Inertial Sub-layer or Log Law Region

Observations

• This region lies $y^{+}\ge 40$ and $y/\delta \le 0.1$
• Turbulent Kinetic energy production and dissipation are almost equal in this region.
• Since there is almost no diffusion, the viscosity doesn’t effect the flow.

The Log Law

The Velocity Derivative

$$\frac{\partial u^{+}}{\partial y^{+}}=\frac{1}{y^{+}}\Phi \left(\frac{y}{{\delta }_v}\right)⟹\frac{\partial u^{+}}{\partial y^{+}}=\frac{1}{y^{+}}\frac{1}{\kappa }$$

With solution,

$$\boxed{u^{+}=\frac{1}{\kappa }\ln \left(y^{+}\right)+B}$$

Known as the Log law where,

• $\kappa =0.41,B=5.2\to$ from experiments
• $\Phi \to$ Wake Function