Law of the Wall and Wall Units

Assumption

The turbulence near the a wall/solid boundary is only affected by the flow conditions at the wall and is independent of the flow conditions far away.

$$\text{or}$$

In the Outer Layer, the Turbulence is not affected by the the turbulence in the inner layer due to the huge difference in the Length scales ($\delta$)

Wall Units / Plus Values

Using the above assumption, a few flow variables are identified.

• $y\to$ distance from the wall $[L]$.
• $\bar{u}(y)\to$ Mean velocity (or velocity profile) $[L/T]$.
• ${\tau }_W\to$ Shear stress at the wall $\left[M/L{T}^2\right]$.
• $\rho \to$ Fluid Density $\left[M/L^3\right]$.
• $\nu \to$ Fluid kinematic viscosity $\left[L^2/T\right]$. After performing dimensional analysis we derive two new dimensionless quantities

Dimensionless Distance $y+$

$$\boxed{y^{+}\equiv \frac{y}{\nu }\sqrt{\frac{{\tau }_W}{\rho }}=\frac{y{u}_{\tau }}{\nu }=\frac{y}{{\delta }_v}}$$

Easier to remember

$$\boxed{y^{+}=\frac{y}{{\delta }_v}=\frac{y{u}_{\tau }}{\nu }=\frac{y}{\nu }\sqrt{\frac{{\tau }_W}{\rho }}}$$

Dimensionless Velocity $u+$

$$\boxed{u^{+}\equiv \frac{u}{u_{\tau }}}$$

Where,

• $y\to$ Distance from the wall
• $u_{\tau }=\sqrt{\tau /\rho }\to$ Friction Velocity
• ${\delta }_v\to$ Length Scale in the Inner layer

Note: In theory, the $u^{+}$ is only a function of $y+$, i.e, the assumption.

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References

1. http://brennen.caltech.edu/fluidbook/basicfluiddynamics/turbulence/lawofthewall.pdf