A combined VOF-RANS approach for studying the evolution of incipient two-dimensional wind-driven waves over a viscous liquid. (English) Zbl 07837901
Summary: Recent laboratory experiments have revealed that important insights into the physical processes involved in the wind-driven generation of surface waves may be obtained by varying the viscosity of the carrying liquid over several orders of magnitude. The present paper reports on the development of a companion approach aimed at studying similar phenomena through numerical simulation, a way expected to remove some of the experimental limitations, especially in the near-interface region, and to allow the relative influence of several physical processes to be assessed by disregarding or inactivating arbitrarily some of them. After reviewing available options, we select and approach based on the combination of a volume of fluid technique to track the evolution of the air-liquid interface, and a two-dimensional Reynolds-averaged version of the Navier-Stokes equations supplemented with a turbulence model to predict the velocity and pressure fields in both fluids. We examine the formal and physical frameworks in which such a time-dependent two-dimensional formulation is meaningful, and close the governing momentum equations with the one-equation Spalart-Allmaras model which directly solves a transport equation for the eddy viscosity. For this purpose, we assume the interface to behave as a rigid wall with respect to turbulent fluctuations in the air, and implement a versatile algorithm to compute the local distance to the interface whatever its shape. We first assess the performance of this model in unseparated and separated single-phase flows over a wavy rigid wall, which are of specific relevance with respect to wind-wave generation. Then, we discuss the initialization protocol used in two-phase simulations, which involves an impulse disturbance with a white noise distribution applied to the interface position. We finally present some examples of interface evolutions obtained at several wind speeds with liquids of various viscosities, and discuss the underlying physics revealed by the associated statistics of interface disturbances, streamline patterns and energy spectra.
MSC:
| 76D33 | Waves for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76F25 | Turbulent transport, mixing |
| 76M99 | Basic methods in fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
Reynolds-averaged Navier-Stokes equations; air-liquid interface; volume-of-fluid method; one-equation Spalart-Allmaras turbulence model; eddy viscosity transportReferences:
| [1] | Komen, G. J.; Cavaleri, M.; Donelan, M.; Hasselmann, K.; Hasselmann, S.; Janssen, P. A.E. M., Dynamics and Modelling of Ocean Waves, 1994, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0816.76001 |
| [2] | Wind Stress over the Ocean, 2001, Cambridge University Press: Cambridge University Press Cambridge · Zbl 1072.86500 |
| [3] | Janssen, P., The Interaction of Ocean Waves and Wind, 2004, Cambridge University Press: Cambridge University Press Cambridge |
| [4] | Phillips, O. M., On the generation of waves by turbulent wind, J. Fluid Mech., 2, 417-445, 1957 · Zbl 0078.18102 |
| [5] | Kahma, K. K.; Donelan, M. A., On the generation of waves by turbulent wind, J. Fluid Mech., 192, 339-364, 1988 |
| [6] | Miles, J. W., On the generation of surface waves by shear flows, J. Fluid Mech., 3, 185-204, 1957 · Zbl 0078.40705 |
