Refining the connection between the logarithmic velocity profile and energy spectrum based on eddy's inclination angle

Although the “log” law of the mean-velocity profile (MVP) and the “−5/3 or −1” power law of the energy spectrum have been well studied in wall-bounded turbulent flows over a long period of time, a possible connection between them is still worth exploring. In this paper, an eddy's inclination angle is introduced into a model of a “turbulence eddy” to refine the connection. An expression for the inclination angle depending on both wall-normal height and Reynolds number is proposed. We find that the predicted MVP can follow the logarithmic law excellently, which is also in good agreement with direct numerical simulation results even at much higher Reynolds numbers. Our refined model may provide a deeper relationship between the mean velocity and the turbulence intensity through the eddy's inclination angle. Moreover, through our refined model, the hardly measured inclination angle can be obtained according to the given MVP and streamwise turbulence intensity.

Figure 1

Schematic diagram of the relationship between turbulence shear stress and eddy velocity. $S_y$ is the center of an eddy at a distance $y$ from the wall, ${\tau }_t$ is the turbulence shear stress, θ is the eddy's inclination angle, and S is the characteristic size of the eddy. $u\left[y+s\left|sin\left(\theta \right)\right|\right]$ is the mean velocity at a distance $y+s\left|sin\left(\theta \right)\right|$ from the wall; $u\left[y-s\left|sin\left(\theta \right)\right|\right]$ is the mean velocity at a distance $y-s\left|sin\left(\theta \right)\right|$ from the wall.

Figure 2

(a) Predictions of the MVP using different inclination angles $\left(\theta ={90}^{\circ },{40}^{\circ },{20}^{\circ },{10}^{\circ },5^{\circ }\right)$ at $\mathrm{R}{\mathrm{e}}_{\tau }=4179$, compared with the DNS data of channel flow [42]. (b) The predictions of the MVP at different Reynolds numbers ($\mathrm{R}{\mathrm{e}}_{\tau }=934$, 2004, 4179, 20 000, and 100 000) using an inclination angle $\theta ={10}^{\circ }$, compared with the DNS data of channel flow [42].

Figure 3

Predictions of MVP (plot by ○) at different Reynolds numbers ($\mathrm{R}{\mathrm{e}}_{\tau }=934$, 2004, 4179, 20 000, and 100 000) using the refined expression for the inclination angle [Eq. (9)], compared with the DNS data (solid lines) of channel flow [42]. The red lines represent ${u^{+}}^{\prime }\left(y^{+}\right)=y^{+}$. The green lines represent ${u^{+}}^{\prime }\left(y^{+}\right)=\left(1/\phantom{1k}\kappa \right)ln\left(y^{+}\right)+B$, $\kappa =0.4$, and $B=5.0$. The $y$ coordinates are correspondingly shifted by 5 per case.

Figure 4

(a) Predictions of the MVP (solid lines) at different Reynolds numbers ($\mathrm{R}{\mathrm{e}}_{\tau }=934$, 2004, 4179, 20 000, and 100 000) using Eq. (9), compared with the predictions (red solid lines) of Gioia et al. [15]. (b) The velocity of turbulence eddies $v_s$ (regarded as the streamwise turbulence intensity $u_{\mathrm{rms}}$) normalized by the local mean streamwise velocity U ($\mathrm{R}{\mathrm{e}}_{\tau }=2004$, plot by ○), compared with the results of Gioia et al. [15] (plot by □) and the DNS data [42] (solid line).

Figure 5

Predictions of inclination angles (solid points) under different experimental parameters using Eq. (10), compared with the results of Wu and Christensen [44] (hollow points), Del Alamo et al. [45] (red line), as well as Tang and Akhavan [46] (asterisk symbol).