THERMAL SCIENCE

International Scientific Journal

PERSISTENCE OF THE LAMINAR REGIME IN A FLAT PLATE BOUNDARY LAYER AT VERY HIGH REYNOLDS NUMBER

ABSTRACT
Starting from the Navier-Stokes and the continuity equations of a viscous incompressible fluid, a statistical theory is developed for the prediction of transition and breakdown to turbulence in a laminar boundary layer exposed to small, statistically stationary axisymmetric disturbances. The transport equations for the statistical properties of the disturbances are closed using the two-point correlation technique and invariant theory. By considering the local equilibrium to exist between production and viscous dissipation, which forces the energy of the disturbances in the boundary layer to be lower than that of the free stream, the transition criterion is formulated in terms of the anisotropy of the disturbances and a Reynolds number based on the intensity and the length scale of the disturbances. The transition criterion determines conditions that guarantee maintenance of the laminar flow regime in a flat plate boundary layer. It is shown that predictions of the transition onset deduced from the transition criterion yield the critical Reynolds number, which is in good agreement with the experimental data obtained under well-controlled laboratory conditions reported in the literature. For the preferred mode of the axisymmetric disturbances, for which the intensity of the disturbances in the stream wise direction is larger than in the other two directions, the analysis shows that the anisotropy increases the critical Reynolds number. Theoretical considerations yield the quantitative estimate for the minimum level of the anisotropy of the free stream required to prevent transition and breakdown to turbulence. The numerical databases for fully developed turbulent wall-bounded flows at low and moderate Reynolds numbers were utilized to demonstrate the stabilizing and destabilizing role of the anisotropy in the disturbances on the development of the transition process in wall-bounded flows. The stabilizing role of increased anisotropy in a free stream on the boundary layer development was successfully tested experimentally in a large wind tunnel by maintaining the stable laminar regime in a flat plate boundary layer up to (Rex)T 4·106.
KEYWORDS
PAPER SUBMITTED: 2006-05-30
PAPER REVISED: 2006-06-03
PAPER ACCEPTED: 2006-06-15
DOI REFERENCE: https://doi.org/10.2298/TSCI0602063J
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2006, VOLUME 10, ISSUE 2, PAGES [63 - 96]
REFERENCES
  1. Taylor, G. I., Statistical Theory of Turbulence. Part V. Effects of Turbulence on Boundary Layer. Theoretical Discussion of Relationship between Scale of Turbulence and Critical Resistance of Spheres, Proc. Roy. Soc. Lond. A, 156 (1936), pp. 307-317
  2. Dryden, H. L., Air Flow in the Boundary Layer Near a Plate, NACA Report No. 562, USA, 1936
  3. Schubauer, G. B., Skramstad, H. K., Laminar-Boundary-Layer Oscillation and Transition on a Flat Plate, NACA Report No. 909, USA, 1943
  4. Wells, C. S., Effects of Free Stream Turbulence on Boundary-Layer Transition, AIAA J., 5, (1967), pp. 172-174
  5. Spangler, J. G., Wells, C. S., Effects of Free Stream Disturbances on Boundary-Layer Transition, AIAA J., 6 (1968), pp. 534-545
  6. Saric, W. S., Reynolds, G. A., Experiments on the Stability and Nonlinear Waves in a Boundary Layer, in: Laminar-Turbulent Transition (Eds. E. Eppler, H. Fasel), Springer-Verlag, Berlin, 1980, pp. 125-134
  7. Kachanov, Y. S., Kozlov, V. V., Levchenko, V. Y., The Origin of Turbulence in the Boundary Layer (in Russian), Nauka, Novosibirsk, USSR, 1982
  8. Pfenninger, W., Transition Experiments in the Inlet Length of Tubes at High Reynolds Numbers (Ed. G. V. Lachmann), in: Boundary Layer and Flow Control, Vol. 2, Pergamon Press, Oxford, UK, 1961, pp. 970-980
  9. Hinze, J. O., Turbulence, 2nd ed., McGraw-Hill, New York, USA, 1975
  10. Jovanović, J., Pashtrapanska, M., Frohnapfel, B., Durst, F., Koskinen, J., Koskinen, K., On the Mechanism Responsible for Turbulent Drag Reduction by Dilute Addition of High Polymers: Theory, Experiments, Simulations and Predictions, J. Fluids Eng., 128 (2006), pp. 118-130
  11. Jovanović, J., Pashtrapanska, M., On the Criterion for the Determination Transition on set and Break down to Turbulence in Wall-Bounded Flows, J. Fluids Eng., 126 (2004), pp. 626-633
  12. Rotta, J. C., Turbulent Flows: An Introduction into Theory and Its Application (in German), Teubner, Stuttgart, Germany, 1972, pp. 120-127
  13. Chou, P. Y., On the Velocity Correlation and the Solution of the Equation of Turbulent Fluctuation, Q. Appl. Math., 3 (1945), pp. 38-54