| [7] | Miles, J. W., On the generation of surface waves by shear flows. Part 2, J. Fluid Mech., 6, 568-582, 1959 · Zbl 0092.44102 |
| [8] | Miles, J. W., On the generation of surface waves by shear flows. Part 4, J. Fluid Mech., 13, 423-448, 1962 · Zbl 0106.41101 |
| [9] | Lighthill, M. J., Physical interpretation of the mathematical theory of wave generation by wind, J. Fluid Mech., 14, 385-398, 1962 · Zbl 0116.43301 |
| [10] | Miles, J. W., On the generation of surface waves by shear flows. Part 5, J. Fluid Mech., 30, 163-175, 1967 · Zbl 0153.30303 |
| [11] | Jacobs, S. J., An asymptotic theory for the turbulent flow over a progressive water wave, J. Fluid Mech., 174, 69-80, 1987 · Zbl 0616.76073 |
| [12] | van Duin, C. A.; Janssen, P. A.E. M., An analytic model of the generation of surface gravity waves by turbulent air flow, J. Fluid Mech., 236, 197-215, 1992 · Zbl 0746.76011 |
| [13] | Miles, J. W., Surface-wave generation revisited, J. Fluid Mech., 256, 427-441, 1993 · Zbl 0783.76035 |
| [14] | Belcher, S. E.; Hunt, J. C.R., Turbulent shear flow over slowly moving waves, J. Fluid Mech., 251, 109-148, 1993 · Zbl 0787.76032 |
| [15] | Miles, J. W., Surface-wave generation: a viscoelastic model, J. Fluid Mech., 322, 131-145, 1996 · Zbl 0881.76008 |
| [16] | Francis, J. R.D., Wave motion and the aerodynamic drag on a free oil surface, Phil. Mag., 45, 695-702, 1954 |
| [17] | Gottifredi, J. C.; Jameson, G. J., The growth of short waves on liquid surfaces under the action of a wind, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 319, 373-397, 1970 |
| [18] | Paquier, A.; Moisy, F.; Rabaud, M., Viscosity effects in wind wave generation, Phys. Rev. Fluids, 1, Article 083901 pp., 2016 |
| [19] | Aulnette, M.; Rabaud, M.; Moisy, F., Wind-sustained viscous solitons, Phys. Rev. Fluids, 4, Article 084003 pp., 2019 |
| [20] | Aulnette, M.; Zhang, J.; Rabaud, M.; Moisy, F., Kelvin-Helmholtz instability and formation of viscous solitons on highly viscous liquids, Phys. Rev. Fluids, 7, Article 014003 pp., 2022 |
| [21] | Moisy, F.; Rabaud, M.; Salsac, K., A synthetic Schlieren method for the measurement of the topography of a liquid interface, Exp. Fluids, 46, 1021-1036, 2009 |
| [22] | Fulgosi, M.; Lakehal, D.; Banerjee, S.; De Angelis, V., Direct numerical simulation of turbulence in a sheared air-water flow with a deformable interface, J. Fluid Mech., 482, 319-345, 2003 · Zbl 1119.76343 |
| [23] | Komori, S.; Kurose, R.; Iwano, K.; Ukai, T.; Suzuki, N., Direct numerical simulation of wind-driven turbulence and scalar transfer at sheared gas-liquid interfaces, J. Turbul., 11, 1-20, 2010 |
| [24] | Yang, D.; Shen, L., Direct-simulation-based study of turbulent flow over various waving boundaries, J. Fluid Mech., 650, 131-180, 2010 · Zbl 1189.76267 |
| [25] | Zonta, F.; Soldati, A.; Onorato, M., Growth and spectra of gravity-capillary waves in countercurrent air/water turbulent flow, J. Fluid Mech., 777, 245-259, 2015 |
| [26] | Hao, X.; Shen, L., Wind-wave coupling study using LES of wind and phase-resolved simulation of nonlinear waves, J. Fluid Mech., 874, 391-425, 2019 · Zbl 1419.76366 |
| [27] | Li, T.; Shen, L., The principal stage in wind-wave generation, J. Fluid Mech., 934, A41, 2022 · Zbl 07460291 |
| [28] | Tsai, W. T., A numerical study of the evolution and structure of a turbulent shear layer under a free surface, J. Fluid Mech., 354, 239-276, 1998 · Zbl 0920.76059 |
