Skip to content
Licensed Unlicensed Requires Authentication Published online by De Gruyter January 13, 2022

Testing of logarithmic-law for the slip with friction boundary condition

Özgül İlhan and Niyazi Şahin

Abstract

Large eddy simulation (LES) seeks to predict the dynamics of the organized structures in the flow, that is, local spatial averages u ̄ of the velocity u of the fluid. Although LES has been extensively used to model turbulent flows, very often, the model has difficulty predicting turbulence generated by interactions of a flow with a boundary. A critical problem in LES is to find appropriate boundary conditions for the flow averages, which depend on the behavior of the unknown flow near the wall. In the light of the works of Navier and Maxwell, we use boundary conditions on the wall. We compute the appropriate friction coefficient β for channel flows and investigate its asymptotic behavior as the averaging radius δ → 0 and as the Reynolds number Re → ∞. No-slip conditions are recovered in the first limit, and free-slip conditions are recovered in the second limit. This study is not intended to develop new theories of the turbulent boundary layer; we use available boundary layer theories to improve numerical boundary conditions for flow averages.


Corresponding author: Özgül İlhan, Department of Mathematics, Faculty of Science, Muğla Sıtkı Koçman University, Muğla, Turkey, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

Some integral formulas have been used in the derivation and the analysis of the friction coefficients. It holds,

(14) 0 b e a x 2 d x = 1 2 π a e r f ( b a ) , a > 0

where erf(.) is the error function. Recall that

(15) 0 e a x 2 d x = 1 2 π a , e a x 2 d x = π a

and

(16) 0 e a ( x b ) 2 d x = 1 2 π a ( 1 + e r f ( b a ) )

The derivative of the error function is easily computed from Eq. (14) as follows,

(17) d d x e r f ( a x + b ) = 2 a π e a ( x b ) 2

References

[1] K. S. Abbot and D. E. Kline, “Experimental investigations of subsonic turbulent flow over single and double backward-facing steps,” Trans. ASME, Ser. D, vol. 84, pp. 317–325, 1962. https://doi.org/10.1115/1.3657313.Search in Google Scholar

[2] W. F. Bradshaw and P. Wong, “The reattachment and relaxation of a turbulent shear layer,” J. Fluid Mech., vol. 52, pp. 113–135, 1972. https://doi.org/10.1017/s002211207200299x.Search in Google Scholar

[3] J. Kim, S. J. Kline, and J. P. Johnston, “Investigation of a reattaching turbulent shear layer: flow over a backward-facing step,” J. Fluid Eng., vol. 102, pp. 302–308, 1980. https://doi.org/10.1115/1.3240686.Search in Google Scholar

[4] F. Durst and C. Tropea, “Turbulent, backward-facing step flows in two-dimensional ducts and channels,” in Proceedings of the Fifth International Symposium on Turbulent Shear Flows, Davis, Cornell University, 1981, pp. 181–185.Search in Google Scholar

[5] B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Schönung, “Experimental and theoretical investigation of backward-facing step flow,” J. Fluid Mech., vol. 127, pp. 473–496, 1983. https://doi.org/10.1017/s0022112083002839.Search in Google Scholar

[6] J. J. Adams and E. W. Johnston, “Effects of the separating shear layer on the reat- tachment flow structure. part 1: pressure and turbulence quantities. part 2: reattachment length and wall shear stress,” Exp. Fluid, vol. 6, pp. 400493–408499, 1988. https://doi.org/10.1007/bf00196485.Search in Google Scholar

[7] S. Jovic and D. Driver, “Backward-facing step measurements at low Reynolds number,” NASA Tech. Memorand., p. 108807, 1994.Search in Google Scholar

[8] P. G. Spazzini, G. Iuso, M. Oronato, N. Zurlo, and G. M. Di Cicca, “Unsteady behaviour of back-facing step flow,” Exp. Fluid, vol. 30, pp. 551–561, 2001. https://doi.org/10.1007/s003480000234.Search in Google Scholar

[9] S. Yoshioka, S. Obi, and S. Masuda, “Turbulence statistics of periodically perturbed separated flow over backward-facing step,” Int. J. Heat Fluid Flow, vol. 22, pp. 393–401, 2001. https://doi.org/10.1016/s0142-727x(01)00079-0.Search in Google Scholar

[10] S. D. Hall, M. Behnia, C. A. J. Fletcher, and G. L. Morrison, “Investigation of the secondary corner vortex in a benchmark turbulent backward-facing step using cross- correlation particle imaging velocimetry,” Exp. Fluid, vol. 35, pp. 139–151, 2003. https://doi.org/10.1007/s00348-003-0626-9.Search in Google Scholar