  14. Lumley, J. L., Newman, G., The Return to Isotropy of Homogeneous Turbulence, J. Fluid Mech., 82 (1977), pp. 161-178
  15. Kolovandin, B. A., Vatutin, I. A., Statistical Transfer Theory in Non-Homogeneous Turbulence, Int. J. Heat. Mass Transfer, 15 (1972), pp. 2371-2383
  16. Jovanović, J., Ye, Q.-Y., Durst, F., Statistical Interpretation of the Turbulent Dissipation Rate in Wall-Bounded Flows, J. Fluid Mech., 293 (1995), pp. 321-347
  17. Jovanović, J., Ye, Q.-Y., Durst, F., Refinement of the Equation for the Determination of Turbulent Micro-Scale, Universität Erlangen-Nürnberg Rep., Germany, LSTM 349/T/92, 1992
  18. Jovanović, J., Otić, I., Bradshaw, P., On the Anisotropy of Axisymmetric Strained Turbulence in the Dissipation Range, J. Fluids Eng., 125 (2003), pp. 410-413
  19. Jovanović, J., The Statistical Dynamics of Turbulence, Springer-Verlag, Berlin, 2004
  20. Lumley, J. L., Computational Modeling of Turbulent Flows, Adv. Appl. Mech., 18 (1978), pp. 123-176
  21. Jovanović, J., Otić, I. On the Constitutive Relation for the Reynolds Stresses and the
  22. Prandtl-Kolmogorov Hypothesis of Effective Viscosity in Axisymmetric Strained Turbulence, J. Flu ids Eng., 122 (2000), pp. 48-50
  23. Rotta, J. C., Statistical Theory of Non-Homogeneous Turbulence (in German), Z. Phys., 129 (1951), pp. 547-572
  24. Kolmogorov, A. N., On Degeneration of Isotropic Turbulence in an Incompressible Viscous Liquid (in Russian), Dokl. Akad. Nauk SSSR, 6 (1941), pp. 538-540
  25. Sreenivasan, K. R., On the Scaling of the Turbulence Energy Dissipation Rate, Phys. Fluids, 27 (1984), pp. 1048-1051
  26. Schlichting, H., Boundary-Layer Theory, 6th ed., McGraw-Hill, New York, USA, 1968
  27. Jovanović, J., Hillerbrand, R., On Peculiar Property of the Velocity Fluctuations in Wall-Bounded Flows, Thermal Science, 9 (2005), 1, pp. 3-12
  28. Kim, J., Moin, P., Moser, R., Turbulence Statistics in a Fully Developed Channel Flow at Low Reynolds Numbers, J. Fluid Mech., 177 (1987), pp. 133-166
  29. Antonia, R. A., Teitel, M., Kim, J., Browne, L. W. B., Low-Reynolds Number Effects in a Fully Developed Channel Flow, J. Fluid Mech., 236 (1992), pp. 579-605
  30. Horiuti, K., Miyake, Y., Miyauchi, T., Nagano, Y., Kasagi, N., Establishment of the DNS Data base of Turbulent Transport Phenomena, Rep. Grants-in-aid for Scientific Research, No. 02302043, 1992
  31. Kuroda, A., Kasagi, N., Hirata, M., Direct Numerical Simulation of the Turbulent Plane Couette-Poiseulle Flows: Effect of Mean Shear on the Near Wall Turbulence Structures, Proceedings, 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, 1993, 8.4.1-8.4.6
  32. Moser, R. D., Kim, J., Mansour, N. N., Direct Numerical Simulation of Turbulent Channel Flow up to Ret = 590, Phys. Fluids, 11 (1999), pp. 943-945
  33. Eggels, J. G. M., Unger, F., Weiss, M. H., Westerweel, J., Adrian, R. J., Friedrich, R., Nieuwstadt, F. T. M., Fully Developed Turbulent Pipe Flow: A Comparison between Direct Numerical Simulation and Experiment, J. Fluid Mech., 268 (1994), pp. 175-209
  34. Gilbert, N., Kleiser, L., Turbulence Model Testing with the Aid of Direct Numerical Simulation Results, Proceedings, 8th Symposium on Turbulent Shear Flows, Munich, Germany, 1991, 26.1.1-26.1.6
  35. Spalart, P. R., Numerical Study of Sink-Flow Boundary Layers, J. Fluid Mech., 172 (1986), pp. 307-328
  36. Spalart, P. R., Direct Simulation of a Turbulent Boundary Layer up to RQ = 1410, J. Fluid Mech., 187 (1988), pp. 61-98
  37. Lammers, P., Jovanoviĥ, J., Durst, F., Numerical Experiments on Wall-Turbulence, Thermal Science, 10 (2006), pp. 33-62 (in this issue)
  38. ***, DANTEC In struction Manual for DISA CTA, 1976
  39. Bradshaw, P., An Introduction to Turbulence and Its Measurements, Pergamon Press, Oxford, UK, 1971
  40. Q.-Y., Ye, The Turbulent Dissipation of Mechanical Energy in Shear Flows (in German), Ph. D. thesis, University Erlangen-Nurenberg, Germany, 1996
  41. Schenck, T. C., Measurements of the Turbulent Dissipation Rate in Plane and Axisymmetric Wake Flows, University Erlangen-Nurenberg, Ph. D. thesis, Germany, 1999
  42. Fischer, M., Turbulent Wall-Bounded Flows at Low Reynolds Numbers (in German), Ph. D., thesis, University Erlangen-Nurenberg, Germany, 1999
  43. Tollmien, W., On the Formation of Turbulence (in German), 1. Mitteilungen, Nachr. Ges. Wiss. Göttingen, Math. Phys. Klasse, pp. 21-44, 1929
  44. Schlichting, H., Amplitude Distribution and Energy of Small Disturbances in a Flat Plate Boundry Layer (in German), Nachr. Ges. Wiss. Göttingen, Math. Phys. Klasse, Fachgruppe I, 1, pp. 47-78, 1935

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