| [29] | Lin, M.; Moeng, C.; Tsai, W.; Sullivan, P. P.; Belcher, S. E., Direct numerical simulation of wind-wave generation processes, J. Fluid Mech., 616, 1-30, 2008 · Zbl 1160.76023 |
| [30] | Yang, Z.; Deng, B.-Q.; Shen, L., Direct numerical simulation of wind turbulence over breaking waves, J. Fluid Mech., 850, 120-155, 2018 · Zbl 1415.76412 |
| [31] | Wu, J.; Deike, L., Wind wave growth in the viscous regime, Phys. Rev. Fluids, 6, Article 094801 pp., 2021 |
| [32] | Wu, J.; Popinet, S.; Deike, L., Revisiting wind wave growth with fully coupled direct numerical simulations, J. Fluid Mech., 951, A18, 2022 |
| [33] | Sullivan, P. P.; Edson, J. B.; Hristov, T.; McWilliams, J. C., Large-eddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves, J. Atmos. Sci., 65, 1225-1245, 2008 |
| [34] | Zhang, W.-Y.; Huang, W.-X.; Xu, C.-X., Very large-scale motions in turbulent flows over streamwise traveling wavy boundaries, Phys. Rev. Fluids, 4, Article 054601 pp., 2019 |
| [35] | Sullivan, P. P.; McWilliams, J. C.; Patton, E. G., Large-eddy simulation of marine atmospheric boundary layers above a spectrum of moving waves, J. Atmos. Sci., 71, 4001-4027, 2014 |
| [36] | Sullivan, P. P.; Banner, M. L.; Morison, R. P.; Peirson, W. L., Turbulent flow over steep steady and unsteady waves under strong wind forcing, J. Phys. Oceanogr., 48, 3-27, 2018 |
| [37] | Deskos, G.; Lee, J. C.Y.; Draxl, C.; Sprague, M. A., Review of wind-wave coupling models for large-eddy simulation of the marine atmospheric boundary layer, J. Atmos. Sci., 78, 3025-3045, 2021 |
| [38] | Labourasse, E.; Lacanette, D.; Toutant, A.; Lubin, P.; Vincent, S.; Lebaigue, O.; Caltagirone, J.-P.; Sagaut, P., Towards large eddy simulation of isothermal two-phase flows: Governing equations and a priori tests, Int. J. Multiph. Flow., 33, 1-39, 2007 |
| [39] | Mastenbroek, C.; Makin, V. K.; Garat, M. H.; Giovanangeli, J. P., Experimental evidence of the rapid distortion of turbulence in the air flow over water waves, J. Fluid Mech., 318, 273-302, 1996 |
| [40] | Meirink, J. F.; Makin, V. K., Modelling low-Reynolds-number effects in the turbulent air flow over water waves, J. Fluid Mech., 415, 155-174, 2000 · Zbl 0956.76035 |
| [41] | Li, P. Y.; Xu, D.; Taylor, P. A., Numerical modelling of turbulent airflow over water waves, Bound.-Lay. Meteorol., 95, 397-425, 2000 |
| [42] | Gent, P. R.; Taylor, P. A., A numerical model of the air flow above water waves, J. Fluid Mech., 77, 105-128, 1976 · Zbl 0339.76011 |
| [43] | Belcher, S. E.; Hunt, J. C.R., Turbulent flow over hills and waves, Ann. Rev. Fluid Mech., 30, 507-538, 1998 · Zbl 1398.86006 |
| [44] | Benilov, A. Y.; Filyushkin, B. N., Application of methods of linear filtration to an analysis of fluctuations in the surface layer of the sea, Izv. Acad. Sci. USSR, Atmos. Oceanic Phys., 6, 477-482, 1970 |
| [45] | Thais, L.; Magnaudet, J., A triple decomposition of the fluctuating motion below laboratory wind water waves, J. Geophys. Res., 100, 741-755, 1995 |
| [46] | Hristov, T.; Friehe, C.; Miller, S., Wave-coherent fields in air flow over ocean waves: Identification of cooperative behavior buried in turbulence, Phys. Rev. Lett., 81, 5245-5248, 1998 |
| [47] | Phillips, O. M., The Dynamics of the Upper Ocean, 1977, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0368.76002 |
