Effect of wall slip on laminar flow past a circular cylinder

A numerical study of two-dimensional flow past a confined circular cylinder with slip wall is performed. A dimensionless number, Knudsen number (Kn), is used to describe the slip length of the cylinder wall. The Reynolds number (Re) and Knudsen number (Kn) ranges considered are Re = [1, 180] and Kn = [0, ∞), respectively. Time-averaged flow separation angle (θs¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\theta_{{\text{s}}} }}$$\end{document}), dimensionless recirculation length (Ls¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{L_{{\text{s}}} }}$$\end{document}), and tangential velocity (uτ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{u_{\tau } }}$$\end{document}) distributed on the cylinder’s wall, drag coefficient (Cd¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{{C_{{\text{d}}} }}$$\end{document}) and drag reduction (DR) are investigated. The time-averaged tangential velocity distributed on the cylinder’s wall fits well with the formula uτ¯=U∞·α1+βe-γπ-θ+δ·sinθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{u_{\tau } }} = U_{\infty } \cdot \left[ {\frac{\alpha }{{1 + \beta e^{{ - \gamma \left( {\pi - \theta } \right)}} }} + \delta } \right] \cdot \sin \theta$$\end{document}, where the coefficients (α, β, γ, δ) are related to Re and Kn, and U∞ is the incoming velocity. Several scaling laws are found, log(uτmax¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{u_{\tau \max } }}$$\end{document}) ~ log(Re) and uτmax¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{u_{\tau \max } }}$$\end{document} ~ Kn for low Kn (uτmax¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{u_{\tau \max } }}$$\end{document} is the maximum tangential velocity on the cylinder’s wall), log(DR) ~ log(Re) (Re ≤ 45 and Kn ≤ 0.1) and log(DR) ~ log(Kn) (Kn ≤ 0.05). At low Re, DRv (the friction drag reduction) is the main source of DR. However, DRp (the differential pressure drag reduction) contributes most to DR at high Re (Re > ∼60\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim60$$\end{document}) and Kn over a critical number. DRv is found almost independent of Re.