Home Technology Testing of logarithmic-law for the slip with friction boundary condition
Article
Licensed
Unlicensed Requires Authentication

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

  • Özgül İlhan EMAIL logo and Niyazi Şahin
Published/Copyright: January 13, 2022
Become an author with De Gruyter Brill
Author Information
International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 8

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.

Keywords: boundary layers; Leray-α model; logarithmic law; near wall models (NWM); turbulence
Corresponding author: Özgül İlhan, Department of Mathematics, Faculty of Science, Muğla Sıtkı Koçman University, Muğla, Türkiye, 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

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Testing of logarithmic-law for the slip with friction boundary condition
  4. A new clique polynomial approach for fractional partial differential equations
  5. The modified Rusanov scheme for solving the phonon-Bose model
  6. Delta-shock for a class of strictly hyperbolic systems of conservation laws
  7. The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations
  8. Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method
  9. The simulation of two-dimensional plane problems using ordinary state-based peridynamics
  10. Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format
  11. Simulation of the crystallization processes by population balance model using a linear separation method
  12. PS and GW optimization of variable sliding gains mode control to stabilize a wind energy conversion system under the real wind in Adrar, Algeria
  13. Characteristics of internal flow of nozzle integrated with aircraft under transonic flow
  14. Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field
  15. The influence pulse-like near-field earthquakes on repairability index of reversible in mid-and short-rise buildings
  16. Intelligent controller for maximum power extraction of wind generation systems using ANN
  17. A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems
  18. Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions
  19. A study on solvability of the fourth-order nonlinear boundary value problems
  20. Adaptive control for position and force tracking of uncertain teleoperation with actuators saturation and asymmetric varying time delays
  21. Framing the hydrothermal significance of water-based hybrid nanofluid flow over a revolving disk
  22. Catalytic surface reaction on a vertical wavy surface placed in a non-Darcy porous medium
  23. Carleman framework filtering of nonlinear noisy phase-locked loop system
  24. Corrigendum
  25. Corrigendum to: numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
https://doi.org/10.1515/ijnsns-2021-0184
Keywords for this article
boundary layers; Leray-α model; logarithmic law; near wall models (NWM); turbulence

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Testing of logarithmic-law for the slip with friction boundary condition
  4. A new clique polynomial approach for fractional partial differential equations
  5. The modified Rusanov scheme for solving the phonon-Bose model
  6. Delta-shock for a class of strictly hyperbolic systems of conservation laws
  7. The Cădariu–Radu method for existence, uniqueness and Gauss Hypergeometric stability of a class of Ξ-Hilfer fractional differential equations
  8. Novel periodic and optical soliton solutions for Davey–Stewartson system by generalized Jacobi elliptic expansion method
  9. The simulation of two-dimensional plane problems using ordinary state-based peridynamics
  10. Reduced basis method for the nonlinear Poisson–Boltzmann equation regularized by the range-separated canonical tensor format
  11. Simulation of the crystallization processes by population balance model using a linear separation method
  12. PS and GW optimization of variable sliding gains mode control to stabilize a wind energy conversion system under the real wind in Adrar, Algeria
  13. Characteristics of internal flow of nozzle integrated with aircraft under transonic flow
  14. Magnetogasdynamic shock wave propagation using the method of group invariance in rotating medium with the flux of monochromatic radiation and azimuthal magnetic field
  15. The influence pulse-like near-field earthquakes on repairability index of reversible in mid-and short-rise buildings
  16. Intelligent controller for maximum power extraction of wind generation systems using ANN
  17. A new self-adaptive inertial CQ-algorithm for solving convex feasibility and monotone inclusion problems
  18. Existence and Hyers–Ulam stability of solutions for nonlinear three fractional sequential differential equations with nonlocal boundary conditions
  19. A study on solvability of the fourth-order nonlinear boundary value problems
  20. Adaptive control for position and force tracking of uncertain teleoperation with actuators saturation and asymmetric varying time delays
  21. Framing the hydrothermal significance of water-based hybrid nanofluid flow over a revolving disk
  22. Catalytic surface reaction on a vertical wavy surface placed in a non-Darcy porous medium
  23. Carleman framework filtering of nonlinear noisy phase-locked loop system
  24. Corrigendum
  25. Corrigendum to: numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 5.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2021-0184/html
Scroll to top button