| [48] | Ayet, A.; Chapron, B., The dynamical coupling of wind-waves and atmospheric turbulence: A review of theoretical and phenomenological models, Bound.-Lay. Meteorol., 183, 1-33, 2022 |
| [49] | Caulliez, G.; Ricci, N.; Dupont, R., The generation of the first visible wind waves, Phys. Fluids, 10, 757-759, 1998 |
| [50] | Veron, F.; Melville, W. K., Experiments on the stability and transition of wind-driven water surfaces, J. Fluid Mech., 446, 25-65, 2001 · Zbl 1107.76308 |
| [51] | Benjamin, T. B., Shearing flow over a wavy boundary, J. Fluid Mech., 6, 161-205, 1959 · Zbl 0093.19106 |
| [52] | Townsend, A. A., Flow in a deep turbulent boundary layer over a surface distorted by water waves, J. Fluid Mech., 55, 719-735, 1972 · Zbl 0243.76034 |
| [53] | Ó Náraigh, L.; Spelt, P. D.M.; Zaki, T. A., Turbulent flow over a liquid layer revisited: multi-equation turbulence modelling, J. Fluid Mech., 683, 357-394, 2011 · Zbl 1241.76384 |
| [54] | Brackbill, J.; Kothe, D.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys., 100, 335-354, 1992 · Zbl 0775.76110 |
| [55] | Spalart, P. R.; Allmaras, S., A one-equation turbulence model for aerodynamic flows, Rech. Aérosp., 1, 5-21, 1992 |
| [56] | Nee, V. W.; Kovaznay, L. S.G., Simple phenomenological theory of turbulent shear flows, Phys. Fluids, 12, 473-484, 1969 · Zbl 0176.54603 |
| [57] | Magnaudet, J.; Rivero, M.; Fabre, J., Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow, J. Fluid Mech., 284, 97-135, 1995 · Zbl 0848.76063 |
| [58] | Calmet, I.; Magnaudet, J., Large-eddy simulation of high-Schmidt number mass transfer in a turbulent channel flow, Phys. Fluids, 9, 438-455, 1997 |
| [59] | Bonometti, T.; Magnaudet, J., An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics, Int. J. Multiph. Flow., 33, 109-133, 2007 |
| [60] | Noh, W. F.; Woodward, P., SLIC (simple line interface calculation), (van Dooren, A. I.; Zandbergen, P. J., Lect. Notes Phys., Vol. 59, 1976, Springer: Springer New York), 330-340 · Zbl 0382.76084 |
| [61] | Zilker, D. P.; Cook, G. W.; Hanratty, T. J., Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 1. Non-separated flows, J. Fluid Mech., 82, 29-51, 1977 |
| [62] | Thorsness, C.; Morrisroe, P. E.; Hanratty, T. J., A comparison of linear theory with measurements of the variation of shear stress along a solid wave, Chem. Eng. Sci., 33, 579-592, 1978 |
| [63] | Frederick, K. A.; Hanratty, T. J., Velocity measurements for a turbulent nonseparated flow over solid waves, Exp. Fluids, 6, 477-486, 1988 |
| [64] | Zilker, D. P.; Hanratty, T. J., Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 2. Separated flows, J. Fluid Mech., 90, 257-271, 1979 |
| [65] | Buckles, J.; Hanratty, T. J.; Adrian, R. J., Turbulent flow over large-amplitude wavy surfaces, J. Fluid Mech., 140, 27-44, 1984 |
| [66] | Kuzan, J. D.; Hanratty, T. J.; Adrian, R. J., Turbulent flows with incipient separation over solid waves, Exp. Fluids, 7, 88-98, 1989 |
| [67] | Abrams, J.; Hanratty, T. J., Relaxation effects observed for turbulent flow over a wavy surface, J. Fluid Mech., 151, 443-455, 1985 |
| [68] | Kruse, N.; Günther, A.; von Rohr, P. R., Dynamics of large-scale structures in turbulent flow over a wavy wall, J. Fluid Mech., 485, 87-96, 2003 · Zbl 1103.76025 |
| [69] | Wagner, C.; Kuhn, S.; von Rohr, P. R., Scalar transport from a point source in flows over wavy walls, Exp. Fluids, 43, 261-271, 2007 |