[11] A. M. R. Friedrich, “Analysing turbulent backward-facing step flow with the lowpass-filtered Navier-Stokes equations,” J. Wind Eng. Ind. Aerod., vol. 35, pp. 101–228, 1990. https://doi.org/10.1016/0167-6105(90)90212-u.Search in Google Scholar

[12] A. Silveira Neto, D. Grand, O. Metais, and M. Lesieur, “A numerical investigation of the coherent vortices in turbulence behind a backward-facing step,” J. Fluid Mech., vol. 256, pp. 1–25, 1993. https://doi.org/10.1017/s0022112093002691.Search in Google Scholar

[13] L. C. Berselli, W. J. Layton, and T. Iliescu, Mathematics of Large Eddy Simulation of Turbulent Flows, Berlin, Springer, 2006.Search in Google Scholar

[14] P. Sagaut, Large eddy Simulation for Incompressible Flows, Berlin, Springer-Verlag, 2001.10.1007/978-3-662-04416-2Search in Google Scholar

[15] U. Piomelli and E. Balaras, “Wall-layer models for large-eddy simulations,” Annu. Rev. Fluid Mech., vol. 34, pp. 349–374, 2002. https://doi.org/10.1146/annurev.fluid.34.082901.144919.Search in Google Scholar

[16] H. Werner and H. Wengle, “Large eddy simulation of turbulent flow around a cube in a plane channel,” in Selected Papers From the 8th Symp. on Turb. Shear Flows, F. Durst, B. Launder, U. Schumann, and J. Whitelaw, Eds., New York, Springer, 1993, pp. 155–168.10.1007/978-3-642-77674-8_12Search in Google Scholar

[17] J. W. Deardoff, “A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers,” J. Fluid Mech., vol. 41, pp. 453–480, 1970.10.1017/S0022112070000691Search in Google Scholar

[18] U. Schumann, “Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli,” J. Comput. Phys., vol. 18, pp. 376–404, 1975. https://doi.org/10.1016/0021-9991(75)90093-5.Search in Google Scholar

[19] G. Grötzbach, “Direct numerical and large eddy simulation of turbulent channel flows,” in Encyslopedia of Fluid Mechanics, N. Cheremisinoff, Ed., West Orange, NJ, Gulf, 1987.Search in Google Scholar

[20] U. Piomelli, P. Moin, and J. Kim, “New approximate boundary condition for large eddy simulation of wall-bounded flows,” Phys. Fluids A, vol. 1, pp. 1061–1068, 1989. https://doi.org/10.1063/1.857397.Search in Google Scholar

[21] W. H. Cabot, “Large-eddy simulations with wall models,” in Annual Research Briefs, Stanford, Center for Turbulence Research, 1995, pp. 41–58.Search in Google Scholar

[22] W. H. Cabot, “Near-wall models in large eddy simulations of flow behind a backward-facing step,” in Annual Research Briefs, Stanford, Center for Turbulence Research, 1996, pp. 199–210.Search in Google Scholar

[23] T. Bagwell, “Stochastic estimation of Near Wall Closure in Turbulence Models,” PhD thesis, University of Illinois at Urbana-Champaign, 1994.Search in Google Scholar

[24] T. G. Bagwell, R. D. Moser, and J. Kim, “Improved approximation of wall shear stress boundary conditions for large eddy simulation,” Near-Wall Turbulent Flows, New York, Elsevier Science, 1993.Search in Google Scholar

[25] I. Marusic, G. J. Kunkel, and F. Porte-Agel, “Experimental study of wall boundary conditions for large-eddy simulation,” J. Fluid Mech., vol. 446, pp. 309–320, 2001. https://doi.org/10.1017/s0022112001005924.Search in Google Scholar

[26] S. Bose and P. Moin, “A dynamic slip boundary condition for wall-modeled large-eddy simulation,” Phys. Fluids, vol. 26, no. 1, p. 015104, 2014. https://doi.org/10.1063/1.4849535.Search in Google Scholar

[27] E. Boström, “Boundary Conditions for Spectral Simulations of Atmospheric Boundary Layers,” PhD thesis, Department of Mechanics, Royal Institute of Technology, Stockholm, Sweden, 2017.Search in Google Scholar

[28] A. T. Anum Shafiq and Z. Hammouch, “Impact of radiation in a stagnation point flow of Walters? B fluid towards a riga plate,” Therm. Sci. Eng. Prog., vol. 6, pp. 27–33, 2018. https://doi.org/10.1016/j.tsep.2017.11.005.Search in Google Scholar

