Derivation of Zagarola-Smits scaling in zero-pressure-gradient turbulent boundary layers

This Rapid Communication derives the Zagarola-Smits scaling directly from the governing equations for zero-pressure-gradient turbulent boundary layers (ZPG TBLs). It has long been observed that the scaling of the mean streamwise velocity in turbulent boundary layer flows differs in the near surface region and in the outer layer. In the inner region of small-velocity-defect boundary layers, it is generally accepted that the proper velocity scale is the friction velocity, $u_{\tau }$, and the proper length scale is the viscous length scale, $\nu /u_{\tau }$. In the outer region, the most generally used length scale is the boundary layer thickness, $\delta$. However, there is no consensus on velocity scales in the outer layer. Zagarola and Smits [ASME Paper No. FEDSM98-4950 (1998)] proposed a velocity scale, $U_{{}_{\mathrm{ZS}}}=({\delta }_1/\delta )U_{\infty }$, where ${\delta }_1$ is the displacement thickness and $U_{\infty }$ is the freestream velocity. However, there are some concerns about Zagarola-Smits scaling due to the lack of a theoretical base. In this paper, the Zagarola-Smits scaling is derived directly from a combination of integral, similarity, and order-of-magnitude analysis of the mean continuity equation. The analysis also reveals that $V_{\infty }$, the mean wall-normal velocity at the edge of the boundary layer, is a proper scale for the mean wall-normal velocity $V$. Extending the analysis to the streamwise mean momentum equation, we find that the Reynolds shear stress in ZPG TBLs scales as $U_{\infty }{V}_{\infty }$ in the outer region. This paper also provides a detailed analysis of the mass and mean momentum balance in the outer region of ZPG TBLs.

Figure 1

(a) Boundary layer thickness, $\delta$, and displacement thickness, ${\delta }_1$, as a function of distance in the streamwise direction, $x$. (b) $\alpha -1=\frac{\frac{1}{{\delta }_1}\frac{d{\delta }_1}{dx}}{\frac{1}{\delta }\frac{d\delta }{dx}}-1$ as a function of $x$. ZPG TBLs data are from experiments of Osterlund [21].

Figure 2

Balance of the mean continuity equation, $\eta \frac{\partial U^{*}}{\partial \eta }-\left[\alpha -1\right]U^{*}+\frac{\partial V^{*}}{\partial \eta }=0$. Note the prefactor for the second term is not multiplied (if multiplied, the term would be too small to see). (a) DNS data of Ref. [22]. (b) Experimental data at higher Reynolds numbers. Only the first two terms are available $(V$ is not measured). Solid circles data points are from experiments by Osterlund [21] $({\text{Re}}_{\theta }=2500$ to 27 000). Open asterisk data points are from experiments by Vallikivi et al. [23] $({\text{Re}}_{\theta }=8400$ to ${\text{Re}}_{\theta }=235000)$. Curves: DNS data at $\text{Re}=4060$ [22].

Figure 3

Ratio of the displacement thickness $\left({\delta }_1\right)$ to the boundary layer thickness $\left(\delta \right)$, ${\delta }_1/\delta$. Data points are from three experiments by DeGraaff and Eaton (DE) [3], Osterlund [21], Vallikivi et al. (VHS) [23], and the DNS of Schlatter [22].

Figure 4

Force terms in the streamwise mean momentum equation (22). The data are from DNS of Schlatter et al. [22].