| [70] | Hamed, A. M.; Kamdar, A.; Castillo, L.; Chamorro, L. P., Turbulent boundary layer over 2D and 3D large-scale wavy walls, Phys. Fluids, 27, Article 106601 pp., 2015 |
| [71] | Segunda, V. M.; Ormiston, S. J.; Tachie, M. F., Experimental and numerical investigation of developing turbulent flow over a wavy wall in a horizontal channel, Eur. J. Mech. - B/Fluids, 68, 128-143, 2018 |
| [72] | Maass, C.; Schumann, U., Direct numerical simulation of separated turbulent flow over wavy boundary, (Hirschel, E. H., Notes Numer. Fluid Mech., 52, 1996), 227-241 · Zbl 0875.76368 |
| [73] | De Angelis, V.; Lombardi, P.; Banerjee, S., Direct numerical simulation of turbulent flow over a wavy wall, Phys. Fluids, 9, 2429-2442, 1997 |
| [74] | Cherukat, P.; Na, Y.; Hanratty, T. J.; McLaughlin, J. B., Direct numerical simulation of a fully developed turbulent flow over a wavy wall, Theor. Comput. Fluid Dyn., 11, 109-134, 1998 · Zbl 0920.76066 |
| [75] | Yoon, H. S.; El-Samni, O. A.; Huynh, A. T.; Chun, H. H.; Kim, H. J.; Pham, A. H.; Park, I. R., Effect of wave amplitude on turbulent flow in a wavy channel by direct numerical simulation, Ocean Eng., 36, 697-707, 2009 |
| [76] | Henn, D. S.; Sykes, R. I., Large-eddy simulation of flow over wavy surfaces, J. Fluid Mech., 383, 75-112, 1999 · Zbl 0931.76035 |
| [77] | Cui, J.; Patel, V. C.; Lin, C. L., Prediction of turbulent flow over rough surfaces using a force field in large eddy simulation, Trans. ASME, J. Fluids Eng., 125, 2-9, 2003 |
| [78] | Chang, Y. S.; Scotti, A., Modeling unsteady turbulent flows over ripples: Reynolds-averaged Navier-Stokes equations (RANS) versus large-eddy simulation (LES), J. Geophys. Res.-Oceans, 109, C09012, 2004 |
| [79] | Wagner, C.; Kenjeres, S.; von Rohr, P. R., Dynamic large eddy simulations of momentum and wall heat transfer in forced convection over wavy surfaces, J. Turbul., 12, 1-27, 2011 |
| [80] | Patel, V. C.; Chon, J. T.; Yoon, J. Y., Turbulent flow in a channel with a wavy wall, Trans. ASME, J. Fluids Eng., 113, 579-586, 1991 |
| [81] | S. Knotek, M. Jicha, Simulation of flow over a wavy solid surface: comparison of turbulence models, in: EPJ Web Conf. - EFM11, Vol. 25, 2012, p. 01040. |
| [82] | Chaib, K.; Nehari, D.; Sad Chemloul, N., CFD simulation of turbulent flow and heat transfer over rough surfaces, (Energy Procedia, 74, 2015), 909-918 |
| [83] | V.M. Segunda, S. Ormiston, M. Tachie, Numerical analysis of turbulent flow over a wavy wall in a channel, in: Proc. ASME 2016 Fluids Eng. Div. Summer Meet., Vol. 1A, 2016, FEDSM2016-7712, V01AT03A014. |
| [84] | Lake, B. M.; Yuen, H. C.; Rungaldier, H.; Ferguson, W. E., Experiments on nonlinear instabilities and evolution of steep gravity-wave trains, J. Fluid Mech., 83, 49-74, 1977 |
| [85] | Su, M. Y.; Bergin, M.; Marler, P.; Myrick, R., Experiments on nonlinear instabilities and evolution of steep gravity-wave trains, J. Fluid Mech., 124, 45-72, 1982 |
| [86] | Huang, N. E.; Long, S. R.; Zheng, S., The mechanism for frequency downshift in nonlinear wave evolution, Adv. Appl. Mech., 32, 59-117, 1996 · Zbl 0870.76011 |
| [87] | Melville, W. K., The instability and breaking of deep-water waves, J. Fluid Mech., 115, 165-185, 1982 |
| [88] | Trulsen, K.; Dysthe, K. B., Frequency downshift in three-dimensional wave trains in a deep basin, J. Fluid Mech., 352, 359-373, 1997 · Zbl 0898.76011 |
| [89] | Dias, F.; Kharif, C., Nonlinear gravity and capillary-gravity waves, Annu. Rev. Fluid Mech., 31, 301-346, 1999 |
