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

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