Velocity profile near smooth boundary

Prandtl's mixing length theory

In turbulent flow, lumps of fluid are moved across layers by eddies. Prandtl assumed a lump keeps its momentum over a distance $l$ (mixing length) before mixing. The velocity fluctuations are then $u' \approx v' \approx l\,\dfrac{du}{dy}$, and the turbulent shear stress is

$$\tau_t = -\rho\overline{u'v'} = \rho\,l^2\left(\frac{du}{dy}\right)^2$$

Near a wall the mixing length increases with distance from it: $l = \kappa y$ with $\kappa \approx 0.4$ (von Karman's constant).

Velocity distribution near a smooth boundary

Let $u_* = \sqrt{\tau_0/\rho}$. Close to the wall the shear stress is practically constant, $\tau = \tau_0$, and the viscous stress is negligible outside the laminar sub-layer:

$$\tau_0 = \rho\kappa^2y^2\left(\frac{du}{dy}\right)^2 \;\Rightarrow\; \frac{du}{dy} = \frac{u_*}{\kappa y}$$

Integrating:

$$u = \frac{u_*}{\kappa}\ln y + C \quad\cdots(1)$$

Constant from the laminar sub-layer. In the laminar sub-layer of thickness $\delta'$, $\tau_0 = \mu u/y$, so $u = \dfrac{\tau_0y}{\mu} = \dfrac{u_*^2y}{\nu}$. At its edge $y = \delta'$, experiments give $\delta' = \dfrac{11.6\nu}{u_*}$, so

$$u_{\delta'} = \frac{u_*^2}{\nu}\cdot\frac{11.6\nu}{u_*} = 11.6\,u_*$$

Applying (1) from $y = \delta'$ to $y$:

$$u - 11.6u_* = \frac{u_*}{\kappa}\ln\frac{y}{\delta'} = \frac{u_*}{\kappa}\ln\frac{u_*y}{11.6\nu}$$

With $\kappa = 0.4$:

$$\frac{u}{u_*} = 11.6 + 2.5\ln\frac{u_*y}{\nu} - 2.5\ln 11.6 = 2.5\ln\frac{u_*y}{\nu} + 11.6 - 6.13$$

Convert $\ln$ to $\log_{10}$: $2.5\ln x = 2.5(2.303)\log_{10}x = 5.75\log_{10}x$:

$$\boxed{\frac{u}{u_*} = 5.75\log_{10}\left(\frac{u_*y}{\nu}\right) + 5.5}$$

(the exact constant 5.47 is rounded to 5.5, a value that agrees with Nikuradse's measurements).