NUMERICAL EXPERIMENTS ON WALL TURBULENCE AT LOW REYNOLDS NUMBER

Abstract

A direct numerical simulation of a turbulent channel ?ow, with regularly spaced two-dimensional roughness elements mounted at the wall and perpendicular to the ?ow direction, was performed at a very low Reynolds number of Re approx. 940 based on the centerline velocity and the full channel height. Using the lattice Boltzmann numerical algorithm, all essential scales were resolved with about 19 x 106 grid points (1155 x 129 x 128 in the x1 , x2 and x3 directions). The computed results con?rm the existence of turbulence at such a low Reynolds number. Turbulence persisted over the entire computation time, which was suffciently long to prove its self-maintenance. By examination of statistical features of the ?ow across the anisotropy-invariant map, it was found that these coincide with conclusions emerging from the analysis of transition and breakdown to turbulence in a laminar boundary layer exposed to small, statistically stationary, neutrally stable axisymmetric disturbances with the streamwise intensity component (u1) lower than the intensities in the normal (u2) and spanwise (u3) directions, u < u 2 = u3 . To further support the concept and the results of theoretical considerations of the laminar to turbulent transition process in wall-bounded ?ows using statistical techniques and to demonstrate its great potential for engineering, an additional simulation was performed of a plane channel ?ow with regularly spaced riblet elements mounted at the wall and aligned parallel with the ?ow direction. The supplementary simulation was done at a Reynolds number of Re ? 6584 using about 250 x 106 grid points (4096 x 257 x 240). Analysis of the simulation results carried out across the ?ow region located in the midplane between the riblet elements con?rms the central result which lies in the root of the statistical dynamics of the velocity ?uctuations in wall-bounded ?ows: when the velocity ?uctuations close to the wall tend towards the one-component state, so that the streamwise intensity component is much larger than the intensities in the normal and span-wise directions, u1 >> u 2 = u3 , the turbulent dissipation rate vanishes at the wall, leading to a signi?cant reduction of the wall shear stress. For the simulated ?ow case the local value of wall shear stress reduction SR approx. 92% was found to exceed the wall shear stress reduction SR approx. 85% which corresponds to a fully developed laminar channel ?ow with smooth walls at the same Reynolds number.

Keywords

Dates

  • Submission Date2006-01-28
  • Revision Date2006-05-23
  • Acceptance Date2006-06-15

