47] | Morinishi, Y.; Lund, T. S.; Vasilyev, O. V.; Moin, P., Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys., 143, 90-124, (1998) · Zbl 0932.76054 |

[48] | Nakagawa, H.; Nezu, I., Prediction of the contributions to the Reynolds stress from bursting events in open channel flow, J. Fluid Mech., 80, 99-128, (1977) · Zbl 0358.76042 |

[49] | Nickels, T. B.; Marusic, I.; Hafez, S.; Hutchins, N.; Chong, M. S., Some predictions of the attached eddy model for a high Reynolds number boundary layer, Phil. Trans. R. Soc. A, 365, 807-822, (2007) · Zbl 1152.76414 |

[50] | Panton, R. L.; Lee, M.; Moser, R. D., Correlation of pressure fluctuations in turbulent wall layers, Phys. Rev. Fluids, 2, (2017) |

[51] | Perry, A. E.; Henbest, S.; Chong, M. S., A theoretical and experimental study of wall turbulence, J. Fluid Mech., 165, 163-199, (1986) · Zbl 0597.76052 |

[52] | Pujals, G.; Manuel Garcia-Villalba, M.; Cossu, C.; Depardon, S., A note on optimal transient growth in turbulent channel flows, Phys. Fluids, 21, (2009) · Zbl 1183.76425 |

[53] | Rajagopalan, S.; Antonia, R. A., Some properties of the large structure in a fully developed turbulent duct flow, Phys. Fluids, 22, 614-622, (1979) |

[54] | Robinson, S. K.1991 The kinematics of turbulent boundary layer. NASA TM 103859. |

[55] | Smith, C. R.; Metzler, S. P., The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer, J. Fluid Mech., 129, 27-54, (1983) |

[56] | Spalart, P. R., Direct simulation of a turbulent boundary layer up to R_𝜃 = 1410, J. Fluid Mech., 187, 61-98, (1988) · Zbl 0641.76050 |

[57] | Toh, S.; Itano, T., Interaction between a large-scale structure and near-wall structures in channel flow, J. Fluid Mech., 524, 249-262, (2005) · Zbl 1065.76553 |

[58] | Tomkins, C. D.; Adrian, R. J., Spanwise structure and scale growth in turbulent boundary layers, J. Fluid Mech., 490, 37-74, (2003) · Zbl 1063.76514 |

[59] | Tomkins, C. D.; Adrian, R. J., Energetic spanwise modes in the logarithmic layer of a turbulent boundary layer, J. Fluid Mech., 545, 141-162, (2005) · Zbl 1085.76519 |

[60] | Townsend, A. A., Equilibrium layers and wall turbulence, J. Fluid Mech., 11, 97-120, (1961) · Zbl 0127.42602 |

[61] | Townsend, A. A.1976The Structure of Turbulent Shear Flow, vol. 2. Cambridge University Press. · Zbl 0325.76063 |

[62] | Waleffe, F., On a self-sustaining process in shear flows, Phys. Fluids, 9, 883-900, (1997) |

[63] | Waleffe, F. & Kim, J.1997How streamwise rolls and streaks self-sustain in a shear flow. In Self-Sustaining Mechanisms of Wall Turbulence (ed. Panton, R. L.), pp. 309-332. Computational Mechanics Publications. |

[64] | Wei, T.; Willmarth, W. W., Reynolds-number effects on the structures of a turbulent channel flow, J. Fluid Mech., 204, 57-95, (1989) |

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