| [90] | Hara, T.; Mei, C. C., Frequency downshift in narrow-banded surface waves under the influence of wind, J. Fluid Mech., 230, 429-477, 1991 · Zbl 0728.76019 |
| [91] | Hara, T.; Mei, C. C., Wind effects on the nonlinear evolution of slowly varying gravity-capillary waves, J. Fluid Mech., 267, 221-250, 1994 · Zbl 0808.76012 |
| [92] | Longuet-Higgins, M. S., The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 360, 489-505, 1978 · Zbl 0497.76025 |
| [93] | Longuet-Higgins, M. S., The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 360, 471-488, 1978 · Zbl 0497.76024 |
| [94] | Savelsberg, R.; Van de Water, W., Turbulence of a free surface, Phys. Rev. Lett., 100, Article 034501 pp., 2008 |
| [95] | Giamagas, G.; Zonta, F.; Roccon, A.; Soldati, A., Propagation of capillary waves in two-layer oil-water turbulent flow, J. Fluid Mech., 960, A5, 2023 · Zbl 1520.76087 |
| [96] | Spalart, P. R., Strategies for turbulence modelling and simulations, Int. J. Heat Fluid Flow, 21, 252-263, 2000 |
| [97] | S.R. Allmaras, F.T. Johnson, P.R. Spalart, Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model, in: Proc. 7th Int. Conf. Comput. Fluid Dyn., 2012, pp. ICCFD7-1902. |
| [98] | Spalart, P. R.; Shur, M., On the sensitization of turbulence models to rotation and curvature, Aerosp. Sci. Technol., 5, 297-302, 1997 · Zbl 0941.76566 |
| [99] | Shur, M. L.; Strelets, M. K.; Travin, A. K.; Spalart, P. R., Turbulence modeling in rotating and curved channels: Assessing the Spalart-Shur correction, AIAA J., 8, 784-792, 2000 |
| [100] | A. Hellsten, Some improvements in Menter’s k-\( \omega\) SST turbulence model, in: Proc. 29th AIAA Fluid Dyn. Conf., 1998, pp. A98-32817. |
| [101] | Zhang, Q.; Yang, Y., A new simpler rotation/curvature correction method for Spalart-Allmaras turbulence model, Chin. J. Aeronaut., 26, 326-333, 2013 |
| [102] | McLeish, W.; Putland, G. E., Measurements of wind-driven flow profile in the top millimeter of water, J. Phys. Oceanogr., 5, 515-518, 1975 |
| [103] | Tsai, W.-T.; Chen, S.-M.; Moeng, C.-H., A numerical study on the evolution and structure of a stress-driven free-surface turbulent shear flow, J. Fluid Mech., 545, 163-192, 2005 · Zbl 1085.76518 |
| [104] | Enstad, L. I.; Nagaosa, R.; Alendal, G., Low shear turbulence structures beneath stress-driven interface with neutral and stable stratification, Phys. Fluids, 18, Article 055106 pp., 2006 · Zbl 1185.76710 |
| [105] | Cess, R. D., A Survey of the Literature on Heat Transfer in Turbulent Tube FlowTech. Rep. 8-0529-R24, 1958, Westinghouse Research |
| [106] | Reynolds, W. C.; Tiederman, W. G., Stability of turbulent channel flow, with application to Malkus’s theory, J. Fluid Mech., 27, 253-272, 1967 |
| [107] | Pirozzoli, S., Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: Mean flow and wall friction, J. Fluid Mech., 745, 378-397, 2014 |
| [108] | El Telbany, M. M.M.; Reynolds, A. J., Velocity distributions in plane turbulent channel flows, J. Fluid Mech., 100, 1-29, 1980 |
| [109] | Del Alamo, J. C.; Jiménez, J., Spectra of the very large anisotropic scales in turbulent channels, Phys. Fluids, 15, L41-L44, 2003 · Zbl 1186.76136 |
| [110] | Hoyas, S.; Jiménez, J., Scaling of the velocity fluctuations in turbulent channels up to \(Re{}_\tau =2003\), Phys. Fluids, 18, Article 011702 pp., 2006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
