Spectral Theory of the Turbulent Mean-Velocity Profile

It has long been surmised that the mean-velocity profile (MVP) of pipe flows is closely related to the spectrum of turbulent energy. Here we perform a spectral analysis to identify the eddies that dominate the production of shear stress via momentum transfer. This analysis allows us to express the MVP as a functional of the spectrum. Each part of the MVP relates to a specific spectral range: the buffer layer to the dissipative range, the log layer to the inertial range, and the wake to the energetic range. The parameters of the spectrum set the thickness of the viscous layer, the amplitude of the buffer layer, and the amplitude of the wake.

Figure 1

Log-linear plots of the MVPs in the “wall variables” $\tilde{u}$ and $\tilde{y}$ for four values of Re. The symbols are from experiments [5] and the solid lines from simulations [7]. Each MVP extends from the wall (which corresponds always to $\tilde{y}=0$) to the centerline of the pipe (which corresponds to a value of $\tilde{y}$ that depends on Re); the MVPs collapse on a single MVP close to the wall. Focusing on any one MVP, we can parse the horizontal axis from left to right to find the “viscous layer” (where the MVP has a positive curvature), the “buffer layer” (where the MVP has a noticeable negative curvature), the “log layer” [where the MVP follows the log law of the wall (1)], and a “wake” (where the MVP overshoots the log law of the wall close to the centerline of the pipe) [18].

Figure 2

Schematic for the derivation of the turbulent shear stress. $u(y+s)$ is the mean velocity at a distance $y+s$ from the wall; $u(y-s)$ is the mean velocity at a distance $y-s$ from the wall.

Figure 3

(a) Plots of (7) for $\kappa =0.42$ and three values of ${\beta }_d$. (b) Plots of the thickness of the viscous layer ${\tilde{y}}_v$ as a function of ${\beta }_d$ for three values of $\kappa$.

Figure 4

(a) The MVPs computed using $\kappa =0.42$, ${\beta }_e=7$, and ${\beta }_d=8$. The thick dot indicates the point of contact between the viscous layer and the buffer layer. (b) The same as in (a) but using ${\beta }_e=0$ (dashed lines) and ${\beta }_e=10$ (solid lines). Inset: detail of the wakes. (c) The same as in (a) but using ${\beta }_d=0$ (bottom), 2, 6, and 12 (top). Inset: A plot of (8) for $\kappa =0.42$. (d) A plot of $B$ as a function ${\beta }_d$ for $\kappa =0.42$.