On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

TL;DR

This work introduces the Angular Momentum Integral (AMI) equation as a first-moment, angular-momentum balance for boundary layers, yielding a laminar Blasius term that depends only on $Re_\ell$ and a set of torque-like contributions from turbulence, growth, mean-wall-normal flux, and pressure gradients. By linking AMI to the classical FIK framework for internal flows, the authors provide a unified interpretation that remains valid for external, spatially developing boundary layers, and they demonstrate its utility with Falkner–Skan laminar flows and four DNS datasets spanning natural transition, bypass-transition, and fully turbulent states. A key finding is that during transition the peak skin friction is driven by the Reynolds-stress torque, modulated by negative wall-normal flux regions, while in fully turbulent flows the Reynolds-stress torque dominates and other torques largely cancel when analyzed with $\ell\sim\theta$. The AMI framework thus offers a physically intuitive, extensible tool for diagnosing, predicting, and potentially controlling boundary-layer skin friction in engineering applications.

Abstract

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as freestream pressure gradients and streamwise boundary layer growth also strongly influence the skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows [Fukagata et al. 2002, Phys. Fluids, \textbf{14}, L73-L76]. Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner-Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer DNS datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully-turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Figures (15)

• Figure 1: For a fixed $U_\infty$, a torque that (a) redistributes momentum toward the wall will act to increase the skin friction, or (b) redistributes momentum away from the wall will act to decrease the skin friction.
• Figure 2: Comparison of Falkner-Skan and Blasius boundary layers at: (a,b) fixed $Re_x$; (c,d) fixed $Re_{\delta^*}$; and (e,f) fixed $Re_\theta$. Panels (a,c,e) compare the velocity profiles while panels (b,d,f) show each term in the AMI equation. The vertical dashed gray lines in (a,c,e) indicate $\ell$ chosen for each respective comparison.
• Figure 3: The wall-normal integrand of the flux of streamwise momentum deficit contribution to the skin friction coefficient, $\theta_v$, as a function of the freestream pressure gradient, $m$.
• Figure 4: Wall-normal velocity at $y=0.02 \delta_{inlet}$ for the H-type transition.
• Figure 5: Skin friction coefficient, $C_f$, as a function of both $Re_\theta$ and $Re_x$ for the three transitional boundary layers considered. ($\Box$) H-type simulation; ($\circ$) JHTDB BL; ($\bigtriangleup$) Wu BL; (- -) Laminar solution $C_f/2 = 0.332/\sqrt{Re_x} = 0.221/Re_\theta$; (--) Turbulent correlation $C_f/2 \approx 0.029/Re_x^{1/5} \approx 0.013/Re_\theta^{1/4}$.