Generalized Scaling and Model for Friction in Wall Turbulence

We present a novel generalized scaling framework and predictive model for wall friction in turbulent flows. The scaling is derived from the dynamical equations, and total mean-flow kinetic energy and the velocity profile shape factor are used as surrogates for dynamical and boundary condition effects. Veracity of the present approach is assessed using data from the literature spanning unprecedented ranges of flow types, Reynolds numbers, accelerations, and history effects. Unlike previous models that solely apply to standard flows, the present framework reconciles nonstandard flows with standard flows and enables accurate estimates of wall friction in numerical simulations and experiments without resolving the viscous sublayer or using the law of the wall.

Figure 1

Smooth-wall turbulent flows covered by this study. All flows are two dimensional in the mean. Internal flows are fully developed. External flows are different types of attached TBLs. Thick black lines indicate solid wall(s) and arrows indicate nominal flow direction. Thin red lines schematically indicate turbulent-nonturbulent interface, a measure of the thickness of TBLs.

Figure 2

Traditional scaling of friction ($f_0≔U_{\tau }/U_{\infty }$ and ${\mathrm{Re}}_0≔\delta U_{\infty }/\nu$) in smooth-wall turbulent flows. Circles—pipes, squares—ZPG TBLs, diamonds—channels, triangles—APG TBLs (self-similar and non-self-similar), inverted triangles—FPG TBLs (self-similar and non-self-similar), and pentacles—compressible TBLs. Further details of symbols and datasets are in Supplemental Material Part IIC—Data Tables [22]. Data cover Reynolds number range $95\le {\mathrm{Re}}_{\tau }\le 528860$ (more than three decades in ${\mathrm{Re}}_{\tau }$) and dimensionless pressure gradient range $-1.5\le \beta \le 35.9$ (from strong FPG to strong APG). Note that, ${\mathrm{Re}}_{\tau }≔\delta U_{\tau }/\nu$ is the friction Reynolds number and $\beta ≔{\delta }^{*}/\tau (\mathit{d}p/\mathit{d}x)$ is the Rotta-Clauser pressure gradient parameter [62, 63]; ${\delta }^{*}$ is displacement thickness and $\mathit{d}p/\mathit{d}x$ is mean streamwise pressure gradient.

Figure 3

Schematic depicting roles of dynamics and BCs towards setting up and modulating eddies of turbulence that contribute to friction experienced by the flow at the wall. Dynamics contributes $M\text{−}\nu$ scaling and effects of BCs are taken into account by Clauser’s shape factor $G$, both together yielding generalized $M\text{−}\nu \text{−}G$ scaling.

Figure 4

(a) Generalized universal $M\text{−}\nu \text{−}G$ scaling of friction in smooth-wall turbulent flows. Solid line shows generalized finite-Re friction model Eq. (4) fitted to all data. (b) Percentage deviation of predicted $U_{\tau }$ from its actual value. Dotted, dashed, and dashed-dotted horizontal lines, respectively, indicate deviation bands of $\pm 2.5\%$, $\pm 5\%$, and $\pm 7\%$. Inset shows histogram of the fraction of data points contained in these bands. For symbols, see caption of Fig. 2.