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DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow

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Abstract

The Lie group, or symmetry approach, developed by Oberlack (see e.g. Oberlack [26] and references therein) is used to derive new scaling laws for various quantities of a zero pressure gradient turbulent boundary layer flow. The approach unifies and extends the work done by Oberlack for the mean velocity of stationary parallel turbulent shear flows. From the two-point correlation (TPC) equations the knowledge of the symmetries allows us to derive a variety of invariant solutions (scaling laws) for turbulent flows, one of which is the new exponential mean velocity profile that is found in the mid-wake region of flat-plate boundary layers. Further, a third scaling group was found in the TPC equations for the one-dimensional turbulent boundary layer. This is in contrast to the Navier–Stokes and Euler equations, which have one and two scaling groups, respectively. The present focus is on the exponential law in the outer region of turbulent boundary layer corresponding new scaling laws for one- and two-point correlation functions. A direct numerical simulation (DNS) of a flat plate turbulent boundary layer with zero pressure gradient was performed at two different Reynolds numbers Reθ=750,2240. The Navier–Stokes equations were numerically solved using a spectral method with up to 140 million grid points. The results of the numerical simulations are compared with the new scaling laws. TPC functions are presented. The numerical simulation shows good agreement with the theoretical results, however only for a limited range of applicability.

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References

  1. Barenblatt, G.I., Chorin, A.J., Prostokishin, V.M.: Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers. J. Fluid Mech. 410, 263–283 (2000)

  2. Bertolotti, F.P., Herbert, T., Spalart, P.R.: Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474 (1992)

  3. Bluman, G.W., Anco, S.C.: Symmetry and integration methods for differential equations. In: Applied Mathematical Sciences154. Springer, Berlin Heidelberg New York (2002)

  4. Canuto, C., Hussaini, M.Y., Quareteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin Heidelberg New York (1988)

  5. Carminati, J., Vu, K.: Symbolic Computation and differential equations: Lie symmetries. J. Sym. Comp. 29, 95–116 (2000)

  6. DeGraaff, D.B., Eaton, J.K.: Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–349 (2000)

  7. Fernholz, H.H., Finley, P.J.: The incompressible zero-pressure-gradient turbulent boundary layer: an assessment of the data. Prog. Aerosp. Sci. 32, 25–311 (1996)

  8. George, W.K., Castillo, L.: Zero-pressure-gradient turbulent boundary layer. Appl. Mech. Rev. 50, 689–729 (1997)

  9. Gottlieb, D., Orszag, S.A.: Numerical analysis of spectral methods: theory and applications. Regional Conference Series in Applied Mathematics, 28, SIAM, Philadelphia (1977)

  10. Ibragimov, N,H.: CRC Handbook Of Lie Group Analysis of Differential Equations, vol. 3. CRC Press, Boca Raton, Florida, 33431 (1996)

  11. Kármán, Th. von: Mechanische Ähnlichkeit und Turbulenz. Nach. Ges. Wiss. Göttingen, 86 (1930)

  12. Khujadze, G., Oberlack, M.: DNS and scaling laws from new symmetry groups of turbulent boundary layer with zero pressure gradient. Proceedings of ETC9, Southampton, UK, 2–5 July, pp. 497–500 (2002)

  13. Kim, J., Moin, P., Moser, R.: Turbulence statistics in fully developed channel flow. J. Fluid Mech. 177, 133–166 (1987)

  14. Kreiss, Oliger: J. Fluid Mech. 177, 133–166 (1972)

  15. Lindgren, B., Österlund, J. M., Johansson, A.V.: Evaluation of scaling laws derived from Lie group symmetry methods in zero pressure-gradient turbulent boundary layers. J. Fluid Mech. 502, 127–152 (2004)

  16. Lundbladh A., Henningson, D.S., Johansson, A.V.: An Efficient Spectral Integration Method for the Solution of the Navier–Stokes Equations. FFA-TN 1992-28, Aeronautical Research Institute of Sweden, Bromma (1992)

  17. Lundbladh A., Berlin, S., Skote, M., Hildings, C., Choi, J., Kim, J., Henningson, D.S.: An Efficient Spectral Method for Simulation of Incompressible Flow over a Flat Plate. Tech. Rep. 1999:11. KTH, Stokholm (1999)

  18. Metzger, M.M., Klewicki, J.C., Bradshaw K.L., Sadr, R.: Scaling the near-wall axial turbulent stress in the zero pressure gradient boundary layer. Phys. Fluids 13, pp. 1819–1821 (2001)

  19. Millikan, C.B.: A critical discussion of turbulent flows in channels and circular tubes. Proc. 5th Int. Congr. Applied Mechanics, Cambridge (1939)

  20. Mochizuki, S., Nieuwstadt F.T.M.: Reynolds-number-dependence of the maximum in the streamwise velocity fluctuations in wall turbulence. Exp. Fluids 21, 218–226 (1996)