References

  1. J. W. Deardor?, A numerical study of three-dimensional turbulent channel ?ow at large Reynolds numbers. J. Fluid Mech., 41:453?480, 1970.
  2. S. A. Orszag and G. S. Patterson, Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett., 28:76?79, 1972.
  3. [3] U. Schumann, Ein Verfahren zur direkten numerischen Simulation turbulenter Stromungen in Platten-und Ringspaltkan?uber seine Anwendung zur Untersuchung von Turbualen und lenzmodellen. Dissertation, Universitat Karlsruhe, 1973.
  4. R. S. Rogallo, Numerical experiments in homogeneous turbulence. NASA Tech. Memo 81315, 1981.
  5. J. Kim, P. Moin, and R. Moser, Turbulence statistics in fully developed channel ?ow at low Reynolds number. J. Fluid Mech., 177:133?166, 1987.
  6. P. R. Spalart, Numerical study of sink-?ow boundary layers. J. Fluid Mech., 172:307?328, 1986.
  7. P. R. Spalart, Direct simulation of a turbulent boundary layer up to Re? = 1410. J. Fluid Mech., 187:61?98, 1988.
  8. N. Gilbert and L. Kleiser, Turbulence model testing with the aid of direct numerical simulation results. Proc. Eighth Symp. on Turbulent Shear Flows, Munich, pp. 26.1.1? 26.1.6, 1991.
  9. S. L. Lyons, T. J. Hanratty, and J. B. McLaughlin, Large-scale computer simulation of fully developed turbulent channel ?ow with heat transfer. Int. J. Numer. Met. Fluids, 13:999?1028, 1991.
  10. R. A. Antonia, M. Teitel, J. Kim, and L. W. B. Browne, Low-Reynolds-number e?ects in a fully developed turbulent channel ?ow. J. Fluid Mech., 236:579?605, 1992.
  11. A. Kuroda, N. Kasagi, and M. Hirata, Direct numerical simulation of the turbulent plane Couette-Poiseulle ?ows: E?ect of the mean shear on the near wall turbulent structures. In Proceedings of the Ninth Symposium on Turbulent Shear Flows, Kyoto, Japan, 16-18 August 1993, pp. 241?257.
  12. J. Eggels, F. Unger, M. H. Weiss, J. Westerweel, R. J. Adrian, R. Friedrich, and F. T. M. Nieustadt, Fully developed turbulent pipe ?ow: a comparison between direct numerical simulation and experiment. J. Fluid Mech., 268:175?209, 1994.
  13. H. Choi, P. Moin, and J. Kim, Direct numerical simulation of turbulent ?ow over riblets.
  14. J. Fluid Mech., 255:503?539, 1993.
  15. H. Le and P. Moin, Direct numerical simulation of turbulent ?ow over a backward-facing step. Thermoscience Division Dep. Mech. Eng. Rep. TF-58, 1994, Stanford University.
  16. M. M. Rogers and R. D. Moser, Direct simulation of a self-similar turbulent mixing layer. Phys. Fluids, 6:903?923, 1994.
  17. R. Moser, J. Kim, and N. Mansour, Direct numerical simulation of turbulent channel ?ow up to Re? = 560. Phys. Fluids, 11:943?945, 1999.
  18. O. Reynolds, On the dynamical theory of incompressible viscous ?uids and the determination of the criterion. Phil. Trans. R. Soc. Lond. A, 186: 123?164, 1895.
  19. G.I. Taylor, Statistical theory of turbulence. Part V. E?ects of turbulence on boundary layer. Theoretical discussion of relationship between scale of turbulence and critical resistance of spheres. Proc. R. Soc. Lond. A, 156: 307?317, 1936.
  20. J. Jovanovi?c, The Statistical Dynamics of Turbulence. Springer-Verlag, Berlin, 2004.
  21. J. Jovanovi?c and M. Pashtrapanska, On the criterion for the determination transition onset and breakdown to turbulence in wall-bounded ?ows. J. Fluids Eng., 126:626?633, 2004.
  22. J.L. Lumley and G. Newman, The return to isotropy of homogeneous turbulence. J. Fluid Mech., 82:161?178, 1977.
  23. J.L. Lumley, Computational modeling of turbulent ?ows. Adv. Appl. Mech., 18:123?176, 1978.
  24. J. Jovanovi?c, R. Hillerbrand and M. Pashtrapanska, Mit statistischer DNS-Datenanalyse der Entstehung von Turbulenz auf der Spur. KONWIHR Quartl 31, 6-8, 2001.
  25. J. Jovanovi?c, B. Frohnapfel, E. ?
  26. Skalji?c and M. Jovanovi?c, Persistence of the laminar regime in a ?at plate boundary layer at very high Reynolds number. J. Fluids Eng. submitted, 2005.
  27. N. Jovi?ci?c and M. Breuer Separated ?ow past an airfoil at high angle of attack: LES predictions and analysis. In 5th Workshop on DNS & LDES, M?
  28. unchen, August 27-29, 2003, ERCOFTAC Series, Vol. 9, pp. 611-618, DNS and LES V, Eds. R. Friedrich et al.
  29. Kluwer Acad. Publ., Dordrecht.
  30. J. Jovanovi?c and R. Hillerbrand On peculiar property of the velocity ?uctuations in wall-bounded ?ows. Thermal Sci., 9:3?12, 2005.
  31. J. Jovanovi?c, M. Pashtrapanska, B. Frohnapfel, F. Durst, J. Koskinen and K. Koskinen On the mechanism responsible for turbulent drag reduction by dilute addition of high polymers: theory, experiments, simulations and predictions. J. Fluids Eng. (in press), 2005.
  32. R. Benzi, S. Succi and M. Vergassola, The Lattice Boltzmann Equation: Theory and Applications, Physics Reports (Review Section of Physics Letters), 222:145?197, 1992
  33. S. Chen and G. D. Doolen, Lattice Boltzmann method for ?uid ?ows. Annu. Rev. Fluid Mech., 30:329?364, 1998.
  34. D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer-Verlag, Berlin, 2000.
  35. S. Succi, The Lattice Boltzmann Equation ? For Fluid Dynamics and Beyond. Clarendon Press, Oxford, 2001.
  36. P. Bhatnagar, E. P. Gross, and M. K. Krook, A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94: 511?525, 1954.
  37. S. Hou, Q. Zou, S. Chen, G. Doolen and A. C. Cogley, Simulation of cavity ?ow by the lattice Boltzmann method. J. Comput. Phys., 118: 329?347, 1995.
  38. J. Buick and C. Greated, Gravity in lattice Boltzmann model. Phys. Rev. E, 61: 5307?5320, 2000.
  39. X. He and L.-S. Luo, Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E, 56: 6811?6817, 1997.
  40. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge, 1999.
  41. Y. H. Qian, D. d?Humi`eres, and P. Lallemand, Lattice BGK models for Navier-Stokes equation. Europhys. Lett., 17: 479?484, 1992.
  42. X. He, Q. Zou, L.-S. Luo, and M. Dembo, Analytic solutions of simple ?ows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J. Stat. Phys., 87: 115?136, 1997.
  43. T. Zeiser, M. Steven, H. Freund, P. Lammers, G. Brenner, F. Durst, and J. Bernsdorf, Analysis of the ?ow ?eld and pressure drop in ?xed bed reactors with the help of lattice Boltzmann simulations. Phil. Trans. R. Soc. Lond. A, 360: 507?520, 2002.
  44. M. Fischer, J. Jovanovi?c, and F. Durst, Reynolds number e?ects in the near?wall region of turbulent channel ?ows. Phys. Fluids, 13:1755?1767, 2001.
  45. J.O. Hinze Turbulence, 2nd edn. McGraw-Hill, New York, 1975.
  46. H. Schlichting Boundary-Layer Theory, 6th edn. McGraw-Hill, New York, 1968.
  47. J.Jovanovi?c, I.Oti?c and P.Bradshaw On the anisotropy of axisymmetric strained turbulence in the dissipation range. J. Fluids Eng., 125: 401?413, 2003.
Volume 10, Issue 2, Pages33 -62