[29] Z. Hammouch, M. Amkadni, and A. Azzouzi, “On the exact solutions of laminar mhd flow over a stretching flat plate,” Commun. Nonlinear Sci. Numer. Simulat., vol. 13, pp. 359–368, 2008.10.1016/j.cnsns.2006.04.002Search in Google Scholar

[30] M. C. Lombardo and M. Sammartino, “Nonlocal Boundary Conditions for the Navier–Stokes Equations,” in Waves and Stability in Continuous Media, 2006, pp. 340–345.10.1142/9789812773616_0047Search in Google Scholar

[31] Y. Cao, Global Classical Solutions to the Compressible Navier-Stokes Equations with Navier-type Slip Boundary Condition in 2d Bounded Domains, 2021, arXiv:2102.10235v2.Search in Google Scholar

[32] J. Leray, “Sur le mouvemennt d’un fluide visquex emplissant l’espace,” Acta Math., vol. 63, pp. 193–248, 1934. https://doi.org/10.1007/bf02547354.Search in Google Scholar

[33] E. O. A. Cheskidov, D. Holm, and E. Titi, On a Leray-Alpha Model of Turbulence, Royal Society London, Mathematical, Physical and Engineering Sciences, 2005, pp. 629–649.10.1098/rspa.2004.1373Search in Google Scholar

[34] G. Galdi and W. Layton, “Approximation of the larger eddies in fluid motion ii: a model for space filtered flow,” Math. Model Methods Appl. Sci., vol. 10, pp. 343–350, 2000. https://doi.org/10.1142/s0218202500000203.Search in Google Scholar

[35] N. Sahin, “Derivation, Analysis and Testing of New Near Wall Models for Large Eddy Simulation,” PhD thesis, Department of Mathematics, Pittsburgh University, 2003.Search in Google Scholar

[36] W. Layton, V. John, and N. Sahin, “Derivation and analysis of near wall models for channel and recirculating flows,” Comput. Math. Appl., vol. 28, pp. 1135–1151, 2004.10.1016/j.camwa.2004.10.011Search in Google Scholar

[37] S. Pope, Turbulent Flows, New York, Cambridge University Press, 2000.10.1017/CBO9780511840531Search in Google Scholar

[38] C. Navier, “Memoire sur les lois du movement des fluiales,” Mem. Acad. R. Sci., vol. 6, pp. 389–440, 1823.Search in Google Scholar

[39] J. Maxwell, “On stresses in rarefied gases arising from inequalities of temperature,” Philos. Trans. R. Soc., vol. 170, pp. 249–256, 1879.10.1098/rstl.1879.0067Search in Google Scholar

[40] K. R. Mahmud, “Sensitivity analysis of near-wall turbulence modeling for large eddy simulation of incompressible flows,” Master’s thesis, Master Programme in Computer simulation for Science and Engineering, Royal Institute of Technology, School of Engineering Sciences, Stockholm, Sweden, 2014.Search in Google Scholar

[41] T. Knopp, “Finite-element simulation of buoyancy-driven turbulent flows,” PhD thesis, Univ. of Göttingen, Germany, 2003.Search in Google Scholar

[42] V. John, “Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations-numerical tests and aspects of the implementation,” J. Comput. Appl. Math., vol. 147, pp. 287–300, 2002. https://doi.org/10.1016/s0377-0427(02)00437-5.Search in Google Scholar

[43] H. Schlichting, Boundary-Layer Theory, New York, McGraw-Hill, 1979.Search in Google Scholar

[44] G. Barenblatt and A. Chorin, “New perspectives in turbulence: scaling laws, asymptotic, and intermittency,” SIAM Rev., vol. 40, pp. 265–291, 1998. https://doi.org/10.1137/s0036144597320047.Search in Google Scholar

[45] R. G. Hill, “Benchmark testing the alpha-models of turbulence,” Master’s thesis, The Graduate School of Clemson University, 2010.Search in Google Scholar

[46] P. F. Hecht and O. K. Ohtsuka, Freefem++ Manual, 2021.Search in Google Scholar

[47] O. Ilhan, “Turbulansli Sinir Tabakasi icin Duvar Kenari Modeli ve Modelin Logaritmik Kanunla Testleri,” PhD thesis, Mugla Sitki Kocman University, Institute of Science, 2018.Search in Google Scholar

Received: 2021-04-25
Accepted: 2021-12-24
Published Online: 2022-01-13

© 2021 Walter de Gruyter GmbH, Berlin/Boston