  21. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics, vol. 1. MIT Press, Boston (1971)

  22. Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to Reτ=590. Phys. Fluids 11, 943–945 (1999)

  23. Na, Y., Moin, P.: Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379–405 (1998)

  24. Nikuradse, J.: Gesetzmässigkeit der turbulenten Strömung in glatten Rohren. Forsch. Arb. Ing.-Wes. H. 356 (1932)

  25. Nordström, J., Nordin, N., Henningson, D.S.: The fringe region technique and the fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM J. Sci. Comput. 20, 1365–1393 (1999)

  26. Oberlack, M.: A unified approach for symmetries in plane parallel turbulent shear flows. J. Fluid Mech. 427, 299–328 (2001)

  27. Oberlack, M.: Asymptotic Expansion, Symmetry Groups and Invariant Solutions of Laminar and Turbulent Wall-Bounded Flows. Z. Angew. Math. Mech. 80, 11–12, 791–800 (2000)

  28. Oberlack, M., Busse, F.H. (eds.): Theories of turbulence. CISM Courses and Lectures, 442, Springer, Berlin Heidelberg New York (2002)

  29. Oberlack, M., Peters, N.: Closure of the two-point correlation equations as a basis of Reynolds stress models. In: So, R., Speziale, C., Launder, B. (eds.) Near-Wall Turbulent Flows, 85–94. Elsevier, Amsterdam (1993)

  30. Olver, P.J.: Applications of Lie groups to Differential equations. In: Graduate Texts in Mathematics, 107. Springer, Berlin Heidelberg New York (1993)

  31. Orszag, S.A.: Comparison of pseudospectral and spectral approximations. Stud. Appl. Math. 51, 253–259 (1972)

  32. Österlund, J.: Experimental Studies of Zero Pressure-Gradient Turbulent Boundary-Layer Flow. Dissertation, Royal Institute of technology, Stockholm, Sweden (1999)

  33. Österlund, J., Johansson, A., Nagib, H., Hites, M.: A note on the overlap region in turbulent boundary layers. Phys. Fluids 12(1), 1–4 (2000)

  34. Österlund, J., Johansson, A., Nagib, H.: Comment on’a note on the intermediate region in turbulent boundary layers’ [Phys. Fluids 12, 2159]. Phys. Fluids 12, 2360–2363 (2000)

  35. Peyret, R.: Spectral methods for incompressible viscous flow. In: Applied Mathematical Sciences148. Springer, Berlin Heidelberg New York (2002)

  36. Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)

  37. Prandtl, L.: Zur turbulenten Strömung in Rohren und Längs Platten. Ergeb. Aerodyn. Versuchsanst. Göttingen, 4, pp. 18–29 (1932)

  38. Preston, J.H.: The minimum Reynolds number for a turbulent boundary layer and the selection of the transition device. J. Fluid Mech. 3, 373–384 (1958)

  39. Skote, M., Henningson, D.S., Henkes, R.A.W.M.: Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients. Flow Turbul. Combust. 60, 47–85 (1998)

  40. Skote, M.: Studies of Turbulent Boundary Layer Flow Through Direct Numerical Simulation. Dissertation, Royal Institute of Technology, Stockholm, Sweden (2001)

  41. Skote, M., Henningson, D.S.: Direct numerical simulation of separating turbulent boundary layers. J. Fluid Mech. 471, 107–136 (2002)

  42. Skote, M., Haritonidis, J.H., Henningson, D.S.: Varicose instabilities in turbulent boundary layers. Phys. Fluids 14, 2309–2323 (2002)

  43. Spalart, P.R.: Numerical study of sink-flow boundary layers. J. Fluid Mech. 172, 307–328 (1986)

  44. Spalart, P.R.: Direct simulation of a turbulent boundary layer up to Rθ=1410. J. Fluid Mech. 187, 61–98 (1988)

  45. Spalart, P.R., Watmuff, J.H.: Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 249, 337–371 (1993)

  46. Sreenivasan, K.R.: The turbulent boundary layer. Front. Exp. Fluid Mech. 46, 97–120 (1989)

  47. Zagarola, M.V., Perry, A.E., Smits, A.J.: Log laws or power laws: the scaling in the overlap region. Phys. Fluids 9, 2094–2100 (1997)

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Authors and Affiliations

  1. Hydromechanics and Hydraulics Group, Technische Universität Darmstadt, Petersenstr. 13, Darmstadt, 64287, Germany

    G. Khujadze & M. Oberlack

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  1. G. Khujadze
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  2. M. Oberlack
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Correspondence to G. Khujadze.

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Communicated by

M.Y. Hussaini

PACS

02.20.-a, 47.11.+j, 47.27.Nz, 47.27.Eq

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Khujadze, G., Oberlack, M. DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow. Theor. Comput. Fluid Dyn. 18, 391–411 (2004). https://doi.org/10.1007/s00162-004-0149-x

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