Higher-order homogenised riblet boundary conditions

Abstract

The description of riblets and other drag-reducing devices has long used the concept of longitudinal and transverse protrusion heights, both as a means to predict the drag reduction itself and as equivalent boundary conditions to simplify numerical simulations by transferring the effect of riblets onto a flat virtual boundary. The limitation of this idea is that it stems from a first-order approximation in the riblet-size parameter $s^+$, and as a consequence it cannot predict other than a linear dependence of drag reduction upon $s^+$; in other words, the initial slope of the drag-reduction curve. Here the concept is extended to a full asymptotic expansion using matched asymptotics, which consistently provides higher-order protrusion coefficients and higher-order equivalent boundary conditions on a virtual flat surface. While the majority of our results, though nonlinear in $s^+$, remain linear in velocity, and therefore we shall not directly address the shape of the drag-reduction curve, this procedure will also allow us to explore the way nonlinearities of the Navier-Stokes equations first enter the $s^+$-expansion, with somewhat surprising negative results.

Figures (11)

• Figure 1: Setup: $s^*$ is the riblet period/spacing, $L^* \gg s^*$ is the upper boundary condition plane (measured from the tips at $z^*=0$) where a Neumann boundary condition on length scales O$(L^*)$ is imposed.
• Figure 2: Setup: $s^*$ is the riblet period/spacing, $L^* \gg s^*$ is the upper boundary condition plane (measured from the tips at $z^*=0$) where flow on length scales O$(L^*)$ is imposed.
• Figure 3: (a) First-order streamwise velocity $\overline{U_{11}}=\overline{\varPhi_{11}}$ of \ref{['Phi1']}, responsible for the $h_1$ longitudinal protrusion height, (b) spanwise velocity $\overline{V_{11}}=\overline{\varPsi_{21}}_{,Z}$ of \ref{['Psi21']}, responsible for the $a_1$ transverse protrusion height, (c) wall-normal velocity $\overline{W_{11}}=-\overline{\varPsi_{21}}_{,Y}$, and (d) connected zeroth-order pressure $\overline{P_{01}}$ of \ref{['CauchyRiemann']}, for the six considered geometries. The corresponding riblet profile is drawn on top of each figure.
• Figure 4: Second-order contributions to the streamwise velocity. (a) $\overline{\varPhi_{21}}$ of \ref{['Phi20']}, producing coefficient $h_2$, (b) $\overline{U_{22}}$ of \ref{['Upx']}, producing coefficient $h_2^{(p_x)}$, (c) $\overline{U_{23}}$ of \ref{['U22']}, producing coefficient $h_2^{(v_{xz})}$.
• Figure 5: Second-order contributions to the spanwise velocity. (a) $\overline{\varPsi_{31}}_{,Z}$ of \ref{['Psi31']}, producing $a_2$ and $c_3$, (b) $\overline{\varPsi_{32}}_{,Z}$ of \ref{['Psi32']}, producing $b_2$ and $d_3$, (c) $\overline{V_{23}}$ of \ref{['V23']}, producing